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Fast-track approach to mining construction projects Bissiri, Yassiah 1999

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F A S T - T R A C K A P P R O A C H TO M I N I N G CONSTRUCTION PROJECTS by  Y A S S I A H BISSIRI B. Sc. (Mathematics and Physiques) University of Ouagadougou, Burkina Faso  A THESIS SUBMITTED IN P A R T I A L F U L F I L L M E N T OF THE REQUIREMENTS FOR THE D E G R E E OF M A S T E R OF APPLIED SCIENCE in THE F A C U L T Y OF G R A D U A T E STUDIES D E P A R T M E N T OF MINING A N D M I N E R A L PROCESS ENGINEERING We accept this thesis as conforming to the required standard  T H E UNIVERSITY OF BRITISH C O L U M B I A December 1999 © YASSIAH BISSIRI,  1999  I declare that this thesis is my own, unaided work. Any errors or omissions contained in this thesis are entirely my own. It is being submitted in partial fulfilment of the requirements for the degree of Master of applied Science in Engineering at the University of British Columbia, Vancouver, Canada. It has not been submitted for any degree or examination in any other University. I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by my supervisor or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission.  Department of Mining and Mineral Process Engineering The University of British Columbia 6350 Stores Road, Room 517 Vancouver, B . C . Canada V6T 1Z4  Date:  Abstract  Traditional mining engineering procedure can essentially be decomposed into three separate stages at the engineering level; the design phase; the construction phase; and finally the production and processing phase. The important characteristic of this procedure lies in the sequential manner in which each phase is implemented during a project life cycle. In comparison, fast-tracking involves the overlap of design and construction, the overlap of construction and production, and finally the reduction of the execution time of each phase. The fast-track technique may be tempting to many project managers as it provides a theoretical mechanism for an overall reduction in project duration and probably costs. However, the implementation of the technique itself may be very difficult. For example the fact that some portion of design of overall design is executed simultaneously with construction may put a lot of pressure of the design team in terms of time and accuracy of drawings in order to minimize late changes. The fast-track technique has already been implemented in some civil construction projects with some stories of failure and success. In the mining however, although the fast-track technique is known, very few companies have applied it in the past. The application of the technique is delicate and requires a better understanding of the parameters that are involved in fast-tracking a project in order to control or at least predict their impact in the success of the project. This thesis introduces an approach to fast-tracking mining projects. The traditional and the fast-track approach are presented to highlight their similarities and differences. An economic model, based on the traditional definition of the net present value, is built for each approach. Then an upper bound cost of construction function for the fast-track approach is derived from the two models. This upper bound cost function is then used to conduct a sensitivity analysis with respect to the parameters involved in fasttracking a mining project. The sensitivity analysis provides us with a better understanding of the premium a company is willing to invest in order to fast-track a project. Throughout the analysis of the function, several business strategies known to be used in the mining industry were justified. The factors that may affect the application of the fast-track technique are also discussed in this thesis. These factors are identified, analysed and then inserted into the models in order to minimize the risk of failure. A resources allocation model is developed for fast-tracked project. The model is divided into two parts; a deterministic linear programming part that assumes that all the parameters are deterministic and a U  stochastic linear programming part that takes into account the probabilistic nature of the variables involve in fast-tracking a mining project. Finally, the two models are applied to a real mining project and the results are presented in graphs and tables. The objective of the case study was to demonstrated that  iii  Table of Contents Abstract  i[  Table of Contents  iv  List of Figures  viii  List of Tables  x  Acknowledgements  xi  CHAPTER 1  1  Introduction  1  Example of a Mine Construction Project  2  1.1 Purpose of the Analysis  3  Reduction in interest costs  3  Reduction in costs related to inflation  4  Positive Net Present Value  5  Climatic conditions  5  Market competition  5  1.2.1 Traditional Approach  6  1.2.2 Fast- tracking Approach  6  1.3 Literature Review  7  1.4 Thesis Objectives  10  1.5 Thesis Overview  10  CHAPTER 2  12  Fast-track Economic Model  12  2.1 General  12  2.2 Assumption  12  2.3 The Model  13  2.3.1  Cost parameters  15  2.3.2  Time and duration  16  Lv  2.3.3  Present Value (PV) Formulation for the Traditional Approach  17  2.3.4  The upper bound of the cost function for the fast-track approach  19  2.4 Analysis of the upper bound cost function 2.4.1  20  Results of the Sensitivity Analysis  21  CHAPTER 3  28  Analysis of Factors that may affect the Fast-track Procedure  28  3.1 Factors and Risks  29  3.1.1 Long lead equipment delivery  29  3.1.2 Poor planning  29  3.1.3 Availability of resources  30  3.1.4 Changes made to project  31  3.1.5 Weather  31  3.1.6 Risks associated with contracts  32  3.1.7 Productivity  33  3.2 Analysis  33  3.2.1 Scheduling a project  33  3.2.2 Parameter estimates  34  Duration, time and resources estimates  34  Costs estimates  35  CHAPTER 4  36  Resource Allocation Model for Fast-tracked Construction Projects  36  4.1 First Approach to the Resource Allocation Model  37  4.1.1 The single critical path case  37  4.1.2 The multiple critical paths case  38  4.2 Formulation of a Deterministic Mathematical Model  40  4.4 Discussion  43  4.4.1 Linearity of the crash costs  43  4.4.2 From Deterministic Parameters to Random parameters  44  4.4.3 Application of the Deterministic Model to the Example  47  4.5 Formulation of the Stochastic LP Model  49  4.6 summary  52  •v  CHAPTER 5  53  CASE STUDY  53  5.1 Description of the project  53  5.2 Project Data Summary  55  5.2.1 Mine  55  Tailing Storage and Heap Leach Facilities  55  5.2.2 Plant  55  5.3 Budget Cost Monitoring  55  5.3.1 Budget Highlights  55  5.3.2 Impacts of changes on the project budget  57  5.3.3 Impact of underestimation on the project budget  57  5.4 Overall Project Performance  57  5.4.1 Performance factors  58  Retaining wall  58  Subprojects with bad performance  59  Supply & Install Security System  59  Supply and installation of buried electrical services  59  5.5 Fast- tracking the project  60  5.5.1 Costs estimation  63  5.5.2 Deterministic L P problem  64  CHAPTER 6  71  Conclusions and Recommendations  71  6.1 Thesis review  71  6.2 Findings and conclusions  71  6.2.1 Findings about the economic model  71  6.2.2 Findings about the resource allocation model  72  6.2.3 Findings on the case study  73  6.3 Recommendations for Future Research  74  References  78  Appendix A  80  VL  A l Interest rate  80  A 2 Present Worth and Future Worth  80  A 3 Continuous flows and continuous compounding  80  A 4 The capital expenditure model  81  Cost parameters  83  Time and duration  84  Present Value (PV) Formulation for the Traditional Approach  85  Present Value (PV) Formulation for the Fast-track Approach  87  Expression of factor(x)  88  A 5 Vectors and Matrices used in the 3-D plots  89  APPENDIX B  91  B l Proof of the assertion in Chapter 4  91  B2 Probability distributions  92  B3: Answer and Sensitivity Report of the L P problem  95  vii  List of Figures 1.1- Traditional sequencing of proj ect phases  6  1.2- Fast- track project management overlapping design and construction phases  7  2.1- Continuous cashflow profile of the traditional approach  14  2.2- Continuous cashflow profile for the fast-track approach  15  2.3- Sensitivity analysis of the upper bound cost function  21  2.4- Sensitivity of C f with respect to Xi and x  2  23  2.5- Sensitivity of Co f with respect to Xi and x  3  23  2.6- Sensitivity of Co f with respect to x and x  3  23  0c  c  C  2  2.7- Sensitivity analysis of NPV with respect to variables xi, x and x f  2  2.8- Sensitivity analysis of NPVf with respect to variable X4, x and x 5  24  3  25  5  2.9- Start- start relationship between construction and processing phases  26  2.10- Start- start relationship between production and processing phases  26  2.11- Sensitivity analysis of NPVf  27  with respect to the proposed scenario  3.1- Procedure for designing a network  34  4.1- Illustrative example of a "near critical path"  40  4.2- Linearization of a crash data set  44  4.3- Arrow on arrow representation of the network  45  4.4- Simulation results of the total project length  47  5.1- Bar chart representation of the project construction schedule  54  5.2- Arrow on Arrow representation of activities in Table 5.3  61  5.3- Description of near critical activities  62  5.4- Path # 1 deviation histogram  67  5.5- Path # 2 deviation histogram  68  5.6- Path # 4 deviation histogram  68  6.1- Traditional sequencing of mining operation and processing phases  74  6.2- Representation of the proposed approach  75  6.3- Illustration of a sub-operation phase associated to a block within an ore zone  75  6.4- Description of a Turnkey plant by Summit Valley Inc  77  A 1 - Continuous cashflow representation  80  A 2- Capital expenditure model  81  A 3- Continuous cashflow profile of the traditional approach  82 viii  A 4- Continuous cashflow profile for the fast-track approach  82  B l - Illustration of an activity shared by more than one path  91  B2- Histogram of the project completion time  94  ix  List of Tables 1.1- Typical Mine Construction Schedule2 2.1- Description of the constant dollar cost parameters  15  2.3- Time and duration parameters for the traditional approach  16  2.4- Time and duration parameters for the fast-track approach  17  4.1- Data set of "crash costs"  43  4.2- Activity Duration  45  4.3- Assignment of duration and probability to activities  46  4.4- Paths Information  48  4.5- Activities Information  48  4.6- Results of the Deterministic LP problem  48  5.1- Summary of the Construction Schedule  53  5.2- Summary of Project Budget Variances by Area  56  5.3- Detailed description of activities involved in area 10  60  5.4- Simplification of Table 5.2  61  5.5- Summary of activities characteristics  62  5.6- Results of the estimation of the "crash" costs  63  5.7- Description of variables' upper limit  64  5.8- Results of the LP problem  64  5.9- Activity probability distribution  66  5.10- Crash time probability distribution  67  5.11- Results of the stochastic LP problem  69  5.12- Comparison between the stochastic and the deterministic LP results  69  A 1 - Description of the constant dollar cost parameters  83  A 2- Description of the current cost parameters for the traditional approach  83  A 3 - Time and duration parameters for the traditional approach  84  A 4- Time and duration parameters for the fast-track approach  84  B1 - Cumulative probability distribution of activities  92  B 2- Simulation results of activity times  93  B 3- Simulation replication table  93  B4- Descriptive statistic  94  x  Acknowledgements This thesis has benefited from the assistance of many people. In particular, I would like to thank Doctor Scott Dunbar and Dr. Allan Hall of the Department of Mining and Mineral Process Engineering at the University of British Columbia, Vancouver, Canada for providing me with financial and academic tools needed throughout my studies. Doctor Dunbar took the time to answer my queries about the applicability of the fast-track concept to mining projects and through several observations, he helped me refine my approach to building models related to engineering projects in general. Dr. Allan Hall allowed me to share his great experience about mining projects. I would also like to thank Professor Allan Russell of the Department of Civil Engineering at the University of British Columbia, Vancouver, Canada for having inspired me in choosing the topic of this thesis and mostly to have taken the time to help me refines the mathematical models in this thesis.  Finally, I would to thank Professor Malcolm Scoble of the Department of Mining and Mineral Process Engineering at the University of British Columbia, Vancouver, Canada for reviewing the thesis.  xl  Chapter 1- Introduction  CHAPTER 1 Introduction Fast- tracking can be generally defined as the process of overlapping the design and construction phases of a project. In this thesis, fast- tracking is defined as the process of overlapping and accelerating, by reducing the duration, the design and the construction phase of a project. Fast- tracking a mine construction project is related to the scope of the project itself, the productivity level, and the resources consumed by the project so that an attempt to fast-track a project involves the consideration and the analysis of these three general factors (Reda and Carr, 1989). The idea of developing and analysing a fast- track economic model in this thesis for mine construction projects was formulated based on a civil construction management course (CIYL 520). In this course, it was shown how the inclusion of the parameters involved in a fast- track civil construction project in a mathematical model may help understand better the risks associated with fast- tracking civil construction projects and probably control the degree of uncertainties of these parameters. Through some literature search, it appeared despite the similarities existing between mine and civil construction projects that no attempt has been made to mathematically model a fast-track mine construction project, although the technique is sometimes empirically used by some mines without proper risk analysis. Hence the decision to focus this work on a mathematical model for fast-track mine construction projects is the first attempt of its kind in the mining industry. This thesis analyses the advantages that mining companies can gain, in both Net Present Value (NPV) and revenue start up time, by fast-tracking a mining project. A methodology is developed to describe how fast- tracking can be applied successfully to a mining construction project while highlighting the risks and uncertainties associated with the fast- track approach. The concept of fast- track mine construction is very appealing because of its ability to reduce project duration. Such a reduction may be essential to ensure economic and/or market feasibility. At the very least, a reduction in project duration has the potential to produce savings in the cost of inflation and financing charges. However, along with the possible benefits, one also inherits a greater degree of uncertainty due to the inherent risks of fast tracking. Given a decision to adopt a fast-track strategy, a major risk is that, despite the expenditure of greater resources, the reduced duration may not be achieved.  1  Chapter 1- Introduction  A further risk is that the actual costs associated with fast tracking will greatly exceed estimated costs (Russell and Ranasinghe, 1991). In this thesis, methods for managing a fast- track mine construction project in order to minimize the risk of excessive costs are described. This chapter presents the purpose and background of the analysis, an example of the fast- track process for a mine construction project, and a comparison between a mine and a civil construction project. It then presents a review of literature relating to fast- track projects, the objectives of the thesis, and a thesis overview.  Example of a Mine Construction Project Table 1.1 shows an example of a mine construction schedule. All the activities described in this example are similar to those found in a civil construction project. The only difference between a civil construction project and a mining construction project is the operation stage. In mining, operation consists of depleting as much as possible of a resource to generate revenues whereas in civil the product generating revenue is the sales or leasing of the building. In civil projects, revenue can be generated while the building is under construction whereas in mining, construction has to be completed prior to generating any revenue. Table 1.1 Typical Mine Construction Schedule Area 10 11 12 15 16 18 19 21 22 23 24 25 26 27 28 26 36  Sub- project Description Plantsite Primary Crusher Coarse Ore Stockpile Conveying Grinding Facilities Leaching Facilities Refinery Shop and Warehouse Security/Change House General office Assay Lab. Cold Storage Building Geology/Core Logging Power Supply Tailing/Heap Leach Open Pit Dewatering System Security System Start-up  Early Start 11 Jan. 1995 03 Jun 96 11 Apr. 96 17 Jun. 96 17 Apr. 96 22 Apr. 96 15 May. 96 08 Apr. 96 17 Jun. 96 09 Jul. 96 26 Aug. 96 09 May 96 06 May 96 14 Nov. 95 02 Apr. 96 24 Jan. 96 29 Mar. 96 18 Nov. 96 20 Feb. 97  Early Finish 28 Mar. 1997 24 Jan. 97 22 Dec. 96 19 Feb. 97 22 Feb. 97 08 Feb. 97 28 Feb. 97 14 Jan. 97 22 Dec. 96 20 Dec. 96 14 Feb. 97 01 Aug. 96 31 Jul. 96 01 Oct. 96 01 Apr. 97 30 Jun.97 13 Dec. 96 11 Apr. 97 28 Mar. 97  2  Chapter 1- Introduction  1.1  Purpose of the Analysis  The purpose of this work is to develop a quantitative model, which includes the analysis of uncertainties to explore the behaviour of fast- tracked projects in mining. Within the model, relationships between project performance parameters are developed to account for changes to basic project variables caused by overlapping and compressing the design and construction phases. These relationships are manipulated to shape the economic model to not only show that the fast- track process can be managed, but also to reveal the difficulties nature of implementing the process. Mining engineering projects consume large quantities of time, resources, and capital. They are executed in highly uncertain environments. Each project, even if identical in design and scope, is a unique entity simply by the fact that the execution environment varies with time, location, current economic situation, perceived future economic situation, and financing strategy. Exposure to economic, social, or political uncertainty, the need to reduce inflationary and interest costs, market commitments, and market competition have resulted in several strategies aimed at reducing the time to start a mining operation conditioned by the execution phases of construction (Atkinson, 1987). Reduction in interest costs The majority of mining companies obtain their capital from shareholders equity and a bank loan. For example, a company might receive 30% of its capital from shareholders equity and the other 70% from a bank loan. Interest on the loan, and the method of repayment is negotiated with the bank. The usual scenario is that payments start as soon as the company begins generating revenue. The longer it takes to start generating revenue, the higher the costs of the loan. A fast- tracked mine construction project offers a possibility of reducing the costs related to the loan. Reduction of uncertainty Mining is a risky industry and two particular risks should be considered. Political risks: Investing in some regions of the world can be very risky for mining companies because of political uncertainties that can lead to civil unrest. When investing in regions with high risk of instability, mining companies are preoccupied with the mine life. Mining companies with mine developments in a politically unstable environment prioritize faster revenue start up in order to minimize the risk of loss due to production interference. Most companies investing outside the western nations should certainly consider fast tracked projects as an option of risk minimization.  3  Chapter I- Introduction  •  Market uncertainties: Mining companies should always consider fast tracking because the development of all projects is subject to metal price cycles. Financing decisions are only completed as metal prices peak. This leaves project managers struggling to bring in the project before the next price trough (Swallow et al, 1998). The impact of severe fluctuation of gold price on the market can be used as an example to show why  mining companies should consider fast-tracked projects. The gold price at the beginning of January 1996 was about $ 420 per ounce and plunged in free fall thirteen months later when it was traded at $401 per ounce. A Company that based its feasibility studies on 1996 prices, and estimated its annual production at 120,000 ounces/year, has lost about $2 millions per year. Projects that can withstand such losses are rare. Companies can start generating revenue, before prices fluctuate dramatically, if they choose to fast-track projects. Reduction in costs related to inflation An inflation rate is usually applied to project cashflows in order to reflect the difference between constant dollars (dollars used today) and current dollars (value of dollars in the future with respect to consumer price indices). Companies usually include inflation in their cashflows, but since the estimation or prediction of inflation rates is subject to significant error, a company may want to use all the methods available to minimize the impact of inflation rate fluctuation on their cashflows. Reducing project life is a good approach in reducing the effect of inflation on cashflows. Following are three scenarios that explain better why fast- tracking projects minimises the impacts of inflation on cashflows: A mining company has decided that the design-construction phase of the mining project will take 18 months to be completed. Scenario 1: The inflation rate will remain constant at 4% per annum over a 24-month period. Scenario 2: The inflation rate will remain constant and equal to 4% per annum over a period of 15 months and will jump to 8% for the next 6 months because of an economic crisis. Scenario 3: Thejnflation rate will remain constant and equal to 4% per annum over a period of 15 months and then decrease to 0.9% per annum for the next 10 months.  4  Chapter 1- Introduction  Scenarios 1 and 3 do not justify fast-tracking the project since the effects of inflation will be constant or insignificant over the duration of the project. In the case of scenario 3, the inflation rate decreases from 4% to 0.9% and therefore fast track is not only unjustified but may be costly. However, the other advantages of fast- tracking may offset these losses. Scenario 2 justifies fast tracking because if the project can be reduced by at least 3 months, the increase of inflation from 4% to 8% will not have any significant impact on the cashflows of the project. Positive Net Present Value The Net Present Value (NPV) is the most commonly used economical tool to judge the viability of a project. Usually a project will be given priority if its NPV is positive. Early operation means early production of metals and therefore early generation of positive cashflows that will increase the value of the NPV. Climatic conditions Climatic conditions may be factors why a construction project is fast-tracked. In fact, production and productivity level may decrease considerable due to hard working conditions and/or multiple interruptions in the work process. Overlapping design and construction and reducing each duration to finish construction or part of construction before severe winter conditions has the advantage of allowing indoor works to be carried on while waiting for favourable weather conditions to advance the progress of the project. This method was adopted at Brewery Creek Mine, Located at about 75 km from Dawson City, Yukon. Winter conditions are very severe in the Yukon territories (- 40° C) and management decided to undertake a very aggressive construction project in order to complete mine construction before the beginning of the next winter season. By the beginning of winter, the two cranes for the processing plant were erected, allowing indoor construction to continue even with extremely cold temperatures. Market competition Consider two mines A and B that each discovered a copper deposit at approximately the same time. A smelter company wants a certain amount of concentrate to be delivered at regular intervals starting at time T and is willing to pay a premium to a company that can satisfy such demand. The smelter can only deal with one mine at a time. Suppose that neither mine could begin production before time T if they both chose to adopt the traditional approach to construction. It is obvious in this case that the two mines may consider the fast track approach to remain competitive and gain the market for the concentrate.  5  Chapter 1- Introduction  1.2.1 T r a d i t i o n a l A p p r o a c h The traditional approach for a mining construction project consists of distinct sequential phases: design, construction, commission and operation as shown in Figure 1.1. The guiding economic principle is to continue spending as long as the return on incremental investment equals or exceeds the minimum attractive rate of return.  TRb  Loan  Sb ^  :  Revenue  w  Design  Time  Loan repayment Construction Production  y  Principal  Processing  Tpb  T b m  1.1: Traditional sequencing of project phases. T = design duration, T = construction duration, Sb = time between construction and revenue, t = time between production and processing, T = project life, T = revenue duration Figure  db  b  cb  b  Rb  In the traditional approach, production is usually overlapped with construction, e.g. the ore is stockpiled. It has been ignored here for simplicity and because the overlap may be short in comparison to other duration.  1.2.2 F a s t - t r a c k i n g A p p r o a c h The fast- tracking approach for a mining construction project consists of overlapping construction and design as shown in Figure 1.2. At the same time construction time and possibly the production phase may be reduced. These procedures can be completely and successfully achieved by proper management of the resources required for the execution of the project.  6  Chapter 1- Introduction  tdf  H-  • T  Loan  Revenue  df  Design Loan Repayment Construction 'ml 1^  m ™  Production  tcf  —  Tf P  Processing T  f  ^mf Fast- track project management overlapping design and construction phases. T = design duration, T f = fast track construction duration, Tiy = fast track revenue duration, T = fast track project duration, tf = fast track time between production and processing, Sf = fast track time between construction and revenue. Figure 1.2:  df  c  Note the overlap between construction and production. This overlap involves earlier ore stockpiling than might occur in the traditional case. It is assumed that processing cannot commence until construction is completed, although this may not be the case in heap leaching operations where processing may start earlier with the completion of the leach pad as the refining complex construction is still underway.  1.3 Literature Review In a search through the literature, quantitative treatments of fast- tracked mining projects and hard data, which could be used to derive empirical relationships between fast- track techniques and project variables, were sought. The search was not successful although some mining companies described very superficially their successes in fast-tracking some of their construction projects. Fast- tracking appears to be a new technique used in the mining industry with little known projects that have succeeded in the  7  Chapter I- Introduction  application of the technique. Most of the articles found throughout the search were related to civil construction projects, which in return happen to be very useful and enriching for this work. In fact the economic model for fast- tracked mine construction projects is derived based on the similarities existing between mining construction projects and civil construction. A search through the literature highlighted an example of a civil construction project where fasttracking was a success. It is in fact a litigation case involving two malls. The plaintiff was the owner of a retail mall that was built in a non- fast- track manner. When construction of the first mall was well underway, work began on a competing mall. The second mall, which began construction well after the first mall but was fast- tracked, opened a few weeks before the first mall due to the compressed design/construction schedule. The first mall argued that the fast- tracked mall took an equal market share. The fact that the second mall adopted the fast-track techniques helped achieve the market share it did. This example shows how fast-tracking a construction project can be beneficial although it does not apply directly to a mine construction project. The search for articles specific to fast- tracked mine construction projects has not been very successful. Only one article was found without any description of the techniques used to fast- track the mine construction project. The article focuses more on the early completion of the construction project without really describing the challenges encountered during the execution phases along with data supporting the technique. However, several articles were found about civil fast- tracked projects. These articles fell into three categories: the methods employed on a successful fast-tracked project, the failure resulting from the adoption of the decision to fast- track a project, and the theoretical benefits of fasttracking construction projects. They mainly highlight the uncertainties surrounding the decision to fasttrack a project. Finally they all agree that in order to succeed in fast-tracking construction projects, more data are needed in order to control and better predict the outcomes of any action taken in the procedure of fast-tracking and that fast-tracking a project is very dependent on its environment. Also throughout the search, several articles related to overlapping activities in the manufacturing industry were found. These articles, although not related to mine or civil construction were relevant to this work. Russell and Ranasinghe (1991) in their article build the NPV model for a fast- track construction project. The model of the NPV is then compared with the NPV of a non fast-track project. They show that the decision to fast-track a construction project although it may be successful incurs some risks. Throughout the article, the upper bound of the fast-track construction cost is expresses and analyzed. The authors show how the upper bond of the fast-track construction cost can be used to determine if fasttracking a project can be satisfactory by comparing its construction cost estimates with the bound. They  8  Chapter 1- Introduction  finally emphasize the fact that although this technique does not provide optimal solutions, it can help make appropriate decisions. Terwiesh and Loch (1998) in their article confirm statistically the acceleration impact of overlap on project completion time. They operationalized the concept of uncertainty resolution and statistically showed how it influences the effectiveness of overlap using 140 completed projects in the manufacturing industry. The authors outlined that a compression of the development process through overlapping requires a situation with limited uncertainty where changes are foreseeable and can be kept under control. Otherwise, overlapping may cause substantial rework outweighing the time gain from overlapping. Fazio et al (1988) cites increased technical complexity, government regulations, inflation, and political pressures as contributors to the increased cost of construction. The reaction to this increase has been fasttracked and/ or phased construction techniques. The case being examined in the article had several delays, many of which involved drawing production and co-ordination. Initial delays were attributed to obtaining vendor information, which further complicated design co-ordination. Initial drawings were not complete for the schedule bid dates and the owner/ engineer delayed and compressed the tendering of the main construction packages. Since the project was not behind schedule, they also actively rescheduled, accelerated, and overlapped activities. As a result of the incomplete bid package, numerous drawing revisions, and accelerated and rescheduled activities, initial contractor productivity values were low and the construction period was also lengthened. Although most of the delays were attributed to fast-tracking, incorrect, incomplete drawings and low initial productivity due to drawing revisions and rescheduled activities are to blame. Although fast-tracking exacerbated the delays on this project, the authors do not prove that fasttracking caused the faulty design packages. They state that more effort should have been spent earlier in the project due to far- reaching effects of mistakes made during the project. The low initial construction productivity is the result of unending effect decisions and changes, made in an attempt to recoup time lost. A non-fast-tracked project would have been delayed if the same problems had been encountered. Malcom and Henry (1998) in their article described how Mount Polley (an open pit mine) was completed ahead of schedule as the result of fast- tracking its construction project. They briefly mentioned how operating costs have decreased as a result on early operation start up. The authors attributed the success of the fast-tracking procedure to the communication and tremendous teamwork effort developed by the construction crews on site and the fantastic co-ordination between engineering managers at each level of the project. Although this article is relevant to fast-tracked mine construction projects, no details analysis of the fast-track techniques used was described.  9  Chapter 1- Introduction  Whitcomb and Kliment (1973) discuss how the fast- track process affects architects. According to these authors, fast- tracking and/ or phased construction both increases and decreases pressures on architects. Certain decisions must be made and become irrevocable. Other decisions, usually involving details, can be deferred to a later date. There is an increase in pressure while fast-tracking, as the construction schedule now governs the design process. Multiple bid packages, bid submissions, and estimates produce an environment where each work package contract comes under scrutiny; the architect must expend more effort to control this process. Also, an owner is operating in a riskier mode by being required to commit money before the project has a final total cost. These authors suggest pre- engineered components as a method to reduce the cost risk as these components can be more easily estimated than site- built components. James Sproul (1986) in his Master of Applied Science thesis developed an NPV model for a fast-tracked civil project. He tried to demonstrate that fast- tracking could be beneficial for a civil project. He also indicated that fast-tracking incurs additional risks in the project by developing a mathematical relationship between cost parameters and the scope, the reduced completion time, the productivity factors of each component within the project and the overlap factor. He then performed a sensitivity analysis on the cost parameter functions.  1.4  Thesis Objectives  The objectives of this thesis are not to develop a recipe on how to fast- track a mine construction project, but to first show that the fast- track techniques introduced in civil construction projects can also be applied to mine construction projects, second develop an economic model that can respond to changes at each level of the mine construction phases and finally develop a resource allocation model that can support the fast-track techniques ensuring hence greater chances of success. The economic model is applied to an arbitrary example to show how changes to basic variables affect the net present value (NPV) in a deterministic environment. A fast-track vector is introduced for a better understanding of the model. The resource allocation model is in fact a combination of deterministic and probabilistic procedure supported by a simulation process that will account for uncertainties and it is applied to a real mine to show how results from a deterministic environment may differ from a stochastic one.  1.5  Thesis Overview  Chapter 2 develops the NPV model for a fast-tracked mine construction project. The economic model for a mine construction project is defined in a basic way allowing the analysis to focus on the effects of the 10  Chapter I- Introduction  fast- track vector on the project NPV. The model is applied to an example. A series of results showing the sensitivity of the NPV to the fast-track vector are presented. Chapter 3 highlights the important factors that may affect a fast-tracked mine construction project. Solutions are proposed to improve these factors with detailed mathematical analysis of some of the factors. A simulation process is applied to an example to show how a combination of deterministic and probabilistic data can be used to reduce and/ or predict uncertainties caused by the use of resources within a construction project. Chapter 4 develops a resource allocation model and an algorithm that may be a first step for reducing risks associated to the use of resources as a mean to ensure that a project is successfully fasttracked. This model accounts for the existing resource allocation techniques and is specifically applied to fast- tracked projects. Chapter 5 treats a case study of a real mine using real data. The case study illustrates the two models developed and highlights their limitations. Chapter 6 presents conclusions and recommendations for future work.  11  Chapter 2- Fast-track Economic Model  CHAPTER 2 Fast-track Economic Model 2.1  General  Building an economic model for the fast-track approach is one goal of this thesis. An economic model helps in understanding the concept of fast-tracking a mining project while exposing its limitations.lt is usually very difficult to describe the cashflow profiles of a mining project as a preconceived recipe of cashflows; the cashflows are generally discrete and are recorded monthly (most common), quarterly or annually. Since the timing of mining project cashflows is usually not known, it is very difficult to generalise the concept of mining project cashflows. Therefore, continuous cashflow profiles are assumed. The continuous model allows considerable flexibility because basic mathematical tools can be used to conduct very instructive analyses. The use of continuous cashflow profiles generalises the concept of the model allowing in depth sensitivity analyses. Continuous cashflow profiles are discussed in Appendix A.  2.2  Assumption  Several assumptions are made in order to simplify the model: •  The duration and times are assumed to be known although the duration can be expressed as a function of the scope of the job, the productivity and the resource level (Sproule, J . , 1986).  •  The variables are well defined.  •  Interest and inflation rates are considered invariant with time.  •  A portion of the construction costs is financed by a financial institution, for example a bank  •  Design costs is expressed as a portion of construction costs  •  No interest is paid during construction  •  Loan repayment profile is uniform in current dollar and begins at the beginning of revenue.  A mining project is composed of the following phases: •  Permitting + Exploration + commissioning  •  Design = Scoping + Prefeasibility + final feasibility + Detailed engineering  12  Chapter 2- Fast-track Economic Model  •  •  Construction  •  Production  •  Processing  And revenue  Permitting, exploration and commissioning are not considered in this model because they are parameters that are difficult to quantify and therefore control. The design phase considered in this model consists in fact of the detailed engineering part of the design. Financial institutions usually lend money at the end of the final feasibility or at the beginning of the detailed engineering phase and therefore if an overlap of construction and design may occur, it may probably be at the detailed engineering level consisting of drawings and detailed analysis of activities involved in the construction phase of the project.  The assumptions above are considered in order to simplify the model. For example inflation and interest rates in reality vary with time, however defining the actual functions is difficult because dependant of many unknown factors.  2.3  The Model  In the following, variables having a subscript  are associated with the traditional approach while  variables having a subscript ' / ' are associated with the fast-track approach. The model computes the net present value of the traditional approach,  N P V  b  , and the net present of the fast-track approach,  N P V  f  .  Comparison of these two values determines the upper bound of the cost of construction a company would be investing if it decides to adopt the fast-track approach.  Indeed companies are interested in the additional costs the process of fast-tracking a project will incur. However, modelling the construction costs for the fast-track approach is nearly impossible since it requires the knowledge of many parameters (such as scaling functions related to scope and productivity) that are mathematically impossible to quantify. In general, approach is preferred only if  V N P V  >  0 . Setting  V N P V  =  N P V  f  =  N P V  b  + V N P V  and the fast-track  0 means that the project gained in time with  same N P V and each dollar increment returns at least the minimum attractive rate of return M A R R . Solving the equation N  P V  f  — N P V  b  for the construction cost determines therefore the upper bound of  the cost of construction for the fast-track approach. The upper bound provides the maximum constant dollar to be spent to achieve the compression of the original completion time given the components of a  13  Chapter 2- Fast-track Economic Model  vector that is introduced as the "fast-track vector" in this chapter. This number gives a bound against which the additional costs associated with the fast-track strategies to achieve the reduction in time can be compared.  If the costs of construction falls below this value, the condition is satisfactory but not  necessary optimal. In fact, there exists interval with this value as the centre and inside which any costs will be satisfactory. In order to develop the model for realistic situations, a certain number of parameters such as loan interest rate, minimum attractive rate of return (MARR), ...etc are considered. Construction costs are the sum of direct and indirect costs whereas production and processing costs are the expenditures generated during operation. In addition, the relationship among the durations and costs has not been considered. The two approaches are described in Figures 2.1 and 2.2.  14  Chapter 2- Fast-track Economic Model  k  Tf — tdf +  Tf c  _  —  tdf -  T f-  (l-Xl)Tdf  <  9»  (l-Xs)Tcb  c  p  <\  •  —  ^  -v> w  ^  Sf  _i  _ Revenue  Loan \  1 Construction i  Design  — Td/  -(l-x^Tdb  \  /  /  Production expenditure 1  / i / / i / i /  1  \  \  W  Loan repayment Principal  Ii —1t  1  \  \  V  'l-x )T 4  Time  / / /  i i  \\  _ n  \  I J  Processing expenditure  !  cf  i i i i i i  _ w>  T f— m  (l-Xe)T b m  w T f= p  '«  (l-x )T b 5  p  •<  Figure 2.2: Continuous cashflow profile for the fast-track approach  2.3.1 Cost parameters The constant dollar costs are nominal and estimated in terms of today's cost, without consideration of the inflation index relevant to the industry. They are defined for time t (where t is a dummy variable) as follows in Table 2.1: Table 2.1: Description of the constant dollar cost parameters Parameters C c  0  c  o d  ( 0  R (t)  (t)  Description design phase costs construction phase costs revenue  0  E  0  p  ( 0  production expenditure mineral processing expenditure  e e 0p 6 c  R  m  Construction inflation rate Revenue inflation rate Production inflation rate Processing inflation rate  15  Chapter 2- Fast-track Economic Model  The current dollar costs however are estimated in terms of costs as it is applied in the future by taking into account inflation. The current dollar costs, described in Table 2.2, are estimated by applying the inflation rate to the constant dollar costs. Table 2.2: Description of the current cost parameters for the traditional approach Parameters C (t) =  C (t)-e -'  Description design phase costs  C (t) =  C (t)-e ™  construction phase costs  R(t) =  R (t)-e «  revenue  d  c  Definition e  0d  9At+  0c  e  0  it+Si)  E (t) =  production expenditure  E (t) =  mineral processing expenditure  P  m  The current cost parameters for the fast-track approach can be derived from the parameters in Table 2.2 by replacing the subscript "ft" by "f \ indication of the fast-track approach.  2.3.2  Time and duration  Described in Table 2.3 are time parameters for the traditional approach.  Table 2.3: Time and duration parameters for the traditional approach Parameter T*  Description duration of the design phase duration of the construction phase  T  cb  duration of the revenue phase  T  Rb  duration of the production phase T mb  duration of the mineral processing phase  1  T  h  Sb  Duration of design + construction Start- start duration between processing and revenue  For the fast-track approach, we may define a six dimensional vector x = ( X , , - - - , J C ) 0 < x < 1, called 6  i  the "fast-track vector" in the rest of this chapter, that transforms the time variables of the traditional approach to those of the corresponding fast track approach as shown in Table 2.4:  16  Chapter 2- Fast-track Economic Model  Table 2.4: Time and duration parameters for the fast-track approach Parameters tdf  Definition (l-x,)T  =  Td  f  =  (l-*2)T  Tc  f  =  0-xs)T  db  tcf —  (l-x )T  Tf  0-*5)T  4  =  P  TmfT*r  V-XJ  Overlap of construction and production duration of the production phase  c f  p b  duration of the mineral processing phase  mb  T  m  —  duration of the revenue phase Total duration of design and construction Start- start duration between processing and revenue  T j) m  tfTf +  =  c  c  T f-  f  s  duration of the construction phase  c b  TRb — (T b  =  Tf  Description Overlap of design and construction duration of the design phase  d f  m  =  Tf R  Variables x x , x , X4, x , and X6 represent respectively the degree of overlapping construction and design u  2  3  5  phase as compared to design duration, the degree of reducing design duration, the degree of reducing construction duration, the degree of overlapping construction and production as compared to construction duration, the degree of reducing production duration and the degree of reducing processing duration. The following constants are also defined in the model: q r r  = fraction to total capital expenditure financed using debt = minimum attractive rate of return (MARR) = Loan interest rate = fraction of construction costs allocated to design  k kd c  2.3.3  = construction unit cost per time = Design unit cost per time  P r e s e n t V a l u e (PV)  Formulation for the Traditional A p p r o a c h  Current construction cost function C (t) = C c  • k (t) • e - '"' ' 9c<  0c  T  + )  c  Current design cost function C (t) = id • C (t)dt d  d  c  17  Chapter 2- Fast-track Economic Model  PV (design) = b  'lib  jc (0 • e~ dt r,  rf  o  Tcb  P V (construction) = e~  rTjt  b  PV (revenue) = e" b  r(7i+7  |C (f) • <T  rt  C  o  ™ " ™ • j^CO-e""* 4  r  r  PV(production) = e~  rTh  o  *  • JE (t)• e~ dt n  o  p  Tb m  P V (processing) = e~  rTb  b  )  • JE (r) -e~ dt n  m  o  P V (loan) = q • P V (construction) b  b  PV (principal) - C b  0 c  . <*<-'X^_) . j » -r), e  g(  c  0 PV (loan repayment) = e~ " • A • ^e~ dt rT  b  rt  b  o  Future value (loan) where A = = h  o and Future value (loan) = q • e '  ri Tcb  b  • | C (t) • e'" ' e  1  o  The present values for the fast-track approach is derived from the present values above by replacing each parameter with a subscript "ft" by its corresponding parameter with a subscript " / ' . The process is described in Appendix A .  The net present value for the traditional approach  NPVb  =  PV (revenue) b  + PVb(loan)  + PVb(production)  (NPVb)  -[(PV (design)  + PVb(loan  b  repayment)  is derived as follows:  + PVb(construction)  +  PVb(processing)  +PVb(principal))J  Assuming that the expenditures and revenue remain the same the net present value for the fast-track approach  (NPVj)  is:  18  Chapter 2- Fast-track Economic Model  NPVf = PV/revenue)  + PV/loan)  + PVf(production)  -[(PV/design)  + PV/construction)  + PVj(loan repayment)  +  PV/processing)  +PV/principal))]  The assumption of keeping the same revenue and expenditures is justified by the fact that the fast-tracking occurs at the construction level and therefore does not have any repercussion on the production, the processing and the revenue phases.  2.3.4  T h e u p p e r b o u n d of the c o s t f u n c t i o n for the fast-track a p p r o a c h  Before a decision is made to fast-track a project, companies are interested in knowing how much it costs to undertake such an aggressive schedule. In other words, what is the premium one is willing to invest to fast-track a project. Therefore understanding the sensitivity of that premium with respect to the variables involved in the process is crucial and may help achieve better plans for the success of the project. However, there is not a mathematical expression that gives the cost of construction for fast-tracked projects because many parameters involved and their correlation to one another are unknown. The upper bound function Co f (X) is introduced in this chapter to overcome this lack of c  mathematical formula. Given a vector X , C f (X ) represents the cost of construction to fast-track the 0  0c  0  project at the same net present value as the traditional approach. Spending more that Co / (XQ) may result c  in lower net present value. C  0c  (XQ) is therefore the upper bound of the cost of construction for the fast-  track approach. The upper bound function, C  0cf  (X), is derived from the equation NPV  b  = NPVf where  NPVb is the net present value for the traditional approach and NPVf is the net present value for the fasttrack approach.  NPVf = PVf(revenue) - PV/production)-  PV/processing)  +(PV/loan) -fPV/design)  + PV/construction) + PV/loan repayment) +PV/principal)]) The part in bold in the NPVf equation are terms that contains the construction cost function and therefore can be factored as follows: NPVf = PV/revenue)  - PV/production)-  PV/processing)+Factor(X).C /X), c  where C f(X) is the expression of the cost function. C  CQC/X) is derived from the equation above by equating NPVf and NPVb as follows:  19  Chapter 2- Fast-track Economic Model  NPV  =NPV  f  o Factor(X)-C (X)  b  0cf  NPV  <=> C  Qcf  {X) =  b  = NPV  b  - PV (revenue) + PV (processing)  - PV'(revenue) + PV(processing)  +  + PV  {production)  PV(production)  Factor{X)  0cf  The derivation of the upper bound cost function is elaborated in details in Appendix A. The upper bound cost function is usually used for comparison purpose with cost estimates. For some given components of vector X, the value of the upper bound cost is calculated and then compared to the estimation of the cost of construction related to the fast-track process. If C , is the cost estimate then es  the following scenarios take place: if C  esl  >C  Qc/  then the fast track technique is not satisfactory  if C  esl  <C  0cf  then the fast track technique is satisfactory  Although C f is called the upper bound cost function, it does not necessarily yield the optimal cost 0c  function. The optimal function could be above or below Co / based on the type of estimators used and the c  goals sought by the company undertaking the fast-track approach.  2.4 Analysis of the upper bound cost function An arbitrary set of data is chosen to simplify the analysis of the models. For the sensitivity analysis the following data for the traditional approach, described below, are used: Parameter Coc Eop Eom R)b r e q r Tdb Tb Tpb Tb TRb L  C  m  Description Constant dollar construction cost Constant dollar production expenditure Constant dollar processing expenditure Constant dollar revenue Minimum attractive rate of return Constant inflation rate Portion of loan Constant loan rate Design duration Construction duration Production duration Processing duration Revenue duration  Value $100 m $60 m $10 m $400 m 14% 1% 20% 8% 0.6 year 1.5 years 8.13 years 8.13 years 7.5 years  The data are chosen arbitrarily just for the purpose of simplification, the sensitivity analysis of the upper bound cost function is not affected by this choice. It is assumed that inflation is the same for all the phases and interest rates are constant over the time they are applied.  20  Chapter 2- Fast-track Economic Model  2.4.1  Results of the Sensitivity  Analysis  The sensitivity analysis of the mathematical model is divided in three parts. The first part consists of analysing the upper bound cost function with respect to the overlap, the design compression and the construction compression variables, the second part consist of analysing the behaviour of the net present value with respect to the three remaining variables (X4, x , and x ) and the last part consists of analysing 5  6  the net present value with modifications made to the way the production, the revenue and the processing phases are sequenced in the entire process. Figure 2.3 shows the plots of the upper bound cost function with respect to variables X i , X2 and x , 3  respectively the overlap, the design and the construction compression factors and the construction cost for the traditional approach, C - C j(X) is an increasing function of the three variables meaning that the 0c  0c  upper bound cost of construction will increase as a result of accelerating the design and the construction phases. As the compression factor relative to the three variables increases, the premium that justifies fasttracking the project increases as well.  I—1—1—1—1—1—1 0.0  0.1  0.2  1  0.3  1  0.4  1  1  0.5  1  1  0.6  1  1  0.7  1  1  0.8  1  1  1 I  0.9  1.0  Variables x , x^, Xj  Figure 2.3: Sensitivity analysis of the upper bound cost function The plot of the upper bound function with respect to the reduction of the duration, x , of construction 3  appears to increase more rapidly as compared to the ones with respect to x and x , meaning that one is 2  3  prepared to spend more financial resources to accelerate the construction phase. Figure2.3 shows for example that starting construction when only 60% of design is completed (variable x i = 0.4 or 40%), reducing design duration by 40% (variable x = 0.4 or 40%) and reducing construction duration by 40% 2  21  Chapter 2- Fast-track Economic Model  (variable x = 0.4 or 40%) justifies spending respectively $900,000, $1.1 million and $5 millions in extra 3  costs. The advantage of the analysis is to highlight a cost target that companies may consider in their decision making. However, the model may provide even better information if the upper bound function is analysed with all the variables at the same time. In this chapter, the upper bound function is analysed with respect to two variables at the same time and the results of the sensitivity are shown in Figure 2.4, 2.5 and 2.6. The graphs indicate that the upper bound cost function increases even more when plotted against two variables at the same time. For example starting construction after 60% of design is completed and reducing the design phase by 60% justifies spending an additional $1.2 million for the construction phase. Companies, provided with the upper bound cost of construction may choose to undertake one of the scenarios (determined by the values of variables Xj, x^,and Xs) according to how much they can afford to invest to fast-track the project. Once the target is chosen, then the costs estimates, for the company to go ahead with their choice must be lower, equal or slightly higher than the upper cost value determined by the chosen values of the three variables. If the estimates is higher than the target value, the net present value might be less than the net present value for the traditional approach and the company might decide weather or not to adopt the fast-track approach.  22  Chapter 2- Fast-track Economic Model  Figure 2.4: Sensitivity of C f with respect to xj and x 0c  Cocf(X2,X3)  Figure 2.6: Sensitivity of C r with respect to x and x 0c  2  3  2  Chapter 2- Fast-track Economic Model  The sensitivity of the net present value for the fast-track approach with respect to the three variables x  h  %2,  and  NPVf,  is shown in Figure 2.7. The net present value increases as the compression and the  overlap factors increase. Figure 2.7 shows that fast-tracking a mining construction project may be beneficial in the point of view of the net present value . Again, the plots show that NPVf is very sensitive to the acceleration of the construction phase. The overlap of design and construction and the acceleration of the design phase have almost the same effect on the net present value.  Figure 2.7: Sensitivity analysis of NPVf with respect to variables Xj, x and x 2  3  Analyses were also conducted on the net present value function for the fast-track approach with respect to the three variables x , x , and x (respectively the overlap factor of construction and production, 4  5  6  the compression factor of production and the compression factor of processing) and the graphs are shown in Figure 2.8. The plots indicate that the net present value of a fast-tracked project increases with increasing value of variable X6 and therefore with increasing compression factor for the processing phase while the net present value decreases with increasing values of variable X4, the overlap factor of production and construction, and increasing values of variable x , the compression factor for the 5  production phase. Hence, there is no economic benefits in overlapping construction and production and/or reducing the production phase duration. However, shortening the processing duration increases the net present value. In practice, this latter remark suggests that mining companies should minimize the process of stockpiling ore if it is not processed immediately, as it is not generating revenue while consuming resources. They should probably consider finding new technologies that could accelerate the process of  24  Chapter 2- Fast-track Economic Model  ore or create a start- start relationship between production and processing so that ore is processed immediately after it is mined. — • —NVP baseline — • — N V P when construction is overlapped with production(x ) 4  — * — NVP whith the compression of production phase(x ) s  70  — • — NPV with the compression of processing phase(x ) 6  Variables x-.x-.x. 4' 5' 6  Figure 2.8: Sensitivity analysis of NPV with respect to variable X4, x and x f  5  5  The process of stockpiling ore may however be a management strategy to benefit from a forecast of future mineral prices. For example low-grade ore may be stockpiled until the period of depressed prices and then processed economically as it no longer has mining cost at that time (Ren Y i and J. R. Sturgul, 1999). Lower-grade material may also be stockpiled for blending purpose. Nevertheless, in all the cases, Figure 2.8 shows that as long as ore is not processed at the same time it is produced, it is uneconomic in the net present value point of view to accelerate production. Ore may also be stockpiled for mines that deal with extremes weather conditions. For mines that operate in severe winter conditions, digging becomes a problem under these conditions and therefore the mines are better off accelerating production during warmer periods. The ore is then stockpiled and processed during the winter, keeping the processing plant running at its required capacity. Overlapping production and construction is justified for heap leach mines. Ore can be mined and sent to the leach pads while the construction of the plant facilities is underway. Figure 2.9 depicts the scenario of the model where the processing phase is in a start-start relationship with the construction phase and Figure 2.10 depicts the proposed scenario where processing and production phases are in a start-start relationship. In the latter, revenue is moved along with  25  Chapter 2- Fast-track Economic Model  processing and production increasing the net present value as revenue is generating earlier than in the initial model. Figure 2.11 shows the economic benefits of the start- start relationship between production and processing on the NPV point of view as compared to the NPV with respect to the process in which the processing phase is in a finish start relationship with the construction phase. In the latter case, the NPV function was a decreasing function of increasing values of x and x , whereas in the new case (Refers to 4  NPV(.) new  5  in Figure 2.11) the NPV Sanction is increasing with increasing values of x and x . Notice that 4  5  variable x , related to the compression of processing alone is replaced by variable x which represents the 6  5  compression of both production and processing and variable x is related to the overlap of both production 4  and processing phase.  Non overlapped  Processing Construction  Production  Construction Production  Overlapped Processing  Figure 2.9: Start- start relationship between construction and processing phases Duration  •  < Processing Construction  Non overlapped  Processing  Construction Production j  Overlapped  Processing New duration  Figure 2.10: Start- start relationship between production and processing phases 26  Chapter 2- Fast-track Economic Model  70  NPV baseline — ® — N P V (x )  r  4  —o—NPV (x ) new 4  — * — NPV (x )  60 h  5  — d — N P V (x ) new 5  0.0  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  1.0  Variables x, and x.  4  5  Figure 2.11: Sensitivity analysis ofNPVfV/ith respect to the proposed scenario An important remark to make about the last part of the analysis is that in order to conduct the sensitivity analysis on the net present value with respect to the last three variables, it was assumed that the capital expenditure function is uniform in terms of constant dollar, however, the model would have been more realistic if the relationships between the expenditures and the variables were known. Since these relationships are unknown, the upper bound expenditure function could have been introduced to study the behaviour of a premium in terms of production and processing expenditures, as production and processing is accelerated. From the sensitivity analysis of the net present value function, it can be predicted that this premium will also increase as demonstrated below: In general,  NPV/ = PV(revenue)  - PV(costs) - PV  (expenditures)  if NPVf = NPVi, and NPV/ increases, because it is assumed that Expenditures remain the same, then for N P V f to be equal to NPVb, PV (expenditures)  should increase. In other words, the expenditure should  increase for the net present value to remain constant. The premium for expenditure costs will therefore increase as a results of the sensitivity of the net present value function.  27  Chapter 3- Analysis of Factor that may affect the Fast-track procedure  CHAPTER 3 Analysis of Factors that may affect the Fast-track Procedure In chapter 2, it was shown that the fast-track method works in theory and a model was developed to quantify the construction costs to bid in order to successfully implement the model. In the real environment however, we are confronted with high risks due to many factors that may cause the method to become unsuccessful and even disastrous for the project. Given a decision to adopt fast-track strategy, a major risk is that, despite the expenditure of greater resources, the reduced duration may not be achieved. A further risk is that the actual costs associated with fast- tracking will greatly exceed estimated costs provided by the model. Considerable resources are used during the fast-track execution of a project and most of the risks lie in the management of these resources. Rational planning and control over the design, construction and operation sub-projects are the key to successfully minimize risks associated with the approach. Rational planning and control of projects is provided by an efficient project management system. As a project environment becomes more complex, so do the requirements of project management. Project management is required if construction projects are complex and subject to demanding constraints of time, cost, and environmental regulations therefore a fast-tracked project is an example of such a project. If several activities or disciplines are to be integrated, and there is a need to co-ordinate the cooperation of various departments within the organisation, project management is a viable approach. Project management is especially appropriate when the project faces changing environmental needs and other external considerations. Planning is one of the key functions of the management process and it is the project manager's prime activity. Planning is selecting objectives and establishing programs and procedures for achieving the objectives. It is decision making for the future of the project (Willis M., 1986). Planning is not only scheduling, which is only one of the many plans that can be put in place. Planning results in setting out milestones, which are time objectives. It clearly establishes the work to be done to achieve certain contract scope and cost objectives. The main purpose of planning is that it reduces the uncertainty that exists before a project or portion of it is launched (such as overlapping design and construction phase). Planning also improves efficiency of execution and it clarifies the objectives.  28  Chapter 3- Analysis of Factor that may affect the Fast-track procedure  In this chapter, factors that may cause a mining project to fail to achieve its fast-tracking goals are identified. In the first part of this chapter these factors will be identified. In the second part it is demonstrated that rational planning and control of the project may reduce the risks related to fasttracking and even make its implementation a success.  3.1 Factors and Risks Several factors may contribute to a delay in mining projects and among them are: •  long lead equipment delivery  •  poor planning/ unnecessary constraints  •  availability of resources  •  changes made to the project during execution  •  weather  •  risks associated with contracts  •  productivity  3.1.1 L o n g l e a d e q u i p m e n t d e l i v e r y  The delivery of equipment on time plays an important role in the planning of mining projects. Unlike most industrial projects, the equipment required by mining projects is scarce and only a few companies manufacture this kind of equipment. Therefore availability is limited. The scarcity of this type of resource makes it difficult for mining companies to "shop around" in case the equipment is not delivered on time. Mining companies can protect themselves from such situations by providing themselves with enough information about the contractors. This information is generally shared among companies or can be directly provided by contractors. Transportation to remote sites may be done by specialist independent contractors. Failure to meet their objectives is as damaging to the project as late delivery by the supplier. Losing booked transport space for large items can require transport air transport at very high cost or a severe delay until alternative space is available on that route. When equipment is scheduled for long distance shipping and delivery, the client should insist to be notified of any delay or non-shipment. Information should immediately be reviewed in the project scheduling and management to minimize the impact on project fast-tracking. 3.1.2 P o o r p l a n n i n g  Poor planning may be the result of an inefficient scheduling process, but may also be related to poor estimation and incoherent assignment of tasks. For example a poor estimation of the geotechnical 29  Chapter 3- Analysis of Factor that may affect the Fast-track procedure  properties of rock structures could cause a temporary shutdown of the development of an open pit for safety reasons. An underestimation of the quantity of earth removal required prior to excavation can cause serious delays. A scheduler may specify completion of one task as a constraint to the initiation of a second task, even though such a relationship is not essential. Often, the simplest and most economical method for expediting a project is to delete non-essential constraints. For example, in the traditional approach to mine construction, construction has to be completed prior to operation. This constraint certainly eliminates any co-ordination between construction and mining engineers and its specification may unnecessarily delay project completion. In fact, in the fast-track approach, some tasks in construction can be done concurrently with others in operation. In general, constraints should be avoided unless they are essential to the project.  3.1.3 Availability of resources A resource is anything that is precisely defined and measurable and required for the completion of a project. Resources may be classified into two types: •  available or unlimited resources  •  limited resources  Available resources are situated in the pool of resources which amount is greater than resource requirements for the project. Examples of these are: •  Capital, which may progressively diminish as work proceeds, but which can be replenished by progressive payments. Space, for maintenance and overhaul work, where rate of progress is controlled by the working area available at any given time.  •  Consumable materials such as gravel may be treated from the point of view both of supply and space.  Limited resources are resources that are scarce or simply not available in sufficient amount to satisfy the project requirements. These include resources with regular rate of supply, but with no capacity for being carried forward from one period to another such as labour. Labour once allocated to a certain area of the project is available for the time period specified, but does not accumulate. Labour in the mining industry is one of the hardest resources to manage, generally because labour is very hard to train and maintain . If 1  not managed correctly, available resources may become limited because of mismanagement and therefore Indeed maintaining labor in mining is one of the greatest challenges that mining human resource managers encountered for mine sites located in remote areas. Managers have to rely on the work force that is willing to stay in 1  30  Chapter 3- Analysis of Factor that may affect the Fast-track procedure  should be managed in an efficient manner to optimise their usage when face with a project that has to be fast-tracked in order to maximize the use of resources within the project pool of resources.  3.1.4 C h a n g e s m a d e to p r o j e c t  Changes during the execution phase may cause some activities to be stopped and rescheduled, thus greatly affecting project completion time and also the overall project costs (Thomas and Napolitan, 1995). For these reasons, it is very important to allocate enough resources during the design phase to ensure minimum changes as the project goes on since late changes is more disruptive to the project overall productivity(Ibbs, 1997) . The fast-track approach proposes a reduction of design time by the use of additional resources in order for the design results to be satisfactory. A project with the least changes has a greater probability of success than the one requiring many changes. Any project is subject to changes during its execution phase but the main goal is to minimize the changes as much as possible.  3.1.5  Weather  Weather plays an important part in outdoor mining activities. Indeed as mentioned in Chapter 1, severe weather conditions may cause expensive and significant delays in open pit mines. For example in countries with severe winter conditions, it is practically impossible to maintain the same level of productivity. This also applies to African and Asian countries during the monsoon season. Therefore, reducing the quantity of the activities during severe weather is recommended and that is what the fasttrack approach proposes: fast-track construction and design during good weather in order to do indoor work during severe weather. The following example will illustrate the need to minimize outdoor work during severe winter conditions: Suppose that the two mines will be open pit copper mines and that the deposit for mine A is located in a location where winter conditions are very severe. The deposit for mine B is situated in a location where winter conditions are acceptable. During severe winter conditions, the structure of the soil to be handled is very hard and causes most of mines to slow down. In fact productivity and even production levels decrease. For example, if during the summer a shovel digs a 9 m by 4 meters bench, the will probably consider for the severe winter conditions reducing the size of the bench to be dug (for example to a 9 m by 2 m bench) and as a direct result, production decreases. Also during severe winter conditions, resources such as labour and equipment are rarely used at optimum level.  the town for as long as possible. Usually people with families are the best work force for a mining company because they tend to settle in order to create financial stability for their families.  31  Chapter 3- Analysis of Factor that may affect the Fast-track procedure  In the case of mine A, something has to be done about the production to maintain the rate of producing copper concentrate constant if it wants to reliably supply a smelter contract. One option is to consider increasing production during summer by directing more resources to the operation phase. Increasing production during summer must raise another problem with respect to the plant capacity, which once designed is inflexible. The mine should consider stockpiling the ore in order to process it during winter, but this also raises the question of space and material re-handling. Very accurate feasibility studies should be conducted in order to decide whether this approach is economically feasible. The studies focus on the space available for stockpile and the cost of re-handling materials versus the cost of material not re-handled. The mine will reduce its resource usage such as equipment, machines and manpower and this constitutes economical advantages because equipment and machines will gain more life and at the same time the mine will save on salary costs for the winter period. Productivity for trucks is lower during winter (snow or icy roads makes it hard to maintain a certain minimum speed needed to keep production rate constant). Hence, weather represents a risk that managers have to account for in their planning by estimating the impact in such situations and by preparing a response to that eventuality. Winter is not the only season at risk, even during the summer the project could be delayed by heavy rain or by violent winds. The fast-track approach proposes a way of decreasing the risk associated with very severe winter conditions.  3.1.6 Risks associated with contracts Risks are inevitable at all stages of a project, especially in contracts. When signing a contract, it is assumed that the client has identified and accepted the major risks and is prepared to proceed. The main goal at this stage is to share the risks with the contractors involved in the projects. Such action can be materialized as penalties for not delivering the results on time or other legal procedures. Mining companies sign numerous contracts with contractors in the construction industry as well as the manufacturing industry and these contracts are not without risks. The risks that affect the project are related to time of deliveries. When scheduling projects, managers usually assume that all the deliveries will happen on time, but as will be highlighted in the case studies, sometimes contracts are violated and the project should hence ensure some guaranties while signing any contracts. Project managers have the duty of the very important task of collecting the lists of "reliable" contractors. Penalties imposed to the contractor for non-respect of the contract is a form of sharing risks, but the project will be better off without any delays, especially when the decision is made to fast-track the project. Indeed care must be taken when signing contracts to ensure minimum risks related to deliveries.  32  Chapter 3- Analysis of Factor that may affect the Fast-track procedure  The deliveries are usually drawings, equipment or even qualified technicians to be sent on the site for consultation.  3.1.7  Productivity  Citing productivity as one the factors that may affect the fast-track procedure leads to the factor that may affect productivity itself. Productivity is usually defined as the ratio of output and input, however depending on whether one is an economist, an accountant or industrial engineer, there is a slight difference in the concept of productivity. In this section, productivity is defined as partial productivity as it quantifies the productivity of each type of resource used during a construction operation making it easier to always refer to in case of bad performance. For example, labour's partial productivity is the ratio of output and labour input, material's productivity is the ratio of output to material input. There is no doubt that poor productivity will affect negatively a fast-track procedure. In fact bad productivity will affect any projects. For a non fast-tracked project, bad productivity may cause delays in the completion time of the project. In a fast-track procedure, the risk related to bad productivity should be minimised first since the fast-track procedure is itself risky. There are several ways of improving productivity and it is the project that has the tasks of ensuring high productivity on sites. For example the introduction of incentives may be one mean of guaranteeing the improvement of productivity. Hiring an experienced worker is another one.  3.2 Analysis 3.2.1  S c h e d u l i n g a project  Scheduling projects is very challenging and more challenging is the decision to reschedule or compress activity duration and times as proposed by the fast track approach. The key factor in the fast track approach is the time between the design and the construction phase. This requires that more energy be spent in the planning because as was previously stated, the decision to fast track a mining project although it can be beneficial results in increasing uncertainties. For this reason, the manager of the project should know and understand each aspect of the project. He has the duty of making the other parties involved familiar with the project. The planning of a project requires not only competency, but it also requires good teamwork and continuous improvement of co-ordination among the teams that are involved in the project. There are some procedures to follow in order to design a very efficient network. Recall that the goal of designing a network is to assign resources to activities and control the execution phases of the entire project. Figure 3.1 summarises one of the procedures used in the industry:  33  Chapter 3- Analysis of Factor that may affect the Fast-track procedure  Divide the project into subprojects  Identification of the subprojects  Estimation of  1  Constraints definition  durations, times  Sub- projects sequencing  Network Analysis  Figure 3.1: Procedure for designing a network  3.2.2 Parameter estimates Estimating is a fundamental part of the construction industry. It is a business skill of utmost importance. The success or failure of a project is dependent on the accuracy of several estimates through the course of the project, that is, from conceptual and feasibility estimates through to the detailed estimates (Ahuja, 1994). Numerous failures of construction projects can be attributed to faulty or inaccurate estimating. Project management must know what parameter information is needed and how to use this information in making decisions. The main parameters involved in a construction project are the duration and times, the costs associated with the activities, and the amount of resources needed. Duration, time and resources estimates Times and duration estimates are crucial to the success of a construction project in that they define the estimated total length of a project. In order to successfully fast- track a project without having to spoil resources, it is important to perform accurate estimates of the total length of the project. Suppose that, for a given project, it is estimated that the total project length is 60 days and a decision is made to bring it to 40 days. If the real length of the project is 45 days instead, it means that ineffective resources are allocated to the project at this phase. In fact these additional resources may even complicate the execution of the project.  34  Chapter 3- Analysis of Factor that may affect the Fast-track procedure  Costs estimates Construction is a unique industry which is inherently risky because most projects must be priced before they are constructed. For this reason, the estimates performed on costs related to activities and resources must be the best estimates. In the fast- track process, the management team must estimates the resources needed to "crash" an activity and since the cost related to "crashing" activities are not linear functions, estimate exercises are required to find the best approximate. The use of a combination of expert system and simulation may be the best method for generating estimates. The use of fitting functions might have proven to be acceptable but they do not take into account uncertainty factor such as weather, productivity or unavailability of resources. The expert system is build based on the point of view of an experienced manager and probability distributions (usually discrete) are derived. The simulation technique analyses all the possibilities or scenarios and then it is the manager's discretion to make a decision with respect to a specific variable. A n example of estimates is treated in appendix B.  35  Chapter 4- Resource Allocation Model for Fast-tracked Construction projects  CHAPTER 4 Resource Allocation Model for Fast-tracked Construction Projects When fast-tracking a project, additional resources have to be used in order to achieve the reduction of the project duration. But the distribution of these additional resources on the network (Activity relationship diagram) related to that project has to be managed rationally in order to minimize the total cost of the project. In fact because of the risky nature of the decision to fast-track, a lot of information should be made available to management. The risks should be identified, quantified and then analyzed before a decision is made to fast-track the project. When building a network for a project, management teams are usually concerned with six main goals (Deckro, 1991): •  Minimisation of the total project duration  •  Minimisation of activity lateness  •  Maximisation of activity earliness  •  Minimisation of maximum resource requirements  •  Minimisation of absolute deviation from target resource levels  •  Minimisation of the cost of project acceleration  Minimising the total project duration is a result of minimising the duration of each activity on the network. A probabilistic approach is considered in this model for the times (duration, earliness, and lateness) assigned to each activity. Minimisation of resource requirement is the result of proper management. Indeed the requirement in resources can be minimised if the management team is provided with enough details on each activity and the environment in which the activity is being executed. Also the minimisation of the use of additional resources is greatly related to the overall productivity of the project. In fact continuous improvement of the productivity at every work level will determine the quality of the management. Minimisation of the cost of fast- tracking is the key factor to the success of this model because a decision to fast-track a project is based on the budget available to fulfil the requirement of additional resources.  36  Chapter 4- Resource Allocation Model for Fast-tracked Construction projects  Different types of resources are used during the progress of a project and the combinations of these are very crucial to the realisation of the project. For example in mine construction, resources used are machines, construction equipment, labour, hours, ...etc. The recipe for good combinations of these resources is the result of a strong feasibility study done by the management team. When considering fasttracking a project, the criteria for combining resources should be very narrow because of the risks associated with fast-tracking projects. For example, when should overtime be used instead of hiring additional labour or renting more equipment or machinery. In fact managers are aware that the productivity factor of a worker decreases after a certain period of time, but does it justify hiring additional workers instead of using overtime? This is one example of analysis managers should make when faced with the decision to accelerate a project.  4.1 First Approach to the Resource Allocation Model 4.1.1 T h e s i n g l e c r i t i c a l p a t h c a s e When the decision is made to accelerate a project, it is the critical activities that are generally targeted first since the critical path determines the total length of the project. The question is how much should each activity be crashed in order to minimize the total cost of accelerating the project? The problem is formulated in a deterministic manner below: Let {ai, a ... a } be the set of critical activities on a network containing only one critical path. 2  n  d = project duration before acceleration d' = project duration after acceleration d -d' = acceleration time c - unit cost of crashing activity i i  M , = maximum crashing time of activity i f,. = amount of time activity i is crashed The total cost of accelerating the project is then  An optimization problem is then posed as follows: Minimize  37  Chapter 4- Resource Allocation Model for Fast-tracked Construction projects  Subject to:  0 < t,•< M, n  d-d'=yt,  4.1.2 The multiple critical paths case There is a possibility that several critical paths exist simultaneously in a network and the summation of the duration of the activities lying on these individual paths will give the same total project length. The same process as in paragraph 4.1.1 is used in this section. Consider a project that has a total of n critical paths. The project expected total duration is T and a decision is made to reduce the duration to T'.  n  number of critical paths number of critical activities, a , on critical path k, 1 < j <m tj  k  Note that critical paths may share activities. For critical path i = \a ,a ,-• n  j2  • ,a  im  the following are  defined: c  ki  /i  k i  a  ki  = cost of crashing critical activity i on critical path k ($/day) = upper limit of" crashability" of critical activity i on critical path k(days) =" crash" variable of critical activity i on critical path k(days)  a = T - T = Expected" crash" days  Cost of crashing Path i The total cost of crashing the critical path i is the sum of the cost of crashing all critical activities on this path:  (1)  38  Chapter 4- Resource Allocation Model for Fast-tracked Construction projects  Cost of fast-tracking the project from T to T The total cost of fast-tracking the entire project is the sum of the total "crash" cost of the n critical paths of the project.  n  n  (=1  i=l  mi  k=l  Equation 2 represents the additional cost of fast-tracking the project. The objective of any project planner who considers fast-tracking a project is to minimize the total cost of using more resources for the project. The problem of allocating the proper "crashing" times becomes an optimization problem. To account for activities that are shared by the critical paths, it suffices to replace the cost of "crashing" each activity by the ratio of that cost by the number of critical paths sharing that particular activity. The proof of this assertion is shown in Appendix B. Q For example if 3 critical paths have in common activity xn, the cost allocated to xn is — where en is 3 the estimated cost of crashing xi i. Optimization problem n  Minimize  mi  ^ ^ i=l  (XaCM  k=l  Subject to: mi  ^^CZki  = a for every i.  (3)  k=\  a>0 otki < f3ki, for any i, k such that 1 < / < n and 1 < k < rrn (4) G = equality constraints The optimization model discussed in this section assigns the proper "crash" time to activities that should be "crashed" in order to successfully fast-track the project length to the new length (T). But this approach does not take into account paths that have their total length lying between the project total  39  Chapter 4- Resource Allocation Model for Fast-tracked Construction projects  length (T) and the indicated length the project is fast-tracked to. These paths are called "near critical path" in the following section Figure 4.1 represents the network of a construction project that is supposed to fast-tracked to a total length of 16 days.  Figure 4.1: Illustrative example of a "near critical path" The critical path of this network is path {1-3 — 4 - 5 } with a total length of 24 days representing the project total length. Path { l - 2 —5} has a total length of 20 days. Fast-tracking the project to 16 days without including path {l — 2 — 5} will be incomplete. Therefore, path {l — 2 — 5} should be taken into account in the model. A "near critical" path in our model is path that is not critical but has its total duration satisfying the following condition: If T* is the "near critical" path length, T the project initial length and T the fast tracked project length, then:  T'<T*<T  (4)  Compressing the project length from T to T' automatically makes the "near critical" paths critical at the end the compression.  4.2 Formulation of a Deterministic Mathematical Model Consider a project that has a total of n critical paths and n* "near critical paths". The project expected total duration is Tand a decision is made to reduce the duration from T to T'. It is important to point out that critical and "near critical" paths may share activities with other paths of the same nature or not. This will not have any impact on the model since the equality of activities  40  Chapter 4- Resource Allocation Model for Fast-tracked Construction projects  is considered as constraints. Activities that are shared by paths should be included in the constraints as equalities. n - number of critical paths m = number of critical activities, k  Critical Path 1 =  , on critical path k, 1 < j <m  k  {a ,a ,---,a } n  n  lmi  Critical Path n = {a , a , • • •, a , } nX  n 2  nn n  Note that critical paths may share activities. For critical path i = \a ,a ,• n  i2  • -,a  im  the following are  defined: c = cost of crashing critical activity i on critical path k ($/day) /? = upper 1 imit of" crashability" of critical activity i on critical path k(days) a =" crash" variable of critical activity i on critical path k(days) or amount "crashed" ki  ki  ki  a = T - T' = "crash" time for the entire project  n = number of "near critical paths" m * = number of "near critical activities", a*, on "near critical" path k, 1 < j <m k  k  Near critical Path 1 = |a ,a H  Near critical Path n* = [a*„'\  X2  ,---,a .*| |m  ,---,a 'm '  ,a\'2  n  H  For "near critical" path i = {a , a , • • •, a n  n  t  }  the following are defined:  c \ i = cost of crashing near critical activity i on near critical path k ($/day) jB* = upper limit of "crashability"of near critical activity i on near critical path k(days) ki  a* ki =" crash" variable of near critical activity i on near critical path k(days) or amount crashed T*k = length of near critical activity k a *=T * - T = number of days near critical path k is expected to be crashed to fast - track the project k  k  n*  m*p  The total cost of fast tracking the "near critical paths" is TC* = ]T £ (cap * -ci *) P=I  i=\  P  (5)  41  Chapter 4- Resource Allocation Model for Fast-tracked Construction projects  Again, the "crash" cost for activities that are shared by different critical and/ or "near critical" paths should be replaced by that cost divided by the number of critical and/ or "near critical" paths that share these activities. Applying equation 5 to the first approach gives the following deterministic LP problem:  mi  n  Minimize  n*  m*p  ^ _ ^ (ocki • cu) + __ __ (O»P * -cip*) i=l  k=\  (6)  p=\ 1=1  Subject to: = a, V i , i > l a n d i integer  (7)  k=\  *  ]T] oup* =  Op  *, V p, p > 1 and p integer  (8)  i=\  a = T- T'  (9)  ctp* = Tp*-T'  (10)  0 < exh < j&i, V i, k / , 1 < i < n and 1 < k < mi  (11)  0 <atp*< / V \ V l , p / l < l < m * a n d l < p < « *  (12)  T'<T *<T  (13)  p  P  This is a deterministic approach to the resource allocation model for fast-tracked projects, in reality many more parameters must be included for the model to be reliable. Equation (7) sets the extent to which a critical path should be crashed and this limit is the same for all the critical paths. Equation (8) sets the amount of days "near critical" paths should be crashed, but this time this amount is not necessary the same for all the "near critical" paths. Equation (9), (10), (11), (12), and (13) set the limits for the parameters and variables. The constraint (14) is the set of equalities among the activities that should be added to the set of constraints.  The solution of this LP is a (nm +nm*)- dimensional vector, 1 < i < n and 1 < p < n*. j  The other important point is that the new length (fast-tracked length) of the project is assumed to be known, however, the new length can be a variable and the problem to pose will then be to determine how far a project can be fast-tracked for the decision to still be feasible based on how much one is willing  42  Chapter 4- Resource Allocation Model for Fast-tracked Construction projects  to invest in extra costs. In that case, one will try different scenarios for the new project length where the cost will be determined instead of being minimized. The appropriate cost will then be used with the model above to determine the "crash" time for each activity. The model above is deterministic with respect to the new length. In this model, it is assumed that the fast- track process is successful and what really matter is to obtain realistic durations and times that will reduce the risks associated with the process.  4.4 Discussion Many assumptions have been made during the process of building this model: •  The different costs are assumed to be linear functions of the time, which is not always the case in the real environment.  •  Most of the parameters are assumed to be deterministic. However, they are random in the real environment.  •  The upper limits of the "crashability" of the activities are assumed to be known.  4.4.1 Linearity of the crash costs In general, the "crash" costs are not linear functions of the number of days. The costs are usually estimated by managers and/or experts who have worked with similar activities in similar environments. The cost data are then collected and then fitted with a linear function. Table 4.1 represents a collection of "crash" cost data of a given activity and figure 4.2 shows how a linear function is fit to the graph of the data set.  Table 4.1 Data set of "crash costs  1  Crash costs ($) 300 200 375 450 175  Equivalent number of days  4 5 7 10 3  43  Chapter 4- Resource Allocation Model for Fast-tracked Construction projects  T h e " C r a s h " c o s t function  ^ o _° j= 2 o  600  y = 49.497X  R = 0.6986 2  400 200  6  8  10  12  Days  Figure 4.2: Linearization of a crash data set  The data in table 4.1 have been fitted with a linear function (y = 49.Sx). The unit cost of crashing this activity is about $49.5. It is certain that the unit cost changes with the availability of more data. The reliability of the unit cost estimated, increases with the amount of data. Although this method of estimating is not very accurate, it brings the problem of non- linearity to a linear problem.  4.4.2 From Deterministic Parameters to Random parameters Construction operations are usually subject to a wide variety of fluctuations and interruptions. Varying weather conditions, learning development on repetitive operations, equipment breakdowns, management interference and other factors may have an impact on construction operations. As a result of such interference, the production process in construction becomes subject to random variations ( Ahuja ,1994). In fact it rarely happens on a construction site that the same tasks consume the same exact duration on successive occurrences. An activity therefore may not be completed at the indicated time. Its completion time becomes random and has to be taken into account when building a model involving the consumption of time. An initial critical path may not be critical at a certain phase of the project because another path such as for example a "near critical" path may take a little longer than the duration assigned to be completed. Since the success of this model is based on the identification and the use of critical and "near critical" paths, the randomness of these parameters should be considered. One way of accounting for randomness is the introduction of Monte Carlo simulation (Winston and Albright, 1997) to the model. Table 4.2 shows the duration of the activities constituting a construction project and figure 4.3 is the Arrow on Arrow diagram of the network. This example illustrates the concerns posed in this section.  44  Chapter 4- Resource Allocation Model for Fast-tracked Construction projects  Table 4.2 Activity Duration Activity Duration (days)  1-2 4  1-4 6  1-7 2  2-3 8  3-6 4  4-5 10  4-8 16  5-6 8  6-9 6  7-8 6  8-9 10  •The network in figure 4.3 is composed of 4 paths with the following characteristics: - Path { 1 - 2 - 3 - 6 - 9 } with a total length of 22 days - Path { 1 - 4 - 5 - 6 - 9 } with a total length o f 30 days - Path { 1 - 4 - 8 - 9 } with a total length o f 32 days - Path { 1 - 7 - 8 - 9 } with a total length o f 18 days - The critical path is {l - 4 - 8 - 9} The durations on a given network are usually the expected durations obtained from past projects and managers' past experiences. Now suppose that the duration of activity 8-9 is known with certainty but instead may take one of the following values: 7days, 8 days, 9 days, 10 days, and 11 days. The following scenarios may occur: If the duration of activity 8-9 is 7 days, then path {1-4-8-9} has a total length of 29 days and is not critical anymore. The new critical path is therefore path {1-4-5-6-9} If the duration of activity 8-9 is 8 days, then path {1-4-8-9} has a total length of 30 days and the network has now 2 critical paths. If the duration of activity 8-9 is 9 days, then path {1-4-8-9} has a total length of 31 days and is still the critical path of the network. But the length of the project has changed from 32 days to 31 days. The same remarks are applied to "near critical" activities and paths when the compression time is determined.  45  Chapter 4- Resource Allocation Model for Fast-tracked Construction projects  The example points out the fact that failing to take into consideration the random nature of the durations associated to each activity may have a negative impact on the model. The example shows that the characteristics of a network may change dramatically with the introduction of random durations, which in fact depicts the behaviour of the project in real environment. The crucial question that arises is related to the collection of these random durations. Data are usually collected from past projects or existing databases. In the mining industry however, little information is available and therefore a procedure for obtaining reliable data should be designed. The Monte Carlo simulation is one way of solving this situation, but it requires the use of discrete probability distribution. Hence not only data are required, but they also have to be assigned a probability distribution to account for environmental uncertainties and its influence on decision making (Laufer, 1990). By consulting with managers and experts that have worked in similar projects, a set of data can be collected and put in a table format that will allow an analysis by descriptive statistics. For example, the data and their equivalent probability may be obtained by asking the following questions for a particular activity to a group of experts and/ or managers: What is the most likely duration? What is the worst case scenario (the longest duration)? What is the best case scenario (the shortest duration)? What the difficulty of executing this activity? The procedure is like building an Expert System with the only difference that we do not associate a degree of belief (DOB) to the answer to the question but instead the frequency of the answer will determine the discrete probability. Table 4.3 sets the format for the data collection and the probability assignment. Table 4.3 Assignment of duration and probability to activities Time for 1-2 Probabilities Time for 1-4 Probabilities Time for 1 -7 Probabilities Time for 2-3 Probabilities Time for 3-6 Probabilities Time for 4-5 Probabilities  2 0.1 4 0.1 1 0.2 6 0.1 3 0.15 8 0.25  3 0.25 5 0.2 2 0.5 7 0.2 4 0.6 9 0.2  4 0.35 6 0.4 3 0.3 8 0.4 5 0.25 10 0.35  5 0.3 7 0.3  Time for 4-8 Probabilities Time for 5-6 Probabilities Time for 6-9 Probabilities 9 Time for 7-8 0.3 Probabilities Time for 8-9 Probabilities 11 0.2  14 0.2 7 0.3 4 0.2 5 0.1 9 0.2  15 0.1 8 0.5 5 0.4 6 0.7 10 0.4  16 0.4 9 0.2 6 0.3 7 0.2 11 0.4  17 0.3 7 0.1  46  Chapter 4- Resource Allocation Model for Fast-tracked Construction projects  In the case of activity 1-2, the interpretation of the data is that out of 100 data collected on the possible duration of activity 1-2, 10 data support the duration of activity to be 2 days, 25 support 3 days, 35 support 4 days and 30 support a duration of 5 days. The probability distribution is then used in the Monte Carlo simulation to repeatedly generate random activity times. One assumption that has to be underlined is that the random times are probabilistically independent of one another. The complete simulation procedure for this example is provided in Appendix B, only the results, their analysis and interpretation are presented in this section. Also i Appendix B provides all the probability distributions for each activity and a detailed explanation of how the simulation is built. Figure 4.4 represents the histogram of the distribution for the total project length for 1000 data simulated.  Histogram  Total length of project  Figure 4.4: Simulation results of the total project length The simulation provides the distribution about the criticality and "near criticality" of each path and activity. From the probability distribution, the planner can conduct a statistical test on chosen samples and then derive the expected values of the durations and then apply them to the network. The primary goal of the simulation is to generate data that could be used to estimate the expected durations and therefore the expected total length of the project. The simulation program provides an analysis of each path and each activity about its criticality and "near criticality" with the corresponding distribution. For example a probability about the criticality of a given path is given by the simulation program. The simulation is run for 1000 data and then the durations and times are assigned to the network. However, this method does not completely take into account the probabilistic nature of the success of fast-tracking the project. The method suggests that a deterministic LP be solved for each scenario.  4.4.3 Application of the Deterministic Model to the Example Suppose that after running the simulation, the scenario described in tables 4.4, and 4.5 seems to occur with a greater confidence limit. The project total length has been found to be 30 days.  47  Chapter 4- Resource Allocation Model for Fast-tracked Construction projects  Table 4.4 Paths Information Paths 1 2 3 4  Length 20 29 ' 30 19  Criticality No No Yes No  Near Criticality No Yes No No  Table 4.5 Activities Information Activity 1-2 1-4 1-7 2-3 3-6 4-5 4-8 5-6 6-9 7-8 8-9  Duration (days) 2 5 3 9 3 10 15 8 6 6 10  Criticality No Yes No No No No yes No No No Yes  Near Criticality No Yes No Yes No No No No Yes No No  Upper limits (days) 2 3 1 3 1 3 5 3 2 3 3  Crash cost (May) 80 98 140 104 120 187 70 250 140 79 153  Activities 1-4, 2-3, 4-5, 6-8, and 8-9 are potential "crashing" candidates. Activity 1-4 at the is shared by 2 paths, one critical and the other "near critical" and therefore its "crash cost is divided by 2 in the expression of the objective function. It was decided that the project be accelerated from the 30 days to 24 days. The results of the deterministic optimisation problem solved with EXCEL SOLVER are presented in table 4.6. The entire optimisation process is described in Appendix B. Table 4.6 Results of the Deterministic L P problem Activity 1-4 2-3 4-8 4-5 6-9 8-9  Crashed (days) 3 0 5 1 2 0 Total cost:  Costs ($) 294 0 350 187 140 0 $1111  48  Chapter 4- Resource Allocation Model for Fast-tracked Construction projects  The results of the LP problem suggests that the total length of the project could be reduced from 30 days to 24 days at a minimum cost $1111 by crashing activities 1-4, 4-8, 4-5, and 8-9 by respectively 3 days, 5 days, 1 day, and 2 days. The results also point out the fact that although activities 2-3 and 4-8 are respectively "near critical" and critical, they are not crashed. Hence, the LP model not only determines the exact amount of days to "crash", but it also determines activities that are effectively crashed. In this example, the assumption that the project is successfully "crashed" from 30 days to 24 days is made. In the real environment, however, the success of reducing the project length is not guaranteed. In fact, the project may even be extended beyond its initial completion time due to the same factors that slow down project schedules (As discussed in chapter 3). Another concern is that even if the length of the project is reduced and the new length is not the one indicated by the fast-tracking length, it means that despite the use of additional resources, the project failed to be fast-tracked to the appropriate new length. In any case, the failure of the project may cause losses to the project. Also suppose that the estimation of the project length is not accurate, suppose for example that the estimated project length is 30 days whereas in reality it is 36 days. If the initial decision was to fast-track the project from 30 days to 24 days, a compression of 6 days is needed whereas with the real event (36 days), a compression of 12 days is needed. Hence the project may fail to be compressed due to bad estimation of the project initial total length. Based on these remarks, it is appropriate to consider cases where the fast-tracked length of the project is not known with certainty. The simulation process used to forecast the duration could be reused in that case to simulate the entire network with this time all the durations considered deterministic (after being each simulated) and only the new total duration is random. Paragraph 4.5 of this chapter described an approach to allocating the appropriate resources to the network with random new project duration.  4.5 Formulation of the Stochastic LP Model The stochastic model described in this chapter takes into account the risks related to fast-tracking a project. The risks are associated with the critical and the "near critical" paths. The move to design a stochastic model is justified by the fact that due to the uncertainties of the parameters, future unfeasibility within the model should be considered. In the stochastic programming literature, two approaches to solving stochastic LP are widely studied: one is based on modelling future recourse (response) and the other restricts the probability of unfeasibility to be no greater that a pre- specified threshold (Sen and Higle, 1999). The uncertainty is related in this case to the length after compression of the project T . If Rand(T) is the random variable associated to T , the planner usually expects that the probability that Rand(T) is greater than T be negligible. Uncertainties on T , means uncertainties for the "crash" variables. In the deterministic model the coefficients in constraints (7) and (8) were considered deterministic and equal to 49  Chapter 4- Resource Allocation Model for Fast-tracked Construction projects  one. But in the stochastic model, these coefficients are replaced by their respective probabilities. To the objective function is added a penalty function to account for not reaching the goal for failing to reach the target. This penalty function depends on the probability distribution of each critical and "near critical" path.  n* m*p  Minimize  £  £ (Oki • CM) + £ £ (cap * -cip*) +CE{a-Y cc ) i  =\ i=\  (=1 4=1  J  ki  C {a -JX)  +  p  p  k=]  P  (15) Subject to: a = T- T'  (16)  ap* = T *-T'  (17)  P  0 < ah < pki, V i, k / , 1 < / < n and 1 < k < mi  (18)  0 < c V ^ A>*,Vl,p/l<l<m * andl<p<«*  (19)  T' < T * < T  (20)  p  p  This model is solved using the simple recourse method. The parameters C and C represent the penalty ;  p  costs of the critical and "near critical" paths. The expression E [.] represents the expected values of the expression in the brackets and the probabilities used the expected values are the joint probabilities associated with the criticality and/ or "near criticality" of the paths. The decision to use the probability distribution of the criticality and/ or the "near criticality" of the paths in the stochastic model is justified by the fact that it is the criticality and/ or "near criticality" of these paths that determines the criticality and/ or "near criticality" of the activities within the network. The stochastic LP model can be rewritten by introducing new variables that represent the simple recourse action to correct the deviations of the variables from their target. The methodology is proposed by Sen and Higle (1999). Additional variables are introduced in the SLP above in order to transform it to a deterministic LP easier to solve. Let S; denote an index set o f all the outcomes of the random vector ( a , a) and ki  p  is  = Pr ob((a , ki  a) = ( a , a). k i  Let S* denote an index set o f all the outcomes of the random  vector (a i , a ) a n d p \ = /?ro6((a/ , a * ) = ( a ^ , a*)). A n y notation w i t h a " ~ " denotes randomness. P  p  P  50  Chapter 4- Resource Allocation Model for Fast-tracked Construction projects  The stochastic model is then formulated as follows: Minimize  Subject to: cc = T- T' a* = T * -T 0<ctk,< fa, V i , k / , 1 < / < n a n d \<k<mi P  (22) (23) (24)  0<ai *<p, *, V l , p / 1 < 1 < m *  and 1 < p < n (25) ^oou + yl -y~ = a, V i , i > 1 andi integer;s e S P  P  p  f  j^ai *  +y*\ -y'.'  P  (26)  = Op*, V p , p > 1 andpinteger;s*  T' < T * < T Set of activities that are equal  (27)  (28) (29)  p  The model above is an SLP with simple recourse and it is solved using a deterministic approach whenever the random variables are discrete. The new variables y and y" are introduced as the recourse variables. +  They in fact represent recourse variables for each outcome and appear in both objective function and some constraints that outline the random nature of the problem, in this case the critical and near critical paths. The penalty function can be decomposed for the critical and "near critical" paths as follows:  C, —> C, and C- to account for positive and negative deviation on the critical paths +  C* —> C* and C*" to account for positive and negative deviation on the "near critical" paths +  Although the model may not be the best solution of solving the problems related to "crashing" activities in order to fast-track a project due to many factors such as weather, it provides a little bit of flexibility in implementing a plan that will ensure that the activities are "crashed" within the paths of interest. The model is applied to a case study in chapter 5. The application shows the use of simulation to derive joint probability distributions.  51  Chapter 4- Resource Allocation Model for Fast-tracked Construction projects  4.6 summary In order to effectively applied the model, several steps must be followed: •  Build the network corresponding to the project  •  Analysis of the paths involved in the project  •  Identification of paths that share the same activities  •  Identification of the activities shared  •  Analysis of the critical and "near critical" paths  •  Estimation of costs related to activities and paths  •  Solution of the deterministicXP  •  Estimation of the probability distribution of the activities  •  Simulation  •  Determination of joint probability distribution  •  Solution of the stochastic LP  •  Conclusion  The accuracy of this model depends on the accuracy of the techniques used to derive estimates and probability distributions. Although in this model the uncertainty related to cost estimates was not considered for simplification, a complete stochastic model will consider introducing probability distributions for costs.  52  Chapter 5- Case Study  CHAPTER 5 CASE STUDY This chapter treats a real mine construction project case. The application of the resource allocation model is applied to emphasise the difficulties related to fast- tracking a mine construction project. This chapter is divided in three parts: •  the description of the project  •  the highlight of the project success and difficulties during its execution  •  And an application of the resource allocation model to fast-track one portion of the project.  5.1 Description of the project The project is divided into subprojects and Table 5.1 gives a summary of the project construction schedule. Table 5.1 Summary of the Construction Schedule Area 10 11 12 15 16 18 19 21 22 23 24 25 26 27 28 29 36  Subproject Description Plantsite Primary Crusher Coarse Ore Stockpile Conveying Grinding Facilities Leaching Facilities Refinery Shop and Warehouse Security/Change House General office Assay Lab. Cold Storage Building Geology/Core Logging Power Supply Tailing/Heap Leach Open Pit Dewatering System Security System Startup  Early Start 11 Jan. 1995 03 Jun 96 11 Apr. 96 17 Jun. 96 17 Apr. 96 22 Apr. 96 15 May. 96 08 Apr. 96 17 Jun. 96 09 Jul. 96 26 Aug. 96 09 May 96 06 May 96 14 Nov. 95 02 Apr. 96 24 Jan. 96 29 Mar. 96 18 Nov. 96 20 Feb. 97  Early Finish 28 Mar. 1997 24 Jan. 97 22 Dec. 96 19 Feb. 97 22 Feb. 97 08 Feb. 97 28 Feb. 97 14 Jan. 97 22 Dec. 96 20 Dec. 96 14 Feb. 97 01 Aug. 96 31 Jul. 96 01 Oct. 96 01 Apr. 97 30 Jun. 97 13 Dec. 96 11 Apr. 97 28 Mar. 97  Areas 10, 28 and 29, the construction of the campsite, the open pit and the tailing facilities certainly lie on the critical path of the project. Figure 5.1 represents the bar chart of the construction schedule.  53  Chapter 5- Case Study  Jan. 95  Jan. 96  Jan. 97  Date  —• AREA 10 AREA 11 AREA 12 M M  AREA 15  M M  AREA 16  M M M M _  AREA 18 AREA 19 AREA 21 AREA 22 AREA 23 AREA 24 AREA 25 AREA 26 AREA 27 AREA 28 M H  AREA 29 AREA 36  Figure 5.1: Bar chart representation of the project construction schedule  The decomposition of the construction project into subprojects defined by areas is the follow up of the work breakdown structure (WBS) and allows a better control over the parameters involved in the entire construction procedure.  54  Chapter 5- Case Study  5.2 Project Data Summary 5.2.1 M i n e  The initial plan included 79 million tons of prestripping occurring over the 15month preproduction period. The initial mining cost was between $0.55/ton and $0.607ton, but was to average $0.75/ ton over the life of the mine (excluding dewatering). The dewatering costs were estimated to be about $0.20/ton. The mine fleet in the initial plans consisted of 15 240- ton trucks, two electric shovels with 43 cubic yard capacity, one hydraulic shovel with 21 cubic yard capacity, and ancillary equipment. The mine and mine maintenance work force was to peak at approximately 450 persons, with a steady decline in the work force after Year 4 of the project. Heap Leach/Tailing Disposal A combined tailing/heap leach facility that will provide additional recovery from heap leaching of lowgrade ore and storage of tailing plant was to be built. Tailing Storage and Heap Leach Facilities The facilities were to be fully lined with a double system.  5.2.2 P l a n t  The processing facility was designed for a milling capacity of 9280 tons/day. Processing the ore consisted initially of primary crushing, Sag and ball mill grinding and classification, gold absorption by carbon-inleach (CIL), carbon stripping/reactivation/refining, CCD tailing wash and tailing treatment.  5.3 Budget Cost Monitoring 5.3.1 B u d g e t  Highlights  Most of the activities in the project were performed under budget except for a few of them that were over budget. The direct costs for example were $US 215,117,106 which are 13% under the estimated budget allocated to direct costs. The indirect costs were 27% under the estimated budget allocated. The activities over budget are the piping installation, the electrical installation, the structural steel and the prefabrication building. During the construction period, new needs and unforeseen items were identified from the feasibility study. Additional funds were required to purchase or performed jobs not included in the original scope of the project. Considering expenses higher than $US 100,000, the total amount approved  55  Chapter 5- Case Study  for items not budgeted was $US 13.7 million. For example additional 10 million tons pre- stripping was required at a cost of $US 6,474,446, the acquisition of a dust controller unforeseen in the scope of the assay lab. Building cost $378,677. A project always includes items that are under or over budget because it is difficult to assess with exactitude the cost of any item. The success of the project feasibility study depends of the efficiency of the team performing the feasibility study. In the case of the additional prestripping, not only some material that (300,000 tons) was initially ore is now pre- stripping material, the project could have been delayed if one additional Dresser 830E 240 t truck has not been purchased to accelerate the schedule of the pre- stripping stage. If the underestimation of costs may not in general impact the schedule of the project, the underestimation of the scope of work assigned to activities may cause a project to be delayed, prompting the use of additional resources to maintain the initial schedule. Table 5.2 shows the variances on the budget per area. Table 5.2 Summary of Project Budget Variances by Area Area 10 11 12 15 16 18 19 21 22 23 24 25 26 27 36 40 50 51 52 53 60  Description Plantsite Primary Crushing Coarse Ore Stockpile Conveying Grinding Facilities Leaching Facilities Refinery Shop and Warehouse ChangeHouse/ Safety Facility General Office Assay Lab. Miscellaneous Building Open Pit Power Supply Dewateriiig System Accommodation Construction overhead Operation overheads Project Management Design and Engineering Warehouse Inventory  Budget 8,029,880 3,966,060 2,050,940 2,327,420 23,162,917 21,385,980 9,350,360 14,620,314 717,345 1,483,420 1,665,600 382,000 109,312,953 7,349,630 15,861,402 2,972,700 11,944,850 2,028,000 10,079,300 12,298,660 7,197,314  (over)/under (200,270) (1,105,698) 262,421 (480,383) 3,246,023 4,474,713 (4,262) (123,002) (397,172) (420,009) (757,133) (32,792) 15,206,061 876,298 3,524,742 2,729,193 7,530,359 623,946 1,478,750 3,570,015 (517,659)  The major over budgeted Area is the construction of the primary crusher (Area 11) and the major under budgeted Area is the open pit (Area 26).  56  Chapter 5- Case Study  5.3.2 Impacts of changes on the project budget One of the reasons why Area 11, the primary crusher, was over budgeted is that some changes has to be made in the design of the retaining which size had to be increased due to. This is one of the reasons why the design team has to be very efficient in order to minimize as much as possible changes that may occur during the execution phase of the project. As a result of the change in the design, an additional $US 1,105, 698 had to be injected into the project to correct the design problem. The relocation of the primary crusher might have even delayed some activities within Area 11.  5.3.3 Impact of underestimation on the project budget Another reason why Area 11 was over budgeted is due to the underestimation of the quantity of the platework and the cost for installation of the cable tray in the conveyor. For the platework the initial budget predicted a quantity of 43 tons of platework as opposed to 124 tons almost three times the initial estimate. The cost for the installation of the cable tray in the conveyor was initially estimated to $US4, 510 versus $US 47,310 for the actual cost. One a main reason why a project may fail after its execution phase has commenced is underestimation of costs and quantities. In fact a project underestimated may present in theory high returns and at a general surprise produces high losses in return. That is one of the reasons why chapter 3 of this thesis highlighted the necessity of performing good estimates. At the over end, overestimation although it limits the project financial flexibility may not impact the project and turn it to losses. The project may not start because of the theoretical high costs cause underestimation. In the case of Area 26, the open pit, the under budgeted situation appeared mainly because the cost of some items such as the cost of equipment.  5.4 Overall Project Performance The project overall performance was excellent. The construction project was completed three months ahead of schedule allowing operation to fully start three months earlier. Several factors contributed to this early completion but not without additional costs. In fact accelerating the completion time of a project involves bringing more resources into the project as mentioned in chapter 3. If the overall performance of the project was excellent, some activities within the project did not go well as forecasted by the initial design. Table 5.3 describes the performance level of some crucial activities involved in the construction phase.  57  Chapter 5- Case Study  5.4.1 Performance factors The majority of the subprojects were delayed mainly because of delays in the delivery of construction equipment and material. Despite the delays noticed the project was completed well ahead of schedule. This is first due to the fact that the delays except for a few of the activities were not greater that 14 days which were acceptable. Another explanation of the early completion of the construction project despite the delays resides in the estimation of the activities duration that may have been over estimated or the allocation of higher floats balancing therefore the delays, assuming that the contractors were efficient. Subprojects with good performance  Among the subprojects that performed very well, the following are highlighted: •  Construction of the dewatering well  •  Design/Supply and Installation of Temporary Power  •  And Retaining Wall  Construction of the Dewatering well The work or activities performed under this subproject consist of drilling and installing 21 production dewatering wells ranging in depth from 455 feet to 1090 feet. The construction of the dewatering wells was completed in only 8 months. This early completion not only allowed the next subproject, the installation of the security system, to start 2 months earlier, but also reduced the cost of using labour within this time frame. The subproject was completed with very little variation of its allocated budget. The original budget allocated to the subproject was estimated to be $US 4,361,741 and the final budget was $US 4,400,441 which represents a variation of $US 38,700. The contractor who was awarded this job had excellent equipment and support and its labour were very experienced. Retaining wall The retaining wall subproject was initiated for the crusher Area. It ultimately included the tailing pipe and corridor and final site grading. The subproject was awarded to a contractor which bid a value of $US 2,398,214 for a completion time of 3 months. The subproject was delayed due to a change in the design of the retaining wall. The contractor performed very well despite the delay that was not related to his work. Their workers work in an efficient manner despite the tight schedule. The overall delay due to design was 1 month, which is less greater than the earlier start time allocated to the successor of the retaining wall activity, the installation of stairs in the crusher building. The contractor started the construction of the  58  Chapter 5- Case Study  tailings pipe and corridor 5 days before the completion of the retaining wall. Considering that the labour is mobilized at least 48 hours prior to the actual start of the work, one month represent 4 weeks at a labour cost rate of $US 23.28 per hour for 50 hours a week for a crew of seven. The total cost is estimated to be $US 651.84 that the one month delay costs to the contractor. The best approach to dealing with such delays is to reallocate labour to other area while waiting for the actual start time. Subprojects with bad performance Among the project that did not performed well the following subprojects can be considered as illustrative examples: •  Supply & Install security System  •  Supply and installation of buried electrical services  Supply & Install Security System The contract of this subproject was awarded to a contractor who bid $US 107,790 but was unable to complete the project on schedule. In fact the project was delayed by almost 2 months due to the inability of the contractor to coordinate the work on site. The performance of its labour was poor. Although this delay did not result in high costs, it delayed the Start up.  Supply and installation of buried electrical services  The delays observed for this subproject resulted mainly from the lack of continuity and the late delivery of the cable necessary to perform the work. The contract was awarded to a company that bid $US 298,343 to obtain the contract. The original budget allocated to the subproject was estimated at $US 377,388, and the final contract value was evaluated at $US 403,045 which corresponds to a variation of $25,657 about 68% of the original budget. The delays are certainly the justification of the budget variation. Since there was no co-ordination in the work, the resources were not rationally used.  Most of the delays observed in this case study were due to late delivery of supply and the changes implemented to some activities during the construction phase of the project. In fact care should be taken in order to minimize changes from the beginning of the design phase. The management team should make sure that it deals in the future with contractors that will guarantee the completion on schedule of its  59  Chapter 5- Case Study  project. The risk of not completing the project on time along with the penalties should be shared between the contractors and the company.  5.5 Fast- tracking the project Although the project was not intended to be fast-tracked, it was completed 3 months ahead of scheduled. The early completion of the project was probably due to overestimation of durations or favorable unplanned environments. Now supposed that the project has been actually scheduled to be fast-tracked and that area 10 is to be completed 95 days ahead of its original completion date. This is a hypothetical fast-track of area 10 to illustrate the use of the resource allocation model in real mine construction problems. Table 5.3 describes the activities involved in the construction of area 10. Table 5.3 Detailed description of activities involved in area 10 Activity Description Civil Fencing U/G Pipe Electrical Excavation Concrete U/G Piping Tie- ins Prefabricated Buildings Structural Field Erected Tank Piping Fibre Optics, Phone  Early Start 18MAR96 15JUL96 01APR96 24JUN96 04MAR96 11JAN96 08JUL96 26AUG96 19MAR96 29MAY96 18MAR96 25NOV96  Early Finish 19APR96 02APR97 24JUN96 22DEC96 26JUL96 28MAR97 30AUG96 30AUG96 29MARS96 15NOV96 30AUG96 13JAN97  Table 5.2 is a detailed description of the activities involved in the construction of area 10. The next step involved in crashing the activities is the analysis of activity relationships such finish- finish, finish- start or start- start relationship. Table 5.2 is simplified by identifying activities of the same nature and using the relationship between activities. The simplification is given in table 5.4.  60  Chapter 5- Case Study  Table 5.4 Simplification of Table 5.2 Activity 1-2 3-4 3-5 4-5 4-7 5-6 1-3 2-7 6-7  Activity description Concrete Civil + Excavation + Structural Piping + U/G pipe + Electrical + Prefab. Buldg. Field erected tank Fencing Fiber optics + Phone Dummy Dummy Dummy  Duration (days) 369 134 262 109 257 49 0 0 0  Figure 5.2 is the arrow on arrow representation of the activities involved in area 10, the plantsite construction.  Figure 5.2: Arrow on Arrow representation of activities in Table 5.3  The network given in figure 5.2 is composed of 4 paths {1-2-7, 1-3-4-7. 1-3-4-5-6-7, and 1-3-5-6-7} with respective lengths {369 days, 391 days, 292 days, and 311 days}. The longest path, the critical path, of this project is represented by path {1-3-4-7}, which is composed of civil, excavation, structural and fencing. Hence the total duration of area 10 is 391 days. If area 10 was to be completed 95 days ahead of schedule, then the new task for management is to ensure that the total duration for area 10 be 296 days. In order to achieve this goal, the resource allocation model suggests that the near critical paths and activities be identified. Figure 5.3 shows activities that are "near critical" in dotted dark arrow.  61  Chapter 5- Case Study  For simplicity, The following notation is adopted:  •  Path {1-2-7} = Path # 1  •  Path {1-3-4-7} = Path #2  •  Path {1-3-4-5-6-7} = Path # 3  •  Path {1-3-5-6-7} = Path #4  Table 5.5 is the summary of the criticality and near criticality of the activities and paths involved in area 10. Path #1 and path # 3 are "near critical" whereas path # 2 is critical. Activity 3-4 is critical and "near critical" and path 4 is neither critical nor "near critical". Table 5.5: Summary of activities characteristics  Activity 1-2 3-4 3-5 4-5 4-7 5-6 1-3 2-7 6-7  Critical? No Yes No No Yes No Yes No No  Near critical? Yes No Yes No No Yes Yes Yes Yes  Path# 1 2 and 3 4 3 2 3 and 4 2,3 and 4 1 3 and 4  62  Chapter 5- Case Study  Table 5.4 indicates that activity 1-3 is near critical and critical at the same time. Remark that should be taken into account in the optimization problem in the next section. But activity 1-3 along with activities 27 and 6-7 will not have any effect on the computation because they represent dummy activities (activities with duration equal to zero).  In order to achieve the goal of bringing the project length from an initial plan of 391 days to 296 days, the following actions must be taken:  •  Path # 1 must be "crashed" by (369 - 296) = 73 days  •  Path # 2 must be "crashed" by (391- 296) = 95 days  •  Path # 4 must be "crashed" by (311- 296) = 15 days  Path # 1 is composed of only one activity and therefore the number of days this path must be "crashed" is directly allocated to activity 1-2. The number of days path # 2 and # 3 must be "crashed" is allocated respectively to activities {3-4 and 4-7} and {3-5 and 5-6}. Let X and Y be respectively the number of days path # 2 and path # 4 must "crashed" in order to fasttrack area 10. Then X is decomposed in two values X i and x , and Y in two values y i and y where these 2  2  values correspond respectively to the number of days, activities {3-4, 4-7} and {3-5, 5-6) must be "crashed". If C is the cost of crashing activity 1-2, C , C , C , and C the cost of crashing respectively )2  34  47  35  56  activities 3-4, 4-7, 3-5 and 5-6. Then the expression of the total cost function is given by: TC = 73 C  12  + x,C  34  +x C 2  4 7  + v,C  35  + yC 2  56  5.5.1 Costs estimation The methods and data used to estimate the costs are explain in appendix B and the results are displayed in table 5.6. Table 5.6 Results of the estimation of the "crash" costs Parameter C C  Estimate (US $/Day) 650 1 800 C4.7 1050 C 8 00 C 1600 The maximum number of day activities that can be crashed is displayed in Table 5.7. 12  34  35  56  63  Chapter 5- Case Study  Table 5.7 Description of variables' upper limit Variables  Upper limit (days) 45 85 55 8  Xl  x yi y  2  2  These values as mentioned in chapter 4 are usually obtained from an expert system like questionnaire. In this case study the data were estimated based on the budget and the variances related to each area of the same nature. 5.5.2 Deterministic L P problem Provided with the information above, the deterministic LP problem is formulated as follows: Minimize Subject to:  TC = 73 C  12  +^^34 +x C 2  4 7  + yC x  35  + yC 2  56  x,>0, >0 y j  x, < 45, x < 75, y < 55, and y < 8 2  Xj  l  2  + x = 95 2  Solution of this deterministic LP gives the following results shown in Table 5.8. Table 5.8 Results of the LP problem Variable Xl  x yi  2  yi  Value 20 75 15 0  Activity 3-4 4-7 3-5 5-6  The resource allocation model suggests that in order to bring area 10 to a total length of 296 in the most economic manner, only 4 activities (1-2, 3-4, 4-7 and 3-5) must be "crashed". Activity 3-4 (Civil + Excavation + Structural) should be crashed by 20 days, activity 4-7 (Fencing) must be accelerated by 75 days, activity 1-2 (concrete) must be crashed by 73 days and activity 3-5 by 15 day. Activities 4-5 and 5-6 remain unmodified. The model ensures then that unnecessary jobs are not performed; for example activity 5-6 that was "near critical" has not been crashed. Provided with these values, it is the task of management  64  Chapter 5- Case Study  to allocate the proper resources to each activity. The task is defined by the combination of different resources that is in the limit of the costs estimated previously. For example additional resources may be a combination of more labour + overtime + equipment. The results of the LP model depend greatly on the estimates provided relative to the information collected on each individual activity. That is why estimates play an important role in fast- tracking a project. For this project, it was underlined that some subprojects that were not intended to be fast- tracked were completed ahead of schedule or the budget allocated to them was either over or under the limit. Now suppose that the variables are in fact randomly distributed around the values determined by the deterministic LP. Then the LP problem can be put into a stochastic LP problem that takes into account some of the uncertainties related to the construction process of area 10. The equivalent stochastic LP problem is posed as follows:  Minimize TC — Z j C + x C 1 2  Subject to:  l  34  + I PI( I-^) e  > 0,  C  Y j  +x C 2  4 7  + C (x  Z  p2  x  + yC x  + yC  35  2  +x -e ) 2  56  + C ^y  2  p  x  +y  2  -e )] 4  >0  < 45, x < 75, y, < 40, and y < 8 2  2  + x = 95 2  +y  2  =15  = 73 The introduction of the variable ei, e and e4 is to account for the respective deviation path #1, # 2, # 4 2  from their target determine by the deterministic LP and C i, C and C represent the penalties associated p  p2  p4  with the failure to "crash" respectively path # 1, # 2 and # 4. E [.] is the expected value function. The introduction of variable Z] in this equation is to account for the fact although path # 1 is composed of only one activity, the crash time allocated to it is note known with certainty as in the deterministic case where it had a value of 73 days corresponding to the amount days to crash path # 1 in order to fast-track area 10. The problem can be reformulated as follows:  65  Chapter 5- Case Study  Minimize TC = Z , C  ] 2  + *iC  +  + ^2^47 ~*~ 3^1 ^--35 ~*~ ^ 2 ^ 5 6  34  [C P e +C P e +C P e ] p]  ]  ]  p2  2  2  p4  4  4  Subject to: x, >0,y  >0  }  x, < 45, x < 75, y, < 40, and y < 8 2  Xj  2  + x + e = 95 2  2  z, + e, = 73  The scalars P P and P3 represent the respective joint probabilities of path #1, #2 and # 4 with respect to ];  2  their deviations from their initial target (crash time determined by the deterministic LP problem). These probabilities are determined by the use of the Monte Carlo simulation techniques. The simulation techniques take into account the probability distributions of each duration shown in Tables 5.9 and 5.10 that are not generally available but could be obtained by questioning experts. The number shown below are estimates based upon comparison of actual versus forecast times in area 10.  Activity 1-2  3-4  3-5  4-5  4-7  5-6  Duration 338 358 369 385 95 115 134 164 237 262 282 83 95 109 138 230 257 310 40 49 62  Table 5.9 Activity probability distribution Probabilities 0.1 0.25 0.35 0.3 0.1 0.2 0.4 0.3 0.2 0.5 0.3 0.1 0.2 0.4 0.3 0.3 0.5 0.2 0.3 0.4 0.3  66  Chapter 5- Case Study  Table 5.10 Crash time probability distribution Time 62 73 81  Crash # 1 Probability 0.2 0.5 0.3  Time 86 95 110  Crash # 2 Probability 0.15 0.65 0.2  Time 10 15 22  Crash # 4 Probability 0.1 0.6 0.3  Table 5.8 highlights the fact not only the activity times are random, but also the crash times found by solving the deterministic LP problem. Figure 5.4, 5.5 and 5.6 represent respectively the histograms of the deviation observed for path #1, #2 and #4 from their target after the probabilities in Table 5.7 and 5.8 were simulated and Replicated 1000 times. Histogram for Path # 1 220  120.00% 100.00% 80.00% 4- 60.00% 40.00% 4- 20.00% I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I II I I I I I I .00%  Q  <6  <0  $>  r§>  rfi  rfr  &  <V  Jb  >  ^)  N  Deviation (days)  fc  K W Frequency —a—Cumulative %  Figure 5.4: Path # 1 deviation histogram  67  Chapter 5- Case Study  Histogram for Path #2  0  0  1  t  M ' - M ( J ( t l O N l D  h n t N < D t ^ N  O l  O  ^  K  I  1  1  0  120.00%  T - i - i !M i t U) (O  Deviation (days)  Figure 5.5: Path # 2 deviation histogram  Histogram for Path # 4  110.00% 100.00% 90.00% 80.00% 70.00% 60.00% 50.00% 40.00% 30.00% 20.00% 10.00%  o  r-~ CD  I I II I II I I II M IlI11IIII II I I II I I II .00% I II I I II I II I I II  • * i - i r > o t ^ T - c o T - T f c o c N c o o t ^ c o a > ' * •  I  ^  C  M  C  I  M  C  I  O  O  I  O  C  I  M  C  I  N  T  I  -  '  1  T - T - C M C N C O I  Deviation (days)  1  I  I Frequency  —m— Cumulative %  Figure 5.6: Path # 4 deviation histogram  68  Chapter 5- Case Study  The simulation gives the following results: • • •  Probability (Path #1 deviates from target) = 80% Probability (Path #2 deviates from target) = 89% Probability (Path #4 deviates from target) = 87%  The results are dependent of the probability distributions given on Table 5.7 and 5.8. Different values will have certainly given different results. The simulation also provides upper and lower limits for the deviations factor e i , e and e4. The limits also play an important role in the solution of the stochastic L P problem. 2  The solution of the stochastic L P problem is displayed in Table 5.11. The results are slightly different from the ones in the deterministic problem because the stochastic problem takes into account the probability distributions of the activities. The results for the stochastic problem shows that in order to fast- track area 10 activity 3-4 needs, for example, to be "crashed" by only 8 days whereas in the deterministic case, the same activity needed to be "crashed" by 20 days. A comparison between the results of the two LP problems is given in Table 5.12. Table 5.11 Results of the stochastic L P problem Variable  Value 8 75 25 0 63 10 12 -10 158309  Xl  x  2  yi  Y2 Zl  ei e2  e TC 4  Description Activity 3-4 Activity 4-7 Activity 3-5 Activity 5-6 Activity 1-2 Deviation factor for path # 1 Deviation factor for path # 2 Deviation factor for path # 4 Total cost ($US)  Table 5.12 Comparison between the stochastic and the deterministic L P results Variables Xl  Deterministic results 20  Stochastic results 8  x  75  75  15  25  0  0  73 174200  63 158309  2  yi  Zl  TC  69  Chapter 5- Case Study  The only activity the stochastic and the deterministic method agree on is 4-7 with the number of days to crash 4-7 equal to 75 days. The number of days to crash the rest of the activities differs for both techniques. The differences observed in the results depend greatly on the probability assign to each activity. Activity 3-4 in the deterministic case has to be crashed by 20 days whereas in the stochastic approach the probability distribution of the completion time of path # 2 is considered and shows that the completion time of path # 2 may have been overestimated by e = 12 days and therefore activity 3-4 2  should be crashed by only 8 days in order to fast- track the project. The same remarks are applied to activity 1-2 that is to be crashed by 73 days in the deterministic approach and by 63 days by the stochastic method because of an overestimation of the completion time of path # 1 by ei = 10 days. Activity 3-5 is to be crashed by 15 days if the deterministic approach is used whereas it has to be crashed by 25 days if the stochastic method is applied because the introduction of the probability distributions suggest that the completion time of path # 4 is underestimated and will take 10 additional days (e = -10 days) to be 4  completed. A negative deviation value indicates an underestimation and a positive value an overestimation. Deviations will always exist since the methods through which estimates are derived contain margin of errors. In the stochastic techniques, the joint probabilities that the deviations are non zero is used to simplify the problem. The right method for this simple recourse problem is to introduce the joint probabilities for positive and negative deviation that will take into account overestimation an underestimation directly in the equation for the total cost TC. For example the deviation, e, could be divided into e, and e[ respectively the positive and negative deviation of critical or "near critical path /'". +  The deviation factor e is in fact the average deviation factor.  70  Chapter 6- Conclusions and Recommendations  CHAPTER 6 Conclusions and Recommendations 6.1 Thesis review The objective of this thesis was to explore the time and their economic performances of fast-track strategies on mine construction projects and develop a resource allocation model that will support the fast-track strategies. The strategies examined were the overlap of the design and the construction phases of a mine construction project, the compression of its design and construction duration, and the allocation of resources within the project that will ensure the success of the fast- track process. Despite the scarcity of information related to fast-tracked mine construction projects, several points were demonstrated. The work was organised in four parts. The first part defined a mining construction project model and derived the upper bound of the construction cost equation for the model based on a net present value model derived from an arbitrary mine data. The second part identified the factors that may affect the process of fast- tracking a mine construction project. The third part used a resource allocation model to show how resources, in terms of time, can be allocated to a mining project in order to minimize excessive "crashing" costs and unnecessary "crashing" exercises. Finally the fourth part was a case study of real data obtained from a Canadian mining company. The data were used to support the theory developed about fast- tracking mine construction projects. The sensitivity analysis of the upper bound cost functions was performed by plotting graphs (one at- a- time sensitivity and surface plots) and the Monte Carlo simulation technique was used to generate probability distributions of activity times and duration. Matrices were introduced in order to generate values for the multidimensional functions.  6.2 Findings and conclusions Despite the scarcity of information related to fast-tracked mining projects and some assumptions made to develop the economic and the resource allocation model, it was possible to demonstrate several points.  6.2.1 Findings about the economic model Fast- tracking a mine construction project is possible and has been demonstrated to be successful (in theory) under the assumptions made. Moreover, it is possible if provided with enough information to  71  Chapter 6- Conclusions and Recommendations  build an accurate economic model for fast-tracked mine construction projects that are able to respond to any changes at the vector fast-track components level. Today however, the availability of such information is a major challenge in the mining industry. To this date, there is no integrated system that allows a mining engineer to search for information about costs related to a specific activity in a given environment. The economic model has provided us with an upper bound of the cost of construction for a fast-tracked project with respect to the magnitude of the variables of the fast-track vector. The upper bound represents a target that should be aimed at when considering fast-tracking a project. In fact it is a company responsibility to define the interval in which its cost estimates (premium) may fall in. Hence, given an upper bound for a specific duration to achieve, the task becomes to develop a fast-track strategy to achieve the desire duration and estimate its costs. If it falls below the upper bound, it is an economically satisfactory solution, although not necessary an optimal one because some values above the upper may also be feasible in the company point of view and therefore, preferred. It is important to emphasise that the model does not give a recipe for the cost of construction for fasttracked projects but guides in finding better cost estimates that are suitable for the need of the project based its resources. The model has confirmed three specific points: •  The premium requires to fast-track mining projects is an increasing function of the overlap and compression factors (related to variables Xi, X2 and x ) and that the premium function is 3  more sensitive to the compression of construction duration (variable xj). •  Overlapping production and construction phases without processing phase may have a negative impact on the net present value of the fast-tracked project.  •  Anticipating a start-start relationship and then overlapping construction, production and processing phases increase the net present value.  6.2.2 Findings about the resource allocation model The resource allocation model developed in this thesis take into account the other available model for resource allocation such as the heuristic model for resource allocation before a decision is made to fasttrack the project. Once the initial resources have been allocated then the model build in this thesis is used to allocate additional resources necessary to fast- track the project at least costs. The model assumed the knowledge of data related to the cost of activities related to labour, overtime, equipment... etc. Most of the literatures always supported the idea that it is the critical activities that should be "crashed" in order to reduce a project completion time. However, this model has underlined three essential points: 72  Chapter 6- Conclusions and Recommendations  •  It is not only the critical activities that should be crashed, but also the "near critical activities".  •  Not all the critical and "near critical" activities must be "crashed" in order to fast- track a project. The model, based on the information provided discriminate activities to crash. We might think of defining "crashable" activities as activities that are candidate for "crashing".  •  The model allows a project to be fast- tracked at minimum cost by suggesting the activities to crash and the resources allocated to these particular activities.  The stochastic part of the model provides greater flexibility due to the fact that the data are estimates that may be subject to errors. Although most of the cost functions were assumed to be deterministic in the model for simplification, they are in fact random and subject to estimation errors that should be taken into account for a more realistic model. A successful application of the model requires accurate estimates that are nearly impossible to realise.  6.2.3 Findings on the case study The case study supports the fact that the application of these two models to a real mine is not an easy task. In fact the difficulties appear in the estimation of costs. The cost functions are not usually linear and therefore a good estimate relies on the data that are available. The estimates used in this case study are based on the variances observe between the predicted and the actual values of the costs and then the average costs are derived based on the duration of the activity of interest. This way of estimating costs is very poor and does not take into account certain parameters such as productivity, availability, ...etc. The case study has shown that the use of the deterministic approach may lead to overestimation or underestimation of the additional costs needed to fast-track the project. In order to obtain accurate results from the application of the resource allocation model, mining companies should consider keeping records of their daily productivity data such as labour productivity, equipment productivity, ...etc. They may also consider recording all anomalies on the site during the progress of the project. These anomalies may be related to weather conditions, changes during the construction phase, and other unforeseen factors. The data should be recorded for smaller workpackages as they may yield more accurate data information. Recording these types of data should not generate important additional costs since they may be inserted into the every day routine of the engineering team. Data also related to the number of a particular resource needed for a given activity should be logged along with the variation of these data with respect to the amount of the resource. The use of faster computers and the application of simulation techniques may help improve the computation of estimates. The use of simulation techniques may be preferred as oppose to finding a probability distribution that will fit the actual distribution of the data.  73  Chapter 6- Conclusions and Recommendations  6.3 Recommendations for Future Research The mining industry today is being hit by the decrease in metal and other mineral prices so that some mines have no choice other than to close and wait for the prices to return to profitable values. The mines that closed or are on the path of doing so did not in general predict such fluctuation in the mineral prices. Faced with these uncertainties, it is vital for mining companies to find new ways of operating their mines. Mining companies have been operating their mines the same way as they used to when the mineral prices were profitable with some little improvements that failed sometimes to offset the dramatic decreases that are being observed today. Today, new methods of operation and processing are needed to keep the mines operating and the optional sequencing of operation may be a preliminary solution toward the methods of operation of tomorrow. The traditional sequencing, briefly described in Figure 6.1, consists of processing the ore that is being mined at a fixed rate inducing hence the mill to keep operating until the mine is not economically viable independent of the spot price of minerals on the market. In this process, the mine keeps operating as long as the net present value is positive. The plant with its relative fixed processing rate should be run at a maximum capacity in order to maximize the return on the equipment. But is it 1  really necessary to run the mill at maximum capacity? In the traditional case, yes it is important to run the mill at maximum capacity, because the driving factors are the net present value and the return on the cost of building the plant. This approach could have been once, the way to go provided with the fact that the relative high prices of mineral on the market were producing high returns for mining companies.  Operation Figure 6.1: Traditional sequencing of mining operation and processing phases The operation phase in the traditional approach is restricted by the constraints imposed by the processing plant such as ore grade, quantity ...etc. The operation phase is carried out in a way to ensure the plant maximum capacity sustainability. With the fluctuation of the price of minerals on the market, new methods should be sought in order to reduce the operating costs of mining projects. The new approach to mining operation and processing phases proposed is summarized by Figure 6.2.  Some companies have succeeded in increasing their plant production rate making it nearly flexible. But in general, the plant production rate is considered fix and inflexible. 1  74  Chapter 6- Conclusions and Recommendations  Ore Zone  Operation Figure 6.2: Representation of the proposed approach  Processing Plant  The new approach consists of dividing the operation phase into sub- operation phases and the processing phase into processing options and finally assigning to each ore block a sub- operation and a processing option. Figure 6.3 represents blocks within an ore zone. In order to access block K, blocks K l , K2, K3, K4, K5 etc... should be removed. Once block K is removed, it could be sent to the crusher or stockpiled if its grade is above the cut-off grade ore be sent to the waste dump. A sub-operation with the operation phase could be assigned to block K.  Figure 6.3: Illustration of a sub-operation phase associated to a block within an ore zone For example sub-operation K consists of extracting blocks K l , K2, K3, K4, K5, etc... and then extracting block K. The success of sub-operation K depends on how the constraints requirement of the blocks surrounding block K are fulfilled. The introduction of sub-operations into the mining operation phase may help in its optimisation. A sub-operation may be characterized by: •  The nature of the rocks and its geometry  75  Chapter 6- Conclusions and Recommendations  •  The technique available (equipment type)  •  A sub-operating cost (Cost of extracting)  •  The block grade (for its destination)  Dividing the operation phase into sub-operations may help build a better cost estimates because this procedure unlike the others deal with the ore zone at the block level, yielding therefore more accurate information. Once the sub-operations are determined, the block have to be sent to one of the three options (crusher, stockpiles or waste dump). The processing options determine the future of the blocks. These options may be determined by some business strategies in order to optimise return. These business strategies depend greatly on the commodities' market. A processing option for a block that is identified as waste material is "send block to waste". In other word, the option is not to sent this block to the processing plant or stockpiles. If the block is identified as ore, the strategy may dictate that the block be sent to the crusher or be stockpiled for future use according to the mineral prices. The processing options are therefore divided into two categories that are: •  A certain number of blocks are processed because of their grades (for example high grades blocks)  •  Some blocks are stockpiled for future use such blending with other materials or processed when the price of minerals is low.  Introducing processing option into mining processing phase requires the processing plant to display a certain degree of flexibility such as the option to vary production rate. This type of flexibility can be made obtain with the introduction of modular plant components and multiple plant circuits.  In fact the  "inflexibility" of the mill can be resolved by the use of modular equipment that have the flexibility to be added or removed when needed. The compression of the operation phase becomes optional. Based on the price of metals on the market, a mine may want to maximize his concentrate production in order to enjoy a sudden hike in metal price. The option to increase or decrease production by adding or subtracting components, Popplewell (1992), if successfully applied in real life may be a breakthrough in the mining industry. Like modular components used for campsites, the modular milling components may help reduce the construction time of plants. According to Summit Valley Equipment Incorporation, an American engineering company specialised in modular plants, modular plant components are not only easier to install, they may also reduce the construction time of the plant itself while offering greater transportation flexibility. The use of multiple circuits may also contribute to greater flexibility in the operation phase of a mine. A mine may  76  Chapter 6- Conclusions and Recommendations  decide to increase or decrease its processing rate based on the price of metal. Hence, the processing and the operation options are not dependent of the processing rate.  Figure 6.4: Description of a Turnkey plant by Summit Valley Inc. If the modular plants components can transported to the site and the assembled, then not only construction time is saved, but the start- start relationship between processing and production phase can be obtained justifying then the conclusion made in chapter 2 with respect to the sensitivity of the net present value of the fast-tracked project.  77  References  References Ahuja H . N., Dozzi, S.P. and Abourizk, S. M . (1994)- "Project Management second edition(Techniques in planning and controlling construction projects)". John Wiley and Sons, New York, N . Y, pp. 191- 208. Atkinson, T. (1987)- "Risk Analysis Techniques for Mine Project Financial Appraisal". University of Nottingham. Mining Department Magazine, Vol. 34, pp. 71-86. Decko, R. F., Hebert, J. E and Verdini, W. A (1991)- "Cost Based Allocation of Resources in Project Planning". IEEE journal. Fazio, P., Mosheli, O., Theberge, P. and Revay, S (1988)- "Design Impact of Construction Fast-Track", Construction Management and Economics, 5, pp 195- 208. Ibbs, W. and Kim,S. (1991) - "Sequencing Knowledge for Construction Scheduling". Journal of Construction Engineering and Management, Vol. 117, No. 1, March. Laufer, A. and Cohenca (1990)- "Factors Affecting Construction-Planning Outcomes". Journal of construction Engineering and management. Vol. 116, No. 1, March. Malcom, J . A. Swallow, Ewanchuk, H . G. and Kynoch, J. B. (1998)- "Fast-track construction at Mount Polley". Mining Engineering, 50(11), 31- 35. Newman, D., G. (1996)- "Engineering Economic Analysis". Sixth Edition. Engineering Press. San Jose California. Popplewell, G. and Smith, J. (1992)- " The application of modular plants to diamond recovery". CIM Bulletin, Vol. 84. Reda, R. and Carr, R. I. (1989) - " Time-Cost Trade off Among Related Activities". Journal of Construction Engineering and Management, Vol. 115 No. 3, September.  78  References  Russel, A. and Ranasinghe, M . (1990)- "Decision Framework for Fast-track Construction: A Deterministic Analysis." Submitted to the journal of Construction Management and Economics, UK, June. Russel, A. (1984)- "Construction Planning Control - Civil 520 course note". University of British Columbia, Vancouver, B.C. Sen, S. and Higle, J. L . (1999)- "An Introduction Tutorial on Stochastic Linear Programming Models". Institute for operations Research and Management Sciences, Interfaces 29, pp. 33-61. Sproule, J. A. (1992)- " Fast-track benefits: fact or fiction?", Master of Applied Science Thesis, University of British Columbia, Department of Civil Engineering. Terwiesh, C. and Loch, C.  (1998)- "Measuring the Effectiveness of Overlapping Development  Activities". The Wharton School, Department of Operations and Information management, Philadelphia. Thomas, R. and Napolitan, C. L . , (1995)- "Quantitative Effects of Construction Changes on Labor Productivity". Journal of Construction Engineering and Management. Vol.121. No. 3. September. Willis, E. M.(1986)- "Scheduling Construction projects". Englewood Cliffs, N . J. : Prentice Hall. Winston, L . W., albright, S. C. and Broadie, M . (1999)- " Practical Management Science, Spreadsheet Modeling and Applications." Wadsworth Publishing Company, pp. 612-622. Whitcomb, F. and Kliment, S. (1973)- "On Track with Fast-Track." AIA Journal, February, pp. 45- 48. Y i , R. and Sturgul, J . R. (1999)- "Economic significance of sub-cutoff grade in open pits operating with concentrator bottleneck". Mineral Resources Engineering, 8, No. 2, 195- 203.  79  Appendix A  Appendix A At Interest rate Nominal and effective interest rates must be considered. For a given time period (e. g. year) the nominal interest rate, r, is the rate that applies to that time period without any specification of the number of compounding period intervals within that time period. The effective interest rate, i, reflects the number of compounding periods and is equal to the ratio of the interest charged in one year to the principal (Newman, 1996). In equation form, the effective rate i, is equal to: i=a+-> - i m In equation 2.1, r represents the nominal rate and m the number of compounding intervals. When m —> +00 in equation 2.1, z = g -1 for a given time period. The compounding factor in this r  case is continuous.  A 2 Present Worth and Future Worth Let P be the present worth of a cashflow and F its future worth, there exits a relationship P and F: F = P(l + i) , N  where i is the effective interest rate and N the given time period. F = P e , for rN  continuous cashflow profiles.  A 3 Continuous flows and continuous compounding The modeling of projects requires discounting arbitrary cashflow profiles. In practice, one would probably use a time period of a month, estimate each monthly cashflow, and then compute the present worth of a given cashflow using discrete compounding factors. However, the use of the continuous models can provide much greater modeling power and insights into the project behaviour. While not commonly used in practice, continuous models allow the treatment of very general cashflow profiles. Continuous models also allow a broader study of parameters, which is crucial to decision making. $/time c(t) ,  Figure A 1: Continuous cashflow representation  Time  80  Appendix A  Let r be the nominal interest rate such that i = e — 1, where i is the effective interest rate. The incremental contribution of the flow c(t).dt at the time t to the present worth at time zero is given by:  AP = e  • c(t) • dt  n  Summing the increments and proceeding the limit as At—»0, the following fundamental relationship is obtained: P=  \e -c(t)dt n  Then the present worth at the origin (time t = 0) is: Po = ~  rn  e  \ ~ '-c(t)dt r  e  t\ Equation 2.4 gives the present worth (at time t = 0) of the continuous cashflow c(t), that occurs between the time ti and the time t . 2  A 4 The capital expenditure model Let C represent the constant dollar estimate of a given project and T its duration. Assume that 0  inflation and financing rates will be constant during the project (not the case in general). C will 0  be expended according to a function C (t), and at any time t, 0 < t < T, the amount expended to 0  date is given by the cumulative expenditure function To^) :  Fo(^) = ^Co(r)dr  x  «Po(0  Cost ($)  CM/  *  w  T-t t  T-t  Figure A 2: Capital expenditure model  81  Appendix A  In reality, one is interested in actual or current dollar expenditures, as it is these dollars that must be found to finance the project. The actual expenditure at time t is equal to the expenditure Fo(^) v  adjusted for the fact that inflation has been running at a rate 6 for the time period t.  Figure A 3: Continuous cashflow profile of the traditional approach  Figure A 4: Continuous cashflow profile for the fast-track approach 82  Appendix A  C o s t parameters  The constant dollar costs are nominal and estimated in terms of today's cost, without consideration of the inflation index relevant to the industry. They are defined for time t as follows in Table 2.1: Table A 1: Description of the constant dollar cost parameters Parameters C (Y)  Description design phase costs construction phase costs  R (t)  revenue  E (Y)  production expenditure  Q  o  E  (t) 9 0 0 0  mineral processing expenditure Construction inflation rate Revenue inflation rate Production inflation rate Processing inflation rate  C R p  m  The current dollar costs however are estimated in terms of costs as it is applied in the future by taking into account inflation. The current dollar costs, described in Table 2.2, are estimated by applying the inflation rate to the constant dollar costs. Table A 2: Description of the current cost parameters for the traditional approach Parameters c (t) = d  Definition c  M  ( f ) V  C (t) =  C, (t)-e  R(t) =  R (t)-e «  c  e  i,+Sb)  0  E (t) = m  construction phase costs revenue production expenditure  P  E (t) =  ^  9A,+T  c  Description design phase costs  E (t)-e ^ 9  0m  mineral processing expenditure  83  Appendix A  The current cost parameters for the fast-track approach can be derived from the parameters in Table 2.2 by replacing the subscript "b" by "/', indication of the fast-track approach.  Time and duration Described in Table 2.3 are time parameters for the traditional approach. Table A 3: Time and duration parameters for the traditional approach Parameter  Description duration of the design phase  T  db  T  duration of the construction phase  T  duration of the revenue phase  cb  Rb  T b T  duration of the production phase  T  Duration of design + construction Start- start duration between processing and revenue  1  duration of the mineral processing phase  P  b  s  b  For the fast-track approach, we may define a six dimensional vector x = ( X j , • • •, x ) 0 < x, < 1, 6  called the "fast-track vector" in the rest of this chapter, that transforms the time variables of the traditional approach to those of the corresponding fast track approach as shown in Table 2.4: Table A 4: Time and duration parameters for the fast-track approach Parameters tdf =  (l-x,)T  T# =  0-*2)T  Description Overlap of design and construction duration of the design phase  Tc =  a-x )T  duration of the construction phase  tcf Tf =  (l-x )T  f  =  Definition df  db  3  cb  4  Overlap of construction and production duration of the production phase  cf  P  Tf  duration of the mineral processing phase  =  m  T = Tf = Rf  s= f  T — (T — T f) tcf T j T f- T f Rb  mb  +  c  m  R  m  duration of the revenue phase Total duration of design and construction Start- start duration between processing and revenue  84  Appendix A  The following constants are also defined in the model: q r  = fraction of loan used to finance construction = minimum attractive rate of return (MARR) = Loan interest rate  d  = fraction of construction costs allocated to design  k ka  = construction unit cost per time = Design unit cost per time  r  c  P r e s e n t V a l u e (PV) F o r m u l a t i o n f o r t h e T r a d i t i o n a l A p p r o a c h  Current construction cost function C (r) = C c  • k (t)-e  0 c  c  c  To  Current construction cost function C (t) = \d • C (t)dt d  d  c  85  Appendix A  PV (design) =  \C (t)-e' 'dt r  b  d  0 Tcb  PV (construction) = e~  JC (0 • e~  rTdb  n  b  C  o PV (revenue) = ' ^ ' r(Tb  b  • JR(t) • e' 'dt  b TRb)  r  e  0  PV(production) = e'  • JE (t)-e~"dt  rTb  o  p  Tb m  PV (processing) = e~  rTb  b  • J £ (t) • e~ 'dt r  m  o  PV (loan) = q • PV (construction) b  b  Tb C  PV (principal) = C -e '- ™ {e  b  • je °'- '  r)i  l  0c  r)  -k (t) c  o T»b  PV (loan repayment) = e b  rTb  • A • Je "dt b  o where A, =  Future value (loan)  o Tcb  and Future value (loan) = q •e ' • JC (t) • e~ ' rL Tcb  ri  b  C  0  The net present value for the traditional approach (NPV ) is derived as follows: b  NPVb PV (revenue) + PVb(loan) -[(PVb(design) + PVb(construction) + PV/processing) =  b  + PVb(production) + PVb(loan repayment) +PVb(principal))J Assuming that the expenditures and revenue remain the same the net present value for the fasttrack approach (NPVj) is: NPVf = PV/revenue) + PV/loan) -[(PV/design) + PVj(construction) + PV'/processing) + PV/production) + PVfioan repayment) +PV/principal))]  86  Appendix A  The assumption of keeping the same revenue and expenditures is justified by the fact that the fasttracking occurs at the construction level and therefore does not have any repercussion on the production, the processing and the revenue phases.  Present Value (PV) Formulation for the Fast-track Approach The net present values for the fast-track approach is derived from the expressions for the traditional case by simply replacing the original duration and times by the fast-tracked duration and times. P V (design) = f  Jc  (t)- - dt r,  rf  e  o  JC (0• e~  PV (construction) = e""  df  n  C  f  o v • JR(t) • e~ dt T  PVf (revenue) = ~ '* «~ r{T  T  Tv)  r,  e  o  PS  T  PV (production) = e~  • JE (t) • e- 'dt  PVf (processing) = e~  • J X , (t) • e~"dt  rTf  f  r  p  o  rTf  o PVf (loan) = q • P V (construction) f  PVf (principal) = C • 0c  { e e  ^ ' T  ^ . ^~ '  + T  r)  -k (t) c  o C  —rT  PVf (loan repayment) -e where A, V= -  f  • A • \e~"dt f  0  Future value (loan) T„ je^'dt f  0  and Future value (loan) = q-e '  rL Tcf  f  • |C (t) • e~ ' rL  C  Assuming that the expenditures and revenue remain the same the net present value for the fasttrack approach (NPVf) is: 87  Appendix A  NPVf = PV/revenue)  + PVfloan)  + PV/production)  NPVf = PV/revenue)  -[(PV/design)  + PV/construction)  + PVf(loan repayment)  - PV/production)-  +  PV/processing)  +PV/principal))]  PV/processing)  +[ PV/loan) -PV/design)  + PV/construction) + PV/loan repayment) +PV/principal)J The part in bold in the NPVf equation are terms that contains the construction cost function and therefore can be factored as follows:  NPVf = PV/revenue)  - PV/production)-  PV/processing)+Factor(X).C /X), c  where C f(X) is the expression of the cost function for the fast-track approach. C  Expression offactor(x) The expression of Factor (X) is derived by regrouping the terms that contain the construction cost for the fast-track approach as follows:  ]e <-^'k dt {0  qe Factor(X) = J  f  \c  df  0  (t)e' dt + e rt  ]e- C (t)dt rt  cf  c  + e-rt  0  0 0  q\ e ^ ]e^- k dt }  7  r)!  c  o  J  Coc/X) is derived from the equation above by equating NPV and NPV f  b  as follows:  88  Appendix A  NPV = NPV o Factor(X) • C (X) = NPV - PV (revenue) + PV(proces sin g) + PV (production f  b  0cf  <^C  0 c f  b  NPV - PV (revenue) + PV(proces sin g) + PV (production) b  (X)  Factor (X)  A 5 Vectors and Matrices used in the 3-D plots Two vectors, p and q, representing the range of the variables are introduced and then the matrices M , N and P representing the fast-track construction function with respect to respectively X] and x , x and x , and X] and x are calculated. 2  2  3  3  Definitions £ = (A)o<,<9> 9 = (?,)o^<9>  w h  w h e r e  A  ere q  }  =^  M = (m) , where m = C 0  ij  N = (n) , where n = C v  P = (p) , where tJ  y  P i j  0cf  0 c /  (  Pi  (p,, q ,0,0,0,0) }  ,0,?, ,0,0,0)  = C (0, , q, ,0,0,0) 0cf  P i  The values displayed inside the matrices represent the fast-track construction costs (in $million) when the variables take the values of the two vectors p and q. These matrices are all based on the example. The columns of the matrices reflect the costs of construction with respect to the second variable and the rows, the costs with respect to the first variable. For example the cost of construction decreases slightly with the compression of the overlap component xj, as do the values in the matrix M and P, where along the row (from top to bottom) the costs decrease. The same trends are observed for variable x and x . 2  3  Finally the matrices can be represented in a form of tables that are easy to refer to at anytime.  89  Appendix A  0 100 100.209 100.417 100.625 100.831 101.036 101.241 101.445 101.648 101.85  1 100.266 100.453 100.639 100.825 101.01 101.194 101.378 101.561 101.743 101.924  2 100.531 100.696 100.861 101.025 101.189 101.352 101.515 101.677 101.838 101.999  3 100.795 100.939 101.082 101.225 101.368 101.51 101.652 101.794 101.935 102.075  4 101.058 101.181 101.303 101.425 101.547 101.669 101.79 101.911 102.031 102.151  5 101.32 101.422 101.524 101.625 101.726 101.827 101.927 102.028 102.128 102.228  0 100 100.209 100.417 100.625 100.831 101.036 101.241 101.445 101.648 9 101.85  1 101.227 101.434 101.64 101.846 102.05 102.254 102.457 102.659 102.86 103.06  2 102.449 102.655 102.859 103.063 103.266 103.468 103.669 103.869 104.068 104.267  3 103.668 103.871 104.074 104.276 104.477 104.677 104.876 105.075 105.272 105.469  4 104.881 105.083 105.284 105.484 105.683 105.881 106.079 106.275 106.471 106.666  5 106.09 106.29 106.489 106.687 106.884 107.08 107.276 107.471 107.665  0 100 100.266 100.531 100.795 101.058 101.32 101.582 101.843 102.102 102.362  1 101.227 101.492 101.756 102.019 102.282 102.544 102.805 103.065 103.324 103.582  2 102.449 102.714 102.977 103.24 103.502 103.763 104.023 104.282 104.541 104.799  0 1 2 3: 4 5 6 7 8 9  0 1 2 3 4 5 6 7 8  0 1 2  : 4 5 6 7 8 9  3 103.668 103.931 104.194 104.456 104.717 104.978 105.237 105.496 105.754 106.011  4 104.881 105.144 105.406 105.667 105.928 106.187 106.446 106.704 106.961 107.218  6 101.582 101.663 101.744 101.824 101.905 101.985 102.065 102.145 102.225 102.305  7 101.843 101.903 101.963 102.023 102.084 102.144 102.204 102.263 102.323 102.383  8 102.102 102.142 102.182 102.222 102.262 102.302 102.342 102.382 102.422 102.461  9 102.362 102.381 102.401 102.421 102.441 102.461 102.481 102.501 102.52 102.54  6 107.293 107.491 107.688 107.884 108.079 108.274 108.468 108.661 108.853 107.858 109.044  7 108.49 108.686 108.881 109.075 109.269 109.462 109.654 109.845 110.035 110.225  8 109.68 109.874 110.068 110.26 110.452 110.643 110.833 111.022 111.211 111.399  9 110.864 111.057 111.248 111.439 111.628 111.817 112.006 112.193 112.38 112.566  5 106.09 106.352 106.613 106.874 107.133 107.392 107.65 107.907 108.164 108.419  6 107.293 107.554 107.815 108.074 108.333 108.591 108.848 109.105 109.361 109.615  7 108.49 108.75 109.01 109.269 109.527 109.784 110.041  8 109.68 109.94 110.199 110.457 110.714 110.971 111.227 110.296 111.481 110.551 111.736 110.805 111.989  9 110.864 111.123 111.381 111.639 111.895 112.151 112.406 112.66 112.913 113.166  90  Appendix B  APPENDIX B B1 Proof of the assertion in Chapter 4 Assume that a given activity x is shared by n paths (P P , pq  h  2  P„j, as shown in Figure B l , and  that it is the only activity shared by these paths. Suppose that the cost of crashing activity x is pg  Cpq and it is determined by the network analysis that k crash days are required in order to fasttrack the project.  Pi  P„  Figure B l : Illustration of an activity shared by more than one path. The total of crashing path /' for example is given by the following relation: TCj = C, + kC , where C* is the cost of crashing the other activities. pq  The total cost for crashing the project if we are to consider the total cost of crashing each individual activity is therefore:  ,=i  ,=i  But this formula implies that the cost of crashing activity x appeared n times in the expression pq  of the total cost whereas it is crashed only once, which is unrealistic . To account for that, the cost of crashing activity x  pq  (C ), is replaced by a new cost (C ) in the model and it is defined as pil  pq  followed:  91  Appendix B  c pq  c PI n  Replacing the new cost in the total cost formula yields the following more realistic expression of the total crashing cost:  TC = (£c,)  + k.C  p<l  i=\  This expression of the total crashing cost accounts for the fact that activity x  pq  is crashed only  once for the entire project. The same procedures is applied if the paths shares more than one activity.  B2 Probability distributions The probability distributions are used to build the Monte Carlo simulation on Excel Spreadsheet. These numbers may be obtained by investigating the performance of similar past projects or by interviews with experts and managers that have worked with similar projects. In this case, the probabilities are arbitrary and are used to illustrate the use of simulation techniques to solve real problems related to uncertainties in scheduling techniques. Table B l : Cumulative probability distribution of activities Activity 1-2 Cum Prob Time 0 2 0.1 3 0.35 4 0.7 5  Activity 1-4 Cum Prob Time 0 4 0.1 5 0 3 6 0.7 7  Activity 1-7 Cum Prob Time 0 1 0.2 2 0.7 3  Activity 2-3 Cum Prob Time 0 6 0.1 7 0.3 8 0.7 9  Activity 3-6 Cum Prob Time 0 3 0.15 4 0.75 5  Activity 4-5 Cum Prob Time 0 8 0.25 9 0 45 10  Activity 5-6 Cum Prob Time 0 7 0.3 8 0.8 9  Activity 4-8 Cum Prob Time 0 14 0.2 15 0.3 16  Activity 6-9 Cum Prob Time 0 4 0.2 5 06 6  Activity 7-8 Cum Prob Time 0 5 0.1 6 0.8 7  Activity 8-9 Cum Prob Time 0 9 0.2 10 06 11  92  Appendix B  A probability distribution is assigned to each activity on the network to account for randomness related to their durations and the cumulative probability distribution for each individual activity is built. Then using the random number generation function RAND () and the VLOOKUP function available in "MICROSOFT EXCEL ", the duration of the each activity is simulated and replicated 100 times to obtain a convergence in the shape of the time distribution. An algorithm is inserted into the simulation to analyze the criticality of each activity and path. The criticality of an activity is characterized by a yes or no or by 1 or 0 to respectively respond positively or negatively. Tables B2 and B3 represent the simulated data and its replication. Table B 2: Simulation results of activity times M'lillldtlull Activity 1-2 1-4 1-7 2-3 3-6 4-5 4-8 5-6 6-9 7-8 8-9  Time 3 4 1 8 4 9 16 7 5 6 11  Critical? 0 1 0 0 0 0 1 0 0 0 1  Total  Path info Path 1 2 3 4  near critical? 0 1 0 0 0 1 0 1 1 0 0  3  Length 20 25 31 18  Critical?  near critical 0 1 0 0  0 0 1 0  Critical path length 31  Meet deadline ?  4  1  Table B 3: Simulation replication table The average represents in fact the uniform probability about the criticality of each activity. For Data R e p l i c a t i o n table  Critical  Replication  Total Length  1-2  2-3  1-6  6-9  1-4  4-5  5-6  1 2 3 4 5 6 7 8 9 10 92 93 94 95 96 97 98 99 100  31 31 30 32 32 33 31 35 34 34 33 32 31 34 31 33 34 33 34  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0',o  O'o  Average  •  1  5'  0  4-8  1-7  7-8  8-9  1  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  99-.'o  0%  0%  0",  1  1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1  93  Appendix B  example it is predicted that activity 1-4 will be critical all the time, making it a potential candidate for "crashing". The uniform probability distribution of the entire path length is shown in Figure B2. It is predicted that the total length of the project will be 31 days with a probabiltity of 31%.  Histogram  35 30 5*  U  25  c 20 3 CT 15 10 5 0 28  29  30  31  32  33  34  35  36  C o m p l e t i o n time ( d a y s )  Figure B2: Histogram of the project completion time The shape of the histogram is very dependent on the decision of probability assignment. Different probability distributions will lead to different answers and this one the reason why the calculation and the analysis of estimates are crucial to the success of the resource allocation model. The descriptive statistics of the replication is provided in Table B4. Table B4: Descriptive statistic Statistics  of the  rep lication  Mean S t a n d a r d Error M ed ian Mode S t a n d a r d Deviation Sam pie V a r i a n c e Kurtosis Skew n ess Range Minimum Maximum Sum Count Largest(1) Sm allest(1) C o n f i d e n c e Level(95.0%)  32.0 0.0 32.0 32.0 1.6 2.4 -0.3 -0.3 8.0 27.0 35.0 32032.0 1 001.0 35.0 27.0 0.1  94  Appendix B  B3: Answer and Sensitivity Report of the LP problem Microsoft Excel 8.0a A n s w e r Report Worksheet: [simulation4.xls]Sheet2 Report Created: 06/07/99 3:46:51 A M  T a r g e t C e l l (Min) Cell $B$2 9  Name objective function 89  Original Value 1111  Final Value 1111  djustable C e l l s Cell $C$2 $C$3 $C$4 $C$5 $C$9 $C$1 0 $C$1 1 $C$1 2  $C$1 6 $C$1 7  $C$1 8 $C$2 2 $C$2 3 $C$2 4  Name  Original Value  variables variables variables variables variables variables  0 0 0 0 3 1  0 0 0 0 3 1  5-6 v a r i a b l e s  0  0  6-9 v a r i a b l e s  2  2  1-4 v a r i a b l e s  3  3  4-8 v a r i a b l e s  5  5  8-9 v a r i a b l e s  0  0  1-7 v a r i a b l e s  0  0  7-8 v a r i a b l e s  0  0  8-9 v a r i a b l e s  0  0  1-2 2-3 3-6 6-9 1-4 4-5  Final Value  Constraints Cell $C$2 5 $C$1 9  Name  Cell Value  Formula  Status  Length variables  0 $ C $ 2 5 = $ B $ 2 6 Not  Length variables  Binding 8 $ C $ 1 9 = $ B $ 2 0 Not Binding  Slack 0 0  95  Appendix B  $C$1 3 $C$6  Length variables  0 $C$6=$B$7 0 0 0 0 3  $C$2>=0 $C$3>=0 $C$4>=0 $C$5>=0 $C$9>=0  Binding Binding Binding Binding Binding Not  1 $C$10>=0  Binding Not  5-6 v a r i a b l e s  0$C$11>=0  Binding Binding  0  6-9 v a r i a b l e s  2$C$12>=0  Not  2  1-4 v a r i a b l e s  3$C$16>=0  Binding Not  4-8 v a r i a b l e s  5$C$17>=0  Binding Not  8-9 v a r i a b l e s  0$C$18>=0  Binding Binding  1-7 v a r i a b l e s  0 $C$22>=0  Binding  0  7-8 v a r i a b l e s  0 $C$23>=0  Binding  0  8-9 v a r i a b l e s  0 $C$24>=0  Binding  0  1-2 v a r i a b l e s  0 $C$2<=$D$2  Not  2 3  Length variables 1-2 2-3 3-6 6-9 1-4  $C$1 0 $C$1 1 $C$1 2 $C$1 6 $C$1 7 $C$1 8 $C$2 2 $C$2 3 $C$2 4 $C$2  4-5 v a r i a b l e s  variables variables variables variables variables  $C$3  2-3 v a r i a b l e s  0 $C$3<=$D$3  Binding Not  $C$4  3-6 v a r i a b l e s  0 $C$4<=$D$4  Binding Not Binding Not  $C$5  6-9 v a r i a b l e s  0 $C$5<=$D$5  $C$9 $C$1 0 $C$1 1  1-4 v a r i a b l e s 4-5 v a r i a b l e s  3 $C$9<=$D$9 1 $C$10<=$D$1 0 0$C$11<=$D$1 1 2$C$12<=$D$1 2  $C$1 2 $C$1 6 $C$1 7 $C$1 8  0  Binding Not  $C$2 $C$3 $C$4  $C$5 $C$9  6 $ C $ 1 3 = $ B $ 1 4 Not  5-6 v a r i a b l e s 6-9 v a r i a b l e s 1-4 v a r i a b l e s 4-8 v a r i a b l e s 8-9 v a r i a b l e s  Binding Binding Not Binding Not Binding Binding  3 $ C $ 1 6 < = $ D $ 1 Binding 6 5 $ C $ 1 7 < = $ D $ 1 Binding 7  0 $ C $ 1 8 < = $ D $ 1 Not 8 Binding  0 0 0 0 0 3 1  3 5 0  1 2 0 2 3 0 0 0 3  96  Appendix B  $C$2 2 $C$2 3 $C$2 4  1-7 v a r i a b l e s  0 $C$22<=$D$2 Not 2 Binding 0 $C$23<=$D$2 Not 3 Binding 0 $C$24<=$D$2 Not 4 Binding  7-8 v a r i a b l e s 8-9 v a r i a b l e s  1 3 3  Microsoft Excel 8.0a Sensitivity Report Worksheet: [simulation4.xls]Sheet2 Report Created: 06/07/99 3:46:53 A M  Adjustable Cells Cell $C$2 $C$3 $C$4 $C$5 $C$9 $C$1 0 $C$1 1 $C$1 2 $C$1 6 $C$1 7  Name 1-2 2-3 3-6 6-9 1 -4 4-5  Final Value  Reduced Gradient  variables variables variables variables variables variables  0 0 0 0 3 1  2 26 42 0 -138 0  5-6 v a r i a b l e s  0  63  6-9 v a r i a b l e s  2  -47  1 -4 v a r i a b l e s  3  -110  4-8 v a r i a b l e s  5  -89  $C$1 8 $C$2 2 $C$2 3  8-9 v a r i a b l e s  0  0  1-7 v a r i a b l e s  0  61  7-8 v a r i a b l e s  0  0  $C$2 4  8-9 v a r i a b l e s  0  80  Constraints Cell $C$2 5 $C$1  Name  Final Value  Lagrange Multiplier  Length variables  0  79  Length variables  8  159  97  Appendix B  $C$1 3 $C$6  Length variables  6  187  Length variables  0  78  98  

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