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Optimizing earthwork block removal in road construction 2010
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Title | Optimizing earthwork block removal in road construction |
Creator |
Koch, Valentin Raphael |
Publisher | University of British Columbia |
Date Created | 2010-04-16T16:00:10Z |
Date Issued | 2010-04-16T16:00:10Z |
Date | 2010 |
Description | In road construction, earthwork operations account for about 25% of the construction costs. Existing linear programming models for earthwork logistics optimization are designed to minimize the hauling costs and to balance the earth across the construction site. However, these models do not consider the removal of physical blocks that may influence the earthwork process. In this thesis, we extend the linear programming model of Mayer and Stark (1981) with the addition of a block removal schedule. The resulting model is a mixed-integer linear program. We analyze the model size and the schedule search space in order to make conclusion about the use of the model. Based on structural observations, we introduce a set of algorithms that significantly reduce the solving time of the model. Finally, we conduct numerical experiments to compare our solutions with the solutions of a traditional earthwork process that makes use of linear programming. From our numerical results, we conclude that an optimal removal schedule produces solutions that are 4.1% cheaper on average than a traditional method, with savings that can go as high as 19%. We conclude our discussion with possible extensions to the model, that can help an engineer to design roads that are more economical and ecological with respect to the earthwork operations. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | Eng |
Collection |
Electronic Theses and Dissertations (ETDs) 2008+ |
Date Available | 2010-04-16T16:00:10Z |
Rights | Attribution-NonCommercial 2.5 Canada |
DOI | 10.14288/1.0070953 |
Degree |
Master of Science - MSc |
Program |
Mathematics |
Affiliation |
Science, Faculty of |
Degree Grantor | University of British Columbia |
Graduation Date | 2010-05 |
Campus |
UBCO |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/3.0/ |
URI | http://hdl.handle.net/2429/23722 |
Aggregated Source Repository | DSpace |
Digital Resource Original Record | https://open.library.ubc.ca/collections/24/items/1.0070953/source |
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Optimizing Earthwork Block Removal in Road Construction by Valentin Raphael Koch B.Sc. Hons., The University of British Columbia, 2008 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in The College of Graduate Studies (Mathematics) THE UNIVERSITY OF BRITISH COLUMBIA (Okanagan) April 2010 c© Valentin Raphael Koch 2010 Abstract In road construction, earthwork operations account for about 25% of the con- struction costs. Existing linear programming models for earthwork logistics optimization are designed to minimize the hauling costs and to balance the earth across the construction site. However, these models do not consider the removal of physical blocks that may influence the earthwork process. In this thesis, we extend the linear programming model of Mayer and Stark [MS81] with the addition of a block removal schedule. The resulting model is a mixed-integer linear program. We analyze the model size and the schedule search space in order to make conclusion about the use of the model. Based on structural observations, we introduce a set of algorithms that significantly reduce the solving time of the model. Finally, we con- duct numerical experiments to compare our solutions with the solutions of a traditional earthwork process that makes use of linear programming. From our numerical results, we conclude that an optimal removal sched- ule produces solutions that are 4.1% cheaper on average than a traditional method, with savings that can go as high as 19%. We conclude our dis- cussion with possible extensions to the model, that can help an engineer to design roads that are more economical and ecological with respect to the earthwork operations. ii Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi Glossary of notation . . . . . . . . . . . . . . . . . . . . . . . . . . vii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . viii Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Road design . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Road design and construction . . . . . . . . . . . . . 1 1.1.2 Road design optimization . . . . . . . . . . . . . . . . 2 1.1.3 Earthwork optimization . . . . . . . . . . . . . . . . . 4 1.2 Basic linear programming model . . . . . . . . . . . . . . . . 7 1.2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2.2 Model parameters . . . . . . . . . . . . . . . . . . . . 7 1.2.3 Decision variables . . . . . . . . . . . . . . . . . . . . 9 1.2.4 Objective and constraints . . . . . . . . . . . . . . . . 10 2 Mixed integer programming model . . . . . . . . . . . . . . . 13 2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2 Model parameters . . . . . . . . . . . . . . . . . . . . . . . . 14 2.3 Decision variables . . . . . . . . . . . . . . . . . . . . . . . . 15 2.4 Objective and constraints . . . . . . . . . . . . . . . . . . . . 16 2.4.1 Objective function . . . . . . . . . . . . . . . . . . . . 17 2.4.2 Volume constraints . . . . . . . . . . . . . . . . . . . 17 2.4.3 Block removal indicator . . . . . . . . . . . . . . . . . 17 iii Table of Contents 2.4.4 Move feasibility constraint . . . . . . . . . . . . . . . 18 2.4.5 Stockpile constraints . . . . . . . . . . . . . . . . . . 18 2.4.6 Block removal enforcement . . . . . . . . . . . . . . . 20 2.4.7 Observations . . . . . . . . . . . . . . . . . . . . . . . 20 3 Model analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.1 Space complexity . . . . . . . . . . . . . . . . . . . . . . . . 22 3.2 Schedule search . . . . . . . . . . . . . . . . . . . . . . . . . 26 4 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.1 Mass schedule as starting point . . . . . . . . . . . . . . . . . 36 4.2 Maximal allowed moving distance dmax . . . . . . . . . . . . 37 4.3 Warmstarting with increasing dmax . . . . . . . . . . . . . . 39 4.4 Warmstarting with mipgap check . . . . . . . . . . . . . . . 40 5 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . 42 5.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . 42 5.1.1 Problems . . . . . . . . . . . . . . . . . . . . . . . . . 42 5.1.2 Benchmark . . . . . . . . . . . . . . . . . . . . . . . . 42 5.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 5.2.1 Analysis of the dmax-heuristic . . . . . . . . . . . . . 46 5.2.2 Performance . . . . . . . . . . . . . . . . . . . . . . . 48 5.2.3 Economical analysis . . . . . . . . . . . . . . . . . . . 54 6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 6.1 Accomplishments . . . . . . . . . . . . . . . . . . . . . . . . 57 6.2 Research outcomes . . . . . . . . . . . . . . . . . . . . . . . . 57 6.3 Future research . . . . . . . . . . . . . . . . . . . . . . . . . 59 6.3.1 Warmstart schemes and εLP . . . . . . . . . . . . . . 59 6.3.2 Vertical alignment . . . . . . . . . . . . . . . . . . . . 59 6.3.3 Schedule enumeration . . . . . . . . . . . . . . . . . . 60 6.3.4 Lagrangian relaxation . . . . . . . . . . . . . . . . . . 61 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Appendices A Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 B Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 iv List of Tables 1.1 Major components of construction costs . . . . . . . . . . . . 3 2.1 Stockpile example data . . . . . . . . . . . . . . . . . . . . . . 18 3.1 Schedule example data . . . . . . . . . . . . . . . . . . . . . . 27 5.1 Test configurations for dmax . . . . . . . . . . . . . . . . . . . 45 5.2 Economical test and performance test configurations . . . . . 45 A.1 Problem configurations . . . . . . . . . . . . . . . . . . . . . . 66 A.2 Statistics for objective function error with dmax . . . . . . . . 67 A.3 Statistics for solving time with dmax . . . . . . . . . . . . . . 67 A.4 Histogram statistics for 1% tolerance . . . . . . . . . . . . . . 68 A.5 Histogram statistics for 5% tolerance . . . . . . . . . . . . . . 68 A.6 CPU time in seconds for ε = 0.01, problems 1-22 . . . . . . . 69 A.7 CPU time in seconds for ε = 0.01, problems 23-44 . . . . . . 70 A.8 CPU time in seconds for ε = 0.05, problems 1-22 . . . . . . . 71 A.9 CPU time in seconds for ε = 0.05, problems 23-44 . . . . . . 72 A.10 Objective value in $ for direct solve, ε = 0.01 . . . . . . . . . 73 A.11 Objective value in $ for direct solve with x◦, ε = 0.01 . . . . . 74 A.12 Objective value in $ for warmstart (d0max = 25, α = 10), ε = 0.01 75 A.13 Objective value in $ for warmstart with ǫ (d0max = 25, α = 10), ε = 0.01 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 A.14 Objective value in $ for warmstart with ǫ (d0max = 5, α = 5), ε = 0.01 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 A.15 Objective value in $ for direct solve, ε = 0.05 . . . . . . . . . 78 A.16 Objective value in $ for direct solve with x◦, ε = 0.05 . . . . . 79 A.17 Objective value in $ for warmstart (d0max = 25, α = 10), ε = 0.05 80 A.18 Objective value in $ for warmstart with ǫ (d0max = 25, α = 10), ε = 0.05 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 A.19 Objective value in $ for warmstart with ǫ (d0max = 5, α = 5), ε = 0.05 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 v List of Figures 1.1 Example of vertical alignment with mass diagram . . . . . . . 5 1.2 Example of vertical alignment with sections . . . . . . . . . . 8 2.1 Earthwork scheme with removal schedule (black squares in- dicate blocks, gray squares indicate removed blocks). . . . . . 19 3.1 Potential schedules . . . . . . . . . . . . . . . . . . . . . . . . 27 3.2 Earthwork scheme with removal schedule. . . . . . . . . . . . 28 3.3 Schedule tree . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.4 (6× 4)-lattice in Z2 . . . . . . . . . . . . . . . . . . . . . . . 30 5.1 Objective and solving time with increasing dmax . . . . . . . . 47 5.2 Algorithms performance profiles with log2 scale . . . . . . . . 50 5.3 Algorithms performance profiles with small τ . . . . . . . . . 52 5.4 Algorithms performance profiles with large τ . . . . . . . . . 53 5.5 Savings with warmstart with ǫ (d0max = 25, α = 10) . . . . . . 55 5.6 Stratification for warmstart with ǫ (d0max = 25, α = 10) . . . . 56 B.1 Savings with direct solve . . . . . . . . . . . . . . . . . . . . . 83 B.2 Savings with direct solve and x◦ . . . . . . . . . . . . . . . . 84 B.3 Savings with warmstart (d0max = 25, α = 10) . . . . . . . . . . 84 B.4 Savings with warmstart and ǫ (d0max = 5, α = 5) . . . . . . . . 84 B.5 Stratification for direct solve . . . . . . . . . . . . . . . . . . 85 B.6 Stratification for direct solve with x◦ . . . . . . . . . . . . . . 85 B.7 Stratification for warmstart (d0max = 25, α = 10) . . . . . . . . 86 B.8 Stratification for warmstart with ǫ (d0max = 5, α = 5) . . . . . 86 vi Glossary of notation a ∈ A a is an element of set A. 4 R Real numbers. 4 R+ Nonnegative real numbers. 5 φ : A → B The mapping A into B by φ. 5 [a, b] Interval {x ∈ R : a ≤ x ≤ b}. 5 ∞ Infinity. 5 f ′ Derivative of function f . 6∫ Integral. 5 | · | Absolute value. 7 |A| Cardinality of set A. 7 {x : P (x)} The set of all x such that P (x) is satisfied. 8 Z+ Nonnegative integers. 8 A ∪ B The union of sets A and B. 8 dist(i, j) Distance between section i and j. 8 R++ Positive real numbers. 8 min f The minimum of function f . 10∑ Summation. 10 A× B Cartesian product {(x, y) : x ∈ A, y ∈ B}. 12 ∧ Logical connective representing and . 14 ∨ Logical connective representing or . 14 A \ B Set difference {x : x ∈ A and x 6∈ B}. 16 ∅ The empty set. 16 Θ (·) Asymptotic tight bound. 22 O (·) Asymptotic upper bound. 22 Z++ Positive integers. 22 A ⊆ B A is a subset of B. 22( n k ) Binomial coefficient (n choose k). 30 R n Real n-vectors (n× 1 matrices). 35 XT Transpose of matrix X. 35 R m×n Real m× n matrices. 38 a← b Assign value b to a. 37 vii Acknowledgements Many people were involved in the creation of this thesis. First, I would like to express my sincere gratitude to my supervisors, Dr. Yves Lucet and Dr. Warren Hare, for their help and support throughout this research. I greatly enjoyed our lively discussions and conversations. It was a great privilege to have supervisors that helped me to overcome so many challenges, whether they were in mathematics, computer science, or simply personal. I would also like to thank the members of my graduate committee, Dr. Heinz Bauschke and Dr. Shawn Wang, for their earnest support and guid- ance in my graduate studies, and for providing me insight into the beautiful subject of convex analysis. It is a pleasure to thank my colleagues, especially Jim Nastos and Mason Macklem, for their patience to listen and the many ideas and input they contributed. A very special thanks goes out to our industrial partner, for the profes- sional assistance and valuable help on engineering questions. I owe my deepest gratitude to my parents. Without their prayers and support, my academic career would have been impossible. Finally, I would like to thank my wife Delphine, and my children Jordan and Tabitha, for their love and patience during these intense years. This work was partially supported by the Ministry of Advanced Educa- tion of the Province of British Columbia through a Pacific Century Graduate Scholarship, and by the Mathematics of Information Technology and Com- plex Systems network through a MITACS Accelerate internship. viii Dedication To Delphine, Jordan, and Tabitha. ix Chapter 1 Introduction La tendance la plus profonde de toute activité humaine est la marche vers l’équilibre.1 Jean Piaget (1896–1980) 1.1 Road design 1.1.1 Road design and construction Transportation of people and goods over land is, and has always been, an important part of civilization. Construction of the first roadways dates back as early as 5000 BCE [Lay99]. The invention of the wheel created the need for better roads and more sophisticated forms of road construction. Complex road design, including the alignment, earthwork, and pavement was already performed during the Roman Empire. The first scientific approach to road design was by Trésaguet in France at about 1764 [Lay99]; Trésaguet also introduced the first system of continuous road maintenance. The first com- prehensive guide on road design and construction was published by McAdam in 1816 [McA24], introducing the concept of the elevation of roads through embankment. This resulted in much more stable roads, but also increased the cost of construction. Roadways are an essential part of the economy in most countries. For ex- ample, in Canada in 2008, trucking accounted for 54% of the value of trade with the United States and commercial transportation services accounted for 4.1% of Canada’s value-added GDP [Stu09]. At the same time, all levels of Canadian government combined spent $20.9 billion, or 1.6% of Canada’s GDP, to build and maintain its 1 million kilometer road network. In view of these numbers, it is a natural consequence that governments of coun- tries that depend heavily on road networks have a vast interest in keeping a 1The most profound tendency of all Human activity is progression toward equilibrium. 1 1.1. Road design high quality road infrastructure, while at the same time trying to lower the expenditures on construction and maintenance. From an economic perspec- tive, it is therefore important to try to minimize the cost and maximize the quality and safety of the road infrastructure. The design of a road largely depends on safety requirements, which in turn depend on the traffic volume and vehicle speed. In 2007, in Canada, about 140,000 road collisions occurred, with about 16,000 seriously injured persons and about 3,000 fatalities. More than 100 million tonnes of dan- gerous goods were transported on Canadian roads during the same year [Stu09]. While the economical aspects in road construction are important, safety requirements are essential. Depending on the location, the construction of a road can have a great impact on the environment. Maki [MKV02] shows that poorly planned road construction, especially in sensitive areas like the Amazon region, of- ten harms the local economy instead of helping it. According to Maki, it is important to optimize the land use through environmental and social impact analysis. Appropriate classification and zoning of land should therefore be taken into account, and environmental impacts, like deforestation or earth- work, should be minimized when a road is built. It follows from the above discussion that the road infrastructure needs to be optimized with respect to economic and social costs, safety requirements, and environmental impact. 1.1.2 Road design optimization Multiple factors determine the total transportation cost of a road. Jha et. al. [JSJ06] list five main types of costs that affect the total transportation cost: − planning, design, and administrative costs; − construction costs; − operation and maintenance costs; − user costs; and − social and environmental costs. Planning and administrative costs are usually not considered in road de- sign optimization. In [CGF89], Chew et. al. divide construction costs in six major categories, shown in Table 1.1 with an approximate associated 2 1.1. Road design Table 1.1: Major components of construction costs Pavement 30% Earthwork (half of this is haul cost) 25% Bridges 20% Drainage 10% Miscellaneous items 10% Land 5% percentage of the total construction cost. Operation and maintenance costs include roadway surface, roadside features, bridges, tunnels, snow and ice control, and traffic control devices and amount at about 5% of the construc- tion costs, over a period of 30 years. The user costs, which include vehicle operating costs, travel time, and traffic accident costs, range from 300% to 1000% of the construction costs over a period of 30 years [JSJ06]. Finally, social and environmental costs can be the most critical factor in highway design. It is, however, difficult to quantify these costs and they are usually considered before the optimization of the design. The design of the road can roughly be divided into geometric design and pavement design. The geometric design determines the alignment of the road and affects the construction costs, user costs, and social and environmental costs. The pavement design affects operation and maintenance, as well as user costs. Optimization of the geometric design is traditionally done in two stages: the horizontal alignment design and the vertical design. The horizontal alignment is the road trajectory from a satellite’s eye view. The primary objects that influence a horizontal alignment are forbid- den areas like lakes, ocean shores, steep mountains, nature reserves, and pri- vate property. Cost parameters in horizontal alignments might include land costs and road distance. Early attempts at horizontal alignment optimiza- tion were based on dynamic programing [Tri87]. More recent approaches use genetic algorithms [JK06] or local neighborhood heuristics [LTL09]. Once the horizontal alignment is done, one can draw a line from the starting point to the end point of the road and look at the vertical ground profile along this line. The vertical alignment optimization tries to fit a road to the vertical ground profile as close as possible, while respecting grade constraints and other road specifications. The closer the road is to the ground profile, the less earthwork needs to be done in order to cut or fill sections of the road. Usually, vertical alignment optimization includes earthwork allocation and earth-moving optimization. In civil engineering, 3 1.1. Road design the vertical alignment optimization is commonly done through a manual iteration of computations that estimate the earthwork volumes for a given alignment. The alignment is then changed and the volumes are recomputed until a satisfying solution is found. Traditional mathematical models that automate the optimization of the vertical alignment use the method of linear programming (LP) [Eas88b, Mor96, Mor09, KL10]. Recently, the road alignment optimization research community has fo- cused on the simultaneous optimization of horizontal and vertical alignment. In [CGF89], Chew et. al. modeled the road as a three-dimensional cubic spline. Jong and Schonfeld [JS03] use genetic algorithms to first solve for the three-dimensional alignment as a three-dimensional piecewise linear spline, and then fit circular curves to the horizontal and vertical projections of the spline. Parametric curves were studied by Makanae [Mak99] to model the road in three dimensions. Cheng and Lee [CL06] use a heuristic bi-level approach to iteratively optimize horizontal and vertical alignments. All of the above optimization methods depend on the cost evaluation of the earthwork. In this thesis, we focus on the aspect of earthwork optimiza- tion for a given alignment. The optimal cost of earthwork is essential for the assessment of a given road alignment. 1.1.3 Earthwork optimization Earthwork depends significantly on the terrain and the geometric design of the road. A road in a mountainous area requires more cutting and filling of earth than a road on the plain. An optimal vertical grade line minimizes the earthwork and satisfies all the design constraints for the road. To deter- mine the optimality with respect to the earthwork, one needs to be able to compute the earthwork volumes for a given alignment. In order to compute the earthwork volumes for a given grade line, cross- sections of the road profile are taken at regular intervals. The point along the horizontal axis where the cross-section is taken is called a station. Sta- tions are usually arranged at an equal distance from one another along the horizontal axis of the grade line. The distance Lij is the actual length be- tween station i and j along the horizontal center line of the road. The area Ai of a cross-section at station i is determined and the volume V ∈ R be- tween two stations i and j is computed as V = Lij(Ai+Aj)/2. A traditional approach uses the mass diagram to depict the net accumulation of cut or fill between any two stations along the horizontal axis of the grade line. In [GH08], Garber and Hoel describe the mass diagram as follows. The ordinate of the mass diagram is the net accumulation 4 1.1. Road design road profile, g(x̄) ground profile, f(x̄) cumulative mass, h(x̄) = ∫ x̄ 0 f(u)− g(u) du balance point balance point x̄ Figure 1.1: Example of vertical alignment with mass diagram in cubic meter from an arbitrary starting point. Thus, the dif- ference in ordinates between any two stations represents the net accumulation of cut or fill between these stations. If the first station of the roadway is considered to be the starting point, then the net accumulation at this station is zero. For this thesis, we will use the first station as the starting point. We can therefore define the mass diagram as follows. Definition 1.1. Consider a vertical alignment. Let x̄ be a station. If f(x̄) represents the height of the ground profile at x̄, and g(x̄) represents the height of the road profile at x̄, where f : R+ → R and g : R+ → R are piecewise continuous functions on [0,+∞[, then the mass diagram is the graph of the Riemann integral h(x̄) = ∫ x̄ 0 (f(u)− g(u)) du, where h(x̄) represents the cumulative earthwork mass at x̄. In Figure 1.1, we show an example of a vertical alignment with the corresponding mass diagram beneath. From Definition 1.1 and Figure 1.1, we observe the following characteristics. 5 1.1. Road design − If h′(x̄) < 0, then x̄ is the position of a fill. If h′(x̄) > 0, then x̄ is the position of a cut. − Any extremum of h(x̄) represents a switch from cut to fill, or from fill to cut. − The net accumulation between a station x̄i and a station x̄j, where x̄i < x̄j is h(x̄j)− h(x̄i). − For any two stations x̄i and x̄j , if h(x̄i) = h(x̄j), then the net accumu- lation between the two stations is zero. These points are called balance points, since they show points of balance between cut and fill volumes. − A consequence of the last point is that if h(0) = 0 and x̄end is the last station of the road, then an optimal alignment with respect to the balance of earth quantities has h(x̄end) = 0. With the above observations, we can derive a rule of thumb to determine the hauling directions. Remark 1.2. Let h(0) = 0. If h(x̄) > 0 and x̄i < x̄ < x̄j , where x̄i and x̄j are balance points, then earth needs to be hauled ahead. The distance between x̄i and x̄j is the maximum push or haul distance. If h(x̄) < 0 and x̄i < x̄ < x̄j, then earth needs to be hauled back. The values at the extrema represent the volumes of material that must be hauled along the road. The mass diagram and the above rule of thumb have several shortcom- ings. − The mass diagram does not take into account haul costs that are not proportional to the haul distance. − The mass diagram does not take into account different soil character- istics. − The mass diagram does not take into account borrow pits or waste pits that can be used to borrow additional quantities of soil or dispose of excessive material at off-the-roadway sites. − The mass diagram does not take into account blocks like trees, rocks, and rivers that are present in the haul route. To address some of the above shortcomings, Mayer and Stark [MS81] presented a LP model that not only balances the earth quantities, but also 6 1.2. Basic linear programming model minimizes the hauling cost and incorporates soil material distribution, bor- row pits, and waste pits. Based on [MS81], Easa developed a mixed integer linear program (MILP) that uses non-constant unit costs [Eas87], and a quadratic program (QP) that uses linear unit costs [Eas88a]. Jayawardane and Harris [JH90] extended the model in [MS81] by incorporating project duration and dynamic selection of borrow and waste pits. In this thesis, we extend a basic version of the model by Mayer and Stark, in order to overcome the problem of blocks in the haul route. 1.2 Basic linear programming model In order to better explain our model of earth-moving, it is helpful to begin by considering the LP model in Mayer and Stark [MS81]. An extensive description of the model is given in [MS81] and [JH90]. We give a shorter outline in this section with some changes to the notation and refer the reader to the original paper for further details. 1.2.1 Notation For the remainder of this thesis, the cardinality of a set S is denoted |S|. The same notation | · | is used to indicate the absolute value and the meaning of the notation should be clear from the context. 1.2.2 Model parameters In this section we provide information on the basic LP model parameters. Let a section be the region between two fixed stations along the horizontal axis. We divide the vertical alignment into ms sections of equal length. Consider Definition 1.1. If for a section i, x̄i represents the station at the beginning of the section, and x̄i+1 represents the station at the end of the section, then h(x̄) = ∫ x̄i+1 x̄i (f(u)− g(u)) du is the required change in volume between the ground profile and the road profile in the section. The required change in volume for a section is positive if the section represents a cut, negative if the section represents a fill. As in [MS81, Eas87, JH90], we introduce mb borrow pits and mw waste pits in order to obtain a feasible solution in the case of an unbalanced vertical alignment. A borrow pit or a waste pit has a volume that represents the capacity. The following input parameters are therefore given: 7 1.2. Basic linear programming model section road profile ground profile 1 2 3 4 5 6 Figure 1.2: Example of vertical alignment with sections − ms ∈ Z+ is the number of sections in the road; − mb ∈ Z+ is the number of borrow pits; − mw ∈ Z+ is the number of waste pits. To simplify notation, we introduce the following index sets: − S = {1, . . . ,ms} is the set of indices that is used to index sections; − B = {ms + 1, . . . ,ms +mb} is the set of indices that is used to index borrow pits; − W = {ms + mb + 1, . . . ,ms +mb +mw} is the set of indices that is used to index waste pits; − M = S ∪ B ∪W. Finally, we need the required change in volume and the moving costs, which are defined by the following parameters: − Vi ∈ R is the required change in volume of a section i ∈ S in cubic units, or the capacity of a borrow pit i ∈ B or waste pit i ∈ W; − cij ∈ R++ ∪ {+∞} is the cost of moving one cubic unit of material from i to j, where i, j ∈ M, i 6= j. In Figure 1.2, we show the alignment from Figure 1.1 that is divided into ms = 6 sections of equal length. We denote the distance between section i and j as dist(i, j), and evaluate this using dist(i, j) = |x̄i − x̄j|, (1.1) where x̄i is the midpoint of section i and x̄j is the midpoint of section j. 8 1.2. Basic linear programming model Remark 1.3. The distance dist(i, j) can be defined arbitrarily. A more so- phisticated definition than Equation (1.1) would incorporate the distance between the sections on the horizontal alignment as well as the difference of altitude. We leave this as a future research topic. Note that there are many different ways to define the cost cij of moving earth. Examples of non-constant unit costs are given in [Eas87, Eas88a]. We use the following definition. Definition 1.4. The cost cij ∈ R++ ∪ {+∞} to move one cubic unit of material from section i to section j, where i, j ∈ M and i 6= j, is defined as cij = cexc + dist(i, j)chau + cemb, if i, j ∈ S, cbor + dist(i, j)chau + cemb, if i ∈ B, j ∈ S, cexc + dist(i, j)chau + cwas, if i ∈ S, j ∈ W, +∞, otherwise, (1.2) where cexc > 0 is the unit cost of excavation, chau > 0 is the cost to haul one cubic unit of material over a unit distance, cemb > 0 is the unit cost for embankment, cbor > 0 is the unit cost to borrow material from a borrow pit, and cwas > 0 is the unit cost to dump material in a waste pit. The following lemma follows clearly from Definition 1.4. Lemma 1.5. Consider any three sections i, j, k ∈ S. If i 6= k, i 6= j, and j 6= k, then the strict triangle inequality cij < cik + ckj holds whenever cij < +∞. The hauling distance from or to borrow pits or waste pits needs to ac- count for dead-haul distances (the distance of the road from the pit-joining section to the pit). Excavation costs on a section are usually smaller than borrowing costs from a pit. This is usually balanced by the lower embank- ment costs of borrowed material. Also material that is dumped into a waste pit does not need a lot of embankment work and the embankment cost to dump material into a pit is much smaller than the embankment costs on a section. 1.2.3 Decision variables The decision variable xij ∈ R represents the number of cubic units of earth that will be hauled from section i to section j, where i, j ∈ M and i 6= j. 9 1.2. Basic linear programming model 1.2.4 Objective and constraints The minimization of the earth-moving costs is modeled by LP (1.3) below min ∑ i,j∈M i 6=j cijxij (1.3a) s.t. ∑ j∈M j 6=i (xij − xji) = Vi for all i ∈ S, (1.3b) ∑ j∈M j 6=i xij ≤ Vi, for all i ∈ B, (1.3c) ∑ j∈M j 6=i xji ≤ Vi, for all i ∈ W, (1.3d) xij ≥ 0, for all i, j ∈ M, i 6= j. (1.3e) Objective function The objective function in (1.3a) minimizes the total hauling cost. Illogical moves (e.g., moving earth into a borrow pit) are penalized in Equation (1.2) of the cost Definition 1.4 by +∞. In practice, to reduce memory size, decision variables that represent illogical moves are not implemented. Constraints For every road section i that has either a demand or a supply, we want the amount of incoming and outgoing earth movements to balance the demand or supply of the section. Constraints (1.3b) balance the demand and supply between sections and are called balance constraints. The model has |S| balance constraints. We also need to account for the capacity of borrow and waste pits. We enforce the capacity constraints with inequalities (1.3c) and (1.3d). Since one constraint is needed for each borrow and waste pit, we have |B| + |W| capacity constraints. Finally, we can only move earth quantities that are greater or equal to zero. This is ensured by Equation (1.3e). 10 1.2. Basic linear programming model Observations Remark 1.6. Note that shrink and swell factors that are mentioned in [MS81] are not included in this thesis since the model works with banked units and the mentioned factors are only taken into account to compute haul and compaction costs. Proposition 1.7. Let Vs = ∑ i∈S Vi, Vb = ∑ i∈B Vi, and Vw = ∑ i∈W Vi. If Vs ≥ 0 and Vw − Vs > 0, then the LP (1.3) has always a solution. If Vs < 0 and Vb + Vs > 0, then the LP (1.3) has always a solution. Proof. The conditions Vw − Vs > 0 or Vb + Vs > 0 ensure that a feasible point exists, as the waste or borrow pit provide sufficient slack. Since for all i, j ∈ M, cij ∈ R++ ∪ {+∞} and xij ≥ 0, we know that min ∑ i,j∈M i 6=j cijxij ≥ 0. Hence, the LP (1.3) is bounded below. Since there exists a feasible point, and the LP is bounded below, there exists a solution to the LP (1.3). Proposition 1.8. Let x∗ be a feasible solution to the LP (1.3). If x∗ik > 0 and x∗kj > 0, where i 6= k and k 6= j, then x∗ is not an optimal solution. Proof. If x∗ik ≤ x∗kj, consider x̃ik = 0, x̃kj = x∗kj − x∗ik, and x̃ij = x∗ij + x∗ik. Then Vi = x ∗ ij + x ∗ ik + ∑ s∈S s 6=i,j,k x∗is − ∑ s∈S s 6=i x∗si = x̃ij + x̃ik + ∑ s∈S s 6=i,j,k x∗is − ∑ s∈S s 6=i x∗si, (1.4) Vk = x ∗ kj − x∗ik + ∑ s∈S s 6=j,k x∗ks − ∑ s∈S s 6=i,k x∗sk = x̃kj − x̃ik + ∑ s∈S s 6=j,k x∗ks − ∑ s∈S s 6=i,k x∗sk, (1.5) Vj= −x∗ij − x∗kj + ∑ s∈S s 6=j x∗js − ∑ s∈S s 6=i,j,k x∗sj = −x̃ij − x̃kj + ∑ s∈S s 6=j x∗js − ∑ s∈S s 6=i,j,k x∗sj , (1.6) and the volume constraints are satisfied. But from Lemma 1.5 we know that cik + ckj > cij , and thus the total cost is cheaper, because cikx ∗ ik+ckjx ∗ kj+cijx ∗ ij > (cij−ckj)x∗ik+ckjx∗kj+cijx∗ij = cikx̃ik+ckjx̃kj+cijx̃ij . 11 1.2. Basic linear programming model If x∗ik > x ∗ kj, consider x̃ik = x ∗ ik − x∗kj, x̃kj = 0, and x̃ij = x∗ij + x∗kj. Then, the same arguments show that equations (1.4), (1.5), and (1.6) hold, and thus the volume constraints are satisfied, while the total cost is cheaper. Proposition 1.9. The LP (1.3) contains |M|2−|M| decision variables and |M|2 constraints. Proof. The number of moves from any section i ∈ M to any other section j ∈ M is equivalent to the number of pairs of section indecies, (i, j), i, j ∈ M, that are formed by the Cartesian productM×M. Now |M×M| = |M|2. Since we do not count moves where i = j, we have |M|2 − |M| decision variables. The constraint in (1.3b) appears for every section in S. Also, constraints (1.3c), and (1.3d) appear for each borrow pit in B and each waste pit inW, respectively. Constraint (1.3e) appears for each of the |M|2−|M| variables. It follows that we have |S|+ |B|+ |W|+ |M|2 − |M| = |M|2 constraints. Remark 1.10. In practice, since we do not implement illogical moves, the number of decision variables and constraints are less than the number given in Proposition 1.9. If we count only logical moves, we have |S|2−|S| potential moves between sections, |S||B| potential moves between sections and borrow pits, and |S||W| potential moves between sections and waste pits. Thus in the actual implementation of the model, we have |S|(|M| − 1) decision variables. Accordingly, we have |S|+ |B|+ |W|+ |S|(|M| − 1) = |M|(1 + |S|)− |S| constraints in the actual implementation. In the next chapter we will use these results in our discussion of the extension of the basic LP model (1.3). 12 Chapter 2 Mixed integer programming model He has barred my way with blocks of stone; he has made my paths crooked. Lamentations 3:9 (Holy Bible, NIV) The model presented in Section 1.2 is a major improvement over the typical mass diagram due to the minimization of hauling distances and the incorporation of borrow and waste pits. However, in practice, a move from one section to another might not be possible due to natural blocks between the two sections. In this thesis, a block is defined as follows. Definition 2.1. A block is an obstacle like a river, a group of trees, a rock-bed, or some other topographical feature that needs to be removed or otherwise dealt with before earth can be moved over the location of the block. In a traditional earthwork construction process, clearing of blocks is usually treated as a separate task before excavation and embankment occurs [Trb90]. In such a case, the use of a mass diagram or a LP model, as the one described in Section 1.2, is perfectly valid, since after the obstacle removal, all the sections are accessible. This, however, does not account for earthwork that may occur during the obstacle removal (i.e. filling of a ditch or blasting of rock beds). During obstacle removal, stockpiles may be created and material may be borrowed elsewhere. This makes it difficult to assess the overall costs of obstacle removal, excavation, embankment, and haul in a traditional setting. Moreover, the order of removal of blocks might affect the final cost of the earthwork. We therefore extend the model in [MS81] with the introduction of blocks. 13 2.1. Notation 2.1 Notation For the remainder of this thesis, the symbol ∧ is a logical connective repre- senting and, and the symbol ∨ is a logical connective representing or. Hence A ∧B means “A and B”, where A ∨B means “A or B” 2.2 Model parameters We extend the basic LP formulation in Section 1.2 to develop a model that incorporates nb blocks. Definition 2.2. Let I = {1, . . . , nb}. Each block is assigned to a road section. We let ς : I → S be the mapping that maps the block index to the assigned section index. Also, we add the access roads that lead to the construction site of the road. Each access road is connected to one of the ms sections. Access roads are typically attached to sections at the beginning and end of the road, but may also occur at any point along the road and in between blocks. The number and position of access roads influences the optimal schedule for the removal of the blocks. Definition 2.3. Let R = {1, . . . , nr} be the set of all access roads. Each access road is assigned to a road section. We let ̺ : R→ S be the mapping that maps the access road index to the assigned section index. Since each access road allows for transporting earth out to an external waste pit or importing earth from an external borrow site, we attach an external borrow and waste pit to every access road with infinite capacity each. These pits are accounted for in set B and W. We introduce discrete time steps to remove the blocks. The removal of a block requires one time step and the block is removed at the end of the time step. Multiple blocks may be removed within the same time step. In a worst-case scenario, where all the blocks need to be removed in sequential order, nb time steps are needed to remove the blocks. Once all the blocks are removed, we may need an additional time step to finish all the earthwork. We therefore introduce nb + 1 time steps t with t ∈ T = {0, . . . , nb}. 14 2.3. Decision variables The cost coefficients from Equation (1.2) remain the same in our model. However, we need a cost coefficient for a move for every time step. Hence, for each t ∈ T we set cijt = cij . Excavation and embankment costs on external pits should be higher than for sections. Remark 2.4. If more than one access road is present, then any movement of earth between two sections could also go over the access roads. One possible way to allow such moves is to represent each access road as a section. Each section that has an access road attached could be cloned and the clone be added to the set S. In this thesis, however, we do not account for such moves and we leave it as a further research topic. In Section 1.2.2, we stated that the temporary stocking of earth between sections is avoided due to Lemma 1.5. With the introduction of blocks, how- ever, this is not guaranteed anymore (as we demonstrate in Example 2.7). We therefore introduce for each section i a stockpile tolerance parameter ǫstki ∈ R+. The significance and usage of this parameter is shown in Section 2.4.5. 2.3 Decision variables Similar to the cost coefficients, we keep the same variables representing the quantities of earth that are moved between sections. However, since we now have the flexibility of moving earth between any two sections at any time step, we multiply the variables in Section 1.2.3 by the number of time steps. The variable xijt ∈ R represents the quantity of earth that is moved from section i to section j at time step t. A block k examined at time step t is represented with a binary variable. A zero value for a block shows that no or only partial work was done to remove the block during that time step, and a one indicates that the block was cleared at the end of time step t. Hence for block k with k ∈ I, t ∈ T , we define the variable that indicates the removal of block k at time t via ykt ∈ {0, 1}, such that if block k is still present at time t, then ykt = 0. 15 2.4. Objective and constraints 2.4 Objective and constraints The minimization of the earth-moving costs, including the removal of blocks, can therefore be stated as MILP (2.1) below min ∑ t∈T ∑ i,j∈M i 6=j cijtxijt (2.1a) s.t. ∑ t∈T ∑ j∈M j 6=i (xijt − xjit) = Vi for all i ∈ S, (2.1b) ∑ t∈T ∑ j∈M j 6=i xijt ≤ Vi, for all i ∈ B, (2.1c) ∑ t∈T ∑ j∈M j 6=i xjit ≤ Vi, for all i ∈ W, (2.1d) u∑ t=0 ∑ j∈M j 6=ς(k) (xς(k)jt − xjς(k)t) ≥ ykuVς(k) for all k ∈ I, u ∈ T , Vς(k) ≥ 0, (2.1e) u∑ t=0 ∑ j∈M j 6=ς(k) (xς(k)jt − xjς(k)t) ≤ ykuVς(k) for all k ∈ I, u ∈ T , Vς(k) < 0, (2.1f) xijt ≤Myk,t−1, for all t ∈ T \ {0}, i, j ∈ M, k ∈ I, (i ≤ ς(k) ≤ j) ∨ (j ≤ ς(k) ≤ i) (2.1g) xijt ≤Myk,t−1 +Myk+1,t−1, for all t ∈ T \ {0}, i, j ∈ M, k ∈ I \ {nb}, r ∈ R, ς(k) ≤ i, j ≤ ς(k + 1) ∧ {r : ς(k) ≤ ̺(r) ≤ ς(k + 1)} = ∅ (2.1h) u∑ t=0 ∑ j∈M j 6=i (xijt − xjit) ≤ Vi + ǫstki , for all i ∈ S, u ∈ T , Vi ≥ 0 (2.1i) 16 2.4. Objective and constraints u∑ t=0 ∑ j∈M j 6=i (xijt − xjit) ≥ Vi − ǫstki , for all i ∈ S, u ∈ T , Vi < 0 (2.1j) u∑ t=0 ∑ k∈I yki ≥ u, for all u ∈ T , (2.1k) xijt ≥ 0, for all i, j ∈ M, i 6= j, t ∈ T , (2.1l) ykt ∈ {0, 1}, for all k ∈ I, t ∈ T , (2.1m) where M ∈ R is a large constant that exceeds the quantity of any possible earth move. 2.4.1 Objective function As in the LP model, the MILP objective function (2.1a) minimizes the total hauling costs, but this time over all the time steps. Note that the cost coefficients for the block removal variables are set to zero. 2.4.2 Volume constraints The balance constraints (2.1b) and the capacity constraints (2.1c) and (2.1d) have the same form as in the LP model, with an additional summation over all the time steps. 2.4.3 Block removal indicator The decision variable ykt, that indicates the removal of a block k at time step t, is binary. If we want to find an optimal removal schedule, we need to set some of these removal indicators to one. Once we decided to remove block k at time step t, we let ykt = 1. The removal of block k in a section ς(k) may involve earthwork. In order to effectively enforce the removal of the block at time step t, we need to satisfy any potential demand or supply of earth in this section at time t. For a supply, this is done by Equation (2.1e), for a demand, by Equation (2.1f). Therefore, the block removal indicator constraints relate the binary variables with the continuous variables. Remark 2.5. Note that once a block k is removed at time step t, there is no need to enforce the monotonicity ykt ≤ yk,t+1 in order to represent the status of a block in the later time steps. If we want to move earth over the 17 2.4. Objective and constraints block at some later time step u > t, we can just set the variables yk,u−1 to 1 and the corresponding removal indicator constraint will already be satisfied. In fact, numerical tests showed that enforcing the monotonicity slows down the solving process. 2.4.4 Move feasibility constraint Consider a move from section i to j. If there is a block k, such that i ≤ ς(k) ≤ j, then we need to remove the block k before the move can happen. Therefore, if yk,t−1 = 0, then the move is not possible. However, if yk,t−1 = 1, then any (reasonable) quantity can be moved. In order to indicate any reasonable quantity, we use a very large constant M as upper bound for such a move. Equation (2.1g) models this relationship. Consider a move from section i to j. If there are blocks k and k+1, such that ς(k) ≤ i, j ≤ ς(k + 1), and there is no access road between ς(k) and ς(k + 1), then we need to remove either block k or block k + 1 first, before the move can happen. This constraint is modeled in (2.1h). The requirement that there is no access road in between ς(k) and ς(k+1) is shown in the condition of (2.1h) as {r : ς(k) ≤ ̺(r) ≤ ς(k + 1)} = ∅. 2.4.5 Stockpile constraints Remark 2.6. Proposition 1.8 from the LP model does not apply to the block removal MILP model in (2.1). An explanation follows from Example 2.7 below. Example 2.7. Consider the example with sections and volumes given in Table 2.1. The road section 3 represents a block with demand of one cubic unit of earth. Road section 4 has a supply of one, but is not accessible due to the block at section 3. The volume for road sections 1 and 2 is zero. There is one access road at the beginning of the road that joins section 1. For simplicity, let us consider cexc = chau = cemb = cbor = cwas = 1. Since Table 2.1: Road sections and volumes (bold font represents a block, ≈ an access road). Section i ≈1 2 3 4 Volume Vi 0 0 -1 1 there is nb = 1 block, we have 2 time steps. It is easy to see that the optimal solution to this problem is to first remove the block in section 3 by borrowing 18 2.4. Objective and constraints material from the adjacent section 2, followed by filling the newly created demand in section 2 with the cut of section 4. Hence the solution is x230 = 1 x421 = 1 y10 = 1, which is illustrated in Figure 2.1. The squares represent the road sections. The time steps are shown as the space between two layers of squares. A black colored square indicates the presence of a block, a gray colored square shows a block that was removed in a previous step. The arrows in between the layers represent earth movements. With the given cost model, any other 1 2 3 4 1 2 3 4 1 1 2 3 4 1 Figure 2.1: Earthwork scheme with removal schedule (black squares indicate blocks, gray squares indicate removed blocks). solution would be more expensive. If we disregard the time steps and add the quantities for each move over time, that is xij = xij0 + xij1, for all i, j ∈ {1, 2, 3, 4}, then we have x42 = 1 x23 = 1 and the solution moves one cubic unit of earth from section 4 to 2, and from section 2 to 3. Hence, in this example, Proposition 1.8 does not apply to the MILP in (2.1). 19 2.4. Objective and constraints From Remark 2.6, we conclude that a constraint is required to limit the amount of earth that can be stocked or borrowed from a road section. This can be enforced by a similar constraint like the balance constraint (2.1b), where the equality is replaced by a bound that represents the volume of a section plus or minus the stockpile tolerance. However, since stockpiling can occur at any time step, we need the constraint to hold at every time step. The resulting equations are (2.1i) for an upper bound, and (2.1j) for a lower bound. 2.4.6 Block removal enforcement In order to provide the MILP solver with as much information as possible, we add the following constraint. At each time step t = 1, 2, . . . , nb, we want to remove at least one block. This is guaranteed with constraint (2.1k). This constraint also ensures that no more than nb + 1 time steps are needed. 2.4.7 Observations Proposition 2.8. Let Vs = ∑ i∈S Vi, Vb = ∑ i∈B Vi, and Vw = ∑ i∈W Vi. If Vs ≥ 0 and Vw − Vs > 0, then the MILP (2.1) has always a solution. If Vs < 0 and Vb + Vs > 0, then the MILP (2.1) has always a solution. Proof. The conditions Vw − Vs > 0 or Vb + Vs > 0 ensure that constraints (2.1b) to (2.1f), and (2.1i) to (2.1j) can be satisfied, since the waste or borrow pit provide sufficient slack. Since for all i, j ∈M, t ∈ T , cijt ∈ R++∪{+∞} and xijt ≥ 0, we know that min ∑ t∈T ∑ i,j∈M i 6=j cijtxijt ≥ 0. In Section 4.1, we show how to set the binary variables ykt in order to obtain a feasible point. Furthermore, in Section 3.2, we show that there are a finite number of ways to select the binary variables, and therefore a finite number of feasible points to the MILP (2.1). Altogether, since the MILP (2.1) has a finite number of feasible points that are bounded below, and there exists at least one such point, there exists a solution to the MILP (2.1). Proposition 2.9. The MILP (2.1) contains (|I|+ 1)(|M|2 − |M|) contin- uous, and (|I|+ 1)|I| binary decision variables. 20 2.4. Objective and constraints Proof. In order to get the number of continuous variables, we multiply the |M|2 − |M| variables from Proposition 1.9 with |I| + 1 time steps. We have therefore (|I| + 1)(|M|2 − |M|) continuous variables. Since there are |I| blocks and |I| + 1 time steps, we have (|I| + 1)|I| binary variables. Altogether, we have (|I|+ 1)((|M|2 − |M|) + |I|) variables. Remark 2.10. As in Remark 1.10, the number of decision variables in the practical implementation of the model is smaller than the number given in Proposition 2.9. If we multiply the number of continuous variables from Remark 1.10 with |I|+1 time steps, we get |S|(|M|−1)(|I|+1) continuous variables. The (|I|+1)|I| binary variables remain the same as in Proposition 2.9. In the next section we examine some of the properties of Model (2.1). 21 Chapter 3 Model analysis Aut inveniam viam aut faciam.2 Hannibal (247–183 BCE) 3.1 Space complexity In this section, we discuss the space complexity of Model (2.1). In particular, we look at the growth rate of the model size, with respect to the input pa- rameters, that is required to store the model in the memory of a computer. In this context, we employ the usual asymptotic notation for space complex- ity. The following definition is a reminder of the most common asymptotic bounds that we are using. Definition 3.1 (Cormen et al. [CLRS09, p. 43]). For a given function g(n), we denote by Θ (g(n)) the set of functions Θ (g(n)) = {f(n) : there exist c1 > 0, c2 > 0, and n0 ∈ Z++ such that 0 ≤ c1g(n) ≤ f(n) ≤ c2g(n) for all n ≥ n0}, and we say that g(n) is an asymptotic tight bound for f(n). We denote by O (g(n)) the set of functions O (g(n)) = {f(n) : there exist c > 0, and n0 ∈ Z++ such that 0 ≤ f(n) ≤ cg(n) for all n ≥ n0}, and we say that g(n) is an asymptotic upper bound for f(n) and Θ (g(n)) ⊆ O (g(n)). Lemma 3.2. The growth rate of the total number of variables for the model in (2.1) is of order Θ (|I||M|2 + |I|2). 2I shall either find a way or make one. 22 3.1. Space complexity Proof. From Proposition 2.9 we know that there are (|I|+ 1)(|M|2 − |M|) + (|I|+ 1)|I| = |I||M|2 + |I|2 + |I| − |I||M| − |M|2 − |M| variables. Let |I| = 1, |M| = 2. Clearly, 2(|I||M|2 + |I|2) ≥ |I||M|2 + |I|2 + |I| − |I||M| − |M|2 − |M|, and 1 2 (|I||M|2 + |I|2) ≤ |I||M|2 + |I|2 + |I| − |I||M| − |M|2 − |M|. We therefore conclude that the growth rate for the size of the total variables is Θ (|I||M|2 + |I|2). Lemma 3.3. The growth rate of the number of balance constraints (2.1b), borrow pit capacity constraints (2.1c), and waste pit capacity constraints (2.1d) is of order Θ(|M|). Proof. Each section needs a balance constraint (2.1b). Hence there are |S| balance constraints. Each borrow pit and each waste pit needs a capacity constraint. Thus there are |B|+ |W| capacity constraints. Altogether, there are |S|+ |B|+ |W| = |M| balance and capacity constraints. The growth rate of the balance constraints is therefore of order Θ (|M|). Lemma 3.4. The growth rate of the number of removal indicator constraints (2.1e) and (2.1f) is Θ (|I|2). Proof. We have one block removal indicator constraint for each indicator variable. From Proposition 2.9 we know that the number of indicator vari- ables is |I|(|I| + 1). Hence the number of indicator variable constraints is |I|(|I|+ 1) which is in Θ (|I|2). Lemma 3.5. The growth rate of the number of feasible move constraints (2.1g) and (2.1h) is of order O (|M|2|I|2) 23 3.1. Space complexity Proof. We consider the (|M|2 − |M|) potential moves between all the sec- tions over |I|+1 time steps. Each move needs at most a total of |I| feasible move constraints. If the move occurs over one or more blocks, we use con- straint (2.1g) at most |I| times. If the move occurs in between two blocks, we use the constraint (2.1h) at most |I| times. If the move occurs over n blocks and between two blocks at the same time, we use constraint (2.1g) n times, and constraint (2.1h) at most |I| − n times. Altogether, we use a total of at most |I| constraints. We have therefore (|M|2 − |M|)(|I|+ 1)|I| feasible move constraints, which is of order O (|M|2|I|2). Lemma 3.6. The growth rate of the number of stockpile constraints (2.1i) and (2.1j) is of order O (|M||I|). Proof. At each time step, we need two stockpile constraints for each section. Hence we have 2|T ||S| = 2(|I|+ 1)|S| ≤ 2|I||M|+ 2|M| stockpile constraints which is in O (|M||I|). Lemma 3.7. The growth rate of the number of removal enforcement con- straints (2.1k) is of order Θ(|I|). Proof. At each time step, we need a constraint that sets the least number of blocks that must be removed at that time step. We therefore have |T | = |I|+ 1 constraints which is in Θ (|I|). Lemma 3.8. The growth rate of the number of non-negativity constraints (2.1l) is of order Θ (|M|2|I|). Proof. We need a non-negativity constraint (2.1l) for each of the continuous variables. From Proposition 2.9, we know that there are (|I|+1)(|M|2−|M|) continuous variables, which is in Θ (|M|2|I|). Lemma 3.9. The growth rate of the number of binary constraints (2.1m) is of order Θ (|I|2). Proof. We need a binary constraint (2.1m) for each of the binary variables. From Proposition 2.9, we know that there are (|I|+ 1)|I| binary variables, which is in Θ (|I|2). 24 3.1. Space complexity Proposition 3.10. The growth rate of the memory size for the MILP de- scribed in (2.1) is of order O (|M|7). If the number of blocks |I| is fixed, then the memory size for the MILP (2.1) is of order O (|M|4). Proof. We add the asymptotic bounds for all the constraints from Lemma 3.3 to Lemma 3.9. The growth rate of the sum of all the constraints is therefore Θ (|M|) + Θ (|I|2)+O (|M|2|I|2)+O (|M||I|) + Θ (|M|2|I|)+Θ (|I|2) , (3.1) which is of order O (|M|2|I|2 + |I|2). From Lemma 3.2, we know that the number of variables is of order Θ (|I||M|2 + |I|2). Hence, the size of the problem is bounded by O (|M|2|I|2 + |I|2)×Θ (|I||M|2 + |I|2) which is of order O (|M|2|I|4 + |M|4|I|3). Since |B| ≥ 1, and |W| ≥ 1, we know that |I| < |M|, and hence |I| ∈ O (|M|). The asymptotic upper bound for the memory size is therefore O (|M |7). If we fix the number of blocks |I|, the dimension of the problem is bounded by O (|M|2)×Θ (|M|2) , which is of order O (|M|4). Remark 3.11. The asymptotic bound of O (|M |7) in Proposition 3.10 is a worst-case bound, where the number of blocks |I| could grow linearly with |M|, the sum of all sections, borrow pits, and waste pits. We mentioned in Chapter 2 that |I| is typically between 5 and 20. Therefore if |M| is large, then |I| is usually small compared to |M|. In that case, we can fix |I| and, as shown in Proposition 3.10, the memory size of the MILP (2.1) grows with order O (|M|4). Regardless of the fact that we fix the number of blocks in practice, the memory growth rate in O (|M|4) remains a concern for roads with hundreds of sections. Most MILP solvers take advantage of the sparsity of (2.1) and the bound may not be attained. However, in our numerical tests, we still observed a growth rate that quickly exhausted the memory of a standard desktop computer. From the given bound and from practical experience us- ing a 64 bit desktop computer with 8 GB of RAM and the Gurobi Optimizer 2.0 MILP solver, we recommend to keep the number of sections |S| ≤ 200. 25 3.2. Schedule search 3.2 Schedule search In Section 3.1, we showed that the memory size grows in polynomial time with the input. However, the difficult part in our model is the scheduling for the block removal. In this section we analyze the scheduling problem more closely. In particular, we look at the schedule search space and the set of feasible schedules in the schedule search space. For all time steps t ∈ T , we list the variables ykt as a binary string in the order of block position. Since |T | = |I|+1, we have |I|+1 binary strings of length |I|. At time t = 0, all the blocks are in place, thus yk0 = 0, and the binary string contains zeros only. At a time step t > 0, the corresponding binary string contains t or more ones that represent the removal of blocks. A binary string that contains ones only, represents a complete removal. Definition 3.12. A schedule is a set of |I| + 1 binary strings where each string belongs to a distinct time step. A road with five blocks, for example, has an initial binary string of 00000. If one block is removed, say the first one to the left, then we have a new string which is 10000. We continue the removal until the binary string is 11111. Note that complete removal of all blocks may occur before time step t = |I|+ 1. In that case, the digits of the remaining |I|+ 1− t strings can be considered all 1 for all remaining time steps (the constraints affected by the binary variables for the remaining time steps are already satisfied). Definition 3.13. The schedule search space is the number of potential schedules in the MILP (2.1) as defined by the binary selection. This in- cludes infeasible schedules. Lemma 3.14. The schedule search space is asymptotically bounded by O ( 2|I| 2 2|I| ) . Proof. Since there are |I|+ 1 binary strings of length |I|, we have 2|I|(|I|+1) potential schedules. It follows that the schedule search space is in O ( 2|I| 2 2|I| ) . 26 3.2. Schedule search Table 3.1: Road sections and volumes (bold font represents a block, ≈ an access road). Section i 1 2 3 4 ≈5 6 7 Volume Vi 2 -2 -1 -1 5 -1 -2 00000 00100 00110 00111 01111 11111 (a) Feasible 00000 10000 10001 11001 11011 11111 (b) Infeasible Figure 3.1: Potential schedules Not all schedules in the schedule search space are feasible. Blocks can only be removed if they have an access road attached or if an adjacent block that is accessible is removed first. We are interested in the number of feasible schedules that lead to a complete removal. Definition 3.15. The feasible schedules are the schedules that only remove a block if the block is accessible at the appropriate time. Hence, if k ∈ I is a block at time step t ∈ T , then ykt = 1 implies that either there exists an access road r ∈ R, such that ς(k − 1) ≤ ̺(r) ≤ ς(k) ∨ ς(k) ≤ ̺(r) ≤ ς(k + 1), or we have yk−1,u = 1 ∨ yk+1,u = 1, for some u ∈ T , u < t. Example 3.16. Consider the example with the sections given in Table 3.1. Sections 1, 3, 4, 6, and 7 are blocks. There is a single access road that joins section 5. Figure 3.1 shows two potential schedules. The schedule in Figure 3.1a is feasible, with a possible earthwork scheme shown in Figure 3.2. However, the schedule in Figure 3.1b is not feasible, since the blocks at either end of the road are not accessible. There are other blocks between the access road and the end of the road. Blocks can be removed sequentially or simultaneously. From an access road with blocks on either side, there are three possible ways to remove 27 3.2. Schedule search 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 1 2 3 4 5 6 7 1 1 2 3 4 5 6 7 2 1 2 3 4 5 6 7 1 1 2 3 4 5 6 7 2 Figure 3.2: Earthwork scheme with removal schedule. 28 3.2. Schedule search 0000 01100100 0010 11111110011011101100 0111 0110 0111 0011 111101111111111011111110 1111 1110 1111 0111 1111 0111 1111 1111 1111 1111 1111 1111 Figure 3.3: Tree structure of all feasible block removal schedules for 4 blocks with an access road joining between the two middle block adjacent blocks. We can remove the block to the left, to the right, or both at once. If the access road is attached to the block, the block needs to be removed first. Using edges to represent left, right, or simultaneous removal, we can draw a tree that contains all the feasible schedules for a given con- figuration of access points. An example for 4 blocks with an access road attached to a section between the two center blocks is shown in Figure 3.3. We start by approximating the number of all feasible schedules for a sin- gle access road from below. We therefore restrict the set of feasible schedules by analyzing schedules that allow left and right removal only, using one sin- gle access road. Let the access road be at a section between block r and block k = |I| − r. Consider the (k × r)-lattice of integral points in Z2 with coordinate (0,0) at the lower left corner and coordinate (k,r) at the upper right corner. We associate a left block removal to a step upward and a right block removal to a step to the right. We call the vertical steps (1,0)-steps and the horizontal steps (0,1)-steps to represent the increase of the y− and x−axis respectively. A lattice path consisting of (0,1)- and (1,0)-steps that starts at (0,0) and ends at (k,r) is in bijection with a schedule starting at the root node with a zero binary string and ending at a leaf with a binary string of ones. Figure 3.4a depicts such a path. The next Lemma provides a formula to count all the possible lattice paths starting at (0,0) and ending at (k,r). We provide the proof as outlined in [Aig07]. Lemma 3.17 (Aigner [Aig07, p. 17]). Consider a (k× r)-lattice of integral points in Z2. The number of paths consisting of (1,0) and (0,1) steps from 29 3.2. Schedule search (0,0) (6,4) r k (a) (1,0),(0,1)-lattice path (0,0) (6,4) (b) Delannoy path Figure 3.4: (6× 4)-lattice in Z2, representing 10 block configuration with a single access road between block 6 and 4 (0,0) to (k,r) is L(k, r) = ( k + r r ) . (3.2) Proof. Assign the symbol L(eft) to a (1,0)-step and R(ight) to a (0,1)-step. Every lattice path is in bijection with a (k+ r)-word over {L,R} containing exactly r L’s. This corresponds to (k+r r ) possible combinations of (k + r) words containing r number of L symbols. Remark 3.18. For the block removal schedules, since k + r = |I|, for |I| blocks and an access road between block k and r, we have L(k, r) = (|I| r ) (3.3) feasible schedules for the restricted set of feasible schedules for a single access road that contain left and right removals only. Remark 3.19. Equation (3.3) is the formula for the binomial coefficients [Aig07, p. 16]. Consider the first five rows of Pascal’s triangle 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1. 30 3.2. Schedule search Let the rows represent the number of blocks |I|, and the columns represent block r, where a single access road is between block k and r. The binomial coefficients in the triangle represent the number of feasible left and right removal schedules. We notice that the central binomial coeffcients along the central axis of the triangle grow the fastest. They correspond to an access road between k and r, where k = r. If k = r, with the formula given in [Wei02, p. 369], then L(r, r) = ( 2r r ) = (|I| r ) . Using Stirling’s approximation [Wei02, p. 2868] for the factorial function, we have (|I| r ) ≤ 4 r √ 3r + 1 , for all r ≥ 1. Since |I| = 2r, we conclude that the number of feasible schedules for a single access road and left and right removal only is in O (2|I|). We now extend our restricted set of feasible schedules by adding another removal step to the schedules with one access. So far we considered left- removal and right-removal steps only. But if the access road is in between two blocks, we can also remove the left and right block at the same time, as shown in Figure 3.3. Remark 3.19 is therefore a lower bound on the worst- case. Instead of having only (0,1)- and (1,0)-steps in our lattice path, we now add a diagonal (1,1)-step. This results in Delannoy paths as shown in Figure 3.4b. The proof for the next Lemma is already given in [SAV+03]. We prove it in a slightly different way. Lemma 3.20. Consider a (k × r)-lattice of integral points in Z2. The number of Delannoy paths consisting of (1,0),(0,1) and (1,1) steps from (0,0) to (k,r) is D(k, r) = k∑ i=0 ( k i )( k + r − i k ) . (3.4) Proof. Assign the symbol L(eft) to a (1,0)-step, R(ight) to a (0,1)-step, and B(oth) to a (1,1)-step. Every Delannoy path corresponds therefore to a string in the {L,R,B} alphabet. Consider a Delannoy path with i diagonals. Hence the corresponding string contains r−i L’s and k−i R’s. Now project3 the (1,1)-steps onto the horizontal x-axis. The projected {L,R} string has 3The projection of the diagonals onto a lattice is based on an idea by Jim Nastos. 31 3.2. Schedule search now r− i L’s and k− i+ i = k R’s. Using the formula for (1,0),(0,1)-lattice paths in Lemma 3.17, Equation (3.2), we have L(k, r − i) = ( k + r − i r − i ) = ( k + r − i k ) (3.5) (1,0),(0,1)-paths. However, there are ( k i ) possible ways we can choose the diagonals that we project. Also, we have i ∈ {0, . . . , k}. Hence we multiply Equation (3.5) by ( k i ) and sum i from 0 to k in order to get the formula in Lemma 3.20. Proposition 3.21. The number of feasible schedules for a road with a single access road on either end of the road is in O (1). Proof. We use Equation (3.4) with k = |I|, r = 0 and obtain D(|I|, 0) = 1. We are now able to compute the exact number of feasible schedules for a given road with one access road. However, a lot of construction sites have more than one access road. We are particularly interested in the very common case of two access roads, one at the start and one at the end of the road. Lemma 3.22. The number of feasible schedules for a road with two access points, one at the start and one at the end of the road, is S(|I|) = |I|∑ j=1 D(|I| − j, j), (3.6) where D is defined in Equation (3.4). Proof. Consider a binary string with |I| zeros that represents the time step at time t = 0. If we remove a block at the start of the road, we set the left- most digit from 0 to 1. If we remove a block at the end, we set the right-most digit to 1. Likewise, if we simultaneously remove on both sides, we set the outer-most digits to 1. Since there is no access point anywhere else, this procedure continues until all digits are set to 1. The last digit or the last two digits that are set to 1 represent the meeting point of the construction crews, once all the blocks are removed. For any removal schedule with a meeting point j ∈ {1, . . . , |I|}, we can invert the schedule in the sense that zeros are set to one, and ones to zero. A inverted schedule represents a schedule that has the meeting point as a single access road. For a fixed 32 3.2. Schedule search meeting point, there is a bijection between all the schedules with two access roads that meet at the meeting point and the schedules with a single access road attached to the meeting point. In order to get all feasible schedules, we consider all |I| possible meeting points. Hence we sum j over {1, . . . , |I|} and compute the inverse schedules with (3.4), where r = j and k = |I| − j. This leads to the formula in Lemma 3.22. As presented in Banderier and Schwer [BS05], the central Delannoy num- bers along the diagonal from (0,0) to any (r,r) grow the fastest. To get a worst-case scenario for a configuration with |I| blocks and one access road, we let the access road join the center of the binary string, thus k = r = |I|/2. In the same paper [BS05], Banderier and Schwer also give an approximation to compute an upper bound on the central Delannoy numbers. It is given as D(k, k) = 5.82842709k(0.57268163k−1/2 − 0.06724283k−3/2 + 0.00625063k−5/2 +O ( k−7/2 ) ). (3.7) Proposition 3.23. The number of feasible schedules for a road with a single access road between the center blocks of the road is in O (3|I|). Proof. Consider Equation (3.4). If the single access road is between the center blocks, we have r = k. The dominating term in the sum of Lemma 3.22 is the number for the central Delannoy paths D(k, k) when k = |I|/2. We therefore use Equation (3.7) as an approximation. Since 5.83|I|/2(|I|/2)−1/2 . 2.42|I| √ 2|I|−1/2, we conclude that the number of feasible schedules is in O (3|I|). Proposition 3.24. The number of feasible schedules for a road with two access roads, one on each end of the road, is in O (3|I|). Proof. Use Equation (3.6) with |I| = 0, 1, 2, . . .. We obtain the sequence {0, 1, 2, 5, 12, 29, 70, 169, . . .}. These are the Pell-Lucas numbers [Wei02, p. 2185], given by the closed form P (n) = (1 + √ 2)n − (1−√2)n 2 √ 2 . Since P (n) ≤ (1 + √2)n√2/4 ≈ 2.42n√2/4, we conclude that the number of feasible schedules is in O (3|I|). 33 3.2. Schedule search Remark 3.25. Consider Proposition 3.21, 3.23, and 3.24. If there is one access road only and the number of blocks |I| is small, then the number of feasible schedules is clearly tractable. In this case, it is reasonable to enumerate all the feasible schedules and to solve a LP for each of the feasible schedules. The optimal removal schedule is the schedule for the LP solution with the lowest objective value. The enumeration of the feasible schedules can be done with the use of a Depth-First search algorithm [CLRS09]. The Depth-First algorithm runs linearly in the size of the tree. The solving time is therefore exactly proportional to the tree size, or the number of paths. If there is one access road in the middle of the road, then there is an exponential number of feasible schedules. Whenever |I| is large, the enu- meration of the feasible schedules is not tractable and one should solve a single MILP instance of the form in (2.1) instead. If there are two access roads, one on either side of the road, then there is an exponential number of feasible schedules. Again, if |I| is large, one should solve the MILP in (2.1). 34 Chapter 4 Algorithms An algorithm must be seen to be believed. Donald Knuth (1938–) Model (2.1) is sufficient to solve the earth-moving problem. In this sec- tion we propose algorithmic techniques that improve the speed of conver- gence. In order to simplify the notation, we formulate (2.1) in standard form. Let m be the number of constraints in (2.1). Let nc denote the number of continuous variables and n = nc + nb denote the total number of variables in (2.1). We rewrite the inequality constraints and group the left-hand sides in a matrix A ∈ {0,±1}m×n, and the right-hand side in a vector b ∈ Rm, such that we can write (2.1) in the canonical form min cTx s.t. A(x, y)T ≥ b x ≥ 0 y ∈ {0, 1}nb , (4.1) where x ∈ Rnc, c ∈ Rnc++ is the cost vector, and x ≥ 0 means component wise xi ≥ 0, for all i ∈ {1, . . . , nc}. We can solve (4.1) directly as a MILP, using a MILP solver software [Fou05]. Although this is not an algorithm in the common sense, we call it here the direct solve algorithm for future comparison. Algorithm 1: Direct solve Input: x,y,c,A,b (from (2.1)) Output: Global minimum z∗ = cTx∗ for (4.1) begin Build and solve MILP (4.1) for x∗; end 35 4.1. Mass schedule as starting point 4.1 Mass schedule as starting point Definition 4.1. The LP obtained by relaxing the constraints y ∈ {0, 1}nb with y ∈ [0, 1]nb , is called the LP relaxation of the MILP (4.1). Fact 4.2 (Dantzig [Dan98, p. 516]). The LP relaxation is a lower bound on the MILP. Most modern MILP solvers use a branch-and-cut algorithm [Fou05]. Branch-and-cut is a combination of the branch-and-bound method with the cutting-plane algorithm of Gomory [Gom58]. A classical branch-and-bound algorithm solves first the LP relaxation to the MILP. The algorithm then partitions the LP relaxation into subproblems which have one of the pre- viously relaxed integer variables fixed to closest integer lower- and upper- bounds. The continuation of this process results in a branch-and-bound tree with nodes that represent subproblems. The bounding part of the algo- rithm looks for the best complete solution zm in the tree. At each node k, where the solution zk to the subproblem is greater than zm, the algorithm can discard any further subproblem created by this node, since all following solutions will be greater than zm. For a more detailed description of the branch-and-bound algorithm, we refer to [PS98]. A common technique to accelerate the solution process of a MILP is to provide an initial upper bound z◦ on the solution value zm. This allows for early bounds on the branch-and-bound tree. An even better technique is to provide the solver with a feasible starting point (x◦, y◦), where z◦ = cTx◦. The number of nodes that need to be expanded can be reduced significantly with a starting point (x◦, y◦) that produces a sufficiently low upper bound z◦. In this section, we describe a way to find a reasonable starting point. In a traditional mass diagram procedure, work is performed in a unidi- rectional way from the start of the road to the end (see Remark 1.2). For a road with access attached to the starting section, a feasible block removal schedule is obtained by removing the blocks sequentially from the start of the road to the end, one block at each time step. If there is no access at the starting section of the road, we select the section with an attached access that is closest to the starting section and first remove sequentially all the blocks on the left of the access, and then all the blocks on the right. Once we have the schedule, we can fix the binary variables in (4.1) and solve the problem as a LP. This will give us feasible values for the continuous vari- ables, and together with the fixed binary variables, a feasible solution to (4.1). 36 4.2. Maximal allowed moving distance dmax Procedure FindFeasiblePoint(x,y,c,A,b) Input: x,y,c,A,b (from (2.1)) Output: Feasible point (x∗, y∗) for (4.1) begin a← FindFirstAccess(S); for k ← a to 1 do y∗k,a−k+1 ← 1; end for k ← a+ 1 to |I| do y∗kk ← 1; end for k ← 1 to |I| do for t← 1 to |I| do if y∗k,t−1 = 1 then y ∗ kt ← 1; end end Solve (4.1) with fixed y∗kk as LP; end We can now extend Algorithm 1 with procedure FindFeasiblePoint. We call this the direct with x◦ algorithm. Algorithm 2: Direct with x◦ Input: x,y,c,A,b (from (2.1)) Output: Solution z∗ = cTx∗ for (4.1) begin Build MILP (4.1); (x◦, y◦)←FindFeasiblePoint(x,y,c,A,b); Solve MILP (4.1) with (x◦, y◦) for (x∗, y∗); end 4.2 Maximal allowed moving distance dmax To reduce the model size, and hence reduce the solving time of the MILP, we introduce a heuristic that eliminates a significant number of variables. In order to do this, we introduce the parameter dmax. Definition 4.3. The Maximal Allowed Moving Distance, dmax ∈ S, is the largest number of sections over which earth is allowed to be moved. There- 37 4.2. Maximal allowed moving distance dmax fore, for all i, j ∈ M, t ∈ T , xijt = 0, if |i− j| > dmax. (4.2) The motivation behind dmax is that from practical experience, it makes little sense to move earth from a supply at one end of the road to a demand at the other end of the road. Intuition tells us that if there are other supplies along the road in between the two sections, then the closest supply to the demanding section should be chosen. Hence if we disallow moves that go over too long a distance, then we restrict the search space to shorter, and thus probably cheaper moves. We enforce dmax with a set of constraints in the form of Equation (4.2). If there are p such constraints, we can group them in a matrix Rdmax ∈ {0, 1}p×n and reformulate the MILP (4.1) in a dmax-extended form min cTx s.t. A(x, y)T ≥ b Rdmaxx = 0 x ≥ 0 y ∈ {0, 1}nb . (4.3) Remark 4.4. It is clear from (4.2) that if dmax is small, then Rdmax becomes large (although very sparse). So instead of reducing the model size, we increase it. However, with each constraint in Rdmax , we can remove a large number of feasible move constraints in A. This would be done before solving the model; potentially while building the model. Therefore, the model size is actually decreased. In fact, not only is the model size reduced, but also the problem becomes easier to solve due to the sparsity of Rdmax . Lemma 4.5. For any optimal solution value z∗ = cTx∗ to the MILP (4.1) and any optimal solution value z∗dmax = c Tx∗dmax to the extended form (4.3), the inequality z∗ ≤ z∗dmax is always true. Proof. Let C = {(x, y) : A(x, y)T ≥ b, x ≥ 0, y ∈ {0, 1}nb} and Cdmax = {(x, y) : A(x, y)T ≥ b,Rdmaxx = 0, x ≥ 0, y ∈ {0, 1}nb}. Clearly, Cdmax ⊆ C. It follows that z∗ ≤ z∗dmax . From Lemma 4.5 we can see that the optimality of the solution with respect to (4.1) is not guaranteed when using dmax, unless dmax ≥ |S|. For this reason, we call (4.3) the dmax-heuristic to the original problem (4.1). Note that the demands and supplies for a particular vertical alignment are 38 4.3. Warmstarting with increasing dmax approximations. As such, a solution to (4.3) that is not optimal with respect to the original problem, may still be a good enough solution for an earth- moving problem. We have therefore an approximation algorithm that we call direct with x◦ and dmax. Algorithm 3: Direct with x◦ and dmax Input: x,y,c,A,b (from (2.1)) Output: Solution z∗ = cTx∗ for (4.3) begin Build Rd0max using (4.2); (x◦, y◦)←FindFeasiblePoint(x,y,c,A,b,Rd0max ); Solve (4.3) for (x0, y0) with starting point (x◦, y◦); end Note that in this algorithm, we modified FindFeasiblePoint in order to take Rd0max into account. The change is implicit and we do not show the adapted procedure here. 4.3 Warmstarting with increasing dmax In Section 4.2, we mention that the dmax-heuristic does not guarantee an optimal solution for (4.1). However, the solution from (4.3) can be used as a starting point to solve (4.1) using a warmstart procedure. Further im- provement can be made by starting to solve (4.3) with a small dmax and use the solution to warmstart (4.3) with αdmax where α ∈ R++, α > 1. If we continue this procedure, we end up with the following algorithm, which we refer to as warmstart algorithm. 39 4.4. Warmstarting with mipgap check Algorithm 4: Warmstart Input: x,y,c,A,b,d0max ≥ 1,α > 1 Output: Solution z∗ = cTx∗ for (4.1) begin k ← 0; Build Rd0max using (4.2); (x◦, y◦)←FindFeasiblePoint(x,y,c,A,b,Rd0max ); Solve (4.3) for (x0, y0) with starting point (x◦, y◦); while dkmax < ms do k ← k + 1; dkmax ← αdk−1max; Build Rdkmax using (4.2); Solve (4.3) for (xk, yk) with starting point (xk−1, yk−1); end (x∗, y∗)← (xk, yk) end Lemma 4.6. The warmstart method in Algorithm 4 converges to the optimal solution value z∗ of the MILP (4.1). Proof. Recall that ms is the number of sections. Let 1 ≤ d0max < ms and α > 1. Hence there exists k ∈ Z+, such that αkd0max ≥ ms. Therefore, at iteration k of Algorithm 4, we have dkmax = αd k−1 max = αα · · ·αd0max = αkd0max ≥ ms. For this k, the extended MILP (4.3) is equal to the original MILP (4.1). 4.4 Warmstarting with mipgap check Definition 4.7. The mipgap |zk − z∗LP| is the gap between the current best feasible value of the MILP, zk, and the optimal value of its LP relaxation, z∗LP. Lemma 4.8. The optimal value z∗LP of the LP relaxation for to the original MILP (4.1) gives a lower bound for the dmax-extended MILP (4.3). Proof. Combine Fact 4.2 and Lemma 4.5. 40 4.4. Warmstarting with mipgap check Lemma 4.9. Let z∗LP be the optimal value for the LP relaxation of (4.1) and z∗ be the optimal value for the MILP (4.1). For any current best feasible value zkdmax of (4.3), |z∗ − z∗LP| ≤ |zkdmax − z∗LP|. Proof. The result follows from Lemma 4.5 and Lemma 4.8. We can now extend Algorithm 4 with a stopping condition that verifies the mipgap. We call this the warmstart with ǫ algorithm. Algorithm 5: Warmstart with ǫ Input: x,y,c,A,b,d0max ≥ 1,α > 1,ǫ > 0 Output: Solution z∗ = cTx∗ for (4.1), within ǫ begin Solve for the LP relaxation solution z∗LP of (4.1); k ← 0; Build Rd0max using (4.2); (x◦, y◦)←FindFeasiblePoint(x,y,c,A,b,Rd0max ); Solve (4.3) for z0 = cTx0 using starting point (x◦, y◦); while dkmax < ms and ǫ < |zk − z∗LP| do k ← k + 1; dkmax ← αdk−1max; Build Rdkmax using (4.2); Solve (4.3) for (xk, yk) with starting point (xk−1, yk−1); end (x∗, y∗)← (xk, yk) end In the next chapter, we look at the numerical results for Algorithms 1 - 5 with a set of problems based on real road alignment designs. We analyze solving times and quality of solutions for the different algorithms and try to find good values for the parameters α and dmax. 41 Chapter 5 Numerical results Understanding is, after all, what science is all about-and science is a great deal more than mere mindless computation. Roger Penrose (1931–) There are two categories of questions that we seek to statistically examine in this section. First, which algorithm is the fastest, and which algorithm is the most robust to solve our MILP model? Second, we want to know how the solution value of our model compares to the solution value of a LP solution for a given non-optimal removal schedule. 5.1 Experimental setup 5.1.1 Problems In order to compare the algorithms in Chapter 4, we based our test prob- lems on a family of real road alignments. We used a road design software to generate 8 different road alignments with length between 1 and 40 kilo- meters. We denote the alignments with letters from A to H. From the 8 designs, we then created 44 problems with different block and access road configurations. For all of the problems we added two access roads, one at the beginning and one at the end of the road. For about half of all the problems we added two additional access roads in between the start and the end of the road. We further split each group into three subgroups of 5, 10, and 20 blocks. The problem configurations are given in Table A.1 of Appendix A. 5.1.2 Benchmark In order to evaluate the benefits of our model over current practices, we would need to conduct a field study on a sufficiently large set of real con- struction projects. Similar to our theoretical problem set, such projects 42 5.1. Experimental setup would need to have obstacles and access roads in different configurations. The effective earthwork costs could then be compared with the solution of our model. In our research, we did not have access to such data. We there- fore tried to build a reference solution that, to our knowledge, would be as fair as possible to compare with our solution. Reference solution For our reference solution we assume that obstacles are removed in a sequen- tial order. We assume that earthwork tasks can occur at the same time as the obstacle removal process, as long as the sections for these tasks are acces- sible. We therefore use the removal scheme in Procedure FindFeasiblePoint as reference block removal schedule. As in Procedure FindFeasiblePoint, we solve the LP for the fixed schedule. Note that we do not use any dmax restrictions in the reference solution. We call this reference solution the LP solution for mass removal schedule. We are aware that block removal schedules vary for different construction projects and they also depend heavily on the experience of the geotechni- cal engineer. A mass removal schedule, however, is perfectly valid in forest areas or in ground conditions where the operation of ordinary construction equipment is not possible. In very difficult ground settings, tractor mounted equipment is used to push large-sized granular material in front of the equip- ment to build a stable construction platform [Fwa06]. This corresponds to a sequential removal of ground obstacles. Parameters For some of the tests we changed parameters for the algorithms in Chapter 4, such as: − ε is the tolerance for the MILP solver; − εLP is the LP optimality tolerance to solve the LP relaxation; − ǫ is the mipgap defined in Definition 4.7; − dmax ] is the maximal allowed moving distance; − α is the scaling factor for dmax within the warmstarting scheme. 43 5.1. Experimental setup Note that except for the warmstart algorithms, ε = ǫ. For the warmstart algorithms, we use ε = { 0.05 if dkmax < ms, ǫ if dkmax ≥ ms. In some problems, the solving time for the LP relaxation z∗LP (see Algorithm 5) can be quite long. To reduce this solving time, we change the LP solver optimality tolerance εLP. The LP solver in the Gurobi Optimizer has a default LP optimality tolerance of εLP = 10 −6. An increase in the LP optimality tolerance results in a faster solving time for an LP. Now if for the mipgap tolerance ǫ > 0, we have |z∗ − z∗LP| ≤ ǫ, then for some δ > 0, by the triangle inequality, we have |z∗ − z∗LP + δ| ≤ ǫ+ δ. So if we let the new LP optimality tolerance ε̃LP > εLP, with δ = ε̃LP− εLP and the new mipgap tolerance ǫ̃ = ǫ+δ, then the convergence of Algorithm 5 within a mipgap tolerance of ǫ is still guaranteed. For our experiments with the warmstart and ǫ (d0max = 25, α = 10) and warmstart with ǫ (d 0 max = 5, α = 5) algorithm, we used ε̃LP = 10 −5 to solve the LP relaxation and ǫ̃ = ǫ+ δ for the mipgap tolerance. For all the other computations, we used the default LP optimality tolerance. Test configurations In our first set of experiments, our goal is to investigate the behavior of the solution and solving time with respect to dmax. Hence we run all the problems with the direct with x◦ and dmax algorithm (Algorithm 3). The configuration for these tests are given in Table 5.1. In our second set of experiments, our goal is to investigate the economical benefit and the per- formance of the algorithms in Chapter 4. The LP solution for mass removal schedule was used for an economical comparison to the exact solution to (2.1), as given by the algorithms in Chapter 4. Furthermore, a performance analysis was done. The LP with mass removal schedule was not used in the performance comparison since it is a pure LP, and it is of little value to compare the solving time of a single LP with the algorithms in Chapter 4. Moreover, the LP with mass removal schedule is not an optimal solu- tion to (2.1). We therefore focus our performance analysis on the MILP 44 5.1. Experimental setup Table 5.1: Test configurations for dmax with the direct solve algorithm using x◦ (Algorithm 3) ǫ dmax 0.01 5 25 100 200 ǫ dmax 0.05 5 25 100 200 Table 5.2: Economical test and performance test configurations ǫ d0max α Algorithm 0.01 - - direct solve (Algorithm 1) direct with x◦ (Algorithm 2) 25 10 warmstart (Algorithm 4) warmstart with mipgap (Algorithm 5) 5 5 warmstart with mipgap (Algorithm 5) 0.05 - - direct solve (Algorithm 1) direct with x◦ (Algorithm 2) 25 10 warmstart (Algorithm 4) warmstart with mipgap (Algorithm 5) 5 5 warmstart with mipgap (Algorithm 5) 45 5.2. Results algorithms only. In the context of economical analysis and performance, we run the experiments with the configurations shown in Table 5.2. For all our tests, we used a CPU time limit of 12 hours. If an algorithm could not solve the problem within the time limit, we aborted the solving process and considered the problem as not solved. Remark 5.1. In practice, a MILP tolerance of 5% is sufficiently precise for an earth-moving problem. The data for earth quantities is based on approx- imations that mostly use triangular surface grids. The use of triangulariza- tion, measurement errors, and non-constant factors for earth shrinkage and swelling, introduce errors that are in the order of 5%. Test equipment and software All the experiments were solved with the Gurobi Optimizer 2.0.2 MILP solver (http://www.gurobi.com) on a standard 64-bit Linux (Ubuntu 9.10 Server) workstation with an IntelR© 2.4GHz Quad-Core CPU and 8GB of memory. Since the solver makes use of multiple cores, the reported CPU- time may be greater than the wall-clock time in some problems. The algo- rithms were implemented in C++ and the code can be made available upon request. Please contact the author at valentin.koch@gmail.com. 5.2 Results 5.2.1 Analysis of the dmax-heuristic In the top row in Figure 5.1 we show the boxplots of the relative error of the objective value with respect to the solution for dmax ≥ ms and the corresponding MILP solver tolerances of ε=1% and ε=5%. The bottom row contains the relative solving time with respect to the solving time for dmax ≥ ms. In the boxplots, the box represents the interquartile range (IQR) and contains 50% of the sample points. It gives therefore a relatively robust statistic. The vertical bar inside the box represents the median. The ends of the whiskers represent the lowest datum still within 1.5 IQR of the lower quartile, and the highest datum still within 1.5 IQR of the upper quartile. Outliers are depicted as small dots. For convenience, the mean is also plotted as a small cross. We only counted problems that could be solved for both, varepsilon=1% and ε=5%. The numbers for the plots are given in Table A.2 and A.3 in the Appendix. Looking at dmax = 5 in the 1% configuration, we see that the median relative error of 34.84% for the objective value is about 5% smaller than the 46 5.2. Results 0 0.5 1 1.5 2 5 25 100 ms dmax 0 0.5 1 1.5 2 5 25 100 ms R el at iv e T im e dmax -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 5 25 100 ms 5% mipgap -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 5 25 100 ms R el at iv e E rr or 1% mipgap MeanMean MeanMean Figure 5.1: Objective and solving time with increasing dmax and different tolerance 47 5.2. Results median relative error of 40.39% for the 5% configuration. Remember that the reference solution for the relative error is also solved within a tolerance of 1% or 5%, respectively. With regard to Remark 5.1, we conclude that, when using dmax = 5, the relative error is too large for a solution to be acceptable. At dmax = 25, however, the median error is 0% in both cases, with an IQR of 4.6% and 3.3%, respectively. At the same time, we can look at the relative solving time and we can see that for 1% tolerance, at dmax = 5, we need 10.2% of the time that the direct solve with x ◦ algorithm uses to solve the MILP. The time for a 5% tolerance at dmax = 5 is slightly smaller, at 6.4% of the direct solve and x◦, which can be explained with the earlier convergence. For dmax = 25 we ended up with 26.4% and 13.8%, respectively. A quick glance at the figure tells us that there is not much difference at dmax = 100 to the direct solve algorithm. Consider, however, that 80% of all problems have ms ≤ 100 and therefore have the same results with dmax = 100 and dmax ≥ ms. If we add the IQR to the solving tolerance, we see that a choice of dmax = 25 produces a reasonable result that lies usually within 5.6% of the real solution with 1% solving tolerance, and within 4.3% of the solution with 5% solving tolerance. The results with 1% solving tolerance do not lie within the 5% suggested by Remark 5.1. However, a solution from dmax = 25 is sufficiently close to the real solution in order to give a good upper bound for a subsequent MILP with a larger dmax or dmax ≥ ms. The median solving time with dmax = 5 for 1% tolerance is around 10 times faster, and with 1% tolerance around 16 times faster than the direct with x◦ algorithm. In more than half of all the problems, the solving time does not exceed 16% of the solving time with the direct and x◦ algorithm for both tolerances. In the dmax = 25 case, the solving time is about 4 and 7 times faster. We observe that the upper quartile for 1% is at about half and for 5% at about a third of the solving time with the direct with x◦ algorithm. We will use these observations in the analysis of the warmstart scheme in order to derive the effect of the starting point given by dmax = 5 and dmax = 25. 5.2.2 Performance In order to compare the performance of the algorithms, we use a performance profile as developed in [DM02]. For each algorithm, we plot the number of problems that are solved within a factor of the best CPU time. Let P be the set of all problems, and A be the set of algorithms. We compute the 48 5.2. Results performance ratio by rp,a = tp,a min{tp,a : a ∈ A} , where p ∈ P, and tp,a is the solving time of algorithm a for problem p. The percentage of problems that are solved within a factor τ ∈ R are computed by ρa(τ) = |{p ∈ P : rp,a ≤ τ}| |P| . Hence for a particular problem p, an algorithm a needs τ times longer than the best algorithm for p, if τ > 1 when algorithm a is able to solve it. If algorithm a needs only τ = 1, the algorithm is considered as the best, or among the best algorithms for problem p. The percentage ρa of all problems solved by algorithm a at τ gives us an overall assessment of the performance of algorithm a. We conclude that the algorithm with the highest ρ value is likely to perform best on a given problem with a probability of ρa(τ). For a good algorithm, the plot of ρ(τ) should therefore be found above the other algorithms. If we increase τ to allow the algorithms to be within a larger factor of the best algorithm, then the algorithm with the highest ρ is the most robust algorithm. Hence an algorithm with the highest ρ value at a large τ , solves most of the problems. For a more detailed description of performance profiles, we refer the reader to [DM02]. Figure 5.2 shows the performance profiles for Algorithms 1 to 5 on a log2 scale, that is, ρa(τ) = |{p ∈ P : log2(rp,a) ≤ τ}| |P| . Figure 5.2a shows the profile for ε=1% and Figure 5.2b shows the profiles with ε=5%. In Figure 5.3a and 5.3b, the profiles are shown on a normal scale with a short τ interval. Figures 5.4a and 5.4b show the profiles on a normal scale with a long τ interval. For all profiles, we have |A| = 5 and |P| = 44. For a ε=1%, we can see that the direct solve with x◦ algorithm outper- forms the other algorithms with a probability of about 32%, and the proba- bility that the warmstart with ǫ (d0max = 25, α = 10) algorithm outperforms all other algorithms is at about 27%. If we increase τ to let the algorithms be within a factor of 4 of the direct solve with x◦ algorithm, we can see that the warmstart with ǫ (d0max = 25, α = 10) algorithm wins with a high probability of about 78% and solves up to 80% of the problems when τ is extended to 5. We can see that the warmstart with ǫ (d0max = 25, α = 10) is the most robust 49 5.2. Results 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 6 7 ρ a (l og 2 (τ )) log2(τ) direct solve direct with x◦ warmstart (d0max = 25, α = 10) warmstart with ǫ (d0max = 25, α = 10) warmstart with ǫ (d0max = 5, α = 5) (a) 1% MILP tolerance 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 6 7 ρ a (l og 2 (τ )) log2(τ) direct solve direct with x◦ warmstart (d0max = 25, α = 10) warmstart with ǫ (d0max = 25, α = 10) warmstart with ǫ (d0max = 5, α = 5) (b) 5% MILP tolerance Figure 5.2: Algorithms performance profiles with log2 scale 50 5.2. Results algorithm (82% of all problems solved), followed by the warmstart with not mipgap check and the direct solve with x◦ algorithm (79.5% of all problems solved). We conclude that the warmstart with ǫ (d0max = 25, α = 10) algo- rithm is a good choice to solve most of the problems relatively fast. Clearly, the direct solve algorithm is not very competitive in these experiments. The warmstart with ǫ (d0max = 25, α = 10) converged with a dmax ≤ 25 on about 34% of all solved problems. Recall that in Section 5.2.1, for ε=1%, we found that the median and lower quartile of the relative solution error for dmax = 25 is 0% with an upper quartile of 4.6%. Therefore half of the solutions were above 4.6% of the real solution. From Lemma 4.9 we expect that the warmstart with ǫ algorithm will use at least one more iteration for these solutions. Furthermore, due to the IQR of 4.6% for the solution error, there is also a significant number of solutions within the IQR that are above the required ε=1% tolerance. Altogether, these problems account for the 66% where the warmstart with ǫ algorithms were required to increment dmax from 25 to 125 or 250, depending on α. For ε=5%, the warmstart with ǫ (d0max = 25, α = 10) algorithm dom- inates all other algorithms on 46% of all problems. The algorithm clearly stays ahead of the others until τ ≈ 35. At that point, either the warm- start with ǫ (d0max = 25, α = 10) or the warmstart without mipgap check would suffice to solve about 97.7% of all the problems. The warmstart with ǫ (d0max = 25, α = 10) algorithm is therefore the clear winner in the ε=5% experiments. An explanation why the warmstart with ǫ (d0max = 25, α = 10) algorithm performs always better or equal to the other algorithms when ε=5% lies in the mipgap check. While the median and lower quartile for the solution error at dmax = 25 are, similar to the ε=1% experiments, at 0%, the upper quartile is only at 3.3%. Therefore half of all the problems that lie in the IQR at dmax = 25 are already within ε=5%. From the results, we know that the algorithm converged on 61% of all solved problems with a dmax ≤ 25. This is a notable result that confirms our dmax-analysis. The warmstart with ǫ (d0max = 5, α = 5) algorithm shows a mixed per- formance. It starts better than the simple warmstart with no mipgap check algorithm and if τ ≈ 3, it also outperforms the direct with x◦ algorithm with a probability of over 70%. However, after τ = 6, the warmstart with ǫ (d0max = 5, α = 5) algorithm performs only better than the direct solve algorithm and finished by solving 88.6% of all the problems. The direct solve algorithm solved only 81%. In the next section, we will analyze the quality of the solutions with respect to this LP with mass removal schedule. 51 5.2. Results 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 2 4 6 8 10 12 ρ a (τ ) τ direct solve direct with x◦ warmstart (d0max = 25, α = 10) warmstart with ǫ (d0max = 25, α = 10) warmstart with ǫ (d0max = 5, α = 5) (a) 1% MILP tolerance 0 0.2 0.4 0.6 0.8 1 0 2 4 6 8 10 12 ρ a (τ ) τ direct solve direct with x◦ warmstart (d0max = 25, α = 10) warmstart with ǫ (d0max = 25, α = 10) warmstart with ǫ (d0max = 5, α = 5) (b) 5% MILP tolerance Figure 5.3: Algorithms performance profiles with normal scale and small intervale for τ 52 5.2. Results 0 0.2 0.4 0.6 0.8 1 0 20 40 60 80 100 ρ a (τ ) τ direct solve direct with x◦ warmstart (d0max = 25, α = 10) warmstart with ǫ (d0max = 25, α = 10) warmstart with ǫ (d0max = 5, α = 5) (a) 1% MILP tolerance 0 0.2 0.4 0.6 0.8 1 0 20 40 60 80 100 ρ a (τ ) τ direct solve direct with x◦ warmstart (d0max = 25, α = 10) warmstart with ǫ (d0max = 25, α = 10) warmstart with ǫ (d0max = 5, α = 5) (b) 5% MILP tolerance Figure 5.4: Algorithms performance profiles with normal scale and large interval for τ 53 5.2. Results 5.2.3 Economical analysis In Section 5.1.2, we defined the LP with mass removal schedule as our ref- erence solution. Our reference solution is based on a LP model and does not necessarily represent the real cost of an actual earthwork construction project with current techniques. Let z∗p,ref be the solution for problem p using the LP with mass removal schedule. Let z∗p,a be the solution for problem p using algorithm a ∈ A. We define the gain gp,a by gp,a = 1− z∗p,a z∗p,ref . For all problems that have been solved, we plot the gain in a histogram. The number of bars in the histogram is the square root of the number of problems that have been solved. We also plot a normal curve over the histogram that fits the data. Note that we observed that in some instances where the problems were not solved, the current solution was still better than the solution of the LP with mass removal schedule. It is not clear if one should count these solutions in the histogram plots. We decided to keep these solutions in our economical analysis on the following grounds. In practice, if a solver is interrupted with an intermediate solution that is better than a traditional LP solution, the intermediate solution will most likely be used. In Section 5.2.2 we identified the best three performing algorithms as the warmstart with ǫ (d0max = 25, α = 10) algorithm (Algorithm 5), the direct with x◦ algorithm (Algorithm 2), and the warmstart (d0max = 25, α = 10) algorithm (Algorithm 4). Since the warmstart with ǫ (d0max = 25, α = 10) algorithm solved more problems than the other two algorithms, we focus our economical analysis on the results of the warmstart with ǫ (d0max = 25, α = 10) algorithm. Figure 5.5 shows the histogram for the warmstart with ǫ (d0max = 25, α = 10) algorithm. Histograms for the other two algo- rithms are shown in Appendix B. The statistics for the histograms can be found in Appendix A, Tables A.4 and A.5. For the problems that have been solved, the use of the warmstart with ǫ (d0max = 25, α = 10) algorithm and ε=1% produced an average gain of 4.1%. The same algorithm delivered an average of 3.3% gain if ε=5%. From the plot we see that the negative gains in the ε=5% histogram, that are caused by the MILP tolerance, have a much greater influence on the average than in the ε=1% case, where almost no negative gains are encountered. The maximum gains are 17.9% in the ε=1% histogram, and 19% in the ε=5% histogram. 54 5.2. Results 0 5 10 15 20 25 -0.2-0.15-0.1-0.05 0 0.050.1 0.150.2 5% mipgap 0 5 10 15 20 25 -0.2-0.15-0.1-0.05 0 0.050.1 0.150.2 P ro b le m s 1% mipgap Figure 5.5: Savings with warmstart with ǫ (d0max = 25, α = 10), compared to mass block removal schedule Overall we can observe that the average gains for ε=1% are not signifi- cantly higher than for ε=5%. This is surprising since in general, the MILP solver stops improving the solution in the ε=5% case much earlier than in the ε=1% case. From the tables in the appendix we can see that the solving time for the ε=1% case is much higher than for ε=5%. The qualitative merit of our model can be assessed when looking at the best gains. For each of the two MILP tolerances, there were 8 problems with gains of 10% and more. This accounts for about 18% of all the problems. In the context of a large highway project, this is a significant amount of gain over the LP with mass schedule. We are interested in the problems that produced the biggest gains. In order to see if there are correlations between problem types and gains, we did a stratification analysis with the use of a scatter plot. The scatter plot is shown in Figure 5.6. The scatter plots for the other algorithms can be found in Appendix B. We first grouped the problems into the number of blocks. Each group is shown with a different color and symbol. Green stars show five block problems, blue circles show ten block problems, and red triangles show twenty block problems. Each group was internally sorted by the number of sections and plotted from left to right in ascending order. Hence, problem 1 in the scatter plot represents the five block problem with the least number of sections, whereas problem 17 represents a five block problems with the largest number of sections. Small points represent problems with two access roads, large points represent problems with four access roads. In the stratification analysis, we could not identify any visibly striking pattern. The ε=1% and the ε=5% results have a similar distribution. One might observe that in the groups of five block problems, there are less solu- 55 5.2. Results -0.05 0 0.05 0.1 0.15 0.2 0 10 20 30 40 50 5% mipgap -0.05 0 0.05 0.1 0.15 0.2 0 10 20 30 40 50 1% mipgap Figure 5.6: Stratification of gains for warmstart with ǫ (d0max = 25, α = 10); * are 5, ⊙ are 10, N are 20 blocks; small symbols are 2, big symbols are 4 access roads; within the block groups, points are sorted from left to right in ascending order of section numbers tions with significant gains than in the other two groups. For these problems, it seems, the LP with mass removal schedule provides often a good enough solution. If we group the results with the highest gains, we can observe a majority of problems with four access roads. Intuition tells us that with more access roads, there is more flexibility and hence a potential for cheaper solutions. We conclude that for a better stratification analysis, experiments with a larger set of problems would certainly give more insight. 56 Chapter 6 Conclusion Frustra fit per plura quod potest fieri per pauciora.4 William of Ockham (1287-1347) 6.1 Accomplishments We started the thesis with the description of traditional methods to balance earthwork and minimize hauling costs in the context of road construction. Two main methods were presented in Chapter 1. First, we explained the mass diagram, a simple tool that helps to balance the earthwork and to obtain an estimate on the directions of haul. Second, we showed a more detailed model of a LP by Mayer and Stark [MS81] that not only balances the earthwork, but also minimizes the hauling cost. We then pointed out that neither method captures the problem of blocks that restrict some of the earth movements to balance earth across the construction site. To address this problem, we presented an extension to the model of [MS81] in Chapter 2. In Chapter 3, we analyzed the model with respect to space complexity and we investigated the growth of the number of feasible block removal schedules. To improve the computation time for solving the MILP, we outlined a set of algorithms in Chapter 4. In Chapter 5, we presented the numerical results of the algorithms, together with an analysis of performance, robustness, and accuracy. 6.2 Research outcomes In the model analysis, we first identified a potential issue of memory growth. With a fixed number of blocks, we found that the theoretical upper bound on the growth of memory for the MILP model in (2.1) is O (|M|4). In 4It is pointless to do with more what can be done with less. 57 6.2. Research outcomes our experiments, we use problems with |M| . 200. However, models with |M| ≈ 200 required relatively large amounts of the available memory. The solver failed to solve some of these problems due to memory exhaustion. A second result of our model analysis was the enumeration of the feasible block removal schedules. We concluded that for a road with one access road at either end, there is no need to solve for the MILP, and one can just solve a LP for a fixed removal schedule that removes the blocks sequentially. If the access road is at some other place, we can quickly compute the number of feasible schedules with equation (3.4). If the number of feasible schedules is small, we can enumerate the schedules and solve a LP for every schedule. The schedule that produces the smallest objective value is the optimal re- moval schedule. We also showed that the worst case of number of schedules for one access road is bounded by O (3|I|). Therefore, if |I| is large, we recommend to solve the MILP instead of a schedule enumeration. The same observations were made for the typical configuration of two access roads, one on each end. In our numerical analysis, we showed that the MILP can be solved effi- ciently with the help of the algorithms from Chapter 4. In the experiments, we demonstrated the strength of the dmax-heuristic, and showed that there is a significant performance benefit when using the warmstart with ǫ, the simple warmstart, or the direct with x◦ algorithm, instead of solving the MILP directly. Furthermore, we showed that these three algorithms are considerably more robust than a method that solves directly for the MILP. As a last observation in our analysis, we showed the economic benefits of solving the MILP instead of using a LP with a fixed mass removal schedule. If we solve a problem to an accuracy of ε=1%, the MILP produces solutions that are about 4.1% cheaper in the average. If we aim for ε=5% accuracy, the MILP provides us with a solution that saves roughly 3.3% on a LP solution for a fixed mass removal schedule. At a first sight, these benefits may not seem very high. However, in the context of a highway project that involves earthwork costs of millions of dollars, 4.1% is a significant figure. In summary, we started by presenting a MILP model that overcomes the problems of the mass diagram and the LP model by Mayer and Stark. We pointed out potential issues with the model and efficient ways to solve the MILP. The result of the model presents an engineer with a good estimate on earthwork costs. The model is therefore a good tool to quantify a given vertical alignment, and therefore provides a valuable tool to find the most economic alignment. As already pointed out by Mayer and Stark [MS81], the objective of an earthwork contractor is to maximize its profit. A contract that is based on 58 6.3. Future research an engineer’s mass diagram, and that fixes the costs for excavation, borrow, and haul, may not reflect additional costs that result from a haul scheme that is not possible due to blocks. The presented MILP enables a contractor to estimate the real earthwork costs of a project, and at the same time provides the contractor with the cheapest haul scheme in order to maximize its profit. 6.3 Future research 6.3.1 Warmstart schemes and εLP In Section 5.2.2, we explained the problems that we encountered when solv- ing the LP relaxation with the default LP optimization tolerance of the Gurobi Optimizer. At the writing of this thesis, preliminary results of tests with εLP = 10 −5 suggest an even better performance for the warmstart with ǫ (d0max = 25, α = 10) and warmstart with ǫ (d 0 max = 5, α = 5) algorithms. We also would like to investigate additional warmstart schemes with differ- ent d0max- and α-values. 6.3.2 Vertical alignment The above model can easily be extended to the vertical alignment. Based on a vertical optimization model by Easa [Eas87], Moreb introduced a LP model that combines the earthwork model by Mayer and Stark with roadway grade selection by using a piecewise linear approximation of the road profile [Mor96]. In [MA04], Moreb and Aljohani extended the model by using a quadratic approximation. The quadratic representation was then further extended in [Mor09] with a spline approximation, with improvements by Koch and Lucet [KL10]. The result was an improved and efficient linear model that avoids sharp connectivities and scaling problems of piecewise linear representations. The main difference between the LP model of Mayer and Stark, and the vertical alignment model by Moreb, is the earth volume. In the earthwork LP and in our MILP model, we treat the volume of a section i as a fixed constant Vi. We recall the balance constraint (1.3b) from the LP as∑ j∈M j 6=i (xij − xji) = Vi for all i ∈ S. However, in the vertical alignment LP by Moreb, the volume is a linear approximation. The volume is computed as the product of the surface area 59 6.3. Future research Ai of section i, with the average height of earth that is cut from the section or filled into that section. The cut or fill is represented with LP decision variables, namely ui for the height of the cut, and vi for the height of the fill in that section. The balance constraint then becomes∑ j∈M j 6=i (xij − xji) = Ai(ui − vi) for all i ∈ S. The ground profile can then be modeled with spline segments. The spline segments do not need any additional decision variables besides ui and vi. We refer the reader to [Mor09, KL10] for the details. Since the model for vertical alignment needs only two additional continuous decision variables per section, the MILP model in (2.1) can readily be combined with the LP model in [Mor09, KL10]. 6.3.3 Schedule enumeration In Remark 3.25, we discussed the use of feasible schedule enumeration in the case of a small number of blocks. The use of a Depth-first search algo- rithm allows an enumeration in linear time. However, the number of feasible schedules itself is exponential. A natural question is what a small number of blocks means in this context. It would be interesting to compare the solving time for the enumeration of schedules, together with solving the LP for each schedule, and the solving time for the MILP. We believe that up to a certain number of blocks, the enumeration of all LP solutions would be significantly faster than solving the MILP. The decision process between the two meth- ods would be simplified significantly with the identification of a threshold that indicates the number of blocks where the MILP will outperform an exhaustive enumeration. An additional topic of interest, with regard to the schedule enumeration, is the use of dynamic programming. In Section 3.2, we described a removal schedule in terms of binary strings, one binary string for each time step. An interesting observation is that the same binary string can be used in multiple schedules. In fact, after a certain time step, multiple schedules may even end with the same set of binary strings. We can therefore look at a binary string as a state in the removal process5. Once two schedules share the same state, they need to solve the same subproblem to finish the removal process. Dynamic programming is particularly well suited for optimization problems with overlapping subproblems [Bel52]. It would be interesting to formulate 5This observation was made by Jim Nastos. 60 6.3. Future research the feasible schedule enumeration as a dynamic program, and to compare the method with exhaustive enumeration, as well as the MILP method. If a threshold was found for the exhaustive enumeration and the MILP, it would be interesting to see if the threshold changes with dynamic programming. 6.3.4 Lagrangian relaxation The final topic for future research is the use of Lagrangian relaxation [Fis04, Lem07]. Let x ∈ Rn. Consider the MILP z = min cTx s.t. Ax = b Dx ≤ e x ≥ 0 and integral, where b ∈ Rm, e ∈ Rk, A ∈ Rm×n, and D ∈ Rk×n. The Lagrangian problem of the above MILP is zD(λ) = min c Tx+ λ(Ax− b) s.t. Dx ≤ e x ≥ 0 and integral, where λ = (λ1, . . . , λm) is a vector of Lagrange multipliers. In [Fis04], it is shown how to choose appropriately λ in order to approach z. 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An exception are the designs C and G with 20 blocks, where the use of 4 access roads was too hard to solve. Problems 23 and 24 are configurations were the blocks are mostly clustered together in adjacent sections. The data for the problems is available upon request. Please contact the author at valentin.koch@gmail.com. Table A.1: Problem configurations Problem Design Length (km) Sections Access roads Blocks 1,25 A 4 101 2,4 5 2,26 B 40 201 2,4 5 3,27 C 1 51 2,4 5 4,28 D 1 100 2,4 5 5,29 E 1 52 2,4 5 6,30 F 1 103 2,4 5 7,31 G 9 100 2,4 5 8,32 H 9 201 2,4 5 9,33 A 4 101 2,4 10 10,34 B 40 201 2,4 10 11,35 C 1 51 2,4 10 12,36 D 1 100 2,4 10 13,37 E 1 52 2,4 10 14,38 F 1 103 2,4 10 15,39 G 9 100 2,4 10 16,40 H 9 201 2,4 10 17,41 A 4 101 2,4 20 18 C 1 51 2 20 19,42 D 1 100 2,4 20 20,43 E 1 52 2,4 20 21,44 F 1 103 2,4 20 22 G 9 100 2 20 23 H 9 201 2 5 24 F 1 103 2 10 66 Appendix A. Tables Table A.2: Statistics for objective function error with dmax ǫ dmax = 5 dmax = 25 dmax = 100 dmax = ms 0.01 Minimum 0.03344 -0.00907 -0.00351 0.00000 Lower quartile 0.21869 0.00000 0.00000 0.00000 Median 0.34841 0.00000 0.00000 0.00000 Upper quartile 0.60876 0.04625 0.00000 0.00000 Maximum 1.08125 0.18926 0.00000 0.00000 Lower conf. for med. 0.23842 0.00000 0.00000 0.00000 Upper conf. for med. 0.45841 0.01304 0.00000 0.00000 0.05 Minimum -0.01434 -0.02768 -0.04701 0.00000 Lower quartile 0.22466 0.00000 0.00000 0.00000 Median 0.40394 0.00000 0.00000 0.00000 Upper quartile 0.60948 0.03376 0.00000 0.00000 Maximum 1.08494 0.20007 0.01690 0.00000 Lower conf. for med. 0.30959 0.00000 0.00000 0.00000 Upper conf. for med. 0.49830 0.00828 0.00000 0.00000 Table A.3: Statistics for solving time with dmax ǫ dmax = 5 dmax = 25 dmax = 100 dmax = ms 0.01 Minimum 0.00000 0.00016 0.00051 1.00000 Lower quartile 0.03820 0.11630 0.59407 1.00000 Median 0.10172 0.26389 0.99488 1.00000 Upper quartile 0.16049 0.54474 1.00000 1.00000 Maximum 0.44444 1.42857 1.50000 1.00000 Lower conf. for med. 0.06544 0.13677 0.87444 1.00000 Upper conf. for med. 0.13801 0.39101 1.00000 1.00000 0.05 Minimum 0.00000 0.00968 0.21221 1.00000 Lower quartile 0.03800 0.09709 0.86429 1.00000 Median 0.06395 0.13793 1.00000 1.00000 Upper quartile 0.15395 0.28175 1.00000 1.00000 Maximum 0.33205 1.27652 1.74622 1.00000 Lower conf. for med. 0.03800 0.09709 0.96588 1.00000 Upper conf. for med. 0.09310 0.18435 1.00000 1.00000 67 Appendix A. Tables Table A.4: Histogram statistics for 1% tolerance direct direct x◦ warmstart warmstart ǫ, 25 warmstart ǫ, 5 Solved 30 35 35 36 33 Min -0.00674 0.00000 -0.00674 -0.00198 -0.00674 1st qrt. 0.00041 0.00000 0.00000 0.00000 0.00000 Median 0.02386 0.01878 0.01878 0.02232 0.01696 3rd qrt. 0.05308 0.04827 0.05051 0.05079 0.04658 Max 0.17902 0.17902 0.18987 0.17902 0.15804 Mean 0.04341 0.03950 0.04122 0.04088 0.03265 Std. dev 0.05429 0.05132 0.05519 0.05154 0.04227 Skew 1.13166 1.29910 1.35167 1.24205 1.34292 Table A.5: Histogram statistics for 5% tolerance direct direct x◦ warmstart warmstart ǫ, 25 warmstart ǫ, 5 Solved 36 42 43 43 39 Min -0.01722 0.00000 -0.02907 -0.02907 -0.04251 1st qrt. 0.00000 0.00000 0.00000 0.00000 -0.00115 Median 0.01537 0.00000 0.00000 0.00000 0.00615 3rd qrt. 0.04733 0.04055 0.04290 0.04290 0.03903 Max 0.17873 0.17815 0.18987 0.18987 0.15804 Mean 0.03865 0.03176 0.03309 0.03309 0.02331 Std. dev 0.05471 0.05305 0.05527 0.05526 0.04658 Skew 1.18703 1.46810 1.35854 1.35773 1.17794 68 A p p en d ix A . T a b les Table A.6: CPU time in seconds for ε = 0.01, problems 1-22 Problem direct suc- direct x◦ suc- warmstart suc- warmstart ǫ suc- dkmax warmstart ǫ suc- d k max cess cess cess (25, 10) cess (5, 5) cess 1 79 1 77 1 82 1 73 1 250 68 1 125 2 911 1 1199 1 986 1 2622 1 250 3249 1 625 3 21 1 16 1 30 1 38 1 250 37 1 125 4 26 1 18 1 23 1 9 1 25 11 1 25 5 14 1 15 1 15 1 12 1 25 14 1 25 6 55 1 21 1 24 1 55 1 250 29 1 125 7 14 1 14 1 18 1 23 1 250 26 1 125 8 316 1 316 1 332 1 357 1 250 285 1 125 9 7104 1 1603 1 1507 1 1380 1 250 1171 1 125 10 43200 0 22338 1 43200 0 17274 1 250 18214 1 625 11 603 1 412 1 793 1 816 1 250 1472 1 125 12 2936 1 7522 1 1528 1 28593 1 250 3473 1 125 13 606 1 134 1 100 1 16 1 25 49 1 25 14 107860 0 43256 0 468 1 540 1 250 43231 0 125 15 376 1 146 1 206 1 285 1 250 223 1 125 16 15777 0 4864 1 5533 1 5629 1 250 11257 1 625 17 43200 0 84261 0 43200 0 87130 0 250 43337 0 5 18 43200 0 43200 0 43200 0 43200 0 25 43200 0 125 19 43324 0 43335 0 44453 0 43224 0 25 43176 0 5 20 43200 0 43200 0 43200 0 43200 0 25 43200 0 125 21 28 1 22 1 49 1 45 1 250 56 1 125 22 490 1 692 1 506 1 549 1 250 576 1 625 69 A p p en d ix A . T a b les Table A.7: CPU time in seconds for ε = 0.01, problems 23-44 Problem direct suc- direct x◦ suc- warmstart suc- warmstart ǫ suc- dkmax warmstart ǫ suc- d k max cess cess cess (25, 10) cess (5, 5) cess 23 2 1 2 1 4 1 5 1 250 2 1 25 24 184 1 17 1 22 1 9 1 25 28 1 125 25 3 1 3 1 4 1 2 1 25 3 1 25 26 24 1 25 1 28 1 9 1 25 12 1 25 27 17 1 16 1 18 1 8 1 25 10 1 25 28 302 1 298 1 315 1 337 1 250 537 1 625 29 8531 1 15307 1 5369 1 10121 1 250 9506 1 125 30 43200 0 43200 0 43200 0 43200 0 25 43200 0 625 31 33 1 20 1 36 1 15 1 25 16 1 25 32 328 1 188 1 207 1 39 1 25 403 1 125 33 91 1 31 1 40 1 35 1 250 18 1 25 34 31175 1 22377 1 22458 1 8180 1 250 11565 1 125 35 197 1 182 1 196 1 38 1 25 54 1 25 36 43200 0 6486 1 7245 1 7362 1 250 3874 1 125 37 43200 0 43200 0 1536 1 7163 1 250 6891 1 125 38 2200 1 1425 1 2221 1 1430 1 250 43256 0 5 39 43200 0 9448 1 9628 1 11723 1 250 43342 0 25 40 43327 0 12576 1 43259 0 43202 0 250 43176 0 5 41 43200 0 44587 0 48538 0 49588 0 250 43200 0 125 42 56509 0 43208 0 43207 0 43200 0 25 43208 0 125 43 319 1 270 1 383 1 353 1 250 586 1 625 44 268 1 173 1 215 1 221 1 250 222 1 125 70 A p p en d ix A . T a b les Table A.8: CPU time in seconds for ε = 0.05, problems 1-22 Problem direct suc- direct x◦ suc- warmstart suc- warmstart ǫ suc- dkmax warmstart ǫ suc- d k max cess cess cess (25, 10) cess (5, 5) cess 1 59 1 19 1 23 1 26 1 250 40 1 125 2 886 1 601 1 1013 1 1044 1 250 107 1 25 3 20 1 14 1 29 1 30 1 250 38 1 125 4 27 1 18 1 23 1 9 1 25 15 1 25 5 14 1 15 1 13 1 13 1 25 15 1 25 6 56 1 19 1 23 1 8 1 25 12 1 25 7 14 1 14 1 18 1 8 1 25 14 1 25 8 323 1 322 1 338 1 367 1 250 386 1 125 9 1232 1 182 1 150 1 163 1 250 242 1 125 10 43200 0 2414 1 3356 1 1051 1 25 885 1 25 11 609 1 411 1 801 1 809 1 250 1559 1 125 12 235 1 145 1 155 1 30 1 25 53 1 25 13 596 1 134 1 101 1 88 1 25 61 1 25 14 1862 1 140 1 155 1 31 1 25 840 1 125 15 383 1 150 1 169 1 33 1 25 62 1 25 16 43200 0 4972 1 5635 1 5755 1 250 8530 1 250 17 43200 0 5967 1 1416 1 1738 1 250 43248 0 5 18 43200 0 43200 0 18899 1 19198 1 25 6564 1 25 19 33900 1 43338 0 43318 0 43308 0 250 42989 0 5 20 43200 0 1187 1 1295 1 636 1 25 3106 1 25 21 28 1 22 1 27 1 32 1 250 39 1 125 22 182 1 181 1 189 1 30 1 25 69 1 25 71 A p p en d ix A . T a b les Table A.9: CPU time in seconds for ε = 0.05, problems 23-44 Problem direct suc- direct x◦ suc- warmstart suc- warmstart ǫ suc- dkmax warmstart ǫ suc- d k max cess cess cess (25, 10) cess (5, 5) cess 23 2 1 2 1 3 1 2 1 25 3 1 25 24 154 1 18 1 21 1 8 1 25 12 1 25 25 3 1 3 1 4 1 2 1 25 3 1 25 26 25 1 26 1 28 1 8 1 25 14 1 25 27 17 1 16 1 19 1 8 1 25 12 1 25 28 309 1 305 1 323 1 343 1 250 686 1 625 29 186 1 169 1 204 1 228 1 250 351 1 125 30 5154 1 5117 1 5169 1 151 1 25 319 1 25 31 33 1 20 1 27 1 15 1 25 18 1 25 32 189 1 192 1 212 1 37 1 25 55 1 25 33 91 1 31 1 37 1 24 1 25 22 1 25 34 1536 1 198 1 214 1 41 1 25 60 1 25 35 200 1 187 1 200 1 37 1 25 64 1 25 36 14951 1 6628 1 7412 1 7513 1 250 5244 1 125 37 9991 1 1244 1 1418 1 1533 1 250 5606 1 125 38 1846 1 330 1 307 1 302 1 25 43051 0 5 39 43200 0 1130 1 1289 1 2144 1 250 43200 0 25 40 8412 1 1859 1 2333 1 3782 1 250 42900 0 5 41 43200 0 1732 1 5447 1 409 1 25 5617 1 25 42 57882 0 6651 1 7637 1 36483 1 25 38204 1 25 43 327 1 276 1 364 1 392 1 250 276 1 125 44 273 1 167 1 211 1 64 1 25 46 1 25 72 Appendix A. Tables Table A.10: Objective value in $ for direct solve, ε = 0.01 Problem Objective success ε LP mass value 1 454426 1 0.00680 455181 2 9351412 1 0.00549 9974094 3 316359 1 0.00000 329040 4 489574 1 0.00446 505297 5 60684 1 0.00833 68898 6 209271 1 0.00540 211057 7 77114005 1 0.00198 76961263 8 56893108 1 0.00714 56926279 9 455163 1 0.00338 472891 10 10000000 0 0.08160 10204508 11 376427 1 0.00123 394819 12 493254 1 0.00982 519193 13 60687 1 0.00396 72079 14 212554 0 0.01895 212554 15 77114005 1 0.00198 76961263 16 0 0 1.00000 56975485 17 0 0 1.00000 477309 18 0 0 1.00000 516378 19 70262 0 0.02304 79597 20 0 0 1.00000 217812 21 451922 1 0.00983 448897 22 9351412 1 0.00966 9512758 23 297389 1 -0.00000 300402 24 482635 1 0.00000 505297 25 60253 1 0.00124 67322 26 209129 1 0.00559 209244 27 61940820 1 0.00000 61940820 28 55096639 1 0.00737 55096639 29 452486 1 0.00905 468367 30 9480000 0 0.04150 9620263 31 297389 1 0.00000 339047 32 484506 1 0.00539 482671 33 60686 1 0.00395 68391 34 210088 1 0.01000 211502 35 61940820 1 0.00000 61940820 36 56100000 0 0.02480 55132871 37 454000 0 0.01480 475342 38 300248 1 0.00952 360154 39 0 0 1.00000 488042 40 63540 0 0.04868 77395 41 0 0 1.00000 217519 42 0 0 1.00000 64716813 43 56487153 1 0.00000 56487153 44 208258 1 0.00073 214188 73 Appendix A. Tables Table A.11: Objective value in $ for direct solve with x◦, ε = 0.01 Problem Objective success ε LP mass value 1 454426 1 0.00861 455181 2 9351412 1 0.00366 9974094 3 316359 1 0.00205 329040 4 488986 1 0.00327 505297 5 60684 1 0.00813 68898 6 209271 1 0.00627 211057 7 76961263 1 0.00000 76961263 8 56926279 1 0.00771 56926279 9 455163 1 0.00584 472891 10 9498097 1 0.00866 10204508 11 376427 1 0.00591 394819 12 493254 1 0.00995 519193 13 60687 1 0.00396 72079 14 212554 0 0.01170 212554 15 76961263 1 0.00000 76961263 16 56975485 1 0.00857 56975485 17 462423 0 0.03118 477309 18 495000 0 0.01470 516378 19 70262 0 0.10922 79597 20 218000 0 0.04520 217812 21 448897 1 0.00316 448897 22 9351412 1 0.00000 9512758 23 297389 1 0.00000 300402 24 482635 1 0.00092 505297 25 60253 1 0.00124 67322 26 209244 1 0.00614 209244 27 61940820 1 0.00000 61940820 28 55096639 1 0.00737 55096639 29 452486 1 0.00933 468367 30 9430000 0 0.03620 9620263 31 297389 1 0.00000 339047 32 482671 1 0.00160 482671 33 60687 1 0.00396 68391 34 210088 1 0.00993 211502 35 61940820 1 0.00000 61940820 36 55132871 1 0.00802 55132871 37 455000 0 0.01560 475342 38 300248 1 0.00952 360154 39 485640 1 0.00771 488042 40 63540 1 0.00821 77395 41 217519 0 0.04395 217519 42 64716813 0 0.04289 64716813 43 56487153 1 0.00000 56487153 44 210165 1 0.00975 214188 74 Appendix A. Tables Table A.12: Objective value in $ for warmstart (d0max = 25, α = 10), ε = 0.01 Problem Objective success ε LP mass value 1 454426 1 0.00843 455181 2 9351412 1 0.00000 9974094 3 316359 1 0.00000 329040 4 489574 1 0.00446 505297 5 60684 1 0.00833 68898 6 209271 1 0.00627 211057 7 76961263 1 0.00000 76961263 8 56982170 1 0.00869 56926279 9 455163 1 0.00452 472891 10 9500000 0 0.03160 10204508 11 376427 1 0.00346 394819 12 493254 1 0.00982 519193 13 60687 1 0.00396 72079 14 212554 1 0.00996 212554 15 77114005 1 0.00198 76961263 16 56975485 1 0.00857 56975485 17 459000 0 0.02130 477309 18 496000 0 0.01730 516378 19 64484 0 0.05960 79597 20 218000 0 0.04510 217812 21 451922 1 0.00983 448897 22 9351412 1 0.00984 9512758 23 297389 1 0.00000 300402 24 483187 1 0.00206 505297 25 60253 1 0.00124 67322 26 209244 1 0.00614 209244 27 61940820 1 0.00000 61940820 28 55096639 1 0.00737 55096639 29 452486 1 0.00999 468367 30 9430000 0 0.03620 9620263 31 297389 1 0.00000 339047 32 482671 1 0.00160 482671 33 60687 1 0.00396 68391 34 210088 1 0.00993 211502 35 61940820 1 0.00000 61940820 36 55132871 1 0.00802 55132871 37 451067 1 0.00795 475342 38 300248 1 0.00952 360154 39 485640 1 0.00771 488042 40 63540 0 0.04868 77395 41 217519 0 0.04395 217519 42 64716813 0 0.04289 64716813 43 56487153 1 0.00000 56487153 44 210165 1 0.00975 214188 75 Appendix A. Tables Table A.13: Objective value in $ for warmstart with ǫ (d0max = 25, α = 10), ε = 0.01 Problem Objective success ε ǫ LP mass value 1 454426 1 0.00929 0.01434 455181 2 9351412 1 0.00591 0.02938 9974094 3 316359 1 -0.00000 0.06244 329040 4 488986 1 0.00327 0.00328 505297 5 60684 1 0.00027 0.00840 68898 6 209271 1 0.00542 0.00631 211057 7 76961263 1 0.00000 0.00000 76961263 8 56982170 1 0.00869 0.00876 56926279 9 455163 1 0.00338 0.01598 472891 10 9498097 1 0.00859 0.04553 10204508 11 376427 1 0.00074 0.26417 394819 12 493254 1 0.00989 0.01203 519193 13 60687 1 0.00396 0.00398 72079 14 212554 1 0.00996 0.02209 212554 15 77114005 1 0.00198 0.00198 76961263 16 56975485 1 0.00857 0.00857 56975485 17 458003 0 0.02183 0.02183 477309 18 496000 0 0.01730 0.01730 516378 19 70234 0 0.10560 0.33415 79597 20 218000 0 0.04510 0.04510 217812 21 448897 1 0.00316 0.00317 448897 22 9351412 1 0.00966 0.02938 9512758 23 297389 1 0.00000 0.00000 300402 24 482635 1 0.00092 0.00092 505297 25 60253 1 0.00124 0.00124 67322 26 209244 1 0.00614 0.00618 209244 27 61940820 1 0.00000 0.00000 61940820 28 55096639 1 0.00737 0.00742 55096639 29 452486 1 0.00926 0.01119 468367 30 9480000 0 0.04150 0.04150 9620263 31 297389 1 0.00000 0.00000 339047 32 482671 1 0.00160 0.00161 482671 33 60748 1 0.00496 0.00498 68391 34 210088 1 0.00996 0.01024 211502 35 61940820 1 0.00000 0.00000 61940820 36 55132871 1 0.00802 0.00809 55132871 37 451067 1 0.00795 0.00802 475342 38 300320 1 0.00976 0.00986 360154 39 484188 1 0.00473 0.00473 488042 40 63540 0 0.04868 0.05154 77395 41 217519 0 0.04395 0.04858 217519 42 0 0 1.00000 1.00000 64716813 43 56487153 1 0.00000 0.00000 56487153 44 208258 1 0.00068 0.00106 214188 76 Appendix A. Tables Table A.14: Objective value in $ for warmstart with ǫ (d0max = 5, α = 5), ε = 0.01 Problem Objective success ε ǫ LP mass value 1 454426 1 0.00843 0.01434 455181 2 9351412 1 0.00593 0.02938 9974094 3 316359 1 0.00000 0.06244 329040 4 488986 1 0.00327 0.00328 505297 5 60684 1 0.00833 0.00840 68898 6 209271 1 0.00627 0.00631 211057 7 76961263 1 0.00000 0.00000 76961263 8 56576215 1 0.00157 0.00158 56926279 9 455163 1 0.00600 0.01598 472891 10 9498097 1 0.00952 0.04553 10204508 11 376427 1 0.00154 0.26417 394819 12 493254 1 0.00865 0.01203 519193 13 60687 1 0.00396 0.00398 72079 14 212554 0 0.01436 0.05611 212554 15 77114005 1 0.00198 0.00198 76961263 16 56902685 1 0.00730 0.00730 56975485 17 656851 0 0.05385 0.52950 477309 18 496000 0 0.01710 0.01761 516378 19 82236 0 0.06211 0.55954 79597 20 218000 0 0.04550 0.04771 217812 21 451922 1 0.00983 0.00993 448897 22 9351412 1 0.00000 0.02938 9512758 23 297389 1 -0.00000 0.00000 300402 24 483187 1 0.00206 0.00207 505297 25 60253 1 0.00124 0.00124 67322 26 208491 1 0.00255 0.00256 209244 27 61940820 1 0.00000 0.00000 61940820 28 54767023 1 0.00139 0.00140 55096639 29 452486 1 0.00905 0.01119 468367 30 9430000 0 0.03620 0.03755 9620263 31 297389 1 0.00000 0.00000 339047 32 483954 1 0.00425 0.00427 482671 33 60735 1 0.00474 0.00477 68391 34 210088 1 0.00974 0.01024 211502 35 61940820 1 0.00000 0.00000 61940820 36 55231802 1 0.00980 0.00989 55132871 37 451067 1 0.00795 0.00802 475342 38 403569 0 0.06654 0.60979 360154 39 538301 0 0.10452 1.00000 488042 40 77847 0 0.07685 0.45749 77395 41 218000 0 0.04510 0.04801 217519 42 64716813 0 1.00000 0.04482 64716813 43 56487153 1 -0.00000 0.00000 56487153 44 210165 1 0.00975 0.01023 214188 77 Appendix A. Tables Table A.15: Objective value in $ for direct solve, ε = 0.05 Problem Objective success ε LP mass value 1 457354 1 0.01905 455181 2 9351412 1 0.00549 9974094 3 316359 1 0.04771 329040 4 489574 1 0.00446 505297 5 60684 1 0.00833 68898 6 209271 1 0.00540 211057 7 77114005 1 0.00198 76961263 8 56893108 1 0.00714 56926279 9 458942 1 0.02107 472891 10 10000000 0 0.08160 10204508 11 376427 1 0.02126 394819 12 494620 1 0.01462 519193 13 61213 1 0.01252 72079 14 212554 1 0.01895 212554 15 77114005 1 0.00198 76961263 16 0 0 1.00000 56975485 17 0 0 1.00000 477309 18 0 0 1.00000 516378 19 70262 1 0.02951 79597 20 0 0 1.00000 217812 21 451922 1 0.00983 448897 22 9351412 1 0.02854 9512758 23 297389 1 -0.00000 300402 24 503968 1 0.04316 505297 25 60253 1 0.00124 67322 26 209129 1 0.00559 209244 27 61940820 1 0.00000 61940820 28 55096639 1 0.00737 55096639 29 453208 1 0.01264 468367 30 9556950 1 0.04943 9620263 31 297389 1 0.00000 339047 32 487393 1 0.01128 482671 33 60686 1 0.00395 68391 34 210088 1 0.01013 211502 35 61940820 1 0.00000 61940820 36 56082250 1 0.02481 55132871 37 468035 1 0.04392 475342 38 301565 1 0.01385 360154 39 0 0 1.00000 488042 40 63562 1 0.04901 77395 41 0 0 1.00000 217519 42 0 0 1.00000 64716813 43 56487153 1 0.00000 56487153 44 208258 1 0.00073 214188 78 Appendix A. Tables Table A.16: Objective value in $ for direct solve with x◦, ε = 0.05 Problem Objective success ε LP mass value 1 455181 1 0.01577 455181 2 9351412 1 0.02854 9974094 3 316359 1 0.04304 329040 4 505297 1 0.03544 505297 5 60684 1 0.00813 68898 6 211057 1 0.01468 211057 7 76961263 1 0.00000 76961263 8 56926279 1 0.00771 56926279 9 466699 1 0.03908 472891 10 9498097 1 0.04354 10204508 11 376427 1 0.03718 394819 12 500476 1 0.02615 519193 13 60687 1 0.00396 72079 14 212554 1 0.02161 212554 15 76961263 1 0.00000 76961263 16 56975485 1 0.00857 56975485 17 466407 1 0.03946 477309 18 516000 0 0.05600 516378 19 70262 0 0.10922 79597 20 217812 1 0.04523 217812 21 448897 1 0.00316 448897 22 9512758 1 0.04502 9512758 23 300402 1 0.01003 300402 24 505297 1 0.04573 505297 25 60253 1 0.00124 67322 26 209244 1 0.00614 209244 27 61940820 1 0.00000 61940820 28 55096639 1 0.00737 55096639 29 468367 1 0.04460 468367 30 9556950 1 0.04943 9620263 31 297389 1 0.00000 339047 32 482671 1 0.00160 482671 33 60687 1 0.00396 68391 34 211502 1 0.01675 211502 35 61940820 1 0.00000 61940820 36 55132871 1 0.00802 55132871 37 463946 1 0.03549 475342 38 301010 1 0.01203 360154 39 488042 1 0.01259 488042 40 63607 1 0.04969 77395 41 217519 1 0.04395 217519 42 64716813 1 0.04289 64716813 43 56487153 1 0.00000 56487153 44 214188 1 0.02871 214188 79 Appendix A. Tables Table A.17: Objective value in $ for warmstart (d0max = 25, α = 10), ε = 0.05 Problem Objective success ε LP mass value 1 454426 1 0.01413 455181 2 9351412 1 0.00000 9974094 3 316359 1 0.04485 329040 4 505297 1 0.03544 505297 5 60684 1 0.00833 68898 6 211057 1 0.01468 211057 7 78347224 1 0.01769 76961263 8 56982170 1 0.00869 56926279 9 461811 1 0.02990 472891 10 9498097 1 0.04354 10204508 11 376427 1 0.00346 394819 12 497561 1 0.02044 519193 13 61213 1 0.01252 72079 14 212554 1 0.02161 212554 15 78347224 1 0.01769 76961263 16 56975485 1 0.00857 56975485 17 461947 1 0.03018 477309 18 495973 1 0.01731 516378 19 64484 0 0.05960 79597 20 218355 1 0.04761 217812 21 461946 1 0.03132 448897 22 9512758 1 0.04502 9512758 23 300402 1 0.01003 300402 24 505297 1 0.04573 505297 25 60253 1 0.00124 67322 26 209244 1 0.00614 209244 27 61940820 1 0.00000 61940820 28 55096639 1 0.00737 55096639 29 456838 1 0.02048 468367 30 9425696 1 0.03620 9620263 31 301197 1 0.01264 339047 32 482671 1 0.00160 482671 33 61213 1 0.01252 68391 34 211502 1 0.01675 211502 35 61940820 1 0.00000 61940820 36 55132871 1 0.00802 55132871 37 455873 1 0.01841 475342 38 312180 1 0.04738 360154 39 488042 1 0.01259 488042 40 63540 1 0.04868 77395 41 218062 1 0.04633 217519 42 64716813 1 0.04289 64716813 43 57840191 1 0.02339 56487153 44 210680 1 0.01254 214188 80 Appendix A. Tables Table A.18: Objective value in $ for warmstart with ǫ (d0max = 25, α = 10), ε = 0.05 Problem Objective success ε ǫ LP mass value 1 454426 1 0.01413 0.01434 455181 2 9351412 1 0.00000 0.02938 9974094 3 316359 1 0.04485 0.06244 329040 4 505297 1 0.03544 0.03674 505297 5 60684 1 0.00833 0.00840 68898 6 211057 1 0.01468 0.01489 211057 7 78347224 1 0.00000 0.01801 76961263 8 56982170 1 0.00869 0.00876 56926279 9 461811 1 0.02990 0.03082 472891 10 9498097 1 0.02020 0.04553 10204508 11 376427 1 0.00346 0.26417 394819 12 497561 1 0.02044 0.02087 519193 13 61213 1 0.01252 0.01267 72079 14 212554 1 0.02161 0.02209 212554 15 78347224 1 0.00000 0.01801 76961263 16 56975485 1 0.00857 0.00857 56975485 17 461947 1 0.03018 0.03112 477309 18 495973 1 0.01709 0.01761 516378 19 64484 0 0.05960 0.06679 79597 20 218355 1 0.04761 0.04999 217812 21 461946 1 0.03132 0.03233 448897 22 9512758 1 0.04502 0.04714 9512758 23 300402 1 0.01003 0.01013 300402 24 505297 1 0.04573 0.04792 505297 25 60253 1 0.00124 0.00124 67322 26 209244 1 0.00614 0.00618 209244 27 61940820 1 0.00000 0.00000 61940820 28 55096639 1 0.00737 0.00742 55096639 29 456838 1 0.02048 0.02091 468367 30 9425696 1 0.03620 0.03756 9620263 31 301197 1 0.01264 0.01281 339047 32 482671 1 0.00160 0.00161 482671 33 61213 1 0.01252 0.01267 68391 34 211502 1 0.01675 0.01703 211502 35 61940820 1 0.00000 0.00000 61940820 36 55132871 1 0.00802 0.00809 55132871 37 455873 1 0.01841 0.01876 475342 38 312180 1 0.04738 0.04974 360154 39 488042 1 0.01259 0.01259 488042 40 63562 1 0.04901 0.05154 77395 41 218062 1 0.04633 0.04858 217519 42 64716813 1 0.04289 0.04482 64716813 43 57840191 1 0.02339 0.02395 56487153 44 210680 1 0.01216 0.01270 214188 81 Appendix A. Tables Table A.19: Objective value in $ for warmstart with ǫ (d0max = 5, α = 5), ε = 0.05 Problem Objective success ε ǫ LP mass value 1 454426 1 0.01413 0.01434 455181 2 9351412 1 0.02854 0.02938 9974094 3 316359 1 0.04655 0.06244 329040 4 488986 1 0.00327 0.03674 505297 5 60684 1 0.00833 0.00840 68898 6 211057 1 0.01468 0.01489 211057 7 78347224 1 0.00000 0.01801 76961263 8 56576215 1 0.00157 0.00876 56926279 9 461811 1 0.02990 0.03082 472891 10 9498097 1 0.04340 0.04553 10204508 11 376427 1 0.03643 0.26417 394819 12 501295 1 0.02774 0.02087 519193 13 60687 1 0.00396 0.01267 72079 14 219628 1 0.04992 0.02209 212554 15 78499966 1 0.00195 0.01801 76961263 16 56975485 1 0.00857 0.00857 56975485 17 656851 0 0.05385 0.03112 477309 18 495973 1 0.01709 0.01761 516378 19 82236 0 0.06379 0.06679 79597 20 217881 1 0.04553 0.04999 217812 21 461946 1 0.03132 0.03233 448897 22 9351412 1 0.02811 0.04714 9512758 23 297389 1 -0.00000 0.01013 300402 24 488396 1 0.01271 0.04792 505297 25 60253 1 0.00124 0.00124 67322 26 208491 1 0.00255 0.00618 209244 27 61940820 1 0.00000 0.00000 61940820 28 57438725 1 0.04784 0.00742 55096639 29 465460 1 0.03863 0.02091 468367 30 9425696 1 0.03620 0.03756 9620263 31 297389 1 0.00000 0.01281 339047 32 490197 1 0.01693 0.00161 482671 33 60735 1 0.00474 0.01267 68391 34 211607 1 0.01724 0.01703 211502 35 61940820 1 0.00000 0.00000 61940820 36 55231802 1 0.00980 0.00809 55132871 37 455042 1 0.01662 0.01876 475342 38 403569 0 0.06654 0.04974 360154 39 560000 0 1.00000 0.01259 488042 40 77847 0 0.07685 0.05154 77395 41 217944 1 0.04581 0.04858 217519 42 64716813 1 0.04289 0.04482 64716813 43 57205059 1 0.01255 0.02395 56487153 44 218432 1 0.04759 0.01270 214188 82 Appendix B Figures 0 5 10 15 20 -0.2 -0.1 0 0.1 0.2 0.3 5% mipgap 0 5 10 15 20 -0.2 -0.1 0 0.1 0.2 0.3 P ro b le m s 1% mipgap Figure B.1: Savings with direct solve, compared to mass block removal schedule 83 Appendix B. Figures 0 5 10 15 20 25 30 35 -0.2-0.15-0.1-0.05 0 0.050.1 0.150.2 5% mipgap 0 5 10 15 20 25 -0.2-0.15-0.1-0.05 0 0.050.1 0.150.2 P ro b le m s 1% mipgap Figure B.2: Savings with direct solve and x◦, compared to mass block re- moval schedule 0 5 10 15 20 25 -0.2-0.15-0.1-0.05 0 0.050.1 0.150.2 5% mipgap 0 5 10 15 20 25 -0.2 -0.1 0 0.1 0.2 0.3 P ro b le m s 1% mipgap Figure B.3: Savings with warmstart (d0max = 25, α = 10), compared to mass block removal schedule 0 5 10 15 20 -0.2-0.15-0.1-0.05 0 0.050.1 0.150.2 5% mipgap 0 5 10 15 20 -0.1 -0.05 0 0.05 0.1 0.15 0.2 P ro b le m s 1% mipgap Figure B.4: Savings with warmstart and ǫ (d0max = 5, α = 5), compared to mass block removal schedule 84 Appendix B. Figures -0.05 0 0.05 0.1 0.15 0.2 0 5 10 15 20 25 30 35 40 5% mipgap -0.05 0 0.05 0.1 0.15 0.2 0 5 10 15 20 25 30 35 40 1% mipgap Figure B.5: Stratification of gains for direct solve; * are 5, ⊙ are 10, N are 20 blocks; small symbols are 2, big symbols are 4 access roads; within the block groups, points are sorted from left to right in ascending order of section numbers 0 0.05 0.1 0.15 0.2 0 10 20 30 40 50 5% mipgap 0 0.05 0.1 0.15 0.2 0 10 20 30 40 50 1% mipgap Figure B.6: Stratification of gains for direct solve with x◦; green=5, blue=10, red=20 blocks; * are 5, ⊙ are 10, N are 20 blocks; small symbols are 2, big symbols are 4 access roads; within the block groups, points are sorted from left to right in ascending order of section numbers 85 Appendix B. Figures -0.05 0 0.05 0.1 0.15 0.2 0 10 20 30 40 50 5% mipgap -0.05 0 0.05 0.1 0.15 0.2 0 10 20 30 40 50 1% mipgap Figure B.7: Stratification of gains for warmstart (d0max = 25, α = 10); * are 5, ⊙ are 10, N are 20 blocks; small symbols are 2, big symbols are 4 access roads; within the block groups, points are sorted from left to right in ascending order of section numbers -0.05 0 0.05 0.1 0.15 0.2 0 5 10 15 20 25 30 35 40 5% mipgap -0.05 0 0.05 0.1 0.15 0.2 0 5 10 15 20 25 30 35 40 1% mipgap Figure B.8: Stratification of gains for warmstart with ǫ (d0max = 5, α = 5); * are 5, ⊙ are 10, N are 20 blocks; small symbols are 2, big symbols are 4 access roads; within the block groups, points are sorted from left to right in ascending order of section numbers 86
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