UBC Theses and Dissertations
Lower bounds for production/inventory problems by cost allocation Iyogun, Paul Omolewa
This thesis presents a cost allocation method for deriving lower bounds on costs of feasible policies for a class of production/inventory problems. Consider the joint replenishment problem where a group of items is replenished together or individually. A sequence of reorders for any particular item will incur holding, backorder and set-up costs specific to the item, in addition whenever any item is replenished a joint cost is incurred. What is required of the total problem is the minimization of a cost function of the replenishment sequence or policy. The cost allocation method consists of decomposing the total problem into sub-problems, one for each item, by allocating the joint cost amongst the items in such a way that every item in the group receives a positive allocation or none. The result is that, for an arbitrary feasible cost allocation, the sum of the minimum costs for the subproblems is a lower bound on the cost of any feasible policy to the total problem. The results for the joint replenishment problem follows: For the constant and continuous demand case we reproduce the lower bound of Jackson, Maxwell and Muckstadt more easily than they did. For the multi-item dynamic lot-size problem, we generalize Silver-Meal and part-period balancing heuristics, and derive a cost allocation bound with little extra work. For the 'can-order' system, we use periodic policies derived from the cost allocation method and show that they are superior to the more complex (s,c,S) policies. The cost allocation method is easily generalized to pure distribution problems where joint replenishment decisions are taken at several facilities. For example, for the one-warehouse multi-retailer problem, we reproduce Roundy's bound more easily than he did. For the multi-facility joint replenishment problem (a pure distribution system with an arbitrary number of warehouses), we give a lower bound algorithm whose complexity is dr log r where d is the maximum number of facilities which replenish a particular item and r is the number of items.
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