Efficient Dynamic Simulation Allocation in Ordinal Optimization

Abstract

Safety stocks are set to minimize inventory costs-the cost function in problem (15)-using as input the buyer's demand model, the service level agreement-the constraint in problem (15)-and (a model of) the production capacities B i n . We will explore how demand parameters can be adjusted according to both supplier and buyer cost structures so that any generated savings can be shared among them.We assume that the buyer's demand is an m-state Markov-modulated process (MMP), with transition probability matrix PD and demand levels at each state given by the vector rD. The supplier's production capacity at each node i of the assembly network is modeled by an m i -state MMP with transition probability matrix P B and capacities at each state given by the vector r B , i = 1; . . . ; N . We assume that any mutually agreed adjustments in r D keep the mean demand E [D] constant; otherwise, the buyer would be unable to satisfy demand in the long term. We denote by E[D] this constant value. Let C R (r D ), the buyer cost associated with any change of r D , bewhere i expresses the buyer's cost for changing the demand level in the i th state of the demand process. rD is the vector of demand levels initially determined and to which the buyer associates zero cost. The supplier's inventory cost function, CM (rD), is the optimal value function of problem (15). Our objective is to find the vector rD that solves the following problem:Problem (17) is a nonlinear optimization problem over a convex set and can be solved using the conditional gradient method. To that end, we need the cost function gradient. The gradient of the buyer's cost can be easily derived from (16). We can evaluate the gradient of the supplier inventory costs using finite differences. It is interesting to observe that the above algorithm can be used in a distributed fashion. In particular, the buyer can take charge of finding new demand vectors on feasible descent directions. At each iteration, the buyer presents the supplier with the new demand levels r k D and the supplier responds with the gradient rC M (r k D ) of its cost function.This gradient information can be provided in the form of appropriate incentives to the buyer. The buyer uses this gradient information to find a feasible direction and compute r k+1 D . Note that neither the supplier nor the buyer need to know each other's cost structures. V. CONCLUSIONWe studied the inventory control problem for a single class assembly network which operates under a modified echelon base-stock policy. We developed an approach to find close-to-optimal echelon stock levels that minimize inventory costs while guaranteeing stockout probabilities stay below some predefined levels. Relying upon large deviations techniques, we reduced the safety stock selection to a deterministic nonlinear optimization problem.We also used our inventory control approach to analyze how a supplier can interact with a buyer to reach a mutually beneficial mode of operations. This interaction takes the form of a supply contract that enforces...

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