Deterministic Analytical Models

The supply chain consists of various stages. Generally, multi-stage models for supply chain design and analysis can be divided into four categories, by modeling approach. In the cases included here, the modeling approach is driven by the nature of the inputs and the objective of the study. The four categories are

• Deterministic analytical models, in which the variables are known and specified
• Stochastic analytical models, where at least one of the variables is unknown, and is assumed to
• follow a particular probability distribution
• Economic models
• Simulation models

Williams (1981) presents seven heuristic algorithms for scheduling production and distribution operations in an assembly supply chain network (i.e., each station has at most one immediate successor, but any number of immediate predecessors). The objective of each heuristic is to determine a minimum-cost production and/or product distribution schedule that satisfies final product demand. The total cost is a sum of average inventory holding and fixed (ordering, delivery, or set-up) costs. Finally, the performance of each heuristic is compared using a wide range of empirical experiments, and recommendations are made on the bases of solution quality and network structure. Williams (1983) develops a dynamic programming algorithm for simultaneously determining the production and distribution batch sizes at each node within a supply chain network. As in Williams (1981), it is assumed that the production process is an assembly process. The objective of the heuristic is to minimize the average cost per period over an infinite horizon, where the average cost is a function of processing costs and inventory holding costs for each node in the network.

Cohen and Lee (1989) present a deterministic, mixed integer, non-linear mathematical programming model, based on economic order quantity (EOQ) techniques, to develop what the authors refer to as a global resource deployment policy. More specifically, the objective function used in their model maximizes the total after-tax profit for the manufacturing facilities and distribution centers (total revenue less total before-tax costs less taxes due). This objective function is subject to a number of constraints, including managerial constraints (resource and production constraints) and logical consistency constraints (feasibility, availability, demand limits, and variable non-negativity). The outputs resulting from their model include:

• Assignments for finished products and subassemblies to manufacturing plants, vendors to distribution centers, distribution centers to market regions.
• Amounts of components, subassemblies, and final products to be shipped among the vendors, manufacturing facilities, and distribution centers.
• Amounts of components, subassemblies, and final products to be manufactured at the manufacturing facilities.

Cohen and Moon (1990) extend Cohen and Lee (1989) by developing a constrained optimization model, called PILOT, to investigate the effects of various parameters on supply chain cost, and consider the additional problem of determining which manufacturing facilities and distribution centers should be open. More specifically, the authors consider a supply chain consisting of raw material suppliers, manufacturing facilities, distribution centers, and retailers. This system produces final products and intermediate products, using various types of raw materials. Using this particular system, the PILOT model accepts as input various production and transportation costs, and consequently outputs:

• Which of the available manufacturing facilities and distribution centers should be open.
• Raw material and intermediate order quantities for vendors and manufacturing facilities.
• Production quantities by product by manufacturing facility.
• Product-specific shipping quantities from manufacturing facility to distribution center to customer.

The objective function of the PILOT model is a cost function, consisting of fixed and variable production and transportation costs, subject to supply, capacity, assignment, demand, and raw material requirement constraints. Based on the results of their example supply chain system, the authors conclude that there are a number of factors that may dominate supply chain costs under a variety of situations, and that transportation costs play a significant role in the overall costs of supply chain operations.