Multi-parametric Optimization and Control. Efstratios N. Pistikopoulos

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       Optimal partition invariancy [8, 10–12]: The optimal partition is given by the cone, which is spanned from the solution found in the directions of the inactive constraints.

      In order to illustrate the concepts developed in this chapter, a classical shipping problem is considered (adapted from [13] and modified for illustrative purposes):

      Given a set of plants images and a set of markets images with corresponding supply and demand, and the distances between images and images, minimize the total transportation cost.

Supply Demand
Seattle 350 Chicago 300
San Diego 600 Topeka 275
Chicago Topeka
Seattle 1.7 1.8
San Diego 1.8 1.4

      2.4.1 The Deterministic Solution

      where images is the loading cost, images is the distance related cost, and images is the distance between plant and market according to Table 2.1. Thus, the amount of product shipped for all combinations needs to be determined. As there are two plants and two markets, this results in four variables, and a total cost given by:

      (2.16)equation

      (2.17a)equation

      Additionally, note that transport can only be positive. This results in the LP problem of the form:

      the solution of which features the minimal cost of images, and the corresponding transport amounts as

      (2.19)equation

      2.4.2 Considering Demand Uncertainty

      In reality, the data in Table 2.1 is time‐varying. Thus, the case of demand uncertainty is considered:

      Given a set of plants images with a constant supply and a set of markets images with an uncertain demand bound between 0 and 1000 cases, and the distances between images and images, minimize the total transportation cost as a function of the demand.

      Remark 2.5

      The numbering of the constraints is according to their occurrence in problem (2.20), e.g. images is constraint number 5.