Multi-parametric Optimization and Control. Efstratios N. Pistikopoulos

       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.

2.4 An Example: Chicago to Topeka

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

In particular, the transport to Chicago and Topeka is considered, and the problem‐specific data is given in Tables 2.1 and 2.2. Note that the freight cost is $90 per 1000 miles and case and the freight loading cost is $25 per case.

Table 2.1 The supply and demand of each plant and market in cases.

Supply		Demand
Seattle	350	Chicago	300
San Diego	600	Topeka	275

Table 2.2 The distances between each plant and market in thousands of miles.

‐	Chicago	Topeka
Seattle	1.7	1.8
San Diego	1.8	1.4

      2.4.1 The Deterministic Solution

Equivalently to the beginning of this chapter, we first consider the uncertainty‐free case. Then, the overall transportation cost per case is determined by

(2.15)

      where is the loading cost, is the distance related cost, and 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)

where the coefficients are calculated according to Eq. (2.15). The next step is to formulate the constraints. The constraints are that (i) there cannot be more supply than amount in stock and (ii) the market demands needs to be satisfied. Mathematically, this can be written as

      (2.17a)

(2.17b)

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

(2.18)

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

      (2.19)

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

Based on the LP problem (2.18), the following mp‐LP problem is formulated:

(2.20)

Note that the objective function does not change as the distances between the destinations do not change. However on the contrary to problem (2.18), problem (2.20) now features the uncertain demands as parameters bound between 0 and 1000. The solution of problem (2.20) is reported in Table 2.3, which was obtained based on the solution strategies presented in Chapter 4.

      Remark 2.5

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

Table 2.3 The solution of problem (