Multicommodity Facility Location—Allocation

The multicommodity capacity facility location-allocation problem is of primary importance in transportation of shipments from the original facilities to intermediate stations and then to the destinations. In this illustrative example we will consider such a problem which involves I plants, J distribution centers, K customers, and P products. The commodity flow of product p which is shipped from plant i, through distribution center j to customer k will be denoted by the continuous variable z tp. It is assumed that each customer k is served by only one distribution center j. Data are provided for the total demand by customer k for commodity p, Dkp, the supply of commodity p at plant i denoted as Sip, as well as the lower and upper bounds on the available throughput in a distribution center j denoted by V and Vf7, respectively. 

The set of constraints ensure that supply and demand requirements are met, provide the logical connection between the existence of a distribution, the assignment of customers to distribution centers, and the demand for commodities, and make certain that only one distribution center is assigned to a customer. 

The binary variables correspond to the existence of a distribution center j, and the assignment of a customer A to a distribution center j. These are denoted as Zj and yjk, respectively. The continuous variables are represented by the commodity flows Zjj p. The mathematical formulation of this problem becomes 

Note that in the above formulation the binary variables yjk, Zj appear linearly and separably in the objective function and constraints. Note also that the continuous variables x kp appear linearly in the constraints while we have a nonlinear contribution of such terms in the objective function. 

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Part 3 of this book presents a number of major developments and applications of MINLP approaches in the area of Process Synthesis. The illustrative examples for MINLP applications, presented next in this section, will focus on different aspects than those described in Part 3. In particular, we will consider the binary distillation design of a single column, the retrofit design of multiproduct batch plants, and the multicommodity facility location/allocation problem. 

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