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## Management Problem

In this section, we will look at the following model [2]:

 (3-1)

This model can be interpreted as a multi-item production and inventory management problem with a limited resource, where the are called decision variables [17,20]. In general, the objective function minimizes the cost, which could be the sum of setup (ordering) costs, inventory holding costs, and purchase costs. The constraints provide the restrictions of a shared resource as well as non-shared resources. Moreover, the model represents numerous application problems based on the interpretations of the decision variables. There are at least four different applications using the model (3-1).

Let us consider the continuous relaxation of (3-1). In order to solve it as a second-order cone problem, we must ensure that (3-1) can be cast as SOCP. Let max. Then for all . We introduce a in this inequality and rewrite it as

By using the following fact [5]:

 (3-2)

the previous inequality becomes

for all

Therefore, we can rewrite the inequalities as

for all

Next, define , ,, and as follows,

and

It is then clear that and . Plugging this into our model, we obtain

 (3-3)

In this form, the objective function and constraints, except the last inequalities, are linear. Therefore, we can rewrite them as LP. Each of the last constraints is a quadratic cone with , , and . We use the random problems described in [2] to generate test cases.

Example 1. Given , , , , , , The total machine time available is generated to ensure the feasibility of problems, i.e.,

random positive number

where for all In Chapter 5, we use Example 1 with as part of our testing set.

Next: Facility Location Problem (I) Up: Application Problems Previous: Application Problems
Hans D. Mittelmann 2003-09-10