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

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
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.