4 A company produces two products. Relevant information for each product is shown in Table 58. The company has a goal of $48 in profits and incurs a $1 penalty for each dollar it falls short of this goal. A total of 32 hours of labor are available. A $2 penalty is incurred for each hour of overtime

(labor over 32 hours) used, and a $1 penalty is incurred for each hour of available labor that is unused. Marketing considerations require that at least 10 units of product 2 be produced. For each unit (of either product) by which production falls short of demand, a penalty of $5 is assessed.

a Formulate an LP that can be used to minimize the penalty incurred by the company.

b Suppose the company sets (in order of importance)

the following goals:

Goal 1 Avoid underutilization of labor.

Goal 2 Meet demand for product 1.

Goal 3 Meet demand for product 2.

Goal 4 Do not use any overtime.

Formulate and solve a preemptive goal programming model for this situation

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TABLE 57 Characteristics of a Lot of 100 Chips Price Per 100 Supplier Excellent Good Mediocre Chips (S) 123 60 50 40 222 20 35 20 220 20 400 15 300 40 250 TABLE 58 Labor required Contribution to profit Product 1 Product 2 4 hours 2 hours $4 $2