Question

A Company produces two products. Relevant information for each product is shown in the Table below. The company has a goal of $48 in profits and incurs $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 $1 penalty is incurred for each hour of available labor that is unused. Marketing considerations require at least 5 units of product 1 and 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.
Product 1 - Product 2
Labor Required - 4 hours - 2 hours
Contribution to profit - $4 - $2

1. Formulate a weighted goal programming that can be used to minimize the penalty incurred by the company. Do NOT solve, just formulate.
2. Suppose that company sets (in order of importance) the following goals:
a. Goal 1: Avoid underutilization of labor
b. Goal 2: Meet demand for product 1
c. Goal 3: Meet demand for product 2
d. Goal 4: Do not use overtime
Formulate and solve the preemptive goal programming model for this situation using Excel solver. Describe clearly the optimal solution to this problem using a managerial statement.

Answer 1

1. Weighted goal programming model is following:

Let x1 and x2 be the number of product 1 and 2 to be produced

Let d_i⁺, d_i^- be the positive and negative deviation variables for i-th goal, where goal 1 is for profit, goal 2 is for labor hours (undertime and overtime), goal 3 is for demand of product 1 and goal 4 is for demand of product 2.

Minimize 1d₁^- + 2d₂⁺ + 1d₂^- + 5d₃^- + 5d₄^-

s.t.

4x1+2x2+1d₁^- -1d₁⁺ = 48

4x1+2x2+1d₂^- -1d₂⁺ = 32

x1+1d₃^- -1d₃⁺ = 5

x2+1d₄^- -1d₄⁺ = 10

all variables >= 0

2) The preemptive goal programming model is following:

In this case, the new goals (as given in part 2) are represented by the respective deviation variable

Minimize 4d₁^- + 3d₂^- + 2d₃^- + 1d₄⁺

s.t.

4x1+2x2+1d₁^- -1d₁⁺ = 32

x1+1d₂^- -1d₂⁺ = 5

x2+1d₃^- -1d₃⁺ = 10

4x1+2x2+1d₄^- -1d₄⁺ = 32

all variables >= 0

Solution using Excel Solver is following:

Optimal result:

3 units of product 1 and 10 units of product 2 must be produced.

This will give a profit of 4*3+2*10 = $ 32

There will be no overtime and under time.

All goals except goal 2 will be satisfied.