A company is developing its weekly production plan. The company produces two products, A and B, which are processed in two departments. Setting up each batch of A requires $60 of labor while setting up a batch of B costs $80. Each unit of A generates a profit of $17 while a unit of B earns a profit of $21. The company can sell all the units it produces. The data for the problem are summarized below. The decision variables are defined as
X_i = the amount of product i produced
Y_i = 1 if X_i > 0 and 0 if X_i = 0
Using the approach discussed in the text, what is the appropriate value for M₁ in the linking constraint for product A?

Question

A company is developing its weekly production plan. The company produces two products, A and B, which are processed in two departments. Setting up each batch of A requires $60 of labor while setting up a batch of B costs $80. Each unit of A generates a profit of $17 while a unit of B earns a profit of $21. The company can sell all the units it produces. The data for the problem are summarized below.

The decision variables are defined as
X_i = the amount of product i produced
Y_i = 1 if X_i > 0 and 0 if X_i = 0
Using the approach discussed in the text, what is the appropriate value for M₁ in the linking constraint for product A?

Quizplus · Accepted Answer

To find the appropriate value for M₁, we need to consider the setup costs for each product and department, as well as the maximum number of units that can be produced given the available resources. For product A, each batch requires $60 of labor in department 1 (U1), and we can produce a maximum of 100 batches given the available labor hours (1200). Therefore, the total setup cost for product A is 60*100 = 6000. For product B, each batch requires $80 of labor in department 1 (U1) and $70 of labor in department 0 (U0). We can produce a maximum of 60 batches given the available labor hours (3600 in U1 and 2800 in U0). Therefore, the total setup cost for product B is 80*60 + 70*60 = 9000. The linking constraint for product A should ensure that we only produce product B if we have already produced at least M₁ units of product A. Since each batch of A produces 10 units, the constraint should be: X_i <= 10*M₁*Y_i We want to find the minimum value of M₁ that ensures that product B is not produced until we have produced at least 1 unit of product A. This means that the setup cost for product A must be greater than or equal to the profit from producing 1 unit of A: 6000 >= 17*1 Solving for M₁, we get: 10*M₁ >= 1 M₁ >= 1/10 = 0.1 This means that the appropriate value for M₁ is the smallest integer greater than or equal to 0.1, which is 1. However, note that producing only 1 unit of A will not be a feasible solution given the available resources. We need to produce at least 16 units of A (i.e. M₁ = 2) to fully utilize the available labor hours in department 1. Therefore, the best choice is C.

A Company Is Developing Its Weekly Production Plan