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
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
What is the appropriate value for M₁ in the linking constraint for product A?

Quizplus · Accepted Answer

The linking constraint for product A will link the amount of A produced (XS1U1B10) to the amount of batches set up for A (XS1U0B10). Each batch of A requires $60 of labor, so the total cost of setting up batches for A will be $60 times the number of batches set up for A which is XS1U0B10. Therefore, the linking constraint for product A should be:

XS1U1B10 = 16XS1U0B10      (since setting up each batch of A requires $60 of labor)

Thus, the appropriate value for MS1U1B11S1U1B0 in the linking constraint for product A is 16, which is the coefficient of XS1U0B10 in the equation.

To maximize profit, the objective function would be:

Maximize: 17XS1U1B10 + 21XS1U1B11
Subject to: XS1U1B10 + XS1U1B11 ≤ 2400   (since the company can produce at most 2400 units per week)
XS1U1B10 ≤ 1500                          (since there is a limit on the amount of A that can be produced)
XS1U1B11 ≤ 800                           (since there is a limit on the amount of B that can be produced)
XS1U1B10, XS1U1B11 ≥ 0                   (since the company cannot produce negative amounts of either product)

The best choice would be to produce both products, but to produce more of product A than product B, since the profit per unit for product A is slightly higher and the setup cost is lower.

A Company Is Developing Its Weekly Production Plan