Question

Problem 11-9 (Algorithmic)

Hawkins Manufacturing Company produces connecting rods for 4- and 6-cylinder automobile engines using the same production line. The cost required to set up the production line to produce the 4-cylinder connecting rods is $2300, and the cost required to set up the production line for the 6-cylinder connecting rods is $3100. Manufacturing costs are $14 for each 4-cylinder connecting rod and $18 for each 6-cylinder connecting rod. Hawkins makes a decision at the end of each week as to which product will be manufactured the following week. If there is a production changeover from one week to the next, the weekend is used to reconfigure the production line. Once the line has been set up, the weekly production capacities are 5800 6-cylinder connecting rods and 8400 4-cylinder connecting rods.

Let
x₄ = the number of 4-cylinder connecting rods produced next week
x₆ = the number of 6-cylinder connecting rods produced next week
s₄= 1 if the production line is set up to produce the 4-cylinder connecting rods; 0 if otherwise
s₆ = 1 if the production line is set up to produce the 6-cylinder connecting rods; 0 if otherwise

1. Using the decision variables x₄ ands₄, write a constraint that limits next week's production of the 4-cylinder connecting rods to either 0 or 8400 units.
   
   x₄   s₄
   

2. Using the decision variables x₆ ands₆, write a constraint that limits next week's production of the 6-cylinder connecting rods to either 0 or 5800 units.
   
   x₆   s₆
   

3. Write three constraints that, taken together, limit the production of connecting rods for next week.
   
   x₄   s₄
   
   x₆   s₆
   
   s₄ +s₆   
   

4. Write an objective function for minimizing the cost of production for next week.
   
   Min x₄ + x₆ +s₄ + s₆

Answer 1

a). Constraint for limiting next week's production of 4-cylinder connecting rods is:

x4 <= 8,400s4

b). Constraint for limiting next week's production of 6-cylinder connecting rods is:

x6 <= 5,800s6

c). The first two contraints are the ones shown above:

x4 <= 8,400s4

x6 <= 5,800s6

3rd constraint is: s4 + s6 = 1 (constraint for setting up production line of one type of connecting rod)

d). Objective function to be minimized is:

Min. 14x4 + 18x6 + 2,300s4 + 3,100s6 (This minimizes total cost.)