Question

Jackson Hole Manufacturing is a small manufacturer of plastic products used in the automotive and computer industries. One of its major contracts is with a large computer company and involves the production of plastic printer cases for the computer company’s portable printers. The printer cases are produced on two injection molding machines. The M-100 machine has a production capacity of 25 printer cases per hour, and the M-200 machine has a production capacity of 40 cases per hour. Both machines use the same chemical material to produce the printer cases; the M-100 uses 40 pounds of the raw material per hour and the M-200 uses 50 pounds per hour. The computer company asked Jackson Hole to produce as many of the cases during the upcoming week as possible; it will pay $18 for each case Jackson Hole can deliver. However, next week is a regularly scheduled vacation period for most of Jackson Hole’s production employees; during this time, annual maintenance is performed for all equipment in the plant. Because of the downtime for maintenance, the M-100 will be available for no more than 15 hours, and the M-200 will be available for no more than 10 hours. However, because of the high setup cost involved with both machines, management requires that, if production is scheduled on either machine, the machine must be operated for at least 5 hours. The supplier of the chemical material used in the production process informed Jackson Hole that a maximum of 1000 pounds of the chemical material will be available for next week’s production; the cost for this raw material is $6 per pound. In addition to the raw material cost, Jackson Hole estimates that the hourly cost of operating the M-100 and the M-200 are $50 and $75, respectively.

a. Formulate a linear programming model that can be used to maximize the contribution to profit.

b. Find the optimal solution.

 

Answer 1

a.         Let       M1       = number of hours spent on the M-100 machine

                        M2       = number of hours spent on the M-200 machine

 

 

            

Total Cost

                        6(40)M1 + 6(50)M2 + 50M1 + 75M2 = 290M1 + 375M2

 

           

Total Revenue

                        25(18)M1 + 40(18)M2 = 450M1 + 720M2

            

Profit Contribution

                        (450 - 290)M1 + (720 - 375) M2 = 160M1 + 345M2

 

Max	160 M1	+	345M2
s.t.
	M1			≤	15	M-100 maximum
			M2	≤	10	M-200 maximum
	M1			≥	5	M-100 minimum
			M2	≥	5	M-200 minimum
	40 M1	+	50  M2	≤	1000	Raw material available

M1, M2 ≥ 0

 

 

b.

Objective Function Value =        5450.000

            Variable             Value             Reduced Costs   

            --------------     ---------------      ------------------ 

             M1                    12.500                   0.000

             M2                    10.000                   0.000

 

            Constraint        Slack/Surplus           Dual Prices    

            -----------------     -------------------      --------------------

            1                      2.500                     0.000

            2                      0.000                 145.000

            3                      7.500                     0.000

            4                      5.000                     0.000

            5                      0.000                     4.000

 

 

The optimal the decision is to schedule 12.5 hours on the M-100 and 10 hours on the M-200.