Standard Pump recently won a $14 million contract with the U.S. Navy to supply 2
ID: 3259999 • Letter: S
Question
Standard Pump recently won a $14 million contract with the U.S. Navy to supply 2000 custom-designed submersible pumps over the next four months. The contract calls for the delivery of 200 pumps at the end of May, 600 pumps at the end of June, 600 pumps at the end of July, and 600 pumps at the end of August. Standard’s production capacity is 500 pumps in May, 400 pumps in June, 800 pumps in July, and 500 pumps in August. Management would like to develop a production schedule that will keep monthly ending inventories low while at the same time minimizing the fluctuations in inventory levels from month to month. In attempting to develop a goal programming model of the prob- lem, the company’s production scheduler let xm denote the number of pumps produced in month m and sm denote the number of pumps in inventory at the end of month m. Here, m 5 1 refers to May, m 5 2 refers to June, m 5 3 refers to July, and m 5 4 refers to August. Management asks you to assist the production scheduler in model development. a. Using these variables, develop a constraint for each month that will satisfy the follow-
ing demand requirement:
1Beginning2 1 1 Current 2 2 1 Ending 2 5 1This Month’s2 Inventory Production Inventory Demand
b. Write goal equations that represent the fluctuations in the production level from May to June, June to July, and July to August.
c. Inventory carrying costs are high. Is it possible for Standard to avoid carrying any monthly ending inventories over the scheduling period of May to August? If not, develop goal equations with a target of zero for the ending inventory in May, June, and July.
d. Besides the goal equations developed in parts (b) and (c), what other constraints are needed in the model?
e. Assuming the production fluctuation and inventory goals are of equal importance, de- velop and solve a goal programming model to determine the best production schedule. f. Can you find a way to reduce the variables and constraints needed in your model by elimi-
nating the goal equations and deviation variables for ending inventory levels? Explain.
Explanation / Answer
Solution:
a. May: x1 - s1 = 200
June: s1 + x2 - s2 = 600
July: s2 + x3 - s3 = 600
August: s3 + x4 = 600 (ending inventory not required)
b. May to June: x2 - x1 - d1' + d1 = 0
June to July: x3 - x2 - d2' + d2 = 0
July to August: x4 - x3 - d3' + d3 = 0
c. No, for example there should be atleast 200 pumps in inventory at the end of May to meet the requirement of June of shipping of 600 pumps.
The inventory variables are nonnegative constraints therefore we are concerned only with positive deviations
June: s1 - d4' = 0
July: s2 - d5' = 0
August: s3 - d6' = 0
d. Production capacity constraints are required each month
May: x1 500
June: x2 400
July: x3 800
August: x4 500
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