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Standard Pump recently won a $14 million contract with the U.S. Navy to supply 2,000 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 problem, 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 = 1 refers to May, m = 2 refers to June, m = 3 refers to July, and m = 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 following demand requirement. May Beginning Inventory + Current Production Ending Inventory. This Month's Demand

Standard Pump recently won a $14 million contract with the U.S. Navy to supply 2,000 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 problem, 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 = 1 refers to May, m = 2 refers to June, m = 3 refers to July, and m = 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 following demand requirement. May Beginning Inventory + Current Production Ending Inventory. This Month's Demand

Practical Management Science
Practical Management Science
6th Edition
ISBN:9781337406659
Author:WINSTON, Wayne L.
Publisher:WINSTON, Wayne L.
Chapter2: Introduction To Spreadsheet Modeling
Section: Chapter Questions
Problem 20P: Julie James is opening a lemonade stand. She believes the fixed cost per week of running the stand...
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Standard Pump recently won a $14 million contract with the U.S. Navy to supply 2,000 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 problem, the company's production scheduler let x denote the number of pumps produced in month m and s denote the number of pumps in inventory at the end of month m. Here, m = 1 refers to May, m = 2 refers to June, m = 3 refers to July, and m = 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 following demand requirement. Beginning Current +( Inventory/ Production A ) = (T This Month's Demand May June July August X₁, 5, ≥ 0, 1 = 1, 2, 3, 4 (b) Write goal equations that represent the fluctuations in the production level from May to June, June to July, and July uly to August. (Let d be the deviation variable below the target value of goal i, and dp be the deviation variable above the target value of goal i for i = 1, 2, 3.) May to June June to July July to August XS pidi 20,/= 1, 2, 3, 4 May (c) Inventory carrying costs are high and Standard Pump would like to keep their monthly ending inventories low. Develop goal equations with a target of zero for the ending inventory in May, June, and July. (Let d be the deviation variable below the target value of goal i, and do, be the deviation variable above the target value of goal i for i= 4, 5, 6.) June July May Spjdnj 20, i = 1, 2, 3 and j = 4, 5, 6 (d) In addition to the goal equations developed in parts (b) and (c), develop constraints for the production capacities for each month. June 300 July Ending= Inventory. August x, 20, 1, 2, 3, 4 Min (e) Assuming the production fluctuation and inventory goals are of equal importance, write an objective function for a goal programming model which, when used with the constraints constructed in parts (a) - (d), can be used to determine the best production schedule. Develop and solve a goal programming model to determine the best production schedule. What is the optimal solution to the goal programming model? (X₁ X₂ X3 X41 S₁ S₂, S3, S4) =
Transcribed Image Text:Standard Pump recently won a $14 million contract with the U.S. Navy to supply 2,000 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 problem, the company's production scheduler let x denote the number of pumps produced in month m and s denote the number of pumps in inventory at the end of month m. Here, m = 1 refers to May, m = 2 refers to June, m = 3 refers to July, and m = 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 following demand requirement. Beginning Current +( Inventory/ Production A ) = (T This Month's Demand May June July August X₁, 5, ≥ 0, 1 = 1, 2, 3, 4 (b) Write goal equations that represent the fluctuations in the production level from May to June, June to July, and July uly to August. (Let d be the deviation variable below the target value of goal i, and dp be the deviation variable above the target value of goal i for i = 1, 2, 3.) May to June June to July July to August XS pidi 20,/= 1, 2, 3, 4 May (c) Inventory carrying costs are high and Standard Pump would like to keep their monthly ending inventories low. Develop goal equations with a target of zero for the ending inventory in May, June, and July. (Let d be the deviation variable below the target value of goal i, and do, be the deviation variable above the target value of goal i for i= 4, 5, 6.) June July May Spjdnj 20, i = 1, 2, 3 and j = 4, 5, 6 (d) In addition to the goal equations developed in parts (b) and (c), develop constraints for the production capacities for each month. June 300 July Ending= Inventory. August x, 20, 1, 2, 3, 4 Min (e) Assuming the production fluctuation and inventory goals are of equal importance, write an objective function for a goal programming model which, when used with the constraints constructed in parts (a) - (d), can be used to determine the best production schedule. Develop and solve a goal programming model to determine the best production schedule. What is the optimal solution to the goal programming model? (X₁ X₂ X3 X41 S₁ S₂, S3, S4) =
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