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The L. Young & Sons Manufacturing Company produces two products, which have the following profit and resource requirement characteristics. Characteristic Profit/unit Dept. A hours/unit Product 1 $4 1 Product 2 $2 1

The L. Young & Sons Manufacturing Company produces two products, which have the following profit and resource requirement characteristics. Characteristic Profit/unit Dept. A hours/unit Product 1 $4 1 Product 2 $2 1

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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6 The L. Young & Sons Manufacturing Company produces two products, which have the following profit and resource requirement characteristics. Goal 1 Goal 2 Goal 3 Characteristic $ Profit/unit Dept. A hours/unit Dept. B hours/unit Max s.t. Last month's production schedule used 350 hours of labor in department A and 1000 hours of labor in department B. Young's management has been experiencing workforce morale and labor union problems during the past six months because of monthly departmental workload fluctuations. New hiring, layoffs, and interdepartmental transfers have been common because the firm has not attempted to stabilize workload requirements. Management would like to develop a production schedule for the coming month that will achieve the following goals. Goal 1: Use 350 hours of labor in department A. Goal 2: Use 1,000 hours of labor in department B. Goal 3: Earn a profit of at least $1,300. Product 1 (a) Assuming that goals 1 and 2 are P₁ level goals and goal 3 is a P₂ level goal and that goals 1 and 2 are equally important, what is the objective function of a goal programming model for this problem? (Let x; be the number of units of product i produced for i = 1, 2, let dpj be the deviation variable which is greater than the value of goal j, and let dj be the deviation variable which is less than the value of goal j, for j = 1, 2, 3.) O Min P₁(dp1) + P₁(dn1) + P₁(dp2) + P1(dn2) + P₂(dn3) O Min P₁(dp1) + P₁(dn1) + P₂(dp₂) + P₂(dn2) + P3(dn3) O Min P₁ (dp1) + P₁(dp2) + P₂ (dn3) O Min P₁(dp1) + P₁(dp2) + P₂ (dn1) + P₂(dn2) + P₂ (dn3) O Min P₁(d1) + P₁ (dn2) + P₂(dn3) Formulate the constraints of the goal programming model for the objective function above. $ $4 at (x₁, x₂) = 1 Department A Hours 2 Product 2 Xirdnji dpj ≥ 0, for i = 1, 2 and j = 1, 2, 3 (b) Solve the model formulated in part (a) using the graphical goal programming procedure and determine the profit (in dollars). (C $2 at (x₁, x₂) = 1 5 (c) Suppose that the firm ignores the workload fluctuations and considers the 350 hours in department A and the 1,000 hours in department B as the maximum available. Formulate linear programming problem to maximize profit subject to these constraints. Department B Hours x, ≥ 0, for i = 1, 2 Find the optimal solution (in dollars) to this linear program. =C (d) Reconsider part (a) assuming that the priority level 1 goal is goal 3 and the priority level 2 goals are goals 1 and 2; as before, assume that goals 1 and 2 are equally important. Solve this revised problem using the graphical goal programming procedure and determine the profit (in dollars). at (x₁, x₂) =
Transcribed Image Text:6 The L. Young & Sons Manufacturing Company produces two products, which have the following profit and resource requirement characteristics. Goal 1 Goal 2 Goal 3 Characteristic $ Profit/unit Dept. A hours/unit Dept. B hours/unit Max s.t. Last month's production schedule used 350 hours of labor in department A and 1000 hours of labor in department B. Young's management has been experiencing workforce morale and labor union problems during the past six months because of monthly departmental workload fluctuations. New hiring, layoffs, and interdepartmental transfers have been common because the firm has not attempted to stabilize workload requirements. Management would like to develop a production schedule for the coming month that will achieve the following goals. Goal 1: Use 350 hours of labor in department A. Goal 2: Use 1,000 hours of labor in department B. Goal 3: Earn a profit of at least $1,300. Product 1 (a) Assuming that goals 1 and 2 are P₁ level goals and goal 3 is a P₂ level goal and that goals 1 and 2 are equally important, what is the objective function of a goal programming model for this problem? (Let x; be the number of units of product i produced for i = 1, 2, let dpj be the deviation variable which is greater than the value of goal j, and let dj be the deviation variable which is less than the value of goal j, for j = 1, 2, 3.) O Min P₁(dp1) + P₁(dn1) + P₁(dp2) + P1(dn2) + P₂(dn3) O Min P₁(dp1) + P₁(dn1) + P₂(dp₂) + P₂(dn2) + P3(dn3) O Min P₁ (dp1) + P₁(dp2) + P₂ (dn3) O Min P₁(dp1) + P₁(dp2) + P₂ (dn1) + P₂(dn2) + P₂ (dn3) O Min P₁(d1) + P₁ (dn2) + P₂(dn3) Formulate the constraints of the goal programming model for the objective function above. $ $4 at (x₁, x₂) = 1 Department A Hours 2 Product 2 Xirdnji dpj ≥ 0, for i = 1, 2 and j = 1, 2, 3 (b) Solve the model formulated in part (a) using the graphical goal programming procedure and determine the profit (in dollars). (C $2 at (x₁, x₂) = 1 5 (c) Suppose that the firm ignores the workload fluctuations and considers the 350 hours in department A and the 1,000 hours in department B as the maximum available. Formulate linear programming problem to maximize profit subject to these constraints. Department B Hours x, ≥ 0, for i = 1, 2 Find the optimal solution (in dollars) to this linear program. =C (d) Reconsider part (a) assuming that the priority level 1 goal is goal 3 and the priority level 2 goals are goals 1 and 2; as before, assume that goals 1 and 2 are equally important. Solve this revised problem using the graphical goal programming procedure and determine the profit (in dollars). at (x₁, x₂) =
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