Krajewski TIF Supplement E
January 14, 2017 | Author: jabbott99 | Category: N/A
Short Description
Download Krajewski TIF Supplement E...
Description
Supplement E Linear Programming
Supplement
E
Linear Programming
TRUE/FALSE 1. Linear programming is useful for allocating scarce resources among competing demands. Answer: True Reference: Introduction Difficulty: Easy Keywords: linear, programming, product, mix 2. A constraint is a limitation that restricts the permissible choices. Answer: True Reference: Basic Concepts Difficulty: Moderate Keywords: constraint, limit 3. Decision variables are represented in both the objective function and the constraints while formulating a linear program. Answer: True Reference: Basic Concepts Difficulty: Moderate Keywords: constraint, decision, variable, objective 4. A parameter is a region that represents all permissible combinations of the decision variables in a linear programming model. Answer: False Reference: Basic Concepts Difficulty: Moderate Keywords: parameter, decision, variable, feasible, region 5. In linear programming, each parameter is assumed to be known with certainty. Answer: True Reference: Basic Concepts Difficulty: Moderate Keywords: certainty, assumption
196
Supplement E Linear Programming
6. The objective function Maximize Z = 3x 2 + 4y is appropriate. Answer: False Reference: Basic Concepts Difficulty: Moderate Keywords: linearity, assumption, proportionality 7. One assumption of linear programming is that a decision maker cannot use negative quantities of the decision variables. Answer: True Reference: Basic Concepts Difficulty: Moderate Keywords: nonnegativity, decision, variable 8. Only corner points should be considered for the optimal solution to a linear programming problem. Answer: True Reference: Graphic Analysis Difficulty: Moderate Keywords: corner, point, optimal 9. The graphical method is a practical method for solving product mix problems of any size, provided the decision maker has sufficient quantities of graph paper. Answer: False Reference: Graphic Analysis Difficulty: Moderate Keywords: graphical, method 10. A binding constraint is the amount by which the left-hand side falls short of the right-hand side. Answer: False Reference: Graphic Analysis Difficulty: Moderate Keywords: binding, constraint 11. A binding constraint has slack but does not have surplus. Answer: False Reference: Graphic Analysis Difficulty: Moderate Keywords: binding, slack, surplus 12. The simplex method is an interactive algebraic procedure for solving linear programming problems. Answer: True Reference: Computer Solutions Difficulty: Moderate Keywords: simplex, method
197
Supplement E Linear Programming
MULTIPLE CHOICE 13. A manager is interested in using linear programming to analyze production for the ensuing week. She knows that it will take exactly 1.5 hours to run a batch of product A and that this batch will consume two tons of sugar. This is an example of the linear programming assumption of: a. linearity. b. certainty. c. continuous variables. d. whole numbers. Answer: b Reference: Basic Concepts Difficulty: Moderate Keywords: certainty, assumption 14. Which of the following statements regarding linear programming is NOT true? a. A parameter is also known as a decision variable. b. Linearity assumes proportionality and additivity. c. The product-mix problem is a one-period type of aggregate planning problem. d. One reasonable sequence for formulating a model is defining the decision variables, writing out the objective function, and writing out the constraints. Answer: a Reference: Basic Concepts Difficulty: Moderate Keywords: parameter, decision, variable 15. Which of the following statements regarding linear programming is NOT true? a. A linear programming problem can have more than one optimal solution. b. Most real-world linear programming problems are solved on a computer. c. If a binding constraint were relaxed, the optimal solution wouldn’t change. d. A surplus variable is added to a > constraint to convert it to an equality. Answer: c Reference: Basic Concepts Difficulty: Moderate Keywords: solution, surplus, variable 16. For the line that has the equation 4X1 + 8X2 = 88, an axis intercept is: a. (0, 22). b. (6, 0). c. (6, 22). d. (0, 11). Answer: d Reference: Graphic Analysis Difficulty: Moderate Keywords: axis, intercept
198
Supplement E Linear Programming
17. Consider a corner point to a linear programming problem, which lies at the intersection of the following two constraints: 6X1 + 15X2 < 390 2X1 + X2 < 50 Which of the following statements about the corner point is true? a. X1 < 21 b. X1 > 25 c. X1 < 10 d. X1 > 17 Answer: a Reference: Graphic Analysis Difficulty: Moderate Keywords: corner, point 18. A manager is interested in deciding production quantities for products A, B, and C. He has an inventory of 20 tons each of raw materials 1, 2, 3, and 4 that are used in the production of products A, B, and C. He can further assume that he can sell all of what he makes. Which of the following statements is correct? a. The manager has four decision variables. b. The manager has three constraints. c. The manager has three decision variables. d. The manager can solve this problem graphically. Answer: c Reference: Graphic Analysis Difficulty: Moderate Keywords: decision, variable 19. A site manager has three day laborers available for eight hours each and a burning desire to maximize his return on their wages. The site manager uses linear programming to assign them to two tasks and notes that he has enough work to occupy 21 labor hours. The linear program that the site manager has constructed has: a. slack. b. surplus. c. a positive shadow price for labor. d. no feasible solution. Answer: d Reference: Graphic Analysis Difficulty: Moderate Keywords: shadow, price
199
Supplement E Linear Programming
20. Suppose that the optimal values of the decision variables to a two-variable linear programming problem remain the same as long as the slope of the objective function lies between the slopes of the following two constraints: 2X1 + 3X2 < 26 2X1 + 2X2 < 20 The current objective function is: 8X1 + 9X2 = Z Which of the following statements about the range of optimality on c1 is TRUE? a. 0 < c1 < 2 b. 2 < c1 < 6 c. 6 < c1 < 9 d. 9 < c1 < 12 Answer: c Reference: Sensitivity Analysis Difficulty: Hard Keywords: range, optimality 21. You are faced with a linear programming objective function of: Max P = $20X + $30Y and constraints of: 3X + 4Y = 24 (Constraint A) 5X – Y = 18 (Constraint B) You discover that the shadow price for Constraint A is 7.5 and the shadow price for Constraint B is 0. Which of these statements is TRUE? a. You can change quantities of X and Y at no cost for Constraint B. b. For every additional unit of the objective function you create, you lose 0 units of B. c. For every additional unit of the objective function you create, the price of A rises by $7.50. d. The most you would want to pay for an additional unit of A would be $7.50. Answer: d Reference: Sensitivity Analysis Difficulty: Hard Keywords: shadow, price 22. While glancing over the sensitivity report, you note that the stitching labor has a shadow price of $10 and a lower limit of 24 hours with an upper limit of 36 hours. If your original right hand value for stitching labor was 30 hours, you know that: a. the next worker that offers to work an extra 8 hours should receive at least $80. b. you can send someone home 6 hours early and still pay them the $60 they would have earned while on the clock. c. you would be willing pay up to $60 for someone to work another 6 hours. d. you would lose $80 if one of your workers missed an entire 8 hour shift. Answer: c Reference: Sensitivity Analysis Difficulty: Moderate Keywords: shadow, price
200
Supplement E Linear Programming
FILL IN THE BLANK 23. ____________ is useful for allocating scarce resources among competing demands. Answer: Linear programming Reference: Introduction Difficulty: Easy Keywords: linear, programming 24. The ____________ is an expression in linear programming models that states mathematically what is being maximized or minimized. Answer: objective function Reference: Basic Concepts Difficulty: Moderate Keywords: objective, function 25. ____________ represent choices the decision maker can control. Answer: Decision variables Reference: Basic Concepts Difficulty: Moderate Keywords: decision, variables 26. ____________ are the limitations that restrict the permissible choices for the decision variables. Answer: Constraints Reference: Basic Concepts Difficulty: Moderate Keywords: constraint 27. The ____________ represents all permissible combinations of the decision variables in a linear programming model. Answer: feasible region Reference: Basic Concepts Difficulty: Moderate Keywords: feasible, region 28. A(n) ____________ is a value that the decision maker cannot control and that does not change when the solution is implemented. Answer: parameter Reference: Basic Concepts Difficulty: Moderate Keywords: parameter, value 29. Each coefficient or given constant is known by the decision maker with ____________. Answer: certainty Reference: Basic Concepts Difficulty: Easy Keywords: parameter, certainty, assumption
201
Supplement E Linear Programming
30. If merely rounding up or rounding down a result for a decision variable is not sufficient when they must be expressed in whole units, then a decision maker might instead use ____________ to analyze the situation. Answer: integer programming Reference: Basic Concepts Difficulty: Moderate Keywords: integer, programming 31. ____________ is an assumption that the decision variables must be either positive or zero. Answer: Nonnegativity Reference: Basic Concepts Difficulty: Easy Keywords: nonnegativity, assumption 32. The assumption of ____________ allows a decision maker to combine the profit from one product with the profit from another to realize the total profit from a feasible solution. Answer: additivity Reference: Basic Concepts Difficulty: Easy Keywords: additivity, assumption 33. The ____________ problem is a one-period type of aggregate planning problem, the solution of which yields optimal output quantities of a group of products or services, subject to resource capacity and market demand conditions. Answer: product-mix Reference: Basic Concepts Difficulty: Moderate Keywords: product-mix, product, mix 34. In linear programming, a ____________ is a point that lies at the intersection of two (or possibly more) constraint lines on the boundary of the feasible region. Answer: corner point Reference: Graphic Analysis Difficulty: Moderate Keywords: corner, point, solution 35. A(n) ____________ forms the optimal corner and limits the ability to improve the objective function. Answer: binding constraint Reference: Graphic Analysis Difficulty: Moderate Keywords: binding, constraint, corner 36. ____________ is the amount by which the left-hand side falls short of the right-hand side in a linear programming model. Answer: Slack Reference: Graphic Analysis Difficulty: Moderate Keywords: slack, left-hand, side, right-hand 202
Supplement E Linear Programming
37. ____________ is the amount by which the left-hand side exceeds the right-hand side in a linear programming model. Answer: Surplus Reference: Graphic Analysis Difficulty: Moderate Keywords: surplus, left-hand, side, right-hand 38. A modeler is limited to two or fewer decision variables when using the ____________. Answer: graphical method Reference: Graphic Analysis Difficulty: Easy Keywords: decision, variables, graphical, method 39. The ____________ is the upper and lower limit over which the optimal values of the decision variables remain unchanged. Answer: range of optimality Reference: Sensitivity Analysis Difficulty: Moderate Keywords: range, optimality 40. For an = constraint, only points ____________ are feasible solutions. Answer: on the line Reference: Graphic Analysis Difficulty: Easy Keywords: equal, than, line, feasible, region 41. A(n) ____________ is the marginal improvement in the objective function value caused by relaxing a constraint by one unit. Answer: shadow price Reference: Sensitivity Analysis Difficulty: Moderate Keywords: shadow, price, sensitivity, relax, constraint 42. The interval over which the right-hand-side parameter can vary while its shadow price remains valid is the ____________. Answer: range of feasibility Reference: Sensitivity Analysis Difficulty: Moderate Keywords: range, feasibility 43. ____________ occurs in a linear programming problem when the number of nonzero variables in the optimal solution is fewer than the number of constraints. Answer: Degeneracy Reference: Computer Solution Difficulty: Moderate Keywords: degeneracy
203
Supplement E Linear Programming
SHORT ANSWER 44. What are the assumptions of linear programming? Provide examples of each. Answer: The assumptions are certainty, linearity, and nonnegativity. The assumption of certainty is that a fact is known without doubt, such as an objective function coefficient, or the parameters in the right- and left-hand sides of the constraints. The assumption of linearity implies proportionality and additivity, that is, that there are no cross products or squared or higher powers of the decision variables. The assumption of nonnegativity is that decision variables must either be positive or zero. Examples will vary. Reference: Graphic Analysis Difficulty: Moderate Keywords: assumption, linearity, certainty, nonnegativity 45. What is the meaning of a slack or surplus variable? Answer: The amount by which the left-hand side falls short of the right-hand side is the slack variable. The amount by which the left-hand side exceeds the right-hand side is the surplus variable. Reference: Graphic Analysis Difficulty: Moderate Keywords: slack, surplus 46. Briefly describe the meaning of a shadow price. Provide an example of how a manager could use information about shadow prices to improve operations? Answer: The shadow price is the marginal improvement in Z caused by relaxing a constraint by one unit. Examples will vary. Reference: Sensitivity Analysis Difficulty: Moderate Keywords: shadow, price 47. Provide three examples of operations management decision problems for which linear programming can be useful, and why. Answer: Answers and justifications will vary. Possible answers include aggregate planning, distribution, inventory, location, process management, and scheduling. Reference: Applications Difficulty: Moderate Keywords: linear, programming, application 48. What are some potential abuses or misuses of linear programming (beyond violation of basic assumptions)? Answer: Answers will vary, but may include a discussion of the inability of modeling techniques to capture all of the relevant factors that may be as important as what can be quantified in an LP formulation. Factors such as aesthetics, ethics, civility, character, etc., may be difficult to capture in an LP. Slavish adhesion to the output from a linear programming formulation robs a manager of the freedom to inject reality or personality into a model. The rush to use a tool without understanding fully the workings of it may render the output meaningless. Reference: Applications Difficulty: Basic Concepts Keywords: linear, programming, application 204
Supplement E Linear Programming
PROBLEMS 49. Use the graphical technique to find the optimal solution for this objective function and associated constraints. Maximize: Z=8A + 5B Subject To: Constraint 1 4A + 5B < 80 Constraint 2 7A + 4B < 120 A, B > 0 a. Graph the problem fully in the following space. Label the axes carefully, plot the constraints, shade the feasibility region, identify all candidate corner points, and indicate which one yields the optimal answer.
B
A
205
Supplement E Linear Programming
Answer:
Intersection of Constraint 1 & 2 (7 A 4 B 120) 5 (4 A 5 B 80) 4 19 A 280 A 14.73, B 4.21 Z (0, 0) 8 0 5 0 0 Z (0,16) 8 0 5 16 90 Z (0,17.14) 8 17.14 5 0 137.14 Z (14.73, 4.21) 8 14.73 5 4.21 138.89 optimal Reference: Graphic Analysis Difficulty: Moderate Keywords: graphic, analysis
206
Supplement E Linear Programming
50. A producer has three products, A, B, and C, which are composed from many of the same raw materials and subassemblies by the same skilled workforce. Each unit of product A uses 15 units of raw material X, a single purge system subassembly, a case, a power cord, three labor hours in the assembly department, and one labor hour in the finishing department. Each unit of product B uses 10 units of raw material X, five units of raw material Y, two purge system subassemblies, a case, a power cord, five labor hours in the assembly department, and 90 minutes in the finishing department. Each unit of product C uses five units of raw material X, 25 units of raw material Y, two purge system subassemblies, a case, a power cord, seven labor hours in the assembly department, and three labor hours in the finishing department. Labor between the assembly and finishing departments is not transferable, but workers within each department work on any of the three products. There are three full-time (40 hours/week) workers in the assembly department and one full-time and one half-time (20 hours/week) worker in the finishing department. At the start of this week, the company has 300 units of raw material X, 400 units of raw material Y, 60 purge system subassemblies, 40 cases, and 50 power cords in inventory. No additional deliveries of raw materials are expected this week. There is a $90 profit on product A, a $120 profit on product B, and a $150 profit on product C. The operations manager doesn’t have any firm orders, but would like to make at least five of each product so he can have the products on the shelf in case a customer wanders in off the street. Formulate the objective function and all constraints, and clearly identify each constraint by the name of the resource or condition it represents. Answer: Objective Function: Max P $90 A $120 B $150C Raw Material X: 15 A 10 B 5C 300 Raw Material Y: 0 A 5 B 25C 400 Purge System Subassembly: 1A 2 B 2C 60 Case: 1A 1B 1C 40 Cord: 1A 1B 1C 50 Assembly Department Labor: 3 A 5 B 7C 120 Finish Department Labor: 1A 1.5 B 3C 60 Minimum Production for A: 1A 0 B 0C 5 Minimum Production for B: 0 A 1B 0C 5 Minimum Production for C: 0 A 0 B 1C 5 Reference: Multiple sections Difficulty: Easy Keywords: linear, programming, objective, function, constraint
207
Supplement E Linear Programming
51. A very confused manager is reading a two-page report given to him by his student intern. “She told me that she had my problem solved, gave me this, and then said she was off to her production management course,” he whined. “I gave her my best estimates of my on-hand inventories and requirements to produce, but what if my numbers are slightly off? I recognize the names of our four models W, X, Y, and Z, but that’s about it. Can you figure out what I’m supposed to do and why?” You take the report from his hands and note that it is the answer report and the sensitivity report from Excel’s solver routine. Explain each of the highlighted cells in layman’s terms and tell the manager what they mean in relation to his problem. Microsoft Excel 10.0 Answer Report Worksheet: Supplement D Report Created: 1/26/2004 11:26:50 AM
Target Cell (Max) Cell Name Original Value $AB$12 900
Final Value 88888.88889
Adjustable Cells Cell Name Original Value $X$12 W 0 $Y$12 X 0 $Z$12 Y 1.5 $AA$12 Z 0
Final Value 111.1111111 0 0 0
Constraints Cell Name $AB$15 $AB$14 $AB$16
Cell Value Formula 10000$AB$15 6 4 a.m. to 8 a.m. X2 > 90 8 a.m. to noon X2 + X3 > 85 noon to 4 p.m. X3 + X4 > 55 4 p.m. to 8 p.m. X4 > 20 8 p.m. to midnight X1, X2, X3, X4 > 0 Reference: Basic Concepts Difficulty: Moderate Keywords: objective function, constraint
212
Supplement E Linear Programming
54. The Really Big Shoe Company is a manufacturer of basketball shoes and football shoes. Ed Sullivan, the manager of marketing, must decide the best way to spend advertising resources. Each football team sponsored requires 120 pairs of shoes. Each basketball team requires 32 pairs of shoes. Football coaches receive $300,000 for shoe sponsorship and basketball coaches receive $1,000,000. Ed's promotional budget is $30,000,000. The Really Big Shoe Company has a very limited supply (4 liters or 4,000cc) of flubber, a rare and costly raw material used only in promotional athletic shoes. Each pair of basketball shoes requires 3cc of flubber, and each pair of football shoes requires 1cc of flubber. Ed desires to sponsor as many basketball and football teams as resources allow. However, he has already committed to sponsoring 19 football teams and wants to keep his promises. a. Give a linear programming formulation for Ed. Make the variable definitions and constraints line up with the computer output appended to this exam. b. Solve the problem graphically, showing constraints, feasible region, and isoprofit lines. Circle the optimal solution, making sure that the isoprofit lines drawn make clear why you chose this point. (Show all your calculations for plotting the constraints and isoprofit line on the left to get credit.)
X2
X1 c. Solve algebraically for the corner point on the feasible region. d. Part of Ed's computer output is shown following. Give a full explanation of the meaning of the three numbers listed at the end. Based on your graphical and algebraic analysis, explain why these numbers make sense. (Hint: He formulated the budget constraint in terms of $000.) See the computer printout that follows.
213
Supplement E Linear Programming
Solver—Linear Programming Solution Variable Label
Variable Original Value Coefficient
Coefficient Sensitivity
Var1 Var2
19.0000 17.9167
1.0000 1.0000
0 0
Constraint Label
Original RHV
Slack or Surplus
Shadow Price
Const1 Const2 Const3
19 30000 4000
0 6383 0
0 0.0104
Objective Function Value:
36.91666667
Sensitivity Analysis and Ranges Objective Function Coefficients Variable Label
Lower Limit
Original Coefficient
Upper Limit
Var1 Var2
No Limit 0.8
1 1
1.25 No Limit
Right-Hand-Side Values Constraint Label
Lower Limit
Original Value
Upper Limit
Const1 Const2 Const3
12.2807 23616.67 2280
19 30000 4000
33.33333333 No Limit 4612.8
First Number: The shadow price of 0.0104 for the "Const3" constraint. Second Number: The slack or surplus of 6383 for the "Const1" constraint. Third Number: The lower limit of 12.2807for the "Const1" constraint. Answer: a. Let X1 = the number of football teams sponsored X2 = the number of basketball teams sponsored Max X1 + X2 s.t. X1 > 19 Commitments 300X1 + 1000X2 < 30000 Budget 120X1 + 96X2 < 4000 Flubber 214
Supplement E Linear Programming
X1, X2 > 0
215
Supplement E Linear Programming
b.
Commitments : X 1 19 X 1 19 Budget : 300 X 1 1000 X 2 30000 30000 30 1000 30000 if X 2 0; X 1 100 300 if X 1 0, X 2
Flubber :120 X 1 96 X 2 4000 4000 41.6 96 4000 if X 2 0; X 1 33.3 120 if X 1 0; X 2
216
Supplement E Linear Programming
c.
corner point 120 X 1 96 X 2 4000 ( X 1 19) 120 96 X 2 1720 X 2 17.916 Therefore, X1 = 19 d. First Number: The shadow price of 0.0104 for the "Const3" constraint. Second Number: The slack or surplus of 6300 for the "Const1" constraint. Third Number: The lower limit of 12.3684 for the "Const1" constraint.
The first number is the amount (.0104) by which the objective function will improve with a one-unit decrease in the right-hand-side value. The second number means that 6,300,000 remains in the promised commitment. The third value is the amount by which the constraint can change and still keep the current values of the shadow price. Reference: Multiple sections Difficulty: Moderate Keywords: constraint, objective, function 55. A portfolio manager is trying to balance investments between bonds, stocks and cash. The return on stocks is 12 percent, 9 percent on bonds, and 3 percent on cash. The total portfolio is $1 billion, and he or she must keep 10 percent in cash in accordance with company policy. The fund's prospectus promises that stocks cannot exceed 75 percent of the portfolio, and the ratio of stocks to bonds must equal two. Formulate this investment decision as a linear programming problem, defining fully your decision variables and then giving the objective function and constraints. Answer: Let X1 = the amount invested in bonds X2 = the amount invested in stocks X3 = the amount invested in cash Max: z = .09X1 + .12X2 +.03X3 s.t. X1 + X2 + X3 1,000,000,000 Portfolio value X1 = 100,000,000 10% minimum stock X2 750,000,000 75% maximum cash 2X1 – X2 = 0 2:1 ratio stocks to bonds X1, X2, X3 > 0 Reference: Basic Concepts Difficulty: Moderate Keywords: objective, function, constraint
217
Supplement E Linear Programming
56. A small oil company has a refining budget of $200,000 and would like to determine the optimal production plan for profitability. The following table lists the costs associated with its three products.
Marketing has a budget of $50,000, and the company has 750,000 gallons of crude oil available. Each gallon of gasoline contributes 14 cents of profits, heating oil provides 10 cents, and plastic resin 30 cents per unit. The refining process results in a ratio of two units of heating oil for each unit of gasoline produced. This problem has been modeled as a linear programming problem and solved on the computer. The output follows: Solution Variable Label
Variable Original Coefficient Value Coefficient Sensitivity
Var1 Var2 Var3
0.0000 150000.0000 0.0000
0.1400 0.1000 0.3000
0 0 0
Constraint Label
Original RHV
Slack or Surplus
Shadow Price
Const1 Const2 Const3
200000 50000 750000
185000 42500 0
0 0 0.0200
Objective Function Value:
15000
Sensitivity Analysis and Ranges Objective Function Coefficients Variable Label Var1 Var2 Var3
Lower Original Limit Coefficient No Limit 0.075 No Limit
Upper Limit
0.14 0.1 0.3
0.2 No Limit 0.4
Right-Hand-Side Values Constraint Label
Lower Limit
Original Value
Upper Limit
Const1 Const2 Const3
15000 7500 0
200000 50000 750000
No Limit No Limit 5000000
218
Supplement E Linear Programming
a. Give a linear programming formulation for this problem. Make the variable definitions and constraints line up with the computer output. b. What product mix maximizes the profit for the company using its limited resources? c. How much gasoline is produced if profits are maximized? d. Give a full explanation of the meaning of the three numbers listed following. First Number: Slack or surplus of 42500 for constraint 2. Second Number: Shadow price of 0 for constraint 1. Third Number: An upper limit of "no limit" for the right-hand-side value constraint 1. Answer: a. Let X1 = gallons of gasoline refined X2 = gallons of heating oil refined X3 = gallons of plastic resin refined Max: .14X1 + .10X2 + .30X3 s.t. .40X1 + .10X2 + .60X3 < 200,000 Refining budget .10X1 + .05X2 + .07X3 < 50,000 Marketing budget 10X1 + 5X2 + 20X3 < 750,000 Crude oil available X2 – 2X1 = 0 Ratio X1, X2, X3 > 0 b. X1 = 0 gallons, X2 = 150,000 gallons, and X3 = 0 gallons c. No gasoline is produced if profits are maximized. d. $42,500 remains in the marketing budget. A zero implies that increasing the refining budget will not improve the value of the objective function. A no-limit implies that the right-hand side can be increased by any amount and the shadow price will remain the same. Reference: Multiple sections Difficulty: Moderate Keywords: objective, function, constraint 57. A snack food producer runs four different plants that supply product to four different regional distribution centers. The division operations manager is focused on one product, so he creates a table showing each plant’s monthly capacity and each distribution center’s monthly demand (both amounts in cases) for the product. The division manager supplements this table with the cost data to ship one case from each plant to each distribution center. Formulate an objective function and constraints that will solve this problem using linear programming. Center 1 Center 2 Center 3 Center 4 Monthly Capacity Plant A $2 $7 $5 $4 8000 Plant B $9 $4 $7 $6 12000 Plant C $7 $6 $4 $3 7500 Plant D $4 $8 $3 $5 5000 Monthly 9000 8500 8000 7000 Demand Answer: This is a cost minimization problem with 16 decision variables, one for each combination of plant and center; there are 8 constraints, one for each plant’s capacity and one for each center’s demand.
Objective Min Z $2 x A1 $7 x A 2 $5 x A3 $4 x A 4 $9 xB1 $4 xB 2 $7 xB 3 $6 xB 4 $7 xC1 $6 xC 2 $4 xC 3 $3 xC 4 $4 xD1 $8 xD 2 $3 xD 3 $5 xD 4 219
Supplement E Linear Programming
Subject to Plant A : x A1 x A2 x A3 x A 4 8000 Plant B : xB1 xB 2 xB 3 xB 4 12000 Plant C : xC1 xC 2 xC 3 xC 4 7500 Plant D : xD1 xD 2 xD 3 xD 4 5000 Center 1: x A1 xB1 xC 1 xD1 9000 Center 2 : x A 2 xB 2 xC 2 xD 2 8500 Center 3 : x A3 xB 3 xC 3 xD 3 8000 Center 4 : x A 4 xB 4 xC 4 xD 4 7000 For those of you keeping score at home, the optimal solution is: Center 1 Center 2 Center 3 Center 4 Monthly Capacity Plant A Plant B Plant C Plant D Monthly Demand
8,000 8,500 1,000 9,000
8,500
3,500 500 4,000 8,000
Reference: Basic Concepts Difficulty: Moderate Keywords: objective, function, constraint
220
7,000 7,000
8,000 12,000 7,500 5,000 $113,500
View more...
Comments
