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Supplement D Linear Programming Supplement D 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 the limitation that restricts the permissible choices. Answer: True Reference: Basic Concepts Difficulty: Moderate Keywords: constraint, limit 3. 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 4. In linear programming, each parameter is assumed to be known with certainty. Answer: True Reference: Basic Concepts Difficulty: Moderate Keywords: certainty, assumption 5. 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 452 Supplement D Linear Programming 6. 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 7. 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 8. 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 9. The range of optimality is the upper and lower limit over which the optimal values of the decision variables remain unchanged. Answer: True Reference: Sensitivity Analysis Difficulty: Moderate Keywords: range, optimality 10. The simplex method is an interactive algebraic procedure for solving linear programming problems. Answer: True Reference: Computer Solutions Difficulty: Moderate Keywords: simplex, method MULTIPLE CHOICE 11. 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 453 Supplement D Linear Programming 12. 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 13. 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 14. 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 15. 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 454 Supplement D Linear Programming 16. 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 17. 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 18. 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 a single unit of A would be $7.50. Answer: d Reference: Sensitivity Analysis Difficulty: Hard Keywords: shadow, price 455 Supplement D Linear Programming FILL IN THE BLANK 19. ____________ is useful for allocating scarce resources among competing demands. Answer: Linear programming Reference: Introduction Difficulty: Easy Keywords: linear, programming 20. 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 21. ____________ represent choices the decision maker can control. Answer: Decision variables Reference: Basic Concepts Difficulty: Moderate Keywords: decision, variables 22. ____________ are the limitations that restrict the permissible choices for the decision variables. Answer: Constraints Reference: Basic Concepts Difficulty: Moderate Keywords: constraint 23. 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 24. 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 25. 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 456 Supplement D Linear Programming 26. ____________ is an assumption that the decision variables must be either positive or zero. Answer: Nonnegativity Reference: Basic Concepts Difficulty: Easy Keywords: nonnegativity, assumption 27. 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 28. In linear programming, the ____________ 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: Basic Concepts Difficulty: Moderate Keywords: corner, point, solution 29. A(n) ____________ forms the optimal corner and limits the ability to improve the objective function. Answer: binding constraint Reference: Basic Concepts Difficulty: Moderate Keywords: binding, constraint, corner 30. ____________ is the amount by which the left-hand side falls short of the right-hand side in a linear programming model. Answer: Slack Reference: Basic Concepts Difficulty: Moderate Keywords: slack, left-hand, side, right-hand 31. 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 32. 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 457 Supplement D Linear Programming 33. For an = constraint, only points ____________ are feasible solutions. Answer: on the line Reference: Sensitivity Analysis Difficulty: Easy Keywords: equal, than, line, feasible, region 34. 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 35. ____________ 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: Sensitivity Analysis Difficulty: Moderate Keywords: degeneracy SHORT ANSWER 36. 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 37. 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 38. Briefly describe the meaning of a shadow price. Answer: The shadow price is the marginal improvement in Z caused by relaxing the constraint by one unit. Reference: Sensitivity Analysis Difficulty: Moderate Keywords: shadow, price 458 Supplement D Linear Programming 39. Provide three examples of decision problems for which linear programming can be useful, and why. Answer: Possible answers include aggregate planning, distribution, inventory, location, process management, and scheduling. Reference: Applications Difficulty: Moderate Keywords: linear, programming, application PROBLEMS 40. 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 459 Supplement D Linear Programming Answer: 40 35 30 25 20 1 B 15 2 10 5 0 1 3 5 7 9 11 13 15 17 19 21 23 -5 -10 A Intersection of Constraint 1 & 2 (7 A 4 B 120) 5 (4 A 5B 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 460 Supplement D Linear Programming 41. 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 $120B $150C Raw Material X: 15 A 10B 5C 300 Raw Material Y: 0 A 5B 25C 400 Purge System Subassembly: 1A 2B 2C 60 Case: 1A 1B 1C 40 Cord: 1A 1B 1C 50 Assembly Department Labor: 3A 5B 7C 120 Finish Department Labor: 1A 1.5B 3C 60 Minimum Production for A: 1A 0B 0C 5 Minimum Production for B: 0 A 1B 0C 5 Minimum Production for C: 0 A 0B 1C 5 Reference: Multiple sections Difficulty: Easy Keywords: linear, programming, objective, function, constraint 461 Supplement D Linear Programming 42. 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 Final Value $AB$12 900 88888.88889 Adjustable Cells Cell Name Original Value Final Value $X$12 W 0 111.1111111 $Y$12 X 0 0 $Z$12 Y 1.5 0 $AA$12 Z 0 0 Constraints Cell Name Cell Value Formula Status Slack $AB$15 10000$AB$15<=$AC$15 Binding 0 $AB$14 1111.111111$AB$14<=$AC$14 Not Binding 3888.888889 $AB$16 5000$AB$16<=$AC$16 Not Binding 25000 462 Supplement D Linear Programming Microsoft Excel 10.0 Sensitivity Report Worksheet: Supplement D Report Created: 1/26/2004 11:26:50 AM Adjustable Cells Final Reduced Objective Allowable Allowable Cell Name Value Cost Coefficient Increase Decrease $X$12 W 111.1111111 0 800 1E+30 80 $Y$12 X 0 -933.3333333 400 933.3333333 1E+30 $Z$12 Y 0 -66.66666667 600 66.66666667 1E+30 $AA$12 Z 0 -1055.555556 500 1055.555556 1E+30 Constraints Final Shadow Constraint Allowable Allowable Cell Name Value Price R.H. Side Increase Decrease $AB$15 10000 8.888888889 10000 35000 10000 $AB$14 1111.111111 0 5000 1E+30 3888.888889 $AB$16 5000 0 30000 1E+30 25000 Answer: Answer Report Target Cell Max: The target cell should be maximized, so the manager must have provided the intern with profit information. Final Value: The final value is the greatest amount possible for the situation. If we are working with profit figures, this is the best return possible given what we estimate is on hand and how it is to be produced. This may change if our inventory or recipes are slightly off. The highest profit identified is $88,888.89 Adjustable Cells: The adjustable cells show that we considered any positive quantity of models W–Z as possible outputs for the week. Name: The names are those of the models we produce. Final Value: These are the exact amounts of each of our four models. In this case we are making 111.1 units of model W and none of the other four models. Status: This shows what is limiting our ability to produce the models. A binding constraint directly limits our output although a nonbinding constraint means that factor does not limit us. In this case, the second and third constraints are nonbinding, so producing 111.1 units of model W leaves us with leftovers for whatever scarce resource they represent. The first constraint is binding, so we are using up every bit of that resource. Slack: Slack shows us how much of each resource we have left. Our first constraint is binding, so we have none left over and therefore have 0 slack. Our second and third constraints are not binding, so we have plenty (3,888 and 25,000 units respectively) of these scarce resources left over. Sensitivity Report Adjustable Cells Reduced Cost: This is the change in the optimum objective per unit change in the upper or lower bounds of the variable. The objective function will increase by 0.-66, and so on, per unit increase. 463 Supplement D Linear Programming Allowable Increase: These two (Allowable Increase and Allowable Decrease) provide a range for our current answer and the recipe we used to arrive at it. For model W, we have assumed that each unit gives us $800 profit. If our estimate were too high, and the return were up to $80 less per unit, we would still arrive at the same answer. If it were more than $80 too high, our answer would change. The same holds true for the models we are not making. If model Y made more than $666.66 profit per unit, then our final product mix would change. Allowable Decrease: See analysis for Allowable Increase. Constraints Shadow Price: This is the marginal return for having one more unit of each resource. Here we have a shadow price of $8.88, so if we had one more unit of resource in the first constraint, we could make an additional $8.88. This gives us an idea of the maximum we would be willing to pay for more of that resource. Allowable Increase: These work the same as the allowable increases and decreases for the adjustable cells except they focus on the shadow prices. They indicate how far the RHS of the constraint can change before the shadow price will change. Allowable Decrease: See discussion immediately preceding. Reference: Sensitivity Analysis Difficulty: Moderate Keywords: sensitivity, analysis 43. The CZ Jewelry Company produces two products: (1) engagement rings and (2) jeweled watches. The production process for each is similar in that both require a certain number of hours of diamond work and a certain number of labor hours in the gold department. Each ring takes four hours of diamond work and two hours in the gold shop. Each watch requires three hours in diamonds and one hour in the gold department. There are 240 hours of diamond labor available and 100 hours of gold department time available for the next month. Each engagement ring sold yields a profit of $9; each watch produced may be sold for a $10 profit. a. Give a complete formulation of this problem, including a careful definition of your decision variables. Let the first decision variable, (X1), deal with rings, the second decision variable, (X2), with watches, the first constraint with diamonds, and the second constraint with gold. b. Graph the problem fully in the following space. Label the axes carefully, plot the constraints, shade the feasibility region, plot at least one isoprofit line that reveals the optimal solution, circle the optimal corner point so found, and solve for it algebraically. (Show all your work to get credit.) X2 X1 464 Supplement D Linear Programming Answer: a. Max: 9X1 + 10X2 s.t. 4X1 + 3X2 < 240 hours of diamond work 2X1 + X2 < 100 hours of gold work X1, X2 > 0 b. 120 100 80 60 Diamond X2 40 Gold X1 Profit 20 0 1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 -20 -40 X1 Reference: Multiple sections Difficulty: Moderate Keywords: objective, function, constraint, graphical 465 Supplement D Linear Programming 44. NYNEX must schedule round-the-clock coverage for its telephone operators. To keep the number of different shifts down to a manageable level, it has only four different shifts. Operators work eight- hour shifts and can begin work at either midnight, 8 a.m., noon, or 4 p.m. Operators are needed according to the following demand pattern, given in four-hour time blocks. Time Period Operators Needed midnight to 4 a.m. 4 4 a.m. to 8 a.m. 6 8 a.m. to noon 90 Noon to 4 p.m. 85 4 p.m. to 8 p.m. 55 8 p.m. to midnight 20 Formulate this scheduling decision as a linear programming problem, defining fully your decision variables and then giving the objective function and constraints. Answer: Let X1 = the number of telephone operators starting their shift at midnight. X2 = the number of telephone operators starting their shift at 8 a.m. X3 = the number of telephone operators starting their shift at noon. X4 = the number of telephone operators starting their shift at 4 p.m. Min: X1 + X2 + X3 + X4 subject to X1 > 4 Midnight to 4 a.m. X1 > 6 4 a.m. to 8 a.m. X2 > 90 8 a.m. to noon X2 + X3 > 85 noon to 4 a.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 466 Supplement D Linear Programming 45. 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 c. Solve algebraically for the corner point on the feasible region. X1 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. 467 Supplement D Linear Programming Solver—Linear Programming Solution Variable Variable Original Coefficient Label Value Coefficient Sensitivity Var1 19.0000 1.0000 0 Var2 17.9167 1.0000 0 Constraint Original Slack or Shadow Label RHV Surplus Price Const1 19 0 Const2 30000 6383 0 Const3 4000 0 0.0104 Objective Function Value: 36.91666667 Sensitivity Analysis and Ranges Objective Function Coefficients Variable Lower Original Upper Label Limit Coefficient Limit Var1 No Limit 1 1.25 Var2 0.8 1 No Limit Right-Hand-Side Values Constraint Lower Original Upper Label Limit Value Limit Const1 12.28070176 19 33.33333333 Const2 23616.66667 30000 No Limit Const3 2280 4000 4612.8 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 X1, X2 > 0 468 Supplement D Linear Programming b. 45 40 35 30 25 Budget X2 20 Flubber Teams 15 10 5 0 1 10 19 28 37 46 55 64 73 82 91 100 -5 X1 Commitments : X 1 19 X 1 19 Budget : 300 X 1 1000 X 2 30000 30000 if X 1 0, X 2 30 1000 30000 if X 2 0; X 1 100 300 Flubber :120 X1 96 X 2 4000 4000 if X 1 0; X 2 41.6 96 4000 if X 2 0; X 1 33.3 120 469 Supplement D Linear Programming c. corner point 120 X 1 96 X 2 4000 ( X 1 19) 120 96 X 2 1720 X 2 17.916 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 46. 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 470 Supplement D Linear Programming 47. 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 Variable Original Coefficient Label Value Coefficient Sensitivity Var1 0.0000 0.1400 0 Var2 150000.0000 0.1000 0 Var3 0.0000 0.3000 0 Constraint Original Slack or Shadow Label RHV Surplus Price Const1 200000 185000 0 Const2 50000 42500 0 Const3 750000 0 0.0200 Objective Function Value: 15000 Sensitivity Analysis and Ranges Objective Function Coefficients Variable Lower Original Upper Label Limit Coefficient Limit Var1 No Limit 0.14 0.2 Var2 0.075 0.1 No Limit Var3 No Limit 0.3 0.4 Right-Hand-Side Values Constraint Lower Original Upper Label Limit Value Limit Const1 15000 200000 No Limit Const2 7500 50000 No Limit Const3 0 750000 5000000 471 Supplement D 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 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 472

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