Krajewski TIF Supplement E

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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

Keywords: parameter, decision, variable, feasible, region

5. In linear programming, each parameter is assumed to be known with certainty. Answer: True

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6. The objective function

_{Maximize Z = 3x + 4y}

2 _{is appropriate.} Answer: False

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

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

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

Keywords: binding, constraint

11. A binding constraint has slack but does not have surplus. Answer: False

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

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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

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

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

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

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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

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

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

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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

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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

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

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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

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

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

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

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37. ____________ is the amount by which the left-hand side exceeds the right-hand side in a linear programming model.

Answer: Surplus

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

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

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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.

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.

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.

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

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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.

A

B

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Answer:

(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

Z

 

 



optimal

Keywords: graphic, analysis

Intersection of Constraint 1 & 2

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4 120) 5

(4

5 80) 4

19

280 14.73,

4.21 A

B

A

B

A

B

















 

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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



$150

C

Raw Material X: 15A10B5C300 Raw Material Y:

0 A



5 B



25 C



400

Purge System Subassembly: 1A2B2C60 Case:

1 A



1 B



1 C



40

Cord: 1A1B1C50

Assembly Department Labor:

3 A



5 B



7 C



120

Finish Department Labor: 1A1.5B3C60 Minimum Production for A:

1 A



0 B



0 C



5

Minimum Production for B: 0A1B0C5 Minimum Production for C:

0 A



0 B



1 C



5

Reference: Multiple sections Difficulty: Easy

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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 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

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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 to produce to earn the final value. In this case we would make 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 of 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.

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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 was too high, and the return was 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

52. 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 corner points and highlight the optimal optimal corner point so found, and solve for it

algebraically. (Show all your work to get credit.)

X1

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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.

Reference: Multiple sections Difficulty: Moderate

Keywords: objective, function, constraint, graphical

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53. 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 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

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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.)

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.

X

₁

X

₂

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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 12.2807 19 33.33333333 Const2 23616.67 30000 No Limit Const3 2280 4000 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

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b. 1 1

:

19

19 Commitments X





X



1 2 1 2 2 1

: 300

1000

30000

0,

30 1000

30000

0;

100

300 Budget

X

if X

X

if X

X







1 2 1 2 2 1

Flubber :120

96 4000

4000

0;

41.6

96 4000

0;

33.3

120 X

X

if X

X

if X

X







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c.

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.

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

1 2 1 2 2

corner point

120

96 4000

(

19)

120

96 1720

17.916 X

X





 



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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 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

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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.

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 Demand 9000 8500 8000 7000 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. 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4

$2

$7

$5

$4

$9

$4

$7

$6

$7

$6

$4

$3

$4

$8

$3

$5

A A A A B B B B C C C C D D D D

Objective Min Z

x





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1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 1 1 1 2 2 2 2 3 3

:

8000

:

12000

:

7500

:

5000

1:

9000

2 :

8500

3 :

A A A A B B B B C C C C D D D D A B C D A B C D A B C

Subject to

Plant A x

x

Plant B x

x

Plant C x

x

Plant D x

x

Center

x

Center

x

Center

x



























3 3 4 4 4 4

8000

4 :

7000

D A B C D

x

Center

x









For those of you keeping score at home, the optimal solution is:

Center 1 Center 2 Center 3 Center 4 Monthly Capacity Plant A 8,000 8,000 Plant B 8,500 3,500 12,000 Plant C 500 7,000 7,500 Plant D 1,000 4,000 5,000 Monthly Demand 9,000 8,500 8,000 7,000 $113,500 Reference: Basic Concepts

Difficulty: Moderate