Supplement D

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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




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                                      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




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                                       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




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                                       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




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                                       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

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                                      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




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                                      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



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                                       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




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                              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




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                                      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




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                                       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




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                                       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.
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                                       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




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                             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




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                                      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




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                                        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.




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                               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



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                                      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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