Chapter 3 - Introduction to Optimization Modeling

1. In an optimization model, there can only be one:

  a. decision variable
  b. constraint
  c. objective function
  d. shadow price
ANSWER:  c

2. In using Excel to solve linear programming problems, the changing variable cells represent the:
  a. value of the objective function
  b. constraints
  c. decision variables
  d. total cost of the model
ANSWER:  c

3. The condition of nonnegativity requires that:

  a. the objective function cannot be less that zero
  b. the decision variables cannot be less than zero
  c. the right hand side of the constraints cannot be greater then zero
  d. the reduced cost cannot be less than zero
ANSWER:  b

4. If a manufacturing process takes 4 hours per unit of x1 and 2 hours per unit of x2 and a maximum of 100 hours of
manufacturing process time are available, then an algebraic formulation of this constraint is:
  a.  4x1 + 2x2 ≥ 100
  b. 4x1 − 2x2 ≤ 100
  c.  4x1 + 2x2 ≤ 100
  d. 4x1 − 2x2 ≥ 100
ANSWER:  c

5. The feasible region in all linear programming problems is bounded by:

  a. corner points
  b. hyperplanes
  c. an objective line
  d. all of these options
ANSWER:  b

6. Suppose a company sells two different products, x1 and x2, for net profits of $6 per unit and $3 per unit, respectively.
The slope of the line representing the objective function is:
  a. 0.5
  b. −0.5
  c. 2
  d. −2
ANSWER:  d

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Chapter 3 - Introduction to Optimization Modeling
7. The equation of the line representing the constraint 4x1 + 2x2 ≤ 100 passes through the points:
  a. (25,0) and (0,50)
  b. (0,25) and (50,0)
  c. (−25,0) and (0,−50)
  d. (0,−25) and (−50,0)
ANSWER:  a

8. When the profit increases with a unit increase in a resource, this change in profit will be shown in Solver's sensitivity
report as the:
  a. add-in price
  b. sensitivity price
  c. shadow price
  d. additional profit
ANSWER:  c

9. Linear programming models have three important properties. They are:

  a. optimality, additivity and sensitivity
  b. optimality, linearity and divisibility
  c. divisibility, linearity and nonnegativity
  d. proportionality, additivity and divisibility
ANSWER:  d

10. Consider the following linear programming problem:

Maximize 4x1 + 2x2
Subject to:  
  4x1 + 2x2 ≤ 40
  2x1 + x2≥ 20
  x1, x2 ≥ 0
The above linear programming problem:
  a. has only one feasible solution
  b. has more than one optimal solution
  c. exhibits infeasibility
  d. exhibits unboundedness
ANSWER:  c

11. All linear programming problems should have a unique solution, if they can be solved.
  a. True
  b. False
ANSWER:  False

12. When formulating a linear programming spreadsheet model, there is a set of designated cells that play the role of the
decision variables. These are called the changing variable cells.
  a. True
  b. False
ANSWER:  True

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Chapter 3 - Introduction to Optimization Modeling
13. The feasible region in a graphical solution of a linear programming problem will appear as some type of polygon, with
lines forming all sides.
  a. True
  b. False
ANSWER:  True

14. If the objective function has the equation {4x1 + 2x2}, then the slope of the objective function line is −2.
  a. True
  b. False
ANSWER:  True

15. If a constraint has the equation 5x1 + 2x2 ≤ 60, then the constraint line passes through the points (0,12) and (30,0).
  a. True
  b. False
ANSWER:  False

16. Shadow prices are associated with nonbinding constraints , and show the change in the optimal objective function
value when the right side of the constraint equation changes by one unit.
  a. True
  b. False
ANSWER:  False

17. Suppose the allowable increase and decrease for an objective coefficient of a decision variable that has a current value
of $50 are $25 (increase) and $10 (decrease). If the coefficient were to change from $50 to $65, the optimal value of the
objective function would not change.
  a. True
  b. False
ANSWER:  False

18. Suppose the shadow price for a constraint is $12 and the allowable increase and decrease for the right hand side of the
constraint are 25 (increase) and 10 (decrease). If the right hand side of that constraint were to increase by 10 the shadow
price changes.
  a. True
  b. False
ANSWER:  False

19. The proportionality property of LP models implies that the sum of the contributions from the various activities to a
particular constraint equals the total contribution to that constraint.
  a. True
  b. False
ANSWER:  False

20. Common errors in LP models that exhibit unboundedness are a constraint that has been omitted or an input which is
incorrect.
  a. True
  b. False
ANSWER:  True
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Chapter 3 - Introduction to Optimization Modeling

Exhibit 3-1
A winemaker in California's Napa Valley must decide how much of two types of wine she will produce from a particular
variety of grapes. Each liter of table wine yields $8 profit, while each liter of dessert wine produces $3 profit. The labor
hours and bottling process time used for type of wine are given in the table below. Resources available include 200 labor
hours and 80 hours of bottling process time. Assume the winemaker has more than enough grapes available to supply any
feasible production plan.

  Table Dessert
Labor (Hours) 0.20 0.60
Process Time (Hours) 0.30 0.10

21. [Part 1] Refer to Exhibit 3-1. Formulate a linear programming model that will enable the winemaker to determine the
number of liters of each type of wine to produce in order to maximize her profit.
ANSWER:  Let x1 = liters of table wine and x2 = liters of dessert wine. The linear programming model to maximize the
profit is:

Maximize 8x1 + 3x2

Subject to:  
  0.2x1 + 0.6x2 ≤ 200
  0.3x1 + 0.1x2 ≤ 80
  x1 , x2 ≥ 0

22. [Part 2] Refer to Exhibit 3-1. Using the graphical solution method, find an optimal solution to the model in Part 1 and
determine the maximum profit.
ANSWER:  The optimal solution is (x1, x2) = (175,275), and the maximum profit is $2,225.

23. [Part 3] Refer to Exhibit 3-1. Implement the model in Part 1 in Excel Solver and obtain an answer report. Which
constraint(s) are binding on the optimal solution?
ANSWER:  The constraints section of the answer report (below) shows that the labor and process constraints are
binding.

24. [Part 4] Refer to Exhibit 3-1. Obtain a sensitivity report for the model in Part 1. How much should the winemaker be
willing to pay for an additional labor hour?
ANSWER:  The shadow price for labor (shown below) indicates that an additional hour would increase profit by $0.625.
The winemaker should be willing to pay up to this amount.

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Chapter 3 - Introduction to Optimization Modeling

25. [Part 5] Refer to Exhibit 3-1. Suppose the winemaker can obtain 100 addition labor hours. Can you use the sensitivity
analysis obtained for Part 4 to determine her expected profit? Would her bottling plan change? Explain your answer.
ANSWER:  Using the shadow price for labor, her profit would increase by $0.625*100=$62.5. The 100 extra hours are
within the allowable increase, and would enable her to bottle considerably more (462.5 liters) desert wine,
while still making 112.5 liters of table wine.

Exhibit 3-2
Western Chassis produces high-quality polished steel and aluminum sheeting and two lines of industrial chassis for the
rack mounting of Internet routers, modems, and other telecommunications equipment. The contribution margin
(contribution toward profit) for steel sheeting is $0.40 per pound and for aluminum sheeting is $0.60 per pound. Western
earns $12 contribution on the sale of a Standard chassis rack and $15 contribution on a Deluxe chassis rack. During the
next production cycle, Western can buy and use up to 25,800 pounds of raw unfinished steel either in sheeting or in
chassis. Similarly, 20,400 pounds of aluminum are available. One standard chassis rack requires 16 pounds of steel and 8
pounds of aluminum. A Deluxe chassis rack requires 12 pounds of each metal. The output of metal sheeting is restricted
only by the capacity of the polisher. For the next production cycle, the polisher can handle any mix of the two metals up to
4,000 pounds of metal sheeting. Chassis manufacture can be restricted by either metal stamping or assembly operations;
no polishing is required. During the cycle no more than 2,500 total chassis can be stamped, and there will be 920 hours of
assembly time available. The assembly time required is 24 minutes for the Standard chassis rack and 36 minutes for the
Deluxe chassis rack. Finally, market conditions limit the number of Standard chassis racks sold to no more than 1,200
Standard and no more than 1,000 Deluxe. Any quantities of metal sheeting can be sold.
26. [Part 1] Refer to Exhibit 3-2. Find an optimal solution to Western's problem. What is the production plan, and what is
the total revenue?
ANSWER:  SS = pounds of steel sheeting produced and sold = 1000
AS = pounds of aluminum sheeting produced and sold = 2000
SC = number of Standard chassis racks produced and sold = 800
DC = number of Deluxe chassis racks produced and sold = 1000
Profit = $26,200

27. [Part 2] Refer to Exhibit 3-2. Obtain a sensitivity report for the solution reported in Part 1. Which constraints are
binding?

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Chapter 3 - Introduction to Optimization Modeling
ANSWER:  

Steel available, Aluminum available, Assembly SS, and Market Deluxe SS all have nonzero shadow prices, so
they are binding.

28. Refer to Exhibit 3-2. What is the incremental contribution associated with adding an hour of assembly time? Over
what range of increase is the marginal value valid?
ANSWER:  Incremental contribution is $2 per hour, with an allowable increase of 25 hours.

29. Refer to Exhibit 3-2. What is the value of additional capacity on the polisher? How much increase and decrease in this
capacity is possible before a change occurs in the optimal production schedule?
ANSWER:  The value of additional capacity on the polisher is zero, since it is not binding. The allowable increase is
unbounded, and the allowable decrease is 1000.

30. Refer to Exhibit 3-2. An advertising agency has devised a marketing plan for the Western Chassis Company that will
increase the market for Deluxe chassis. The plan will increase demand by 75 Deluxe chassis per month at a cost of $100
per month. Should Western adopt the plan? Briefly explain why.
ANSWER:  The change in the profit would be $1.8/hour shadow price times 75 hours, or $135, so Valley should adopt the
plan.

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