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- A nonlinear optimization problem is any optimization problem in which at least one term in the objective function or a constraint is nonlinear. a. True b. False ANSWER: True POINTS: 1 DIFFICULTY: Easy LEARNING OBJECTIVES: IMS.ASWC.19.08.01 - 8. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 8.1 A Production Application—Par, Inc., Revisited KEYWORDS: Bloom's: Remember
- A function is considered a quadratic function if its nonlinear terms have a power of 2. a. True b. False ANSWER: True POINTS: 1 DIFFICULTY: Easy LEARNING OBJECTIVES: IMS.ASWC.19.08.01 - 8. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 8.1 A Production Application—Par, Inc., Revisited KEYWORDS: Bloom's: Remember
- Nonlinear programming algorithms are more complex than linear programming algorithms. a. True b. False ANSWER: True POINTS: 1 DIFFICULTY: Easy LEARNING OBJECTIVES: IMS.ASWC.19.08.01 - 8. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 8.1 A Production Application—Par, Inc., Revisited KEYWORDS: Bloom's: Remember
- Many linear programming algorithms such as the simplex method optimize by examining only the extreme points and selecting the extreme point that gives the best solution value. a. True b. False ANSWER: True POINTS: 1 DIFFICULTY: Easy LEARNING OBJECTIVES: IMS.ASWC.19.08.01 - 8. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 8.1 A Production Application—Par, Inc., Revisited KEYWORDS: Bloom's: Understand
- Nonlinear optimization problems can have only one local optimal solution. a. True b. False ANSWER: False POINTS: 1 DIFFICULTY: Easy LEARNING OBJECTIVES: IMS.ASWC.19.08.01 - 8. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 8.1 A Production Application—Par, Inc., Revisited KEYWORDS: Bloom's: Remember
- A feasible solution is a global optimum if no other feasible solutions with a better objective function value are found in the immediate neighborhood. a. True b. False ANSWER: False POINTS: 1 DIFFICULTY: Easy LEARNING OBJECTIVES: IMS.ASWC.19.08.01 - 8. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 8.1 A Production Application—Par, Inc., Revisited KEYWORDS: Bloom's: Remember
- In the case of functions with multiple local optima, most nonlinear optimization software methods can get stuck and terminate at a local optimum. a. True b. False ANSWER: True POINTS: 1 DIFFICULTY: Easy LEARNING OBJECTIVES: IMS.ASWC.19.08.01 - 8. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 8.1 A Production Application—Par, Inc., Revisited KEYWORDS: Bloom's: Remember
- For a minimization problem, a point is a global minimum if there are no other feasible points with a smaller objective function value. a. True b. False ANSWER: True POINTS: 1 DIFFICULTY: Easy LEARNING OBJECTIVES: IMS.ASWC.19.08.01 - 8. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 8.1 A Production Application—Par, Inc., Revisited KEYWORDS: Bloom's: Remember
- It is possible that a nonlinear application in which there is a single local optimal solution also shows that solution as the global optimal solution. a. True b. False ANSWER: True POINTS: 1 DIFFICULTY: Easy LEARNING OBJECTIVES: IMS.ASWC.19.08.01 - 8. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 8.1 A Production Application—Par, Inc., Revisited KEYWORDS: Bloom's: Understand
- Functions that are convex have a single local maximum that is also the global maximum. a. True b. False ANSWER: False POINTS: 1 DIFFICULTY: Easy LEARNING OBJECTIVES: IMS.ASWC.19.08.01 - 8. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 8.1 A Production Application—Par, Inc., Revisited KEYWORDS: Bloom's: Understand
- The function f ( X , Y ) = X^2 + Y^2 has a single global minimum and is relatively easy to minimize. a. True b. False ANSWER: True POINTS: 1 DIFFICULTY: Moderate LEARNING OBJECTIVES: IMS.ASWC.19.08.01 - 8. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 8.1 A Production Application—Par, Inc., Revisited KEYWORDS: Bloom's: Understand
- The problem of maximizing a concave quadratic function over a linear constraint set is relatively difficult to solve. a. True b. False ANSWER: False POINTS: 1 DIFFICULTY: Easy LEARNING OBJECTIVES: IMS.ASWC.19.08.01 - 8. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 8.1 A Production Application—Par, Inc., Revisited KEYWORDS: Bloom's: Understand
- Each point on the efficient frontier is the maximum possible risk, measured by portfolio
feasible solutions is reduced. a. True b. False ANSWER: True POINTS: 1 DIFFICULTY: Easy LEARNING OBJECTIVES: IMS.ASWC.19.08.04 - 8. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 8.4 Blending: The Pooling Problem KEYWORDS: Bloom's: Understand
- The value of the coefficient of imitation, q , in the Bass model for forecasting adoption of a new product cannot be negative. a. True b. False ANSWER: False POINTS: 1 DIFFICULTY: Moderate LEARNING OBJECTIVES: IMS.ASWC.19.08.05 - 8. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 8.5 Forecasting Adoption of a New Product KEYWORDS: Bloom's: Understand
- The Markowitz mean-variance portfolio model presented in the text is a convex optimization problem. a. True b. False ANSWER: True POINTS: 1 DIFFICULTY: Easy LEARNING OBJECTIVES: IMS.ASWC.19.08.03 - 8. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 8.3 Markowitz Portfolio Model KEYWORDS: Bloom's: Understand
- Because most nonlinear optimization codes will terminate with a local optimum, the
solution returned by the codes will be the best solution. a. True b. False ANSWER: False POINTS: 1 DIFFICULTY: Easy LEARNING OBJECTIVES: IMS.ASWC.19.08.01 - 8. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 8.1 A Production Application—Par, Inc., Revisited KEYWORDS: Bloom's: Understand
- It is possible for the optimal solution to a nonlinear optimization problem to lie in the interior of the feasible region. a. True b. False ANSWER: True POINTS: 1 DIFFICULTY: Easy LEARNING OBJECTIVES: IMS.ASWC.19.08.01 - 8. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 8.1 A Production Application—Par, Inc., Revisited KEYWORDS: Bloom's: Understand
- The minimum value for a convex function is 0, and the point (0, 0) gives the minimum value of 0. a. True b. False ANSWER: True POINTS: 1 DIFFICULTY: Easy LEARNING OBJECTIVES: IMS.ASWC.19.08.01 - 8. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 8.1 A Production Application—Par, Inc., Revisited KEYWORDS: Bloom's: Understand Multiple Choice
DIFFICULTY: Moderate LEARNING OBJECTIVES: IMS.ASWC.19.08.03 - 8. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 8.3 Markowitz Portfolio Model KEYWORDS: Bloom's: Understand
- The Bass model is used to forecast the adoption of a new product. Which of the following is NOT a parameter of this Bass model? a. coefficient of innovation b. coefficient of interaction c. coefficient of imitation d. estimated pool of current adopters ANSWER: b POINTS: 1 DIFFICULTY: Moderate LEARNING OBJECTIVES: IMS.ASWC.19.08.05 - 8. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 8.5 Forecasting Adoption of a New Product KEYWORDS: Bloom's: Understand
- When the number of blending components exceeds the number of storage facilities, the number of feasible solutions to the blending problem is a. reduced. b. increased. c. unchanged. d. zero. ANSWER: a POINTS: 1 DIFFICULTY: Moderate LEARNING OBJECTIVES: IMS.ASWC.19.08.04 - 8. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 8.4 Blending: The Pooling Problem KEYWORDS: Bloom's: Understand
- In the Bass model for forecasting the adoption of a new product, the objective function a. minimizes the sum of forecast errors.
b. minimizes the sum of squared forecast errors. c. maximizes the number of adoptions. d. maximizes the number of adoptions and imitations. ANSWER: b POINTS: 1 DIFFICULTY: Moderate LEARNING OBJECTIVES: IMS.ASWC.19.08.05 - 8. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 8.5 Forecasting Adoption of a New Product KEYWORDS: Bloom's: Understand
- Which of the following is NOT true regarding a concave function? a. It is bowl-shaped down. b. It is relatively easy to maximize. c. It has multiple local maxima. d. It has a single global maximum. ANSWER: c POINTS: 1 DIFFICULTY: Easy LEARNING OBJECTIVES: IMS.ASWC.19.08.01 - 8. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 8.1 A Production Application—Par, Inc., Revisited KEYWORDS: Bloom's: Understand
- A convex function is a. bowl-shaped up. b. bowl-shaped down. c. elliptical in shape. d. sinusoidal in shape. ANSWER: a POINTS: 1 DIFFICULTY: Easy LEARNING OBJECTIVES: IMS.ASWC.19.08.01 - 8. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 8.1 A Production Application—Par, Inc., Revisited
DIFFICULTY: Moderate LEARNING OBJECTIVES: IMS.ASWC.19.08.02 - 8. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 8.2 Constructing an Index Fund KEYWORDS: Bloom's: Understand
- It is common that components that are referred to as pooled a. are shared by two or more customers. b. have common ingredients. c. share a storage facility or storage tank. d. are interchangeable. ANSWER: c POINTS: 1 DIFFICULTY: Easy LEARNING OBJECTIVES: IMS.ASWC.19.08.04 - 8. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 8.4 Blending: The Pooling Problem KEYWORDS: Bloom's: Understand Subjective Short Answer
- Investment manager Max Gaines has several clients who wish to own a mutual fund portfolio that matches, as a whole, the performance of the S&P 500 stock index. His task is to determine what proportion of the portfolio should be invested in each of the five mutual funds listed below so that the portfolio most closely mimics the performance of the S&P 500 index. Formulate the appropriate nonlinear program. Annual Returns (Planning Scenarios) Mutual Fund Year 1 Year 2 Year 3 Year 4 International Stock 26.73 22.37 6.46 3. Large-Cap Blend 18.61 14.88 10.52 5. Mid-Cap Blend 18.04 19.45 15.91 1. Small-Cap Blend 11.33 13.79 2.07 6. Intermediate Bond 8.05 7.29 9.18 3. S&P 500 Index 21.00 19.00 12.00 4. ANSWER: IS = proportion of portfolio invested in international stock LC = proportion of portfolio invested in large-cap blend
MC = proportion of portfolio invested in mid-cap blend SC = proportion of portfolio invested in small-cap blend IB = proportion of portfolio invested in intermediate bond R 1 = portfolio return for scenario 1 (Year 1) R 2 = portfolio return for scenario 2 (Year 2) R 3 = portfolio return for scenario 3 (Year 3) R 4 = portfolio return for scenario 4 (Year 4) Min ( R 1 − 21) 2
- ( R 2 − 19) 2
- ( R 3 − 12) 2
- ( R 4 − 4) 2 26.73 IS + 18.61 LC + 18.04 MC + 11.33 SC + 8.05 IB = R 1 22.37 IS + 14.88 LC + 19.45 MC + 13.79 SC + 7.29 IB = R 2 6.46 IS + 10.52 LC + 15.91 MC − 2.07 SC + 9.18 IB = R 3 −3.19 IS + 5.25 LC − 1.94 MC + 6.85 SC + 3.92 IB = R 4 IS + LC + MC + SC + IB = 1 IS , LC , MC , SC , IB ≥ 0 POINTS: 1 DIFFICULTY: Challenging LEARNING OBJECTIVES: IMS.ASWC.19.08.02 - 8. NATIONAL STANDARDS: United States - BUSPROG: Analytic TOPICS: 8.2 Constructing an Index Fund KEYWORDS: Bloom's: Create
- Financial planner Minnie Margin has a substantial number of clients who wish to own a mutual fund portfolio that matches, as a whole, the performance of the Russell 2000 index. Her task is to determine what proportion of the portfolio should be invested in each of the five mutual funds listed below so that the portfolio most closely mimics the performance of the Russell 2000 index. Formulate the appropriate nonlinear program. Annual Returns (Planning Scenarios) Mutual Fund Year 1 Year 2 Year 3 Year 4 International Stock 22.37 26.73 4.86 2. Large-Cap Value 15.48 19.64 11.50 5. Mid-Cap Value 17.42 20.07 4.97 1. Small-Cap Growth 23.18 12.36 3.25 3. Short-Term Bond 9.26 8.81 6.15 4. Russell 2000 Index 20.00 22.00 8.00 2. ANSWER: IS = proportion of portfolio invested in international stock LC = proportion of portfolio invested in large-cap value
SC = proportion of portfolio invested in small-cap blend IB = proportion of portfolio invested in intermediate bond R 1 = portfolio return for scenario 1 (Year 1) R 2 = portfolio return for scenario 2 (Year 2) R 3 = portfolio return for scenario 3 (Year 3) R 4 = portfolio return for scenario 4 (Year 4) AR = expected (average) portfolio return Min
0.25( R 1 − AR )
2
- 0.25( R 2 − AR ) 2
- 0.25( R 3 − AR ) 2
0.25( R 4 − AR )^2 26.73 IS + 8.61 LC + 18.04 MC + 11.33 SC + 8.05 IB = R 1 22.37 IS + 4.88 LC + 19.45 MC + 13. 79 SC + 7.29 IB = R 2 6.46 IS + 10.52 LC + 15.91 MC − 2.07 SC + 9.18 IB = R 3 −3.19 IS + 5.25 LC − 1.69 MC + 3.81 SC + 4.04 IB = R 4 IS + LC + MC + SC + IB = 1 0.25 R 1 + 0.25 R 2 + 0.25 R 3 + 0.25 R 4 = AR AR ≥ 4 IS , LC , MC , SC , IB ≥ 0 POINTS: 1 DIFFICULTY: Challenging LEARNING OBJECTIVES: IMS.ASWC.19.08.03 - 8. NATIONAL STANDARDS: United States - BUSPROG: Analytic TOPICS: 8.3 Markowitz Portfolio Model KEYWORDS: Bloom's: Create
- Financial planner Minnie Margin wishes to develop a mutual fund portfolio based on the Markowitz portfolio model. She needs to determine the proportion of the portfolio to invest in each of the five mutual funds listed below so that the variance of the portfolio is minimized subject to the constraint that the expected return of the portfolio be at least 5%. Formulate the appropriate nonlinear program. Annual Returns (Planning Scenarios) Mutual Fund Year 1 Year 2 Year 3 Year 4 International Stock 22.37 26.73 4.86 2. Large-Cap Value 15.48 19.64 11.50 5. Mid-Cap Value 17.42 20.07 4.97 1. Small-Cap Growth 23.18 12.36 3.25 3. Short-Term Bond 9.26 8.81 6.15 4. ANSWER: IS = proportion of portfolio invested in international stock
LC = proportion of portfolio invested in large-cap value MC = proportion of portfolio invested in mid-cap value SC = proportion of portfolio invested in small-cap growth SB = proportion of portfolio invested in short-term bond R 1 = portfolio return for scenario 1 (Year 1) R 2 = portfolio return for scenario 2 (Year 2) R 3 = portfolio return for scenario 3 (Year 3) R 4 = portfolio return for scenario 4 (Year 4) AR = expected (average) portfolio return Min
0.25( R 1 − AR )^2 + 0.25( R 2 − AR )^2 + 0.25( R 3 − AR )^2 +
0.25( R 4 − AR )^2
22.37 IS + 15.48 LC + 17.42 MC + 23.18 SC + 9.26 SB = R 1
26.73 IS + 19.64 LC + 20.07 MC + 12.36 SC + 8.81 SB = R 2
4.86 IS + 11.50 LC − 4.97 MC + 3.25 SC + 6.15 SB = R 3
2.17 IS − 5.25 LC − 1.69 MC + 3.81 SC + 4.04 SB = R 4
IS + LC + MC + SC + SB = 1
0.25 R 1 + 0.25 R 2 + 0.25 R 3 + 0.25 R 4 = AR
AR ≥ 5
IS , LC , MC , SC , SB ≥ 0
POINTS: 1
DIFFICULTY: Challenging LEARNING OBJECTIVES: IMS.ASWC.19.08.03 - 8. NATIONAL STANDARDS: United States - BUSPROG: Analytic TOPICS: 8.3 Markowitz Portfolio Model KEYWORDS: Bloom's: Create
- Cutting Edge Yard Care is a residential and commercial lawn service company that has been in business in the Atlanta metropolitan area for almost one year. Cutting Edge would like to use its Atlanta service subscription data below to develop a model for forecasting service subscriptions in other metropolitan areas where it might expand. The first step is to estimate values for p (coefficient of innovation) and q (coefficient of imitation). Formulate the appropriate nonlinear program. Month Subscribers St Cum. Subscribers Ct 1 0.53 0. 2 2.04 2. 3 3.37 5. 4 4.85 10.
bat is $100, and the cost to produce a Launcher bat is $120. We cannot assume that MegaSports will sell all the bats it can produce. As the selling price of each bat model—Slugger and Launcher—increases, the quantity demanded for each model goes down. Assume that the demand, S , for Slugger bats is given by S = 640 − 4 PS and the demand, L , for Launcher bats is given by L = 450 − 3 PL where PS is the price of a Slugger bat and PL is the price of a Launcher bat. The profit contributions are PS S − 100 S for Slugger bats and PL L − 120 L for Launcher bats. Develop the total profit contribution function for this problem. ANSWER: Solving S = 640 − 4 PS for PS , we get: PS = 160 − 1/4 S Substituting 160 − 1/4 S for PS in PS S − 100 S , we get: 60 S − 1/4 S^2 Solving L = 450 − 3 PL for PL , we get: PL = 150 − 1/3 L Substituting 150 − 1/3 L for PL in PL L − 120 L , we get: 30 L − 1/3 L^2 Total profit contribution = 60 S − 1/4 S^2 + 30 L − 1/3 L^2 POINTS: 1 DIFFICULTY: Challenging LEARNING OBJECTIVES: IMS.ASWC.19.08.01 - 8. NATIONAL STANDARDS: United States - BUSPROG: Analytic TOPICS: 8.1 A Production Application—Par, Inc., Revisited KEYWORDS: Bloom's: Create
- Pacific-Gulf Oil Company is faced with the problem of refining three petroleum components into regular and premium gasoline in order to maximize profit. Components 1 and 2 are pooled in a single storage tank, and component 3 has its own storage tank. Regular and premium gasolines are made from blending the pooled components and component 3. Prices per gallon for the two products and three components, as well as product specifications, are listed below. Price per Gallon Regular gasoline
Premium gasoline
Component 1 2. Component 2 2. Component 3 2. Product Specifications Regular gasoline At most 25% component 1 At least 40% component 2 At most 30% component 3
Premium gasoline At least 30% component 1 At most 50% component 2 At least 25% component 3 The maximum number of gallons available for each of the three components is 4000, 8000, and 8000, respectively. Formulate a nonlinear program to determine: (1) what percentages of components 1 and 2 should be used in the pooled mixture and (2) how to make regular and premium gasoline by blending the mixture of components 1 and 2 from the pooling tank with component 3. ANSWER: y 1 = gallons of component 1 in the pooling tank y 2 = gallons of component 2 in the pooling tank xpr = gallons of pooled components 1 and 2 in regular gasoline xpp = gallons of pooled components 1 and 2 in premium gasoline x 3 r = gallons of component 3 in regular gasoline x 3 p = gallons of component 2 in premium gasoline Max 2.8( x (^) pr + x 3 r ) + 3.1( x (^) pp + x 3 p ) − 2.4 y 1 − 2.5 y 2 − 2.75( x 3 r + x 3 p ) y 1 + y 2 = x (^) pr + x (^) pp [ y 1 /( y 1 + y 2 )] x (^) pr ≤ 0.25( x (^) pr + x 3 r ) [ y 2 /( y 1 + y 2 )] x (^) pr ≥ 0.4( x (^) pr + x 3 r ) x 3 r ≤ 0.3( x (^) pr + x 3 r ) [ y 1 /( y 1 + y 2 )] x (^) pp ≥ 0.3( x (^) pp + x 3 p ) [ y 2 /( y 1 + y 2 )] x (^) pp ≤ 0.5( x (^) pp + x 3 p ) x 3 p ≥ 0.25( x (^) pp + x 3 p ) y 1 ≤ 4000 y 2 ≤ 8000 x 3 r + x 3 p ≤ 8000 x (^) pr + x 3 r ≤ 8000 x (^) pr , x (^) pp , x 3 r , x 3 p , y 1 , y 2 ≥ 0 POINTS: 1 DIFFICULTY: Challenging LEARNING OBJECTIVES: IMS.ASWC.19.08.04 - 8. NATIONAL STANDARDS: United States - BUSPROG: Analytic TOPICS: 8.4 Blending: The Pooling Problem KEYWORDS: Bloom's: Create
