Chapter 13

11- Implement the linear optimization model that you developed for Valencia Products in Problem 4 on a spreadsheet and use *Solver* to find an optimal solution. Interpret the *Solver* Answer Report, identify the binding constraints, and verify the values of the slack variables by substituting the optimal solution into the model constraints.

**12.**Implement the linear optimization model that you developed for ColPal Products in Problem 5 on a spreadsheet and use *Solver* to find an optimal solution. Interpret the Solver Answer Report, identify the binding constraints, and verify the values of the slack variables by substituting the optimal solution into the model constraints.

**13.**Implement the linear optimization model that you developed for Burger Office Equipment in Problem 6 on a spreadsheet and use*Solver*to find an optimal solution. Interpret the*Solver*Answer Report, identify the binding constraints, and verify the values of the slack variables by substituting the optimal solution into the model constraints.**14.**Implement the linear optimization model that you developed for the investment scenario in Problem 7 on a spreadsheet and use*Solver*to find an optimal solution. Interpret the*Solver*Answer Report, identify the binding constraints, and verify the values of the slack variables by substituting the optimal solution into the model constraints.***15.**Implement the linear optimization model that you developed for Bangs Leisure Chairs in Problem 8 on a spreadsheet and use*Solver*to find an optimal solution.- Interpret the
*Solver*Answer Report, identify the binding constraints, and verify the values of the slack variables by substituting the optimal solution into the model constraints. - Suppose that Mr. Bangs wants to limit the number of Adirondack chairs to at most 20. Modify and re-solve your model to determine the new solution.
- Suppose that Mr. Bangs does not want to spend more than 40 hours each month on any one activity. Modify and re-solve your original model to determine the new solution.

- Interpret the
***16.**Implement the linear optimization model that you developed for the Morton Supply Company in Problem 9 on a spreadsheet and use*Solver*to find an optimal solution. Interpret the*Solver*Answer Report and identify the binding constraints.

17- Implement the linear optimization model that you developed for Malloy Milling in Problem 10 on a spreadsheet and use *Solver* to find an optimal solution. Interpret the *Solver* Answer Report and identify the binding constraints.

How *Solver* Works

**18.**For the Valencia Products model in Problem 4, graph the constraints and identify the feasible region. Then identify each of the corner points and show how increasing the objective function value identifies the optimal solution.**19.**For the ColPal model in Problem 5, graph the constraints and identify the feasible region. Then identify each of the corner points and show how increasing the objective function value identifies the optimal solution.

Chapter 14

Solve Problem 11 in Chapter 13 (Valencia Products) to ensure that the number of units produced is integer-valued. How much difference is there between the optimal integer solution objective function and the linear optimization solution objective function? Would rounding the continuous solution have provided the optimal integer solution?

2.Solve Problem 12 in Chapter 13 (ColPal Products) to ensure that the number of minutes of radio and TV ads is integer-valued. How much difference is there between the optimal integer solution objective function and the linear optimization solution objective function? Would rounding the continuous solution have provided the optimal integer solution?

*3.Solve Problem 15 in Chapter 13 (Bangs Leisure Chairs) to ensure that the number of units produced is integer-valued. How much difference is there between the optimal integer solution objective function and the linear optimization solution objective function? Would rounding the continuous solution have provided the optimal integer solution?

4.For the Brewer Services scenario described in this chapter, suppose that 11 permanent employees are hired. Find an optimal solution to minimize the number of part-time employees needed.

*5.The Gardner Theater, a community playhouse, needs to determine the lowest-cost production budget for an upcoming show. Specifically, they have to determine which set pieces to construct and which, if any, set pieces to rent from another local theater at a predetermined fee. However, the organization has only two weeks to fully construct the set before the play goes into technical rehearsals. The theater has two part-time carpenters who work up to 12 hours a week, each at $10 an hour. Additionally, the theater has a part-time scenic artist who can work 15 hours per week to paint the set and props as needed at a rate of $15 per hour. The set design requires 20 flats (walls), two hanging drops with painted scenery, and three large wooden tables (props). The number of hours required for each piece for carpentry and painting is shown below:

Flats, hanging drops, and props can also be rented at a cost of $75, $500, and $350 each, respectively. How many of each unit should be built by the theater and how many should be rented to minimize total costs?

7.Joe is an active 26-year-old male who lifts weights six days a week. His rigorous training program requires a diet that will help his body recover efficiently. He is also a graduate student who is looking to minimize the cost of consuming his favorite foods. Joe is trying to gain weight, or at least maintain his current body weight, so he is not concerned about calories. His personal trainer suggests at least 300 grams of protein, 95 grams of fat, 225 grams of carbohydrates, and no more than 110 grams of sodium per day. His favorite foods are all items that he is familiar with preparing, as shown in the table Data for Problem 7. He is willing to consume multiple servings of each food per day to meet his requirements, although he cannot eat more than one steak per day and does not want to eat more than three pulled pork sandwiches a day. He needs to consume at least two servings of broccoli and one serving of carrots per day but is willing to eat two servings of carrots if necessary. Joe likes a certain brand of nutrition bars, but he would not eat more than one. Unless previously noted, he does not want more than five servings of any one food. How many servings of each food should he have in an optimal daily diet?

8- Gales Products manufactures ribbon for thermal transfer printing, which transfers ink from a ribbon onto paper through a combination of heat and pressure. Different types of printers use different sizes of ribbons. The company has forecasted demand for seven different ribbon sizes, as shown below.

Ribbon Size | Forecast Demand |

60 mm | 1,620 |

83 mm | 520 |

102 mm | 840 |

110 mm | 2,640 |

120 mm | 500 |

130 mm | 740 |

165 mm | 680 |

The rolls from which ribbons are cut are 900 mm in length. Scrap is valued at $0.07 per millimeter. Generate ten different cutting patterns so that each size can be cut from at least one pattern. Use your data to construct and solve an optimization model for finding the number of patterns to cut to meet demand and minimize trim loss.

10- Fuller Legal Services wants to determine how much time to allocate to four different services: business consulting, criminal work, nonprofit consulting, and wills/trusts. Mr. Fuller has determined the average hourly fees and the minimum and maximum hours (for consulting and criminal work) and cases (for wills/trusts) that he would like to spend on each. He has no shortage of demand for his services. The relevant data are shown in the table *Data for Problem 10*. Develop and solve an integer optimization model to maximize monthly revenue.

***11.**Riesemberg Medical Devices is allocating next year’s budget among its divisions. As a result, the R&D Division needs to determine which R&D projects to fund. Each project requires various software and hardware and consulting expenses, along with internal human resources. A budget allocation of $1,300,000 has been approved, and 35 engineers are available to work on the projects. The R&D group has determined that at most one of projects 1 and 2 should be pursued, and that if project 4 is chosen, then project 2 must also be chosen. Develop a model to select the best projects within the budget.

Project | NPV | Internal Engineers | Additional Costs |

1 | $600,000 | 9 | $196,000 |

2 | 580,000 | 4 | 400,000 |

3 | 550,000 | 7 | 70,000 |

4 | 400,000 | 12 | 180,000 |

5 | 650,000 | 8 | 225,000 |

6 | 725,000 | 10 | 200,000 |

7 | 340,000 | 8 | 130,000 |

Chapter 15:

- For the Valencia Products scenario (Problems 4 and 11 in Chapter 13), use the spreadsheet model to answer the following questions by changing the parameters and re-solving the model. Answer each question independently relative to the original problem.
- If the unit profit for SpeedBuster is decreased to $130, how will the optimal solution and profit change?

- If the unit profit for LaserStop is increased to $210, how will the optimal solution and profit change?

- If an additional 1,500 units of component A are available, can you predict how the optimal solution and profit will be affected?

- If a supplier delay results in only 3,000 units of component B being available, can you predict how the optimal solution and profit will be affected? Can you explain the result?

- For the ColPal Products scenario (Problems 5 and 12 in Chapter 13), use the spreadsheet model to answer the following questions by changing the parameters and re-solving the model. Answer each question independently relative to the original problem.
- Suppose that the exposure for TV advertising was incorrectly estimated and should have been 875. How would the optimal solution have been affected?

- Radio listening has gone down, and new marketing studies have found that the exposure has dropped to 150. How will this affect the optimal solution?

- The marketing manager has increased the budget by $2,000. How will this affect the solution and total exposure?

- For the Burger Office Equipment scenario (Problems 6 and 13 in Chapter 13), use the spreadsheet model to answer the following questions by changing the parameters and re-solving the model. Answer each question independently relative to the original problem.
- If 25% of the pine is deemed to be cosmetically defective, how will the optimal solution be affected?

- The shop supervisor is suggesting that the workforce be allowed to work an additional 50 hours at an overtime premium of $18/hour. Is this a good suggestion? Why or why not?

- If the unit profit for standard desks is increased to $280, how will the optimal solution and total profit be affected?

- If the unit profit of standard desks is only $190, how will the optimal solution and total profit be affected?

- For the Markowitz model in Example 14.10, determine how the minimum variance and stock allocations change as the target return varies between 8% and 12% (in increments of 1%) by re-solving the model. Summarize your results in a table, and create a chart showing the relationship between the target return and the optimal portfolio variance. Explain what the results mean for an investor.
- Figure 15.32 shows the
*Solver*Sensitivity Report after solving the Crebo Manufacturing problem in Chapter 13 (Example 13.10). Using only the information in the Sensitivity Report, answer the following questions.

- Explain the value of the reduced cost (−0.3)(−0.3) for the number of plugs to produce.
- If the gross margin for rails is decreased to $1.05, can you predict what the optimal solution and profit will be?
- Suppose that the gross margin for rivets is increased to $0.85. Can you predict what the optimal solution and profit will be?
- If the gross margin for clips is reduced to $1.10, can you predict what the optimal solution and profit will be? What if the gross margin is reduced to $1.00?
- Suppose that an additional 500 minutes of machine capacity is available. How will the optimal solution and profit change? What if planned maintenance reduces capacity by 300 minutes?