BUS 660 Week 6 Integer Linear Programming and Advanced Optimization Applications – Homework (2 Versions)
(BUS 660 Week 6 Integer Linear Programming)
Problem 11-13: The Martin-Beck Company operates a plant in St. Louis with an annual capacity of 30,000 units. Product is … to regional distribution centers … in Boston, Atlanta, and Houston. Because of an … increase in demand, Martin-Beck plans to increase capacity by constructing a new plant in one or more of the following cities: Detroit, Toledo, Denver, or Kansas City. The estimated annual fixed cost and the annual capacity for the four proposed plants are as follows:
The company’s long-range planning group … forecasts of the anticipated annual … at the distribution centers as follows:
The shipping cost per unit from each plant to each distribution center is shown in table below
Each potential plant location is shown; capacities and demands are shown in thousands of units. This network representation is for a transportation problem with a plant at St. Louis and at all four … sites. However, the decision has not yet been made as to which new plant or plants will be constructed.
- Formulate a model that could … for choosing the best plant locations and for determining how much to ship from each plant to each distribution center. There is a policy restriction that a plant must … either in Detroit or in Toledo, but not both. For those boxes in which you must enter subtractive or negative numbers use a minus sign. (Example: -300)
- Formulate a model that could … for choosing the best plant locations and for determining how much to ship from each plant to each distribution center. There is a policy restriction that no more than two plants can be located in Denver, Kansas City, and St. Louis. For those boxes in which you must enter subtractive or negative numbers use a minus sign. (Example: -300)
Problem 12-23: Markowitz portfolio optimization: Harry Markowitz … the 1990 Nobel Prize for his path-breaking work in portfolio optimization. One version of the Markowitz model is … on minimizing the variance of the portfolio subject to a constraint on return. The below table shows the annual return (%) for five 1-year periods for the six mutual funds with the last row that gives the S&P 500 return for each planning scenario. Scenario 1 represents a year in which the annual returns are good for all the mutual funds. Scenario 2 is also a good year for most of the mutual funds. But scenario 3 is a bad year for the small-cap value fund; scenario 4 is a bad year for the intermediate-term bond fund; and scenario 5 is a bad year for four of the six mutual funds.
Problem 11-3: Consider the following all-integer linear program:
- Choose the correct graph which shows the constraints for this problem and uses dots to indicate all feasible integer solutions.
- Solve the LP Relaxation of this problem.
- Find the optimal integer solution.
Problem 11-11: Hart Manufacturing makes three products. Each product requires manufacturing operations in three departments: A, B, and C. The labor-hour requirements, by department, are as follows:
- During the next production period, the labor-hours available are 450 in department A, 350 in department B, and 50 in department C. The profit contributions per unit are $25 for product 1, $28 for product 2, and $30 for product 3.
- Formulate a linear programming model for maximizing total profit contribution. If required, round your answers to two decimal places. For those boxes in which you must enter subtractive or negative numbers use a minus sign. (Example: -300)
- Solve the linear program formulated in part (a). How much of each product should … , and what is the … total profit contribution?
- After evaluating the solution … in part (b), one of the production supervisors noted that production setup costs had not been taken into account. She noted that setup costs are $400 for product 1, $550 for product 2, and $600 for product 3. If the solution developed in part (b) is to be used, what is the total profit contribution after taking into account the setup costs?
- Management … that the optimal product mix, taking setup costs into account, might be different from the one … in part (b). Formulate a mixed-integer linear program that takes setup costs into account. Management also stated that we should not consider making more than 175 units of product 1, 150 units of product 2, or 140 units of product 3. If required, round your answers to two decimal places. For those boxes in which you must enter subtractive or negative numbers use a minus sign. (Example: -300) Here introduce a 0-1 variable yi that is one if any quantity of product i is produced and zero otherwise.
- Solve the mixed-integer linear program … in part (d). How much of each product should be produced, and what is the projected total profit contribution? Compare this profit contribution to that obtained in part (c).
Problem 11-1: Indicate whether the following linear program is an all-integer linear program or a mixed-integer linear program. Indicate whether the following linear program is an all-integer linear program or a mixed-integer linear program.
Problem 11-23: Roedel Electronics produces a variety of electrical components, including a remote control for televisions and a remote control for DVD players. Each remote control consists of three subassemblies that are … by Roedel: a base, a cartridge, and a keypad. Both remote controls use the same base subassembly, but different cartridge and keypad subassemblies.
Roedel’s sales forecast indicates that 7000 TV remote controls and 5000 DVD remote controls will … to satisfy demand during the upcoming Christmas season. Because only 500 hours of in-house manufacturing time are available, Roedel is considering purchasing some, or all, of the subassemblies from outside suppliers. If Roedel manufactures a subassembly in-house, it incurs a fixed setup cost as well as a variable manufacturing cost. The following table shows the setup cost, the manufacturing time per subassembly, the manufacturing cost per subassembly, and the cost to purchase each of the subassemblies from an outside supplier:
- Determine how many units of each subassembly Roedel should manufacture and how many units Roedel should purchase. What is the total manufacturing and purchase cost … with your recommendation?
- Suppose Roedel is considering purchasing new machinery to produce DVD cartridges. For the new machinery, the setup cost is $3000; the manufacturing time is 2.5 minutes per cartridge, and the manufacturing cost is $2.60 per cartridge. Assuming that the new machinery is … , determine how many units of each subassembly Roedel should manufacture and how many units of each subassembly Roedel should purchase. What is the total manufacturing and purchase cost … with your recommendation?
Problem 12-07: Hanson Inn is a 96-room hotel … near the airport and convention center in Louisville, Kentucky. When a convention or a special event is in town, Hanson increases its normal room rates and takes reservations … on a revenue management system. The Classic Corvette Owners Association … its annual convention in Louisville for the first weekend in June. Hanson Inn … to make at least 50% of its rooms available for convention attendees at a special convention rate in order to … hotel for the convention. Although the majority of attendees at the annual meeting typically request a Friday and Saturday two-night package, some attendees may select a Friday night only or a Saturday night only reservation. Customers not attending the convention may also request a Friday and Saturday two-night package, or make a Friday night only or Saturday night only reservation. Thus, six types of reservations are possible: convention customers/two-night package; convention customers/Friday night only; convention customers/Saturday night only; regular customers/two-night package; regular customers/Friday night only; and regular customers/Saturday night only. The cost for each type of reservation is shown here.
Hanson Inn would like to determine how many rooms to make available for each type of reservation in order to maximize total revenue.
- Define the decision variables and state the objective function.
- Formulate a linear programming model for this revenue management application.
- What is the optimal allocation and the … total revenue?
- Total Revenue = $
- Suppose that one week before the convention, the number of regular customers/Saturday night only rooms that were made available sell out. If another nonconvention customer calls and requests a Saturday only room, what is the value of accepting this additional reservation?
Problem 12-11 (Algorithmic): Consider the problem
- Find the minimum solution to this problem. If … , round your answers to two decimal places.
- If the right-hand side of the constraint is … from 10 to 11, how much do you expect the objective function to change? If … , round your answer to two decimal places.
- Re-solve the problem with a new right-hand side of 11. How does the actual change compare with your estimate? If …, round your answers to two decimal places.