Help with Optimization and Decision Support Modeling for Business HW1

Module 2 Practice Problems OSCM 471/571 Optimization and Decision Support Modeling for Business

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Study Materials: Lecture Note 2: LP Basic Concepts

1. This is your lucky day. You have just won a $20,000 prize. You are setting aside $8,000 for taxes and

partying expenses, but you have decided to invest the other $12,000. Upon hearing this news, two different friends have offered you an opportunity to become a partner in two different entrepreneurial ventures, one planned by each friend. In both cases, this investment would involve expending some of your time next summer as well as putting up cash. Becoming a full partner in the first friend’s venture would require an investment of $10,000 and 400 hours, and your estimated profit (ignoring the value of your time) would be $9,000. The corresponding figures for the second friend’s venture are $8,000 and 500 hours, with an estimated profit to you of $9,000. However, both friends are flexible and would allow you to come in at any fraction of a full partnership you would like. If you choose a fraction of a full partnership, all the above figures given for a full partnership (money investment, time investment, and your profit) would be multiplied by this same fraction.

Because you were looking for an interesting summer job anyway (maximum of 600 hours), you have decided to participate in one or both friends’ ventures in whichever combination would maximize your total estimated profit. You now need to solve the problem of finding the best combination.

a. Describe the analogy between this problem and the Wyndor Glass Co. problem discussed in

Section 2.1. Then construct and fill in a table like Table 2.1 for this problem, identifying both the activities and the resources. Answer:

As in the Wyndor Glass Co. problem, we want to find the optimal levels of two activities that compete for limited resources. Let x1 be the fraction purchased of the partnership in the first friends venture. Let x2 be the fraction purchased of the partnership in the second friends venture. The following table gives the data for the problem:

Resource Usage per Unit of Activity

Amount of

Resource 1 2 Resource Available Fraction of partnership in

first friends venture

1 0 1

Fraction of partnership in second friends venture

0 1 1

Money

$10,000 $8,000 $12,000

Summer Work Hours 400 500 600 Unit Profit $9,000 $9,000

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b. Identify verbally the decisions to be made, the constraints on these decisions, and the overall

measure of performance for the decisions. Answer: The decisions to be made are how much, if any, to participate in each venture. The constraints on the decisions are that you can’t become more than a full partner in either venture, that your money is limited to $12,000, and time is limited to 600 hours. In addition, negative involvement is not possible. The overall measure of performance for the decisions is the profit to be made.

c. Convert these verbal descriptions of the constraints and the measure of performance into quantitative expressions in terms of the data and decisions. Answer:

First venture: (fraction of 1st) ≤ 1 Second venture: (fraction of 2nd) ≤ 1 Money: 10,000 (fraction of 1st) + 8,000 (fraction of 2nd) ≤ 12,000 Hours: 400 (fraction of 1st) + 500 (fraction of 2nd) ≤ 600 Nonnegativity: (fraction of 1st) ≥ 0, (fraction of 2nd) ≥ 0 Profit = $9,000 (fraction of 1st) + $9,000 (fraction of 2nd)

d. Formulate a spreadsheet model for this problem. Identify the data cells, the changing cells, and the objective cell. Also show the Excel equation for each output cell expressed as a SUMPRODUCT function. Then use Solver to solve this model. Answer:

Data cells: B2:C2, B5:C6, F5:F6, and B11:C11 Changing cells: B9:C9 Objective cell: F9 Output cells: D5:D6

e. Indicate why this spreadsheet model is a linear programming model.

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Answer: This is a linear programming model because the decisions are represented by changing cells that can have any value that satisfy the constraints. Each constraint has an output cell on the left, a mathematical sign in the middle, and a data cell on the right. The overall level of performance is represented by the objective cell and the objective is to maximize that cell. Also, the Excel equation for each output cell is expressed as a SUMPRODUCT function where each term in the sum is the product of a data cell and a changing cell.

f. Formulate this same model algebraically. Answer:

Let x1 = share taken in first friend’s venture x2 = share taken in second friend’s venture Maximize P = $9,000x1 + $9,000x2, subject to x1 ≤ 1 x2 ≤ 1 $10,000x1 + $8,000x2 ≤ $12,000 400x1 + 500x2 ≤ 600 hours and x1 ≥ 0, x2 ≥ 0.

g. Identify the decision variables, objective function, nonnegativity constraints, functional

constraints, and parameters in both the algebraic version and spreadsheet version of the model. Answer:

Algebraic Version decision variables: x1, x2 functional constraints: x1 ≤ 1 x2 ≤ 1 $10,000x1 + $8,000x2 ≤ $12,000 400x1 + 500x2 ≤ 600 hours objective function: Maximize P = $9,000x1 + $9,000x2, parameters: all of the numbers in the above algebraic model nonnegativity constraints: x1 ≥ 0, x2 ≥ 0 Spreadsheet Version decision variables: B9:C9 functional constraints: D4:F7 objective function: F9 parameters: B2:C2, B5:C6, F5:F6, and B11:C11 nonnegativity constraints: “Make Unconstrained Variables Nonnegative” in Solver

h. Use the graphical method by hand to solve this model. What is your total estimated profit?

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Answer: Optimal solution = (x1, x2) = (0.667, 0.667). P = $12,000.

2. Independently consider each of the following changes in the Wyndor problem. In each case, apply the graphical method by hand to this new version of the problem, describe your conclusion, and then explain how and why the nature of this conclusion is different from the original Wyndor problem.

a. The unit profit for the windows now is $200.

Answer: When the unit profit for Windows is $300, there are multiple optima, including (2 doors, 6 windows) and (4 doors and 3 windows) and all points inbetween. It is different than the original unique optimal solution of (2 doors, 6 windows) because windows are now more profitable, making the solution of (4 doors and 3 windows) equally profitable.

b. To justify introducing these two new products, Wyndor management now requires that the total number of doors and windows produced per week must be at least 10.

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Answer: There is no feasible solution with the added requirement that there must be a total of 10 doors and/or windows.

c. The functional constraints for Plants 2 and 3 now have been inadvertently deleted from the model. Answer: If the constraints for plant 2 and plant 3 are inadvertently removed, then the solution is unbounded. There is nothing left to prevent making an unbounded number of windows, and hence making an unbounded profit.

3. Weenies and Buns is a food processing plant that manufactures hot dogs and hot dog buns. They grind their own flour for the hot dog buns at a maximum rate of 200 pounds per week. Each hot dog bun requires 0.1 pound of flour. They currently have a contract with Pigland, Inc., which specifies that a delivery of 800 pounds of pork product is delivered every Monday. Each hot dog requires 1/4 pound of pork product. All the other ingredients in the hot dogs and hot dog buns are in plentiful supply. Finally, the labor force at Weenies and Buns consists of five employees working full time (40 hours per week each). Each hot dog requires three minutes of labor, and each hot dog bun requires two minutes of labor. Each hot dog yields a profit of $0.20, and each bun yields a profit of $0.10.

Weenies and Buns would like to know how many hot dogs and how many hot dog buns they should produce each week so as to achieve the highest possible profit.

a. Identify verbally the decisions to be made, the constraints on these decisions, and the overall

measure of performance for the decisions. Answer: The decisions to be made are how many frankfurters and buns should be produced. The constraints are the amounts of flour and pork available, and the hours available to work. In

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addition, negative production levels are not possible. The overall measure of performance for the decisions is the profit to be made.

b. Convert these verbal descriptions of the constraints and the measure of performance into quantitative expressions in terms of the data and decisions. Answer:

flour: 0.1 (# buns) ≤ 200 pork: 0.25 (# frankfurters) ≤ 800 work hours: 3 (# frankfurters) + 2 (# buns) ≤ 12,000 Nonnegativity: (# frankfurters) ≥ 0, (# buns) ≥ 0 Profit = $0.40 (# frankfurters) + $0.20 (# buns)

c. Formulate and solve a linear programming model for this problem on a spreadsheet.

Answer:

d. Formulate this same model algebraically. Answer:

Let F = # of frankfurters to produce B = # of buns to produce Maximize P = $0.40F + $0.20B, subject to 0.1B ≤ 200 0.25F ≤ 800 3F + 2B ≤ 12,000 and F ≥ 0, B ≥ 0.

e. Use the graphical method to solve this model. Decide yourself whether you would prefer to

do this by hand or by using the Graphical Linear Programming and Sensitivity Analysis module in your Interactive Management Science Modules. Answer: Optimal Solution: (F, B) = (x1, x2) = (3200, 1200) and P = $1520.

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4. Nutri-Jenny is a weight-management center. It produces a wide variety of frozen entrées for consumption by its clients. The entrées are strictly monitored for nutritional content to ensure that the clients are eating a balanced diet. One new entrée will be a beef sirloin tips dinner. It will consist of beef tips and gravy, plus some combination of peas, carrots, and a dinner roll. Nutri-Jenny would like to determine what quantity of each item to include in the entrée to meet the nutritional requirements, while costing as little as possible. The nutritional information for each item and its cost are given in the following table.

The nutritional requirements for the entrée are as follows: (1) it must have between 280 and 320 calories, (2) calories from fat should be no more than 30 percent of the total number of calories, and (3) it must have at least 600 IUs of vitamin A, 10 milligrams of vitamin C, and 30 grams of protein. Furthermore, for practical reasons, it must include at least 2 ounces of beef, and it must have at least half an ounce of gravy per ounce of beef.

a. Formulate and solve a linear programming model for this problem on a spreadsheet. by using

the Excel Solver. Answer:

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b. Formulate this same model algebraically. Answer:

Let B = ounces of beef tips in diet, G = ounces of gravy in diet, P = ounces of peas in diet, C = ounces of carrots in diet, R = ounces of roll in diet. Minimize Z = $0.40B + $0.35G + $0.15P + $0.18C + $0.10R subject to 54B + 20G + 15P + 8C + 40R ≥ 280 54B + 20G + 15P + 8C + 40R ≤ 320 19B + 15G + 10R ≤ 0.3(54B + 20G + 15P + 8C + 40R) 15P + 350C ≥ 600 G + 3P + C ≥ 10 8B + P + C + R ≥ 30 B ≥ 2 G ≥ 0.5B and B ≥ 0, G ≥ 0, P ≥ 0, C ≥ 0, R ≥ 0.

5. You are given the following linear programming model in algebraic form, with x 1 and x 2 as the decision variables:

Minimize 𝐶𝑜𝑠𝑡 = 40𝑥 + 50𝑥 subject to Constraint 1: 2𝑥 + 3𝑥 ≥ 30 Constraint 2: 𝑥 + 𝑥 ≥ 12 Constraint 3: 2𝑥 + 𝑥 ≥ 20 and 𝑥 ≥ 0 𝑥 ≥ 0

a. Use the graphical method to solve this model.

Answer: Optimal Solution: (x1, x2) = (7.5, 5) and C = 550.

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b. How does the optimal solution change if the objective function is changed to 𝐶𝑜𝑠𝑡 = 40𝑥 +

70𝑥 ? Answer: Optimal Solution: (x1, x2) = (15, 0) and C = 600.

c. How does the optimal solution change if the third functional constraint is changed to 2𝑥 +

𝑥 ≥ 15? Answer: Optimal Solution: (x1, x2) = (6, 6) and C = 540.

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d. Now incorporate the original model into a spreadsheet and use Solver to solve this model. Answer:

e. Use Excel to do parts b and c. Answer:

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Part b)

Part c)