Help with Optimization and Decision Support Modeling for Business HW3

OSCM 471/571 Optimization and Decision Support Modeling for Business

Homework 3, Spring 2023

Notice for Homework 3

Instructor: Seokjun Youn ( [email protected] )

· Due date: Thursday 3/16, 11:59 pm

· Please submit your files to D2L > Assignments > Homework 3

1. A Word file (or PDF) with your answers combined into a single document.

2. An Excel spreadsheet template with your answers (for some sub-questions).

· This homework is made up of THREE questions (20 pts):

· Lecture 3-2: LP What-if Analysis

· Q1: 5 sub-questions (4 pts)

· Q2: 4 sub-questions (4 pts)

· Q3: 1 sub-questions (2 pts)

· Lecture Note 4: Network Models

· Q4: 1 sub-questions (3 pts)

· Q5: 2 sub-questions (3.5 pts)

· Q6: 2 sub-questions (3.5 pts)

· Students may choose either handwriting or word processing (or both).

· Handwriting: please properly scan or take photos and organize them into one file before uploading in D2L.

· Please write down your solutions step-by-step for partial credit.

· You may use:

· Your textbook and notes from the class.

· Notes or sources from a related class or internet source.

· Discussion with the instructor.

· Voluntary, mutual, and cooperative discussion with other students currently taking the class.

· You may not use:

· Solution manuals (printed or electronic).

· Copying from other students in this class, including expecting them to reveal their solutions in “discussion.”

· It is fine if your answer is not 100% correct. However, if you do not put enough effort to the assignment, your score for this homework will be lower than your expectation. So, please try to convince your logic to instructor.

Your Name:

1. Ken and Larry, Inc., supplies its ice cream parlors with three flavors of ice cream: chocolate, vanilla, and banana. Due to extremely hot weather and a high demand for its products, the company has run short of its supply of ingredients: milk, sugar, and cream. Hence, they will not be able to fill all the orders received from their retail outlets, the ice cream parlors. Due to these circumstances, the company has decided to choose the amount of each flavor to produce that will maximize total profit, given the constraints on the supply of the basic ingredients.

The chocolate, vanilla, and banana flavors generate, respectively, $1.00, $0.90, and $0.95 of profit per gallon sold. The company has only 200 gallons of milk, 150 pounds of sugar, and 60 gallons of cream left in its inventory. The linear programming formulation for this problem is shown below in algebraic form.

Let

C = Gallons of chocolate ice cream produced

V = Gallons of vanilla ice cream produced

B = Gallons of banana ice cream produced

Maximize

subject to

Milk:

Sugar:

Cream:

and

This problem was solved using Solver. The spreadsheet (already solved) and the sensitivity report are shown below. (Note: The numbers in the sensitivity report for the milk constraint are missing on purpose, since you will be asked to fill in these numbers in part f.)

For each of the following parts, answer the question as specifically and completely as possible without solving the problem again with Solver. Note: Each part is independent (i.e., any change made to the model in one part does not apply to any other parts).

a. What is the optimal solution and total profit?

Answer:

b. Suppose the profit per gallon of banana changes to $1.00. Will the optimal solution change and what can be said about the effect on total profit?

Answer:

c. Suppose the profit per gallon of banana changes to 92¢. Will the optimal solution change and what can be said about the effect on total profit?

Answer:

d. Suppose the company discovers that three gallons of cream have gone sour and so must be thrown out. Will the optimal solution change and what can be said about the effect on total profit?

Answer:

e. Fill in all the sensitivity report information for the milk constraint, given just the optimal solution for the problem. Explain how you were able to deduce each number.

Answer:

2. Consider the Union Airways problem presented in Section 3.3, including the spreadsheet in Figure 3.5 showing its formulation and optimal solution.

Management now is considering increasing the level of service provided to customers by increasing one or more of the numbers in the rightmost column of Table 3.5 for the minimum number of agents needed in the various time periods. To guide them in making this decision, they would like to know what impact this change would have on total cost.

Use Solver to generate the sensitivity report in preparation for addressing the following questions.

· Please include the screenshot of your final spreadsheet model here.

a. Which of the numbers in the rightmost column of Table 3.5 can be increased without increasing total cost? In each case, indicate how much it can be increased (if it is the only one being changed) without increasing total cost.

Answer:

b. For each of the other numbers, how much would the total cost increase per increase of 1 in the number? For each answer, indicate how much the number can be increased (if it is the only one being changed) before the answer is no longer valid.

Answer:

c. Do your answers in part b definitely remain valid if all the numbers considered in part b are simultaneously increased by 1?

Answer:

d. Do your answers in part b definitely remain valid if all 10 numbers are simultaneously increased by 1?

Answer:

3. Reconsider the example illustrating the use of robust optimization that was presented in the lecture note. Wyndor management now feels that there is uncertainty in all of the parameters of the problem – the unit profit per door and window ( and ), the hours of production time used for each door or window produced across the three plants ( and ) and the three right-hand-sides representing the hours available at each plant ( and ). The original estimates along with the ranges of uncertainty are shown in the table below. Apply the procedure for robust optimization with independent parameters to find the solution that maximizes profit when the solution also is guaranteed to be feasible.

· Please include the screenshot of your final spreadsheet model here.

Answer:

4. Sarah and Jennifer have just graduated from college at the University of Washington in Seattle and want to go on a road trip. They have always wanted to see the mile-high city of Denver. Their road atlas shows the driving time (in hours) between various city pairs, as shown below. Formulate and solve a network optimization model to find the quickest route from Seattle to Denver?

· Please include the screenshot of your final spreadsheet model here.

Answer:

5. The Makonsel Company is a fully integrated company that both produces goods and sells them at its retail outlets. After production, the goods are stored in the company’s two warehouses until needed by the retail outlets. Trucks are used to transport the goods from the two plants to the warehouses, and then from the warehouses to the three retail outlets.

Using units of full truckloads, the first table below shows each plant’s monthly output, its shipping cost per truckload sent to each warehouse, and the maximum amount that it can ship per month to each warehouse.

For each retail outlet (RO), the second table below shows its monthly demand, its shipping cost per truckload from each warehouse, and the maximum amount that can be shipped per month from each warehouse.

Management now wants to determine a distribution plan (number of truckloads shipped per month from each plant to each warehouse and from each warehouse to each retail outlet) that will minimize the total shipping cost.

a. Draw a network that depicts the company’s distribution network. Identify the supply nodes, transshipment nodes, and demand nodes in this network. Formulate a network model for this problem as a minimum-cost flow problem by inserting all the necessary data into the network drawn in part a. (Use the format depicted in Figure 6.3 to display these data.)

Answer:

b. Formulate and solve a spreadsheet model for this problem.

· Please include the screenshot of your final spreadsheet model here.

Answer:

6. The diagram depicts a system of aqueducts that originate at three rivers (nodes R1, R2, and R3) and terminate at a major city (node T), where the other nodes are junction points in the system.

Using units of thousands of acre feet, the following tables show the maximum amount of water that can be pumped through each aqueduct per day.

The city water manager wants to determine a flow plan that will maximize the flow of water to the city.

a. Formulate this problem as a maximum flow problem by identifying a source, a sink, and the transshipment nodes, and then drawing the complete network that shows the capacity of each arc.

Answer:

b. Formulate and solve a spreadsheet model for this problem.

Answer:

2/8