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Linear Programming Assignment

Problem 1

students are reminded that submitted assignments must be typed (i.e. can NOT be hand written), neat, readable, and well-organized. However, it is ok to plot GRAPHS by hand and to SCAN/INCLUDE them within the PDF document file as long as they are large, legible, and properly labeled and that their calculations are typed within the rest of the assignment. Assignment marks will be adjusted for sloppiness, poor grammar, spelling mistakes, and technical errors. Submitted assignment solutions (if applicable) must include â€œmanagerial statementsâ€ that communicate the results of the analyses in plain language.Â

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The assignment is to be submitted electronically as a single PDF Document file.The front page of the PDF file has to include the title of the assignment, the course code and section, and the student name and number. Â

Problem 1 (32 points)

Consider the following linear programming model:

Maximize X +2Y
Subject to
(1) Â  Â 2X â€“ 2Y ? ?4
(2) Â  Â X + Y ? 4
(3) Â  Â 2X ? 5
(4) Â  Â Y ? 4

X, Y ?0

a. Â  Â Graph the constraint lines and mark them clearly with the numbers (1), (2), (3) and (4) to indicate which line corresponds to which constraint. Darken the feasible region. Is the feasible region unbounded? (12 points).

b. Â  Â Is there any redundant constraint? If so, indicate which one(s). (2 points)
c. Â  Â Determine the optimal solution(s) and the Maximum value of the objective function using the objective function line method (Show your calculation). (6 points)

d. Â  Â What other method can you choose to find the optimal solution without drawing the objective function? Considering the structure of the feasible region, which method is better? Justify your answer. (3 points)

e. Â  Â  Is there more than one optimal solution? If so, give the two alternate solutions. If not, explain using the graphical method why not? (3 points).

f. Â  Â Use the graph of the feasible region drawn in Part (a) to answer the questions below: Suppose we add the constraint 2X + Y ? ? to the linear programming model.Â

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For what values of ?:
i. Â  Â the optimal solution found above (in part (c)) is no longer optimal? Show your work. (3 points)
ii. Â  Â the linear programming model becomes infeasible? Show your work. (3 points)

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Problem 2 (25 points)

eTrade Canada manages funds for Internet clients. Â For a new commercial client, eTrade has set up a \$25 thousand (CDN) account in two investment funds: (1) equity (stock) fund, and (2) money market fund. Each unit of the equity fund costs \$50 and provides an annual expected rate of return of 15%; each unit of the money market fund costs \$100 and provides an annual expected rate of return of 5%. The client wants to minimize risk subject to the requirement that the expected annual rate of return from the investment be at least \$1,500.

## Problem 2

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The eTrade risk measure scores units invested in equity with an index of 12; units invested in money market have an index of 5. The higher risk index indicates a â€œriskierâ€ investment. The client informed eTrade that 150 units are the minimum amount they want to invest in the money market fund. On behalf of the client, eTrade needs to determine the investment units in each fund to achieve the minimum total risk index for the clientâ€™sÂ Let E and M denote the number of units to purchase, respectively, in the equity (stock) fund and the money market fund.
A correct linear programming formulation for the objective function is:Â
Â  Â  Â  Â  Â  Â  Min Â z Â = Â  Â  Â  Â  Â  Â  Â  Â  Â 12E + Â  Â  Â  5M Â  Â  Â  Â  Â (risk score)

a. Â  Â Write down the algebraic formulation of the model constraints. (8 points)

b. Â  Â Using the graphical method, solve the linear programming problem above. Make sure to graph the constraint lines and mark them clearly. Darken the feasible region. Provide all necessary details to justify your answers. Include â€œmanagerial statementsâ€ that communicate the results of the analysis (i.e. describe verbally the results). (17 points)

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Problem 3 (30 points)
Do not answer questions (a) and (b) from the Custom e-book, and instead do the following:

a. Â  Â Formulate algebraically the Linear Programming (LP) model for this problem. (17 points)

b. Â  Â Formulate this same linear programming problem on a spreadsheet and SOLVE using Excel Solver (Provide the corresponding â€œExcel Spreadsheetâ€ and the â€œAnswer Reportâ€). Include â€œmanagerial statementsâ€ that communicate the results of the analyses (i.e. describe verbally the results). (13 points)

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Problem 4 (23 points)

Tommy & Lefebvre is Ottawa's full-line authorized dealer for Atomic snowboards for the 2020-2021 winter season. T&Lâ€™s sales projections for November, December and January are 550, 800, and 300 snowboards for each of these next three months. Â Atomic has agreed to provide T&L with up to 650 boards each month at a unit cost of \$82. Using â€œrush ordersâ€, Atomic can also provide T&L with up to 50 additional boards per month at an increased unit cost of \$97 each. Boards not sold at the end of the month are stored in the T&L store at a cost of \$20 per board per month. It takes the store clerks 0.5 hours per board to set up, sticker, and display the snowboards and clerk capacity is limited to 500, 600, and 400 hours in each of the 3 months respectively. Finally, T&L have 50 Atomic snowboards from last season (2020 model) that they can sell this year and they want to have at least another 75 snowboards (2021 model) left at the end ofÂ

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Write down the algebraic/mathematical formulation of this problem as a linear programming problem to minimize the total cost to T&L of purchasing and stocking the snowboards. (Define the decision variables, objective function, and constraints). DO NOT SOLVE.Â Ignoring production and inventory holding costs, formulate algebraically the linear programming model that will help RJ-filter Inc. determine the number of units to produce in October, November and December that will minimize the cost of changing the production levels while satisfying the monthly demands. DO NOT SOLVE