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## Midterm Exam

Time: 2.5 hours or 150 minutes

### QUESTION 1: Graphical Method (25 points)

The RBC (Royal Bank of Canada) uses online banking to market two new banking products. The first product is a home risk insurance that allows buyers to default for up to 6 months on their mortgage payments. The second is a guaranteed mortgage fund that buyers may purchase to leverage their funds without increasing their debt loads. The RBC expects to make profit contributions of \$20 per unit on the home risk insurance instrument, and \$8 per unit on the guaranteed mortgage fund. The bank has a policy that at least 50% of total sales of the two products are home risk insurance instruments. The bank is now determining sales quotas for its online offerings to maximize total expected contribution to profits based on the product resource requirements, as follows:

Resource Requirements per Product Offering (Hours per Unit)

A correct formulation for this problem is provided below:

Let HRI and GM denote the number of units of Home Risk Insurance instruments and Guaranteed Mortgage units to sell online, respectively.

MAX Z  = 20 HRI  +  8 GM (\$)

subject to,

1) Legal Hours    6 HRI  +  4 GM    ≤  4,800 hrs

2) Data Mgt Hours       1 HRI  +  2 GM    ≤ 2,000 hrs

3) Policy Claims            3 HRI                   ≤ 1,800 hrs

4) Ratio Policy Limits   0.5 HRI – 0.5 GM    0

5) Non-negativity HRI, GM  0

a）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. (12 points)

(b) Determine the optimal solution that will maximize the total expected contribution to profits. Report the solution in a managerial statement (i.e. describe verbally the optimal solution and its profit). Provide all necessary calculations to justify your answers. (7 points).

(c) Which constraint(s) is (are) redundant? (3 points)

(d) Will there be excess capacity in the Data Management resource? Justify. (3 points).

### QUESTION 2: Linear Programming Sensitivity Analysis (22 points)商业分析期中代考

A salad dressing supplier to Ottawa area restaurants has been using Linear Programming (LP) for years to determine how much dressing they should produce for every season. In particular, they specialize in producing three kinds of dressing: Dijon, Classic Vinaigrette, and Roasted Garlic. All these dressings require some use of olive oil to produce. Taking this into account, the management team has formulated the following LP model that determines the optimal amount they should produce for each dressing.

Answer all the questions below ACCORDING TO the above sensitivity report.

(a) What is the optimal profit? (2 points)

(b) Two numbers have been removed from the resource sensitivity table by your professor (the letters A and B appear instead of the numbers).  What are the correct values of A and B?  Justify. (4 points)

(c) The supplier learned recently that the price of Olive oil has been dropping significantly, and is wondering if they should take this opportunity to purchase more oils. Provide your answer and JUSTIFY it. (3 points)

(d) The profit of Dijon dressing has been found overly estimated. The company however has difficulty determining how much lower the real profit should be. Does this impact the LP optimal solutions? Justify your answer. (3 points)

(e) Some workers are not happy with their salaries and ask for a raise; otherwise, they will quit and this would reduce the available labour minutes from 280 to 240. Should the company consider the raise? If so, what will be a reasonable amount to pay (in total) to these workers in addition to their original salaries?  Justify your answer. (5 points)

(f) What if the profit of ALL three dressings (i.e. Dijon, Classic Vinaigrette, and Roasted Garlic) is now \$1.5/unit. Do the optimal values of the decision variables change? What will be the impact on the profit? Justify. (5 points)

### QUESTION 3: Linear Programming Formulation (33 points)商业分析期中代考

Acme sells two sprocket assemblies for escalators and elevators. Each type of sprocket must first be machined and then assembled. Unit machining and assembly times, capacity limitations, demand restrictions and revenue-cost data are as follows:

Let x1 and x2 represent the number of units of sprockets 1 and 2 to produce, respectively.

MAX Z = 32x1 + 36x2

Subject to

Max Sales Sp. 1: x1 ≤ 15

Max Sales Sp. 2:   x2 ≤ 20

Machine Time:           2x1 + 4x2 ≤ 90

Assembly Time: 2x1 + x2 ≤ 50

Non-negativity:            x1, x2  0

a）In November, Acme introduces a third type of sprocket (Sprocket 3).  To produce one sprocket 3 requires 3 hours of machining and three hours of assembly.  The maximum sales of sprocket 3 is 10 units.  The selling price of a sprocket 3 is \$61.  The above formulation can be modified to take account of the new product.  The new formulation is:

Let x1, x2 and x3 represent the number of units of sprockets 1, 2 and 3 to produce, respectively. 商业分析期中代考

MAX Z = 32x1 + 36x2  _____________

Subject to

Max Sales Sp. 1: x1 ≤ 15

Max Sales Sp. 2: x2 ≤ 20

Max Sales Sp. 3: x3 ≤ _________

Machine Time:      2x1 + 4x2  ________  ≤ 90

Assembly Time:         2x1 + x2  ________ ≤ 50

Non-negativity:            x1, x2, x3 0

Complete the formulation above by filling in the four blanks (8 points).

#### (b)

In December, the problem turns out to be the same as the November problem given in part (a), except that a new packaging process has resulted in an additional requirement that at least 2 sprocket 1’s must be produced for every sprocket 2 that is produced.  Either include below the LP formulation of this new problem, or clearly state how the formulation in part (a) could be changed to deal with the December situation. (2 points)

#### (c)

In January, the problem turns out to be the same as the November problem given in part (a), except that the selling price per unit for sprockets 1 and 2 have each decreased by 10%.  Either included below the LP formulation of this new problem, or clearly state how the formulation in part (a) could be changed to deal with the January situation. (3 points)

#### (d) 商业分析期中代考

In February, the problem turns out to be the same as the November problem given in part (a), except that it is now (in February) possible to purchase up to 10 additional hours of machine time at a price of \$6 per hour.  The \$6 per hour cost applies to only the additional 10 hours.  Either include below the LP formulation of this new problem, or clearly state how the formulation in part (a) could be changed to deal with the February situation. (6 points)

Hint: Add a new decision variable (Y: the number of extra machine hours to produce).

#### (e)

Formulate the original linear programming problem on a spreadsheet and SOLVE using Excel Solver (Provide the corresponding “Excel Spreadsheet” and the “Sensitivity Report”). Include “managerial statements” that communicate the results of the analysis. (i.e. describe verbally the results). (8 points) 商业分析期中代考

Let x1 and x2 represent the number of units of sprockets 1 and 2 to produce, respectively.

MAX Z = 32x1 + 36x2

Subject to

Max Sales Sp. 1:           x1  ≤ 15

Max Sales Sp. 2: x2 ≤ 20

Machine Time: 2x1 +   4x2 ≤ 90

Assembly Time: 2x1 + x2 ≤ 50

Non-negativity: x1,  x2  0

#### (f)

Would it be worthwhile to increase the maximum sales level on Sprocket 2 OR to increase the maximum Machine capacity? Justify. (2 points).

#### (g)

ACME is offered \$7 per hour for use of their assembly time by an outside contractor. How much assembly time is it worthwhile to sell? (2 points).

#### (h)

If a machine breakdown reduced machining capacity by 20 hours, how would profits be affected? (2 points)

### QUESTION 4: Linear Programming Formulation (20 points)商业分析期中代考

Niteton Power and Light Company (NPLC) wants to develop an efficient work schedule for its full- and part-time customer service clerks. The number of clerks needed to provide adequate service during each hour the office is open on a weekday is given below:

Hour 8-9AM 9-10 10-11 11-12PM 12-1 1-2 2-3 3-4

Clerks 5 4 6 8 10 9 7 4

A full time clerk works 3 hours, has a 1-hour break, and then works another 3 hours. Part-time clerks work for 4 consecutive hours. Full-timers get paid for their break. All clerks start work on the hour.

NPLC’s office manager insists that at least one full time clerk be on duty during all open hours and that at least 40% of the clerks should be full-time clerks on payroll.

A full-time clerk costs NPLC \$20 per hour, and a part-timer costs \$15 per hour.

Formulate a linear programming model that will provide a schedule that will meet NPLC’s customer service needs at a minimum labor cost. (Define the decision variables, objective function, and constraints). DO NOT SOLVE.

Hint on the decision variables: there are 2 different full-time shifts and 5 different part-time shifts.

The prev: The next:

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