Please complete the three questions by making the required graphs and doing excel work

Assignment #6

The excel document must be submitted along with the solution (as a separate document).

1. Consider the following all-integer linear program:

Max x + y subject to

6x + 4y ≤ 22

5x + y ≤ 12

x + 2y ≤ 8

x, y ≥ 0 and both integers

1. Graph the constraints for this problem. Use dots to indicate all feasible integer solutions. (5 marks)

1. Solve the ILP for this problem when y is the only integer (x is relaxed). (5 marks)

1. Find the optimal integer solution graphically (show your work). (5 marks)

1. Martha 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 330 in department A, 350 in department B, and 200 in department C. The profit contributions per unit are $33 for product1, $28 for product 2, and $31 for product 3.

1. Formulate a linear programming model for maximizing total profit contribution. (5 marks)

1. Solve the linear program formulated in part (a) using excel. How many (note than number of products should be considered as integer value only) of each product should be produced, and what is the projected total profit contribution? (4 marks)

Note that you must submit your excel document as a separate document (no pictures or

pdfs are allowed) in order to get credit for this question.

3.

TourneSol Canada, Ltd. is a producer of high quality sunflower oil. The company buys raw sunflower seeds directly from large agricultural companies, and refines the seeds into sunflower oil that it sells in the wholesale market. As a by-product, the company also produces sunflower mash (a paste made from the remains of crushed sunflower seeds) that it sells into the market as base product for animal feed.

The company has a maximum input capacity of 150 short tons of raw sunflower seeds every day (or 54,750 short tons per year). Of course the company cannot run at full capacity every day as it is required to shut down or reduce capacity for maintenance periods every year, and it experiences the occasional mechanical problem. The facility is expected to run at 90% capacity over the year (or on average 150 x 90% = 135 short tons per day).

TourneSol is planning to purchase its supply of raw sunflower seeds from three primary growers, Supplier A, Supplier B, and Supplier C. Purchase prices will not set until the orders are actually placed so TourneSol will have to forecast purchase prices for the raw material and sales prices for the refined sunflower oil and mash. The contract is written such that TourneSol is only required to commit to 70% of total capacity up front. Any amounts over that can be purchased only as required for the same price. Historical prices for the last 15 years are in the table below (note that year 15 is the most current year).

​

Sunflower oil contains a number of fatty acids, some which are desirable in food products and others that are not. One desirable fatty acid is oleic acid. TourneSol produces high oleic oil for the wholesale market, and requires that the oleic acid content be a minimum of 77%. Sunflower oil also contains trace amounts of iodine. The market requires that that iodine content be a minimum of 0.78% and maximum of 0.88%

The oleic acid and iodine content for the sunflower seeds from the three suppliers is given in the table below.

​

For all three suppliers, it is expected that the average yield of oil from the seeds is 30%. There is no net loss of material, so the yield of mash from the same supply is expected to be 70%.

Because the oleic acid and iodine content varies across the three suppliers, so does the price. It is expected that the cost of supply from the suppliers will be a percentage of the market average price of seeds.

​

The company faces an additional variable production cost of $8.5/short ton and an estimated fixed cost of $1,050,000 over the upcoming production period.

The company is asking you to provide a recommendation on the amount of raw material it should purchase from each supplier to minimize its cost of feedstock.

Management is also looking for an analysis on the profitability of the company in the next production cycle. 

Suggested Approach

This is a fairly complex problem. The following approach is suggested:

• Use the historical price data set as input to a time series forecast model in order to generate forecasted prices for the average price of sunflower seeds, oil, and mash in the next production period. Use standard measures of error to decide between a three-period moving average model or an exponential smoothing model (with α = 0.2). Use the type of model for all three time series forecasts. That is, if you decide to use the moving average model, use a three-period moving average model to fit the relevant data for all three series. Don’t use the moving average for one time series and the exponential smoothing model for another time series.

• Formulate a linear program to minimize the cost of raw sunflower seeds. Use the average price of seeds forecasted from the previous step in order to determine supplier prices.

• Perform a cost-volume-price analysis (review the handout entitled Cost-Volume-Profit Analysis for details) using the average cost per short ton average selling price per short ton.

  • You can generate an effective cost per short ton by dividing the total cost of supply (from the linear program) by the total volume (that you assumed in the linear program).

  • You can generate an effective selling price per short ton from the expected percentage yields and the forecasted average price of sunflower oil and mash.

  • Because of the way that the contract is written, you can assume that the purchase of raw sunflower seeds is a variable cost (you only purchase what you require).

Recall that the cost-volume-price analysis requires you to provide:

• an algebraic statement of the revenue function and the cost function,

• a detailed break-even chart that includes lines for the revenue and for the total cost, fixed cost, and variable cost (a total of four lines), and

• a calculation break-even point expressed in number of short tons and percent of capacity.