Problem 1

The demand for chocolate ice cream during the three summer months (June, July, and August) at All-Flavors Parlor is estimated at 500,600 , and 400 cartons, respectively. Two wholesalers, 1 and 2, supply All-Flavors with its chocolate ice cream. Ice cream from the suppliers is interchangeable and each supplier can provide up to 400 cartons per month. The suppliers charge different amounts per carton from one month to the next as shown in the table below.

To take advantage of price fluctuation, All-Flavors can purchase more than is needed for a particular month and store the surplus to satisfy demand in a later month. The cost of refrigerated storage is $\$ 5$ per carton per month.

1. Formulate a linear programming model to determine a purchasing and storage plan for All-Flavors that minimizes the cost of the meeting the demand over the summer months.
2. How would your formulation from part (a) change if the storage cost is a function of the average number of cartons on hand during the month? Assume that ice cream cartons purchased from the suppliers arrive at the beginning of each month and that the average is calculated based on the number of cartons on hand at the beginning and ending of each month.

Problem 2

Use the graphical method so solve the following linear program (LP):

Problem 3

Each year the Fromage Cheese Company has a sale to celebrate the anniversary of the opening of its first store. This year, the company has decided to offer two gift packages of cheese at a special price. The Fancy Assortment will contain 30 ounces of Cheddar cheese, 10 ounces of Swiss cheese and 4 ounces of Brie. The Deluxe Assortment will be a package with 12 ounces of Cheddar, 8 ounces of Swiss and 8 ounces of Brie. In the past, these two assortments have been very popular and the company is certain that it can sell out its entire stock if they price the Fancy Assortment at $\$ 4.50$ a box and charge $\$ 4$ for the Deluxe Assortment. The company's storage rooms contain 6,000 ounces of Cheddar, 2,600 ounces of Swiss and 2,000 ounces of Brie.

The company uses the AMPL script shown below to decide how many packages of each type to produce in order to maximize revenue. Note that $x$ and $y$ are the number of Fancy Assortment and Deluxe packages, respectively.

Answer the following questions using the AMPL/CPLEX output on the next page. You should consider each question independently.

1. What is the maximum revenue that the company can receive from selling gift packages? How many of each type should they sell?
2. Suppose the price of the Deluxe Assortment is raised to $\$ 5$. What will the optimal solution be? What will the optimal revenue value be?
3. Should the company pay $\$ 0.30$ an ounce for an additional 100 ounces of Cheddar cheese? Explain your answer.
4. Should the company pay $\$ 0.30$ an ounce for an additional 100 ounces of Swiss cheese? Explain your answer.

Problem 4

Three refineries with daily capacities of 6,5 , and 8 million gallons, respectively, supply three distribution areas with daily demands of 4,8 , and 7 million gallons, respectively. Gasoline is transported to the three distribution areas through a network of pipelines. The transportation cost is 10 cents per 1000 gallons per pipeline mile. The table below gives the mileage between the refineries and distribution areas. Refinery 1 is not connected to distribution area 3.

Formulate and solve an appropriate minimum cost network flow problem (MCNFP) to determine a minimum cost plan for transporting the required amounts of gasoline to the distribution areas.

Problem 5

Formulate and solve a network flow model to determine the shortest path from city 1 to city 8.

Problem 6

Chicken feed is transported by trucks from three silos to four farms. Some of the silos cannot ship directly to some of the farms. The capacities of the other routes limited by the number of trucks available and the number of trips made daily. The following table shows the daily amounts of supply at the silos and demand at the farms (in thousands of pounds). The cell entries of the tale specify the daily capacities of the associated routes.

Formulate and solve an appropriate maximum flow or MCNFP to determine a trucking plan that meets as much demand as possible.

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Solution: The downloadable solution consists of 14 pages, 1287 words and 14 charts.
Deliverable: Word Document