Rutgers Find the Optimal Objective Value and The Optimal Solution Point Questions

Homework 3 – Due on Nov.13 23:59
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Total score of this homework is 30 pt. – each of the sub-problems is for 1 pt.
Late homework will be penalized 20% per day for a maximum of one days.
All homework needs to be submitted through Canvas using a PDF file.
Show your work to get full marks.
1. Walsh’s Fruit Company contracts with growers in Ohio, Pennsylvania, and New York to
purchase grapes. The grapes are processed into juice at the farms and stored in refrigerated vats.
Then the juice is shipped to two plants, where it is processed into bottled grape juice and frozen
concentrate. The juice and concentrate are then transported to three food distribution centers. The
transportation costs per ton from the farms to the plants and from the plants to the distributors,
and the supply at the farms and demand at the distribution centers are summarized in the
following tables:
Plant
Farm
1. Ohio
2. Pennsylvania
3. New York
4. Indiana
$18
$21
$22
5. Georgia
$20
$19
$25
Supply (1k tons)
75
120
90
Plant
4. Indiana
5. Georgia
Demand (1k tons)
Distribution Center
6. Virginia
7. Kentucky
8. Louisiana
$23
$15
$29
$20
$17
$24
100
80
120
The manage wants to minimize the total transportation cost, assuming that no stock is allowed to
be stored in any plant.
(a) Find the optimal objective value and the optimal solution point by using the computer.
(b) Draw the optimal solution network based on the solution in (a) – specify the amount of
shipments for each route.
(c) At the optimal solution in (a), how many tons of grapes will be processed by the plant at
Georgia?
(d) What is the new minimal total cost if the transportation cost from New York to Georgia is
increased to $30 given the original problem?
2. 6Pts
3. Andy Mendoza makes handcrafted dolls, which he sells at craft fairs. He is considering massproducing the dolls to sell in stores. He estimates that the initial investment for plant and equipment
will be $15,000, while labor, materials, packaging, and shipping will be about $4 per doll. He has
determined that monthly sales volume is related to price, according to the following equation:
𝑣 = 2000 − 40𝑝
(a) Develop the nonlinear profit function as a function of price.
(b) Find the optimal price, the optimal volume per month, and the maximal profit per month.
(c) Graphically show the nonlinear curve of the profit function in (a) considering the optimal
price and the maximal profit obtained in (b).
(d) Explain the effect on the maximal profit in (b) when a new constraint 𝑝 ≤ 25 is added.
(e) Find the optimal price, the optimal volume, and the maximal profit per month with a new
relationship between volume and price as 𝑣 = 1000 − 20𝑝 +
2
10000
𝑝
.
4. 6Pts
5. 3Pts
6. 6 Pts
3
Self-practice (No need to hand in your answers)
1.
2. Oranges are grown, picked and then stored in warehouses in Tampa and Fresno. Theses
warehouses supply oranges to markets in New York, Chicago, and Boston. The following table
shows the shipping costs per truckload (in $100s), supply, and demand. Because of an agreement
between distributors, shipments are prohibited from Tampa to New York:
To
From
New York
Chicago
Boston
Supply
Tampa
12
16
14
350
Fresno
16
12
20
500
Demand
250
210
180
(a) Formulate a linear programming model for this problem.
(b) Find the optimal objective value and the optimal solution point in (a) by using the computer.
(c) What is the new optimal solution if the prohibition of the shipment from Tampa to New York
were removed from the original model formulation in (a)?
(d) Formulate a linear programming model for the original problem with the amount of orange
supply from Fresno as 200.
Find the optimal objective value and the optimal solution point in (d) by using the computer
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1.
2. (a) 𝑥𝑖𝑗 : # amount shipped from 𝑖 to 𝑗 for 𝑖 = 1,2 and 𝑗 = 𝐴, 𝐵, 𝐶
Min 𝑍 = ∑𝑖 ∑𝑗 𝑐𝑖𝑗 𝑥𝑖𝑗
S. t.
∑𝑗 𝑥𝑖𝑗 ≤ 𝑠𝑖 , for 𝑖 = 1,2
∑𝑖 𝑥𝑖𝑗 = 𝑑𝑗 , for 𝑗 = 𝐴, 𝐵, 𝐶
𝑥1𝐴 = 0
𝑥𝑖𝑗 ≥ 0 and integer ∀(𝑖, 𝑗)
where 𝑠𝑖 = 350, 500 and 𝑑𝑗 = 250, 210, 180
(b) 𝑍 ∗ = 9040 at
(c) 𝑍 ∗ = 8360 at
(d)
Min 𝑍 = ∑𝑖 ∑𝑗 𝑐𝑖𝑗 𝑥𝑖𝑗
S. t.
∑𝑗 𝑥𝑖𝑗 = 𝑠𝑖 , for 𝑖 = 1,2
∑𝑖 𝑥𝑖𝑗 ≤ 𝑑𝑗 , for 𝑗 = 𝐴, 𝐵, 𝐶
𝑥1𝐴 = 0
𝑥𝑖𝑗 ≥ 0 and 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 ∀(𝑖, 𝑗)
where 𝑠𝑖 = 350, 200 and 𝑑𝑗 = 250, 210, 180
(e) 𝑍 ∗ = 8280 at
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