IE202 Fall 2014
Study Set 1
Question 1.)
Willy Wonka’s Candy Company Produces three types of Candy:
1.Wonka Bars
2.Bottle Caps
3.Giant Sweet Tarts
In order to produce the different type of candies, Willy can run three different production processes as described below. Each process involves blending different types of sugars in the
Magical Factories Mixer.
Process 1:
Running Process 1 for one hour:
Costs: $5
Requires: Two barrels of sugar type A and three barrels of sugar type B
Output: Two Wonka Bars and one packet of Bottle Caps
Process 2:
Running Process 2 for one hour:
Costs: $4
Requires: One barrel of sugar type A and three barrels of sugar type B
Output: Three packets of Bottle Caps
Process 3:
Running Process 1 for one hour:
Costs: $1
Requires: Two barrels of sugar type B and three packets of Bottle Caps
Output: Two Wonka Bars and one packet of Giant Sweet Tarts
Incidentally, the barrels are really tiny, and the candy is very sweet.
Each week we can purchase:

200 barrels of sugar type A at $2 Per Barrel

300 barrels of sugar type B at $3 Per Barrel
Assume that they can sell everything that they can produce.

Wonka Bars are sold at $9 per bar.

Bottle Caps are sold at $10 per packet.

Giant Sweet Tarts are sold at $24 per packet.
Assume that 100 hours of mixing time are available.
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a. Formulate an LP whose solution will maximize Willy Wonka’s Profits. Hint : Use decision variable x i
: number of hours process i runs, i=1,2,3 b. Assume that instead of having 200 barrels of sugar type A and 300 barrels of sugar type
B available that you can order a total of 500 Barrels. Show how to modify your LP formulation in Part A to account for this revised problem. c. Suppose that instead of selling the three candies separately, they can only be sold as part of a box consisting of one Wonka Bar, two packets of Bottle Caps, and one pack of Giant Sweet
Tarts. Each Wonka Box sells for $54. Modify your LP formulation in part A to model this new scenario. (You may need to add another decision variable)
Question 2.) All steel manufactured by Steelco must meet the following requirements: 3.2-
3.5% carbon; 1.8-2.5% silicon; 0.9-1.2% nickel; tensile strength of at least 45000 pounds per square inch (psi). Steelco manufactures steel by combining two alloys. The cost and properties of each alloy are given in the table below. Assume that the tensile strength of a mixture of the two alloys can be determined by averaging that of the alloys that are mixed together. For example, a one-ton mixture that is 40% alloy 1 and 60% alloy 2 has a tensile strength of 0.4(42000) + 0.6(50000). Use linear programming to determine how to minimize the cost of producing a ton of steel.
Cost per ton($)
Percent silicon
Percent nickel
Percent carbon
Tensile strength(psi)
Alloy1
190
2
1
3
42000
Alloy2
200
2.5
1.5
4
50000
Question 3.) A company wishes to plan its production of two products with seasonal demands over a 12 month period from the beginning of January. The monthly demand of
Product 1 is 100,000 kg during the months of October, November and December; 10,000 kg during the months of January, February, March and April; and 30,000 kg during the remaining months. The demand of Product 2 is 50,000 kg during the months of October through February and 15,000 kg during the remaining months. Suppose that cost of producing a kg of Product 1 and 2 is $5.00 and $8.00, respectively, provided that these were produced prior to June. After June, the costs are reduced to $4.50 / kg and $7.00 / kg because of the installation of an improved production system. The total amount of Products 1 and 2 that can be produced during any particular month cannot exceed 120,000 kg for Jan-Sept and 150,000 kg for Oct-Dec. Furthermore, at the end of each month products left at the hand are carried to the inventory space in order to be used in the coming months. Each kg of Product 1 occupies
2 cubic-feet and each kg of Product 2 occupies 4 cubic-feet of inventory. Suppose that the maximum inventory space allocated to these products is 150,000 cubic-feet and that the holding cost per cubic foot during any month is $0.10. Formulate the production scheduling problem so that the total production and inventory costs are minimized.
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Question 4.) A hospital wants to make weekly night shift ( 12pm-8am) schedule for its nurses. The demand for nurses for the night shift on day j is an integer dj, j = 1..7. Every nurse works 3 days in a row on the night shift . Formulate an integer linear program to find the minimal number of nurses the hospital needs to hire.
Question 5.) The Southern Confederation of Kibbutzim is a group of three kibbutzim
(farming communities). Overall planning for this group is done by an office and this office currently is planning agricultural production for the coming year. The agricultural output of each kibbutz is limited by both the amount of available irrigable land and water allocated for irrigation. These data are given in Table1. The crops suited for this region include sugar beets, cotton and sorghum. These crops differ primarily in their return per acre and their consumption of water. In addition to that there is a maximum quota for the total acreage that can be devoted to each of these crops (Table2). To ensure equity between the three kibbutzim, it has been agreed that every kibbutz will plant the same proportion of its available irrigable land. For example, if kibbutz will plant 200 of its available 400 acres, then kibbutz 2 must plant 300 of its 600 acres while kibbutz 3 plants 150 acres of its 300 acres. Formulate an LP whose solution will maximize the total net return to the Southern Confederation of Kibbutzim as a whole.
Crop
Kibbutz
1
2
3
Sugar Beets
Cotton
Sorghum
Maximum
Quota(Acres)
600
500
325
Usable Land(Acres)
400
600
300
Table 1
Water Allocation(Acre Feet)
Water Consumption(Acre
Feet/Acre)
3
2
1
Table 2
600
800
375
Net Return($/Acre)
1000
750
250
Question 6.) NASA has 4 astronauts to be employed at a space mission for 4 posts on a space ship. The posts require different qualifications (competencies) but each astronaut can perform each one of these tasks, albeit with varying degree of proficiency. The proficiency rating of astronaut i assigned to post j is measured as p ij
. It is asked to find the optimal assignment of astronauts to posts which gives the highest total proficiency rating while meeting the following requirements: i. Each astronaut should be assigned to a single post. ii. Each post should have a single astronaut.
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Question 7.) A paper manufacturer produces rolls of standard fixed width w and of standard length l . Customers order rolls of width w but varying lengths. In particular, d k
rolls with length l k
and width w are ordered for customer k=1,…,n (Assume l k
≤ l for all k=1,..,n ) What is the minimum number of rolls that should be cut to meet the demand?
Hint: Let M be a large number such that M number of rolls are enough to satisfy all demand
(e.g. M= d k
is such value) Assume the manufacturer has M number of unmanufactured rolls in the stock. Use decision variables: x i
= y ik
= number of rolls of type k (of length l k
) cut from roll i , i=1,…,M, k=1,…,n
Question 8.) For the LP given in the below:
max ax
1
+ bx
2
s.t.
4 x
1
+ x
2
≤ 12
x
1
- x
2
≤ 4
x
1
≥ 0 a) Find possible relationship(s) between a and b that gives an alternate optima for the LP.
Also indicate their signs. b) Give an example a and b which makes point (3.2, -0.8) unique optimal. c) Give an example a and b which makes point (0, 12) unique optimal.
Question 9.) Consider the following two linear programming problems:
LP1 LP2
min c x+c y
1 2 x

2

4
x
 y

3 x

0
max c x+c y
x x x
1


2 y
2



4
3
0 y

0 y

0 a) Find coefficients c
1
and c
2
such that both LP1 and LP2 will be unbounded. (Note that the same coefficients will be used in both models) b) Find coefficients c
1
and c
2
such that LP1 will have alternative optimal solutions and LP2 will be unbounded. (Note that the same coefficients will be used in both models) c) Find coefficients c
1
and c
2
such that LP1 will have a unique optimal solution and
LP2 will be unbounded. (Note that the same coefficients will be used in both models)
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