Problemas De Programacion Lineal

IE 311 - Operations Research I
Spring 2005

Solved Exercises on LP Modeling
1. [Problem 3 on page 64 in the 3rd edition of the textbook] Leary chemical manufactures three chemicals: A, B, and C. These chemicals are produced via two production processes: 1 and 2. Running process 1 for an hour costs $4 and yields 3 units of A, 1 of B and 1 of C. Running process 2 for an hour costs $1 andproduces 1 unit of A and one unit of B. To meet customer demands, at least 10 units of A, 5 of B and 3 units of C must be produced daily. Graphically determine a daily production plan that minimizes the cost of meeting Leary Chemical’s daily demands. Answer: Three types of chemicals are produced: A, B, and C. There are two types of processes (machines) available for production.
A

Process1
ProcessB C

Process 2
Output (units of chemical) A B C 3 1 1 1 1 0 10 5 3

A B

Cost per hour 4 1 Daily Demand

1 2

A. Decision Variables:

X 1 = Number of hours of process 1 used X 2 = Number of hours of process 2 used

B.

Constraints: For chemical A: For chemical B: For chemical C: Non-negativity constraints:

3 X1 + X 2 ≥ 10 X1 + X 2 ≥ 5 X1 ≥ 3 X1, X 2 ≥ 0

(I) (II) (III)C.

Objective function:

Z = Cost of meeting daily demands for chemicals = Cost of operating process 1 and 2
Minimize

Z = 4 X1 + X 2

( I ) X
2

( I I I )

Feasible Region ( I I )
10

5

B A
3 5

Z
X
1

Point A B

X1
5 3

X2
0 2

Z
(4) (5) + 0 = 20 (4) (3) + 2 = 14 Minimum

Point B is the optimal point. Processes 1 and 2 should be used 3 hours and 2 hours,respectively.

2.

[Problem 6 on page 64 in the 3rd edition of the textbook] Farmer Jane owns 45 acres of land. She is going to plant each with wheat or corn. Each acre planted with wheat yields $200 profit; each with corn yields $300 profit. The labor and fertilizer used for each acre are given in the table below. One hundred workers and 120 tons of fertilizer are available. Use linearprogramming to determine how Jane maximizes profits from her land. Crop Labor Fertilizer Answer: A. Decision Variables: Wheat 3 worke rs 2 tons Corn 2 workers 4 tons

W = acres of land planted with wheat C = acres of land planted with corn

B. Constraints: 1) Total acres of land used must equal to the land she owns.

C + W = 45

2) Maximum number of workers to be used is 100.

3W + 2C ≤ 100

3)Maximum tons of fertilizer is 120 tons.

2W + 4C ≤ 120

3) Non-negativity

W ,C ≥ 0
C. Objective function: Maximize

Z = 300C + 200W

3.

[Problem 3 on page 73 in the 3rd edition of the textbook] Peg and Al Fundy have a limited food budget, so Peg is trying to feed the family as cheaply as possible. However, she still wants to make sure her fa mily members meet their dailynutritional requirements. Peg can buy two foods. Food 1 sells for $7 per pound and, each pound contains 3 units of vitamin A and 1 units of vitamin C. Food 2 sells for $1 per pound, and each pound contains 1 unit of each vitamin. Each day, the family needs at least 12 units of vitamin A and 6 units of vitamin C. (a) Verify that Peg should purchase 12 units of food 2 each day and thus oversatisfy the vitaminC requirement by 6 units. (b) Al has put his foot down and demanded that Peg fulfill the family’s daily nutritional requirement exactly by obtaining precisely 12 units of vitamin A and 6 units of vitamin C. The optimal solution to new problem will involve ingesting less vitamin C, but it will be more expensive. Why? Answer: (a) A. Decision Variables: X 1 = pounds of food 1 purchased

X 2 =pounds of food 2 purchased
B. Constraints: 1) Vitamin A requirement

3 X1 + X 2 ≥ 12
X1 + X 2 ≥ 6

2) Vitamin C requirement

3) Non-negativity

X 1, X 2 ≥ 0

C. Objective function: Minimize
(I) X2 (II)

Z = 7 X1 + X 2

12

C

Point
Feasible Region
6

( X1,

X2 )

Z
42 24 12

A B C

(6,0) (3,3) (0,12)

B A
4 6

Z

X1

Minimum is 12, so Peg should purchase...