1. Consider the following linear program.
Max 7x1 + 5x2
Subject to
3x1 + 4x2 ≤ 240
2x1 + x2 ≤ 100
x1 ≤ 45
x2 ≥ 10
x1, x2 ≥ 0
1.a) Write Dual problem.
1. b) The following is the solution for the primal problem.
(x1, x2, s1, s2, s3, s4) = (32, 36, 0, 0, 13, 26), the objective value, z = 404.
Based on this information, solve the dual problem and find the dual prices.
2. A company produces three products (x1, x2, x3) using three materials (A, B and C). The following shows the LP model and its solution. Show all your works how you get the solution for the following questions.
Max 7x1 + 5x2 + 3x3
Subject to
x1 + x2 + 2x3 ≤ 200 //material A
15x1 + 6x2 - 5x3 ≤ 500 //material B
x1 + 3x2 + 5x3 ≤ 300 //material C
x1, x2, x3 ≥ 0
Lindo Output)
LP OPTIMUM FOUND AT STEP 1
OBJECTIVE FUNCTION VALUE: 500.0000
|
VARIABLE
|
VALUE
|
REDUCED COST
|
|
X1
|
50.000000
|
0.000000
|
|
X2
|
0.000000
|
0.400000
|
|
X3
|
50.000000
|
0.000000
|
| ROW |
SLACK OR SURPLUS |
DUAL PRICES |
| 2) |
50 |
0 |
| 3) |
0 |
0.4 |
| 4) |
0 |
1 |
NO. ITERATIONS= 1
RANGES IN WHICH THE BASIS IS UNCHANGED:
|
VARIABLE
|
CURRENT
|
OBJ COEFFICIENT RANGES ALLOWABLE
|
ALLOWABLE
|
|
X1
|
COEF 7.000000
|
INCREASE INFINITY
|
DECREASE 0.711111
|
|
X2
|
5.000000
|
0.400000
|
INFINITY
|
|
X3
|
3.000000
|
32.000000
|
0.820513
|
|
ROW
|
CURRENT
|
RIGHTHAND SIDE RANGES ALLOWABLE
|
ALLOWABLE
|
|
2
|
RHS 200.000000
|
INCREASE INFINITY
|
DECREASE 50.000000
|
|
3
|
500.000000
|
1333.333252
|
800.000000
|
|
4
|
300.000000
|
114.285713
|
266.666656
|
2. a) Write Dual problem.
2. b) Calculate the marginal cost and the marginal revenue for the product x2. Hint) Use the coefficients of x2 in constraints and dual prices.
2. c) For the objective coefficient for x3, find ranges in which the basic variables are unchanged.
2. d) For the RHS value of material C, find ranges in which the basic variables are unchanged.
2. e) The material C is currently 300 units available. Some supplier wants to sell the material C with $2 / unit. What do you want to do? Why? (answer with the given output)
3. Truck Loading Problem
Mr. Steven Goodman owned Goodman Shipping Company in Orlando, FL. One of his trucks, with a weight capacity of 15,000 pounds and a volume capacity of 1,300 cubic feet, is about to be loaded. Awaiting shipment are the items shown in the following table. Each of these six items has an associated dollar value, available weight, and volume per pound that the item occupies.
|
Item
|
Value per Pound
|
Available Weight (Pounds)
|
Volume (Cu. Ft. per Pound)
|
|
1
|
$3.10
|
5000
|
0.125
|
|
2
|
$3.20
|
4500
|
0.064
|
|
3
|
$3.45
|
3000
|
0.144
|
|
4
|
$4.15
|
3500
|
0.448
|
|
5
|
$3.25
|
4000
|
0.048
|
|
6
|
$2.75
|
3500
|
0.018
|
The objective is to maximize the total value of the items loaded onto the truck without exceeding the truck's weight and volume capacities.
3. a) Formulate this problem as a linear program.
3. b) Solve this problem using LINDO or EXCEL Solver and explain your decision. Attach the output.
Hint: The decision variables are the number of pounds of each item that should be loaded on the truck. Define the decision variables first then write objective function and constraints.