Calculate the marginal cost and the marginal revenue

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Reference no: EM13891195 , Length:

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.

Reference no: EM13891195

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