#### Description of this paper

##### Dennis L. Bricker Dept. of Industrial Engineering The University-(Answered)

Description

Question

Hi, I would like assistance with the problem on page 10 in the attached file (Homework #5 Problem 1). Thank you!

Dennis L. Bricker

Dept. of Industrial Engineering

The University of Iowa

56:171 Operations Research

Homework #1 - Due Wednesday, August 30, 2000

In each case below, you must formulate a linear programming model that will solve the problem.

Be sure to define the meaning of your variables! Then use LINDO (or other appropriate

software) to find the optimal solution. State the optimal objective value, and describe in

?layman?s terms? the optimal decisions.

1. Walnut Orchard has two farms that grow wheat and corn. Because of differing soil

conditions, there are differences in the yields and costs of growing crops on the two farms.

The yields and costs are shown in the table below. Farm #1 has 100 acres available for

cultivation, while Farm #2 has 150 acres. The farm has contracted to grow 11,000 bushels of

corn and 6000 bushels of wheat. Determine a planting plan that will minimize the cost of

meeting these contracts.

Farm #1

Farm #2

Corn yield/acre

100 bushels

120 bushels

Cost/acre of corn

\$90

\$115

Wheat yield/acre

40 bushels

35 bushels

Cost/acre of wheat

\$90

\$80

Note: We are assuming that the costs and yields are known with certainty, which is not the

case in the &quot;real world&quot;!

2. A firm manufactures chicken feed by mixing three different ingredients. Each ingredient

contains four key nutrients: protein, fat, vitamin A, and vitamin B. The amount of each

nutrient contained in 1 kilogram of the three basic ingredients is summarized in the table

below:

Ingredient Protein (grams) Fat (grams) Vitamin A (units) Vitamin B (units)

1

25

11

235

12

2

45

10

160

6

3

32

7

190

10

The costs per kg of Ingredients 1, 2, and 3 are \$0.55, \$0.42, and \$0.38, respectively. Each kg

of the feed must contain at least 35 grams of protein, a minimum of 8 grams (and a maximum

of 10 grams) of fat, at least 200 units of vitamin A and at least 10 units of vitamin B.

Formulate an LP model for finding the feed mix that has the minimum cost per kg.

--revised 8/28/00

3. ?Mama?s Kitchen? serves from 5:30 a.m. each morning until 1:30 p.m. in the afternoon.

Tables are set and cleared by busers working 4-hour shifts beginning on the hour from 5:00

a.m. through 10:00 a.m. Most are college students who hate to get up in the morning, so

Mama?s pays \$9 per hour for the 5:00, 6:00, and 7:00 a.m. shifts, and \$7.50 per hour for the

others. (That is, a person works a shift consisting of 4 consecutive hours, with the wages

equal to 4x\$9 for the three early shifts, and 4x\$7.50 for the 3 later shifts.) The manager

seeks a minimum cost staffing plan that will have at least the number of busers on duty each

hour as specified below:

5 am 6 am 7 am 8 am 9 am 10am 11am Noon 1 pm

#reqd

2

3

5

5

3

2

4

6

3

56:171 O.R. -- HW #1

08/28/00

page 1

56:171 Operations Research

Homework #2 -- Due Wednesday, Sept. 6

The Diet Problem. &quot;The goal of the diet problem is to find the cheapest combination of foods

that will satisfy all the daily nutritional requirements of a person.&quot; Go to the URL:

http://www-fp.mcs.anl.gov/otc/Guide/CaseStudies/diet/index.html

and click on &quot;Give it a try.&quot; Then on the next page select &quot;Edit the constraints&quot; and click on

&quot;Go on&quot; .

a. What are the restrictions on calories in the default set of requirements?

Go back to the previous page, where approximately 100 foods are listed for your selection. Choose &quot;Default

requirements&quot;, and select 15 foods which you think would provide an economical menu meeting the requirements.

Then click on &quot;Go on&quot; again.

b. What is the minimum-cost menu meeting the nutritional requirements using the foods you indicated?

Indicate the solution in the left 2 columns of the table below.

Change the default upper limit on calories to 1500/day and solve the problem again. (Be sure that the lower bound ?

upper bound!)

c. What is the minimum-cost menu meeting the nutritional requirements using the foods you indicated?

Indicate the solution in the right 2 columns of the table below.

Quantity

(# servings)

Total Cost:

56:171 O.R. HW#2

Cost

\$

Food

(&amp; serving size)

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

&lt;&gt;&lt;&gt;&lt;&gt;&lt;&gt;&lt;&gt;&lt;&gt;&lt;&gt;

Quantity

(# servings)

Cost

Total Cost:

\$

page 1 of 1

56:171 Operations Research

Homework #3 -- Due Wednesday, Sept. 13

1. Simplex Algorithm: Use the simplex algorithm to find the optimal solution to the following LP:

Maximize z = 4 x1 + x2

2 x1 + x2 ? 9

x ? 5

2

subject to

x1 ? x 2 ? 4

x1 ? 0 , x 2 ? 0

Show the initial tableau, each intermediate tableau, and the final tableau. Explain how you have decided on the

location of each pivot and how you have decided to stop at the final tableau.

2. Below are several simplex tableaus. Assume that the objective in each case is to be minimized. Classify each

tableau by writing to the right of the tableau a letter A through G, according to the descriptions below. Also answer

the question accompanying each classification, if any.

(A) Nonoptimal, nondegenerate tableau with bounded solution. Circle a pivot element which would

improve the objective.

(B) Nonoptimal, degenerate tableau with bounded solution. Circle an appropriate pivot element. Would

the objective improve with this pivot?

(C) Unique nondegenerate optimum.

(D) Optimal tableau, with alternate optimum. State the values of the basic variables. Circle a pivot

element which would lead to another optimal basic solution. Which variable will enter the basis, and

at what value?

(E) Objective unbounded (below). Specify a variable which, when going to infinity, will make the

objective arbitrarily low.

(F) Tableau with infeasible basic solution.

Warning: Some of these classifications might be used for more than one tableau, while others might not

be used at all!

(i)

-z

1

0

0

0

(ii) -z

1

0

0

0

(iii)-z

1

0

0

0

X1

-3

0

-6

4

X2

0

0

0

1

X3

1

-4

3

2

X4

1

0

-2

-5

X5

0

0

1

0

X6

0

1

0

0

X7

2

0

2

1

X8

3

0

3

1

RHS

X1

3

0

-4

-6

X2

0

0

1

0

X3

-1

-4

2

3

X4

3

0

-5

-2

X5

0

0

0

1

X6

0

1

0

0

X7

2

3

-2

-4

X8

-2

0

1

3

RHS

X1

3

0

4

-6

X2

0

0

1

0

X3

1

-4

2

3

X4

1

0

-5

-2

X5

0

0

0

1

X6

0

1

0

0

X7

3

3

2

-4

X8

5

0

1

3

RHS

56:171 O.R. HW#3

-45

9

5

8

-45

9

0

5

-45

3

7

15

______

________

________

page 1 of 2

(iv) -z

1

0

0

0

(v)

-z

1

0

0

0

(vi) -z

1

0

0

0

(vii)-z

1

0

0

0

(viii)-z

1

0

0

0

(ix) -z

1

0

0

0

X1

3

0

4

-6

X2

0

0

1

0

X3

1

-1

-4

3

X4

-3

0

-5

-2

X5

0

0

0

1

X6

0

1

0

0

X7

2

3

2

-4

X8

0

0

1

3

RHS

X1

3

0

4

-6

X2

0

0

1

0

X3

0

-4

2

3

X4

1

0

-5

-2

X5

0

0

0

1

X6

0

1

0

0

X7

0

3

2

-4

X8

12

0

1

3

RHS

X1

3

0

-6

4

X2

0

0

0

1

X3

1

-4

3

2

X4

3

0

-2

-5

X5

0

0

1

0

X6

0

1

0

0

X7

2

3

-4

2

X8

0

0

3

1

RHS

X1

3

4

-6

0

X2

0

1

0

0

X3

1

2

3

-4

X4

1

-5

2

0

X5

0

0

1

0

X6

0

0

0

1

X7

-2

2

-4

3

X8

0

1

3

0

X1

2

0

6

4

X2

0

0

0

1

X3

-1

-4

3

2

X4

3

0

-2

-5

X5

0

0

1

0

X6

0

1

0

0

X7

2

3

-4

2

X8

0

0

3

1

RHS

X1

3

0

4

-6

X2

0

0

1

0

X3

1

-4

2

3

X4

4

0

-5

-2

X5

0

0

0

1

X6

0

1

0

0

X7

-2

-3

2

-4

X8

2

0

1

3

RHS

-45

9

3

5

-45

9

8

5

-45

9

5

8

________

______

________

RHS

-45

5

0

9

-45

9

5

8

-45

3

-8

15

________

________

________

3. LP Model Formulation (from Operations Research, by W. Winston (3rd edition), page 191): Carco uses robots

to manufacture cars. The following demands for cars must be met (not necessarily on time, but all demands

must be met by end of quarter 4):

Quarter #

1

2

3

4

Demand

600

800

500

400

At the beginning of the first quarter, Carco has two robots. Robots can be purchased at the beginning of each

quarter, but a maximum of two per quarter can be purchased. Each robot can build up to 200 cars per quarter.

It costs \$5000 to purchase a robot. Each quarter, a robot incurs \$500 in maintenance costs (even if it is not

being used to build any cars). Robots can also be sold at the beginning of each quarter for \$3000. At the end of

each quarter, a holding cost of \$200 for each car in inventory is incurred. If any demand is backlogged, a cost

of \$300 per car is incurred for each quarter the customer must wait. At the end of quarter 4, Carco must have at

least two robots.

a. Formulate an LP to minimize the total cost incurred in meeting the next four quarters' demands for cars. Be

sure to define your variables (including units) clearly! (Ignore any integer restrictions.)

b. Use LINDO (or other LP solver) to find the optimal solution and describe it briefly in &quot;plain English&quot;. Are

integer numbers of robots bought &amp; sold?

56:171 O.R. HW#3

page 2 of 2

56:171 Operations Research

Homework #4 -- Due Wednesday, Sept. 20

1. LP Duality: Write the dual of the following LP:

Min 3 x1 + 2 x2 ? 4 x3

5 x1 ? 7 x2 + x3 ?12

x1 ? x2 + 2 x3 = 18

subject to 2 x1 ? x3 ? 6

x + 2 x ?10

3

2

x j ? 0, j=1,2,3

2. Consider the following primal LP problem:

Max x1 + 2 x2 ? 9 x3 + 8 x4 ? 36 x5

2 x2 ? x3 + x4 ? 3x5 ? 40

subject to x1 ? x2 + 2 x4 ? 2 x5 ? 10

x ? 0, j=1,2,3,4,5

j

a. Write the dual LP problem

b. Sketch the feasible region of the dual LP in 2 dimensions, and use it to find the optimal solution.

c. Using complementary slackness conditions,

? write equations which must be satisfied by the optimal primal solution x*

? which primal variables must be zero?

d. Using the information in (c.), determine the optimal solution x*.

3. Sensitivity Analysis (based on LP model Homework #3 from Operations Research, by W. Winston (3rd edition),

page 191): Carco uses robots to manufacture cars. The following demands for cars must be met (not

necessarily on time, but all demands must be met by end of quarter 4):

Quarter #

1

2

3

4

Demand

600

800

500

400

At the beginning of the first quarter, Carco has two robots. Robots can be purchased at the beginning of each

quarter, but a maximum of two per quarter can be purchased. Each robot can build up to 200 cars per quarter.

It costs \$5000 to purchase a robot. Each quarter, a robot incurs \$500 in maintenance costs (even if it is not

being used to build any cars). Robots can also be sold at the beginning of each quarter for \$3000. At the end of

each quarter, a holding cost of \$200 for each car in inventory is incurred. If any demand is backlogged, a cost

of \$300 per car is incurred for each quarter the customer must wait. At the end of quarter 4, Carco must have at

least two robots.

Decision Variables :

Rt : robots available during quarter t (after robots are bought or sold for the quarter)

Bt : robots bought during quarter t

St : robots sold during quarter t

It : cars in inventory at end of quarter t

Ct : cars produced during quarter t

Dt : backlogged demand for cars at end of quarter t

Using the LINDO output below, answer the following questions:

a. During the first quarter, a one-time offer of 20% discount on robots is offered. Will this change the

optimal solution shown below?

b. In the optimal solution, is any demand backlogged?

56:171 O.R. HW#4

Fall 2000

page 1 of 4

c. Suppose that the penalty for backlogging demand is \$250 per month instead of \$300. Will this change

the optimal solution? Note: this change applies to all quarters simultaneously!

d. If the demand in quarter #3 were to increase by 100 cars, what would be the change in the objective

function?

e. Suppose that we know in advance that demand for 10 cars must be backlogged in quarter #2. Using the

substitution rates found in the tableau, describe how this would change the optimal solution.

MIN

500 R1 + 500 R2 + 500 R3 + 500 R4 + 200 I1 + 200 I2 + 200 I3

+ 200 I4 + 5000 B1 + 5000 B2 + 5000 B3 + 5000 B4 - 3000 S1 - 3000 S2

- 3000 S3 - 3000 S4 + 300 D1 + 300 D2 + 300 D3 + 300 D4

SUBJECT TO

2)

R1 - B1 + S1 =

2

3) - R1 + R2 - B2 + S2 =

0

4) - R2 + R3 - B3 + S3 =

0

5) - R3 + R4 - B4 + S4 =

0

6)

I1 - D1 - C1 = - 600

7) - I1 + I2 + D1 - D2 - C2 = - 800

8) - I2 + I3 + D2 - D3 - C3 = - 500

9) - I3 + I4 + D3 - D4 - C4 = - 400

10)

R4 &gt;=

2

11) - 200 R1 + C1 &lt;=

0

12) - 200 R2 + C2 &lt;=

0

13) - 200 R3 + C3 &lt;=

0

14) - 200 R4 + C4 &lt;=

0

15)

D4 =

0

END

SLB

R4

2.00000

SUB

B1

2.00000

SUB

B2

2.00000

SUB

B3

2.00000

SUB

B4

2.00000

OBJECTIVE FUNCTION VALUE

1)

9750.000

VARIABLE

R1

R2

R3

R4

I1

I2

I3

I4

B1

B2

B3

B4

S1

S2

S3

S4

D1

D2

D3

D4

C1

C2

C3

C4

ROW

2)

3)

4)

5)

6)

7)

8)

9)

56:171 O.R. HW#4

VALUE

3.000000

4.000000

2.500000

2.000000

0.000000

0.000000

0.000000

0.000000

1.000000

1.000000

0.000000

0.000000

0.000000

0.000000

1.500000

0.500000

0.000000

0.000000

0.000000

0.000000

600.000000

800.000000

500.000000

400.000000

SLACK OR SURPLUS

0.000000

0.000000

0.000000

0.000000

0.000000

0.000000

0.000000

0.000000

REDUCED COST

0.000000

0.000000

0.000000

3500.000000

190.000000

210.000000

202.500000

200.000000

0.000000

0.000000

2000.000000

2000.000000

2000.000000

2000.000000

0.000000

0.000000

310.000000

290.000000

297.500000

300.000000

0.000000

0.000000

0.000000

0.000000

DUAL PRICES

5000.000000

5000.000000

3000.000000

3000.000000

2.500000

12.500000

2.500000

0.000000

Fall 2000

page 2 of 4

10)

11)

12)

13)

14)

15)

0.000000

0.000000

0.000000

0.000000

0.000000

0.000000

0.000000

2.500000

12.500000

2.500000

0.000000

0.000000

RANGES IN WHICH THE BASIS IS UNCHANGED:

VARIABLE

R1

R2

R3

R4

I1

I2

I3

I4

B1

B2

B3

B4

S1

S2

S3

S4

D1

D2

D3

D4

C1

C2

C3

C4

ROW

2

3

4

5

6

7

8

9

10

11

12

13

14

15

CURRENT

COEF

500.000000

500.000000

500.000000

500.000000

200.000000

200.000000

200.000000

200.000000

5000.000000

5000.000000

5000.000000

5000.000000

-3000.000000

-3000.000000

-3000.000000

-3000.000000

300.000000

300.000000

300.000000

300.000000

0.000000

0.000000

0.000000

0.000000

CURRENT

RHS

2.000000

0.000000

0.000000

0.000000

-600.000000

-800.000000

-500.000000

-400.000000

2.000000

0.000000

0.000000

0.000000

0.000000

0.000000

OBJ COEFFICIENT RANGES

ALLOWABLE

ALLOWABLE

INCREASE

DECREASE

62000.000000

500.000000

38000.000000

2500.000000

42000.000000

500.000000

INFINITY

3500.000000

INFINITY

190.000000

INFINITY

210.000000

INFINITY

202.500000

INFINITY

200.000000

62000.000000

500.000000

500.000000

2000.000000

INFINITY

2000.000000

INFINITY

2000.000000

INFINITY

2000.000000

INFINITY

2000.000000

500.000000

2000.000000

3500.000000

500.000000

INFINITY

310.000000

INFINITY

290.000000

INFINITY

297.500000

INFINITY

300.000000

310.000000

190.000000

190.000000

210.000000

210.000000

202.500000

0.000000

17.500000

RIGHTHAND SIDE RANGES

ALLOWABLE

INCREASE

1.000000

1.000000

INFINITY

INFINITY

200.000000

200.000000

100.000000

0.000000

0.000000

200.000000

200.000000

100.000000

0.000000

0.000000

ALLOWABLE

DECREASE

1.000000

1.000000

1.500000

0.500000

200.000000

200.000000

300.000000

0.000000

INFINITY

200.000000

200.000000

300.000000

0.000000

0.000000

THE TABLEAU

ROW (BASIS)

1 ART

2

R1

3

R2

4

S3

5

R3

6

B1

7

B2

8

S4

9 ART

10 SLK

10

11

C1

12

C2

13

C3

14

C4

56:171 O.R. HW#4

R1

0.000

1.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

R2

0.000

0.000

1.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

R3

0.000

0.000

0.000

0.000

1.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

Fall 2000

R4

3500.000

0.000

0.000

0.000

0.000

0.000

0.000

1.000

-200.000

-1.000

0.000

0.000

0.000

-200.000

I1

190.000

-0.005

0.005

0.005

0.000

-0.005

0.010

0.000

0.000

0.000

-1.000

1.000

0.000

0.000

I2

210.000

0.000

-0.005

-0.010

0.005

0.000

-0.005

0.005

0.000

0.000

0.000

-1.000

1.000

0.000

page 3 of 4

15 ART

0.000

0.000

0.000

0.000

0.000

0.000

B2

B3

B4

S1

0.000 2000.000 2000.000 2000.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

-1.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

-1.000

1.000

0.000

0.000

0.000

0.000

0.000

-1.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

ROW

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

I3

202.500

0.000

0.000

0.005

-0.005

0.000

0.000

-0.005

-1.000

0.000

0.000

0.000

-1.000

0.000

0.000

I4

200.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

1.000

0.000

0.000

0.000

0.000

0.000

0.000

B1

0.000

0.000

0.000

0.000

0.000

1.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

ROW

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

S2

2000.000

0.000

0.000

0.000

0.000

0.000

-1.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

S3

0.000

0.000

0.000

1.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

S4

0.000

0.000

0.000

0.000

0.000

0.000

0.000

1.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

D1

310.000

0.005

-0.005

-0.005

0.000

0.005

-0.010

0.000

0.000

0.000

1.000

-1.000

0.000

0.000

0.000

D2

290.000

0.000

0.005

0.010

-0.005

0.000

0.005

-0.005

0.000

0.000

0.000

1.000

-1.000

0.000

0.000

D3

297.500

0.000

0.000

-0.005

0.005

0.000

0.000

0.005

1.000

0.000

0.000

0.000

1.000

0.000

0.000

D4

300.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

-1.000

0.000

0.000

0.000

0.000

0.000

1.000

ROW

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

C1

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

1.000

0.000

0.000

0.000

0.000

C2

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

1.000

0.000

0.000

0.000

C3

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

1.000

0.000

0.000

C4

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

1.000

0.000

SLK

10

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

1.000

0.000

0.000

0.000

0.000

0.000

SLK

11

2.500

-0.005

0.000

0.000

0.000

-0.005

0.005

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

SLK

12

12.500

0.000

-0.005

-0.005

0.000

0.000

-0.005

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

ROW

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

SLK

13

2.500

0.000

0.000

0.005

-0.005

0.000

0.000

-0.005

0.000

0.000

0.000

0.000

0.000

0.000

0.000

56:171 O.R. HW#4

SLK

14

0.000 -9750.000

0.000

3.000

0.000

4.000

0.000

1.500

0.000

2.500

0.000

1.000

0.000

1.000

0.000

0.500

1.000

0.000

0.000

0.000

0.000

600.000

0.000

800.000

0.000

500.000

1.000

400.000

0.000

0.000

Fall 2000

page 4 of 4

56:171 Operations Research

Homework #5 -- Fall 2000

1. Linear Programming sensitivity. A paper recycling plant processes box board, tissue

paper, newsprint, and book paper into pulp that can be used to produce three grades of

recycled paper (grades 1, 2, and 3). The prices per ton and the pulp contents of the four

inputs are:

Input

Cost

Pulp

type

\$/ton

content

Box board

5

15%

Tissue paper

6

20%

Newsprint

8

30%

Book paper

10

40%

Two methods, de-inking and asphalt dispersion, can be used to process the four inputs into

pulp. It costs \$20 to de-ink a ton of any input. The process of de-inking removes 10% of

the input's pulp. It costs \$15 to apply asphalt dispersion to a ton of material. The asphalt

dispersion process removes 20% of the input's pulp. At most 3000 tons of input can be run

through the asphalt dispersion process or the de-inking process. Grade 1 paper can only be

produced with newsprint or book paper pulp; grade 2 paper, only with book paper, tissue

paper, or box board pulp; and grade 3 paper, only with newsprint, tissue paper, or box board

pulp. To meet its current demands, the company needs 500 tons of pulp for grade 1 paper,

500 tons of pulp for grade 2 paper, and 600 tons of pulp for grade 3 paper. The LP model

below was formulated to minimize the cost of meeting the demands for pulp.

Hint:: you may wish to define the variables

BOX = tons of purchased boxboard

TISS = tons of purchased tissue

NEWS = tons of purchased newsprint

BOOK = tons of purchased book paper

BOX1 = tons of boxboard sent through de-inking

TISS1 = tons of tissue sent through de-inking

NEWS1 = tons of newsprint sent through de-inking

BOOK1 = tons of book paper sent through de-inking

BOX2 = tons of boxboard sent through asphalt dispersion

TISS2 = tons of tissue sent through asphalt dispersion

NEWS2 = tons of newsprint sent through asphalt dispersion

BOOK2 = tons of book paper sent through asphalt dispersion

PBOX = tons of pulp recovered from boxboard

PTISS = tons of pulp recovered from tissue

PNEWS= tons of pulp recovered from newsprint

PBOOK = tons of pulp recovered from book paper

PBOX1 = tons of boxboard pulp used for grade 1 paper,

PBOX2 = tons of boxboard pulp used for grade 2 paper, etc.

...

PBOOK3 = tons of book paper pulp used for grade 3 paper.

The LP model using these variables is:

MIN

5 BOX +6 TISS +8 NEWS +10 BOOK +20 BOX1 +20 TISS1 +20 NEWS1

+20 BOOK1 +15 BOX2 +15 TISS2 +15 NEWS2 +15 BOOK2

SUBJECT TO

56:171 O.R. HW#5

Fall 2000

page 1 of 6

2)

3)

4)

5)

6)

7)

8)

9)

10)

11)

12)

13)

14)

15)

16)

17)

18)

-

-

BOX + BOX1 + BOX2 &lt;=

0

TISS + TISS1 + TISS2 &lt;=

0

NEWS + NEWS1 + NEWS2 &lt;=

0

BOOK + BOOK1 + BOOK2 &lt;=

0

0.135 BOX1 + 0.12 BOX2 - PBOX =

0

0.18 TISS1 + 0.16 TISS2 - PTISS =

0

0.27 NEWS1 + 0.24 NEWS2 - PNEWS =

0

0.36 BOOK1 + 0.32 BOOK2 - PBOOK =

0

PBOX + PBOX2 + PBOX3 &lt;=

0

PTISS + PTISS2 + PTISS3 &lt;=

0

PNEWS + PNEWS1 + PNEWS3 &lt;=

0

PBOOK + PBOOK1 + PBOOK2 &lt;=

0

PNEWS1 + PBOOK1 &gt;=

500

PBOX2 + PTISS2 + P

Paper#9209391 | Written in 27-Jul-2016

Price : \$22