Tora Practice-29-3-17.pptx

OPERATIONS RESEARCH EM – 505 Lecture – 11: TORA Dr Muhammad Fahad

Associate Professor/Director Product Development Centre Dept of Industrial & Manufacturing NED University of Engineering & Technology

Slide 1-2

TORA Practice Linear Programming Exercise (Lecture 2 Problem) 

Reddy Mikks company owns a small paint factory that produces both interior and and exterior paints. Two basic raw materials A and B are used in the manufacturing of both paints. Maximum availability of A is 6 tons/day and that of B is 8 tons/day. The daily requirement of raw materials per ton of interior and exterior paints is given as: Exterior

Interior

Maximum Availability

A

1

2

6

B

2

1

8

Raw Material



Tons of Raw materials/Ton of paint

Market survey has established that the daily demand for interior paint can not exceed that of exterior paint by more than 1 ton and the maximum demand for the interior paint is limited to 2 tons daily. The price per ton for exterior paint is $3000 and for interior paint is $2000. How much the company should produce daily to maximize the income?

Slide 1-3

TORA Practice Linear Programming Exercise (Model) 

Objective Function



Constraints

Maximize 3Xe + 2Xi



Raw Material A

Xe + 2Xi ≤ 6



Raw Material b

2Xe + Xi ≤ 8



Demand for interior paint

Xi ≤ 2



Demand for interior paint

Xi – Xe ≤ 1

TORA Practice

Slide 1-4

Linear Programming Exercise (Lecture 2 Problem) 

PROTRAC Inc. produces two lines of heavy equipment, earth moving equipment for construction applications and forestry equipment for lumber industry. The largest member of each line (i.e. E9 and F9) are both produced in the same department and with the same equipment. The marketing dept. has forecasted that during the next month, the company will be able to sell as many E9s or F9s as the firm can produce. Management must now recommend a production target for the next moth (i.e. how many E9s and F9s should be produced).

Slide 1-5

TORA Practice Linear Programming Exercise (Data) 

PROTRAC Inc. will make a profit of $ 5000 on each E9 and $ 4000 on each F9.



Each product is put through production operations in two departments as follows Department







Hours E9

F9

Total Available

A

10

15

150

B

20

10

160

The total labor hours used in the next month’s “testing of finished products” cannot fall more than 10% below a set value of 150 hrs. This testing is done in a third dept. and it takes 30 hrs and 10 hrs for each E9 and F9 respectively.

In order to maintain the current market position, top management has decided that it is necessary to build at least one F9 for every three E9s produced. A major customer has ordered a total of at least five E9s and F9s (in any combination) for next month.

Slide 1-6

TORA Practice Linear Programming (Model) 



Let 

E = Number of E9s to be produced



F = Number of F9s to be produced

The objective function would be:

Maximize Profit Z = 5000 E + 4000 F 

Subject to the following constraints: (Dept. A)

10 E + 15 F ≤ 150

(Dept. B)

20 E + 10 F ≤ 160

(Testing)

135 ≤ 30 E + 10 F ≤ 150

(Marketing) E/3 ≤ F (Customer)

or

E+F≥5

(Non Negativity) E, F ≥ 0

E ≤ 3F

or

E – 3F ≤ 0

Slide 1-7

Modeling Model Building Exercise 



A balance diet requires that all the food taken by a normal healthy person must come from one of the four basic food groups (meals, snacks, drinks and fruits/sweets); and each day, at least 500 calories, 6 oz of fat, 10 oz of carbohydrates, and 8 oz of proteins must be ingested (taken in the form of food group items) by a normal healthy person. At present the following four items are available for consumption: sandwich, potato chips, orange juice, and a dessert. Each sandwich costs Rs.50, each pack of potato chips costs Rs.20, each orange juice costs Rs.30, and each dessert costs Rs.80. The nutritional content per unit of each food is shown in the following table: Item

Calories

Fats

Carbs.

Proteins

Sandwich

400

3

2

2

Chips

200

2

2

4

Juice

150

0

4

1

Dessert

500

0

4

5

Formulate a linear programming model that can be used to satisfy the daily nutritional requirements at minimum cost.

Transportation 

Solve the following transportation problem using: a. North West Corner Method b. Least Cost Method c.

Vogel’s Approximation Method

Slide 1-8

Slide 1-9

Transportation 

Solve the following transportation problem using: a. North West Corner Method b. Least Cost Method c.

Vogel’s Approximation Method

Source

Destinations

Supply

1

2

3

4

5

1

8

6

3

7

5

20

2

5

-

8

4

7

30

3

6

3

9

6

8

30

Demand 25

25

20

10

20

Slide 1-10

Assignment Model 

Karachi Machining Works has four jobs to be completed. Each machine must be assigned to complete one job. The time required to setup each machine for completing each job is shown. Time (Hours)



Job1

Job2

Job3

Job4

Machine 1

14

5

8

7

Machine 2

2

12

6

5

Machine 3

7

8

3

9

Machine 4

2

4

6

10

Minimize the total setup time needed to complete the four jobs.

Slide 1-11

Assignment Model 

JoShop needs to assign 4 jobs to 4 workers. The cost of performing a jobs is a function of the skills of the worker and the table below summarizes the cost of assignment. Worker 1 cannot do job 3 and worker 3 cannot do job 4. Determine the optimal assignment using the Hungarian method. Worker

Job 1

2

3

4

1

50

50

-

20

2

70

40

20

30

3

90

30

50

-

4

70

20

60

70

Slide 1-12

Assignment Model 

Suppose that an additional (fifth) worker becomes available for performing the four jobs at respective costs of 60, 45, 30 and 80. Is it economical to replace one of the current four workers with the new one?

Worker

Job 1

2

3

4

1

50

50

-

20

2

70

40

20

30

3

90

30

50

-

4

70

20

60

70

Slide 1-13

Assignment Model 

Suppose that the JoShop receives a fifth job and the respective costs of performing it by the current four workers are 20, 10, 20 and 80. Should the fifth job take priority over any of the four jobs the shop already has?

Worker

Job 1

2

3

4

1

50

50

-

20

2

70

40

20

30

3

90

30

50

-

4

70

20

60

70