Formulation of Problem (Extended)

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Example 03 (HA-01/Q-02)

There are two products A and B. Cost of production(per unit) of A and B are Rs. 60 and Rs. 80, respectively. The company has to supply at least 200 units of B. One unit of product A requires 1 machine hour whereas B has a machine hours available abundantly within the company. Total machine hour available for product A is 400 hours. One unit of each product requires 1 labour hour. Total labour hour available are 500. Formulate the LP problem to minimize production cost.

Solution:

The given data can be summarized in the following table:

Decision  Product     Hours on           Cost of

Variable           Machine  Labour  Production/unit  Required

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  X         A      1         1         Rs. 60/-          -
  Y         B      -         1         Rs. 80/-       200(Min)
-----------------------------------------------------------------
Hours available    400       500
     (per week)  (Maximum)   (Maximum)

                  (for A)

Let, X and Y be the number of products A and B are to be produced, respectively.

From the table we get,

Objective Function: minimize Z= 60X + 80Y

Subject to,

X     <= 400

X + Y <= 500

    Y >= 200

    X >= 0

Example 04 (HA-01/Q-01)

There are three products P, Q, R from three raw material A, B, C. One unit of P requires 2 units of A and 3 units of B. One unit of Q requires 2 units of B and 5 units of C. One unit of R requires 3 units of A, 2 units of B and 4 units of C. Profit per unit for P, Q, R are Rs. 3, Rs. 5 and Rs. 4, respectively. Availability of the raw material are 8 units of A, 10 units of B and 15 units of C. Formulate the problem mathematically to maximize the profit.

Solution:

The given data can be summarized in the following table:

Decision  Product Raw Material         Profit per

Variable          A    B    C          Production

-----------------------------------------------------------------
  W         P     2    3    0          Rs. 3/-          -
  X         Q     0    2    5          Rs. 5/- 

  Y         R     3    2    4          Rs. 4/-
-----------------------------------------------------------------
Availability      8    10   15

Let, W, X, Y be the number of products P, Q and R are to be produced, respectively.

From the table we get,

Objective Function: maximize Z= 3W + 5X + 4Y

Subject to,

2W + 0X + 3Y <= 8

3W + 2X + 2Y <= 10

0W + 5X + 4Y <= 15

W, X, Y >= 0

Example 05 (HA-01/Q-03)

Consider the following data

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                     Food1    Food2      Minimum Daily Need

                   (per gm)  (per gm)

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Cost(per gm)       Rs. 0.60  Rs. 1.00

Calcium               10        4               20

Protein               5         5               20

Calories              2         6               12

----------------------------------------------------------------

Formulate the LP problem which will satisfy the daily need in minimum cost.

Solution:

Rearranging the table we get the following summary table:

Decision  Food                        

Variable        Calcium Protein Calories  Cost per gm

-----------------------------------------------------------------
  X       Food1    10      5       2       Rs. 0.60

  Y       Food2     4      5       6       Rs. 1.00

-----------------------------------------------------------------
Daily Need(Min)    20     20      12

From the table we get,

Objective Function: minimize Z= 0.60X + Y

Subject to,

10X + 4Y >= 20

 5X + 5Y >= 20

 2X + 6Y >= 12

  X, Y   >= 0

Example 06 

A firm produces two types of gadgets A and B. Both are first processed in foundary and the goes to machine shop for finishing. The number of man hours required in each shop for the production of each unit of A and B are given in the table. Total amount of available man hours of the firm is also given.

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                     Foundary          Machine shop

---------------------------------------------------

Gadget A               10                   5

Gadget B                6                   4

Capacity(per week)   1000                 600

---------------------------------------------------

Profit on the sale of gadget A and B are Rs. 30 and Rs. 20 respectively.

Solution:

Analyzing all the given data, we can formulate the following LP problem on decision variable X and Y, referring to the number of production of gadget A and B respectively,

Objective Function: maximize Z= 30X + 20Y

Subject to,

10X + 6Y <= 1000

 5X + 4Y <= 600

    X, Y >= 0

Example 07

A firm has an advertising budget 7,20,000. They need to allocate the total budget to magazine & television media. Estimated exposure in each page of a magazine is 60,000 and in television it is 1,20,000. Cost of each page in magazine is 9,000 and cost of each spot in television is 12,000. The firm has to advertise in at least two pages in magazine and at least three spots in television. Formulate the problem to utilize the budget with best exposure.

Solution:

Analyzing all the given data, we can formulate the following LP problem on decision variable X and Y, referring to the number of advertising in magazine and television respectively,

Objective Function: maximize Z= 60,000X + 1,20,000Y

Subject to,

9000X + 12000Y <= 720000

             X >= 2

             Y >= 3

Try Your Own: Construct the summary table and explain the solution of example 06 & 07..