**Example 11. ***A manufacturer produces two types of machines M1 and M2. Each M1 requires 4 hrs. of grinding and 2 hrs. of polishing whereas each M, model requires 2 hrs. of grinding and 4 hrs. of polishing. Manufacturer has 2 grinders and 3 polishers. Each grinder works for 40 hrs a week and each polisher works 40 hrs a week. Profit on an M1 model *is *$ 3 and on an M2 model*

*is $ 4. Whatever is produced *is *sold in the market. How should the manufacturer allocate his production capacity to two types of models so that he may make the maximum profit in a week.*

* *

*Formulate the LPP and solve graphically.*

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**Solution. **Let *x *1 = No. of M 1 machines, and

x2 = No. of M2 machines to be produced in a week.

The above data is summarized as follows :

Therefore the LPP can be formulated as follows :

Maximize profit= *3x1 *+ *4x2*

S/t, *4x *1 + *2x2 *≤80 (grinding)

*2x1 *+ *4x2 *≤120 (polishing)

*x1, x2 *≥0

The graphical region is shown below.

The optimal solution is x1 = 3' *X2 *= 3 ' and max. profit = $ 126.67.

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**Example 12. ***A firm can produce three types of woolen clothes, say, A, B and *C *using three kinds of wool, say red wool, green wool and blue wool. One unit of length of type A cloth needs 2 yards o} red wool and 3 yards of blue wool; one unit length of type B cloth needs 3 yards of red wool, 2 yards of green wool and 2 yards of blue wool; and one unit length of type C cloth needs 5 yards of green wool and 4 yards of blue wool. The firm has only a stock of 8 yards of red wool, 10 yards of green wool and 15 yards of blue wool. It is assumed that income obtained from one unit length of type A cloth is $ 3, of type B cloth is $ 5 and that of type C cloth is $ 4. Formulate the above problem as a LP problem.*

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**Solution. **The given data is summarized below :

Suppose that he produces *x1 , *x2 and x3 unit lengths of A, B and C clothes respectively. Then the LPP is :

Maximize income = 3x_{1} +5x2 +4x3

*S/t, 2x1 *+ 3x2 ≤ 8

2x2 + 5x3≤10

3x 1 + *2x2 *+ 5x3 ≤15

*x1, x2, *x3 ≥0

The solution is obtained as *x1* *= 1.67, *x2* *= 1.56, *x3* *= 1.38, *z* *= 18.29 (It 4, Simplex).

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**PROBLEMS**

1. A manufacturer of furniture makes only chair and tables. A chair requires two hours on m/c A and six hours on m/c B. A table requires five hours on m/c A and two hours on m/c B. I6 hours are available on m/c A and 22 hours, on m/c B per day. Profits for a chair and table be $ 1 and $ 5 respectively. Formulate the LPP of finding daily production of these items for maximum profit and solve graphically.

2. A tailor has 80 sq. m. of cotton material and 120 sq. m. of woolen material. A suit requires 1 sq. m. of cotton and 3 sq. m. of woolen material and a dress requires 2 sq. m . . of each. A suit sells for $ 500 and a dress for $ 400. Pose a LPP in terms of maximizing the income.

3. A company owns two mines : mine A produces 1 tonne of high grade ore, 3 tonne of medium grade ore and 5 tonnes of low grade ore each day; and mine B produces 2 tonnes of each of tfie three grades of ore each day. The company needs 80 tonnes of high grade ore, 160 tonnes of medium grade ore and 200 tonnes of low grade ore. If it costs Rs. 200 per day to work each mine, find the number of days each mine has to be operated for producing the required output with minimum total cost.

4. A company manufactures two products A and B. The profit per unit sale of A and B is $ 10 and $ 15 respectively. The company .can manufacture at most 40 units of A and 20 units of B in a month. The total sale must not be below $ 400 per month. If the market demand of the two items be 40 units in all, write the problem of finding the optimum number of items to be manufactured for maximum profit, as a problem of LP. Solve the problem graphically or otherwise.

5. A ·company is considering two types of buses-ordinary and semi deluxe for transportation. Ordinary bus can carry 40 passengers and requires 2 mechanics for servicing. Semi deluxe bus can carry 60 passengers and requires 3 mechanics for servicing. The company can transport at least 300 persons daily and not more than 12 mechanics can be employed. The cost of purchasing buses is to be minimized, given that the ordinary bus costs $ 1,20,000 and semi deluxe bus costs $ 1 ,50,000. Formulate this problem as a LPP.

6. A pharmaceutical company has 100 kg. of ingredient A, 180 kg. of ingredient B and 120 kg. of ingredient C available per month. They can use these ingredients to make three basic pharmaceutical products namely 5-l 0-5; 5-5-10 and 20-5-1 0; where the numbers in each case represent the percentage by weight of A, B and C respectively in each of the products. The cost of these ingredients are given below :

Selling price of these products are $ 40.5, $ 43 and $ 45 per kg. respectively. There is a capacity restriction of the company for the product 5-10-5, so they cannot produce more than 30 kg. per month. Determine how much of each of the products they should produce in order to maximize their monthly profit.

7. A fruit squash manufacturing company manufactures three types of squashes. The basic formula are :

5 litre lemonade
The squashes sell at Lemonade Grape fruit Orangeade |
2 oz. lemons, 2 kg. of sugar, 2 oz. citric acid and water.
11/2 kg of grape fruit, 1112 kg. of sugar, 11/2 oz. citric acid and water. 11/2 dozen oranges, 11/2 kg. of sugar, 1 oz. citric acid and water.
$ 37.50 per 5 litre; $ 40.00 per $ 42.50 per |

In the last week of the season they have in stock 2500 dozen lemons, 2000 kgs. grapefruit, 750 dozen oranges, 5000 kgs. of sugar and 3000 ozs. citric acid. What should be their manufacturing quantities in the week to maximize the turnover ? ·

8. A farmer is raising cows in his farm. He wishes to determine the qualities of the available types of feed that should be given to each cow to meet certain nutritional requirements at a minimum cost.

The numbers of each type . of basic nutritional ingredient contained within a kg. of each feed type is given in the following table, along with the daily nutritional requirements and feed costs.

Formulate a linear programming model for this problem SQ as to determine the optimal mix of feeds.