Exercise 31.3
Question 1
A diet is to contain at least 400 units of carbohydrate, 500 units of fat, and 300 units of protein. Two foods are available : $F_{1}$ which costs $₹ 2$ per unit, and $F_{2}$, which costs $₹ 4$ per unit. A unit of food $F_{1}$ contains 10 units of carbohydrate, 20 units of fat, and 15 units of protein, a unit of food $F_{2}$ contains 25 units of carbohydrate, 10 units of fat, and 20 units of protein. Find the minimum cost for a diet that consists of a mixture of these two foods and also meets the minimum nutrition requirements. Formulate the problem as a linear programming problem.
Sol :
Question 2
To maintain his health a person must fulfil certain minimum daily requirements for several kinds of nutrients. Assuming that there are only three kinds of nutrients ,calcium, protein and calories and the person's diet consists of only two food items, I and II, whose price and nutrient contents are shown in the table :
| Food I (per lb) |
Food II (per lb) |
Minimum daily requirement for the nutrient | |
| Calcium | 10 | 4 | 20 |
| Protein | 5 | 5 | 20 |
| Calories | 2 | 6 | 13 |
| Price | 0.60 | 1.00 |
What combination of two food items will satisfy the daily requirement and entail the least cost ? Formulate this as a LPP.
Sol :
Question 3
Vitamins A and B are found in two different foods $F_{1}$ and $F_{2}$. One unit of food $F_{1}$ contains 2 units of vitamin $A$ and 3 units of vitamin $B$. One unit of food $F_{2}$ contains 4 units of vitamin $A$ and 2 units of vitamin $B$. One unit of food $F_{1}$ and $F_{2}$ cost ₹ 5 and $2.5$ respectively. The minimum daily requirements for a person of vitamin $A$ and $B$ is 40 and 50 units respectively. Assuming that any thing in excess of daily minimum requirement of vitamin $A$ and $B$ is not harmful, find out the optimum mixture of food $F_{1}$ and $F_{2}$ at the minimum cost which meets the daily minimum requirement of vitamin $A$ and $B$. Formulate this as a LPP.
Sol :
Question 4
A company is making two products A and B. The cost of producing one unit of products $A$ and B are ₹ 60 and $₹ 80$ respectively. As per the agreement, the company has to supply at least 200 units of product $B$ to its regular customers. One unit of product $A$ requires one machine hour whereas product $B$ has machine hours available abundantly within the company. Total machine hours available for product $A$ are 400 hours. One unit of each product $A$ and B requires one labour hour each and total of 500 labour hours are available. The company wants to minimize the cost of production by satisfying the given requirements. Formulate the problem as a LPP.
Sol :
Question 5
Two tailors A and B earn ₹ 150 and ₹ 200 per day respectively. A can stitch 6 shirts and 4 pants per day while $B$ can stitch 10 shirts and 4 pants per day. Form a linear programming problem to minimize the labour cost to produce at least 60 shirts and 32 pants.
Sol :
Question 6
A rubber company is engaged in producing three types of tyres A, B and C. Each type requires processing in two plants, Plant I and Plant II. The capacities of the two plants, in number of tyres per day, are as follows :
| Plant | A | B | C |
| I | 50 | 100 | 100 |
| II | 60 | 60 | 200 |
The monthly demand for tyre A, B and C is 2500,3000 and 7000 respectively. If plant $I$ costs $₹ 2500$ per day, and plant II costs $₹ 3500$ per day to operate, how many days should each be run per month to minimize cost while meeting the demand ? Formulate the problem as LPP.
Sol :
Question 7
An airline agrees to charter planes for a group. The group needs at least 160 first class seats and at least 300 tourist class seats. The airline must use at least two of its model 314 planes which have 20 first class and 30 tourist class seats. The airline will allow use some of its model 535 planes which have 20 first class seats and 60 tourist class seats. Each flight of a model 314 plane costs the company ₹ 1 lakh, and each fight of model 535 plane costs ₹ 1.5 lakh. How many of each type of plane should be used to minimize the flight cost ? Formulate this as a LPP.
Sol :
Question 8
A company sells two different products A and B. The two products are produced in a common production process and are sold in two different markets. The production process has a total capacity of 45000 man-hours. It takes 5 hours to produce a unit of $A$ and 3 hours to produce a unit of $B$. The market has been surveyed and company officials feel that the maximum number of units of $A$ that can be sold is 7000 and that of $B$ is 10,000 . If the profit is $₹ 60$ per unit for the product $A$ and $₹ 40$ per unit for the product $B$, how many units of each product should be sold to maximize profit? Formulate the problem as LPP.
Sol :
Question 9
An automobile manufacturer makes automobiles and trucks in a factory that is divided into two shops. Shop $A$, which performs the basic assembly operation, must work 5 man-days on each truck but only 2 man-days on each automobile. Shop $B$, which performs finishing operations, must work 3 man-days for each automobile or truck that it produces. Because of men and machine limitations, ship $A$ has 180 man days per week available while shop $B$ has 135 man-days per week. If the manufacturer makes a profit of $₹ 30000$ on each truck and $₹ 2000$ on each automobile, how many of each should be produce to maximize his profit ? Formulate this as a LPP.
Sol :
Question 10
A small manufacturing firm produces two types of gadgets $A$ and $B$, which are first processed in the foundry, then sent to the machine shop for finishing. The number of man-hours of labour required in each shop for the production of each unit of $A$ and $B$, and the number of man-hours the firm has available per week are as follows :
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The profit on the sale of $A$ is $₹ 30$ per unit as compared with $₹ 20$ per unit of $B$. The problem is to determine the weekly production of gadgets $A$ and $B$, so that the total profit is maximized. Formulate this problem as a LPP.
Sol :
Question 11
A toy company manufactures two types of doll; a basic version doll $A$ and a deluxe version doll $B$. Each doll of type $B$ takes twice as long to produce as one of type $A$, and the company would have time to make a maximum of 2000 per day if it produces only the basic version. The supply of plastic is sufficient to produce 1500 dolls per day (both $A$ and $B$ combined). The deluxe version requires a fancy dress of which there are only 600 per day available. If the company makes profit of $₹ 3$ and ₹ 5 per doll respectively on doll $A$ and doll $B$; how many of each should be produced per day in order to maximize profit ?
Sol :
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