R D Sharma Solutions for Chapter: Linear Programming, Exercise 1: EXERCISE

A manufacturer can produce two products, $A$ and $B$, during a given time period. Each of these products requires four different manufacturing operations: grinding, turning, assembling and testing. The manufacturing requirements in hours per unit of products $A$ and $B$ are given below.

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 . 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 $\mathrm{Rs} 50$ and $25$ 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 $\mathrm{LPP}$.

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, shop $A$ has $180$ man-days per week available while shop $B$ has $135$ man-days per week. If the manufacturer makes a profit of $\mathrm{Rs} 30000$ on each truck and $\mathrm{Rs} 2000$ on each automobile, how many of each should he produce to maximize his profit? Formulate this as a $\mathrm{LPP}$.

A firm manufactures two products, each of which must be processed through two departments, $1$ and $2$. The hourly requirements per unit for each product in each department, the weekly capacities in each department, selling price per unit, labour cost per unit, and raw material cost per unit are summarized as follows:

The problem is to determine the number of units to produce each product so as to maximize total contribution to profit. Formulate this as a $\mathrm{LPP}$.

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 also 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 $\mathrm{Rs} 100,000$ and each flight of a model $535$ plane costs $\mathrm{Rs} 150,000$. How many of each type of plane should be used to minimize the flight cost? Formulate this as a $\mathrm{LPP}$.

A farmer has a $100$ acre farm. He can sell the tomatoes, lettuce, or radishes he can raise. The price he can obtain is $\mathrm{Rs} 1$ per kilogram for tomatoes, $\mathrm{Rs} 0.75$ a head for lettuce and $\mathrm{Rs} 2$ per kilogram for radishes. The average yield per acre is $2000 \mathrm{kgs}$ for radishes, $3000$ heads of lettuce and $1000$ kilograms of radishes. Fertilizer is available at $\mathrm{Rs} 0.50$ per $\mathrm{kg}$ and the amount required per acre is $100 \mathrm{kgs}$ each for tomatoes and lettuce and $50$ kilograms for radishes. Labour required for sowing, cultivating and harvesting per acre is $5$ man-days for tomatoes and radishes and $6$ man-days for lettuce. A total of $400$ man-days of labour are available at $\mathrm{Rs} 20$ per man-day. Formulate this problem as a $\mathrm{LPP}$ to maximize the farmer's total profit.

A firm has to transport at least $1200$ packages daily using large vans which carry $200$ packages each and small vans which can take $80$ packages each. The cost of engaging each large van is $₹ 400$ and each small van is $₹ 200$. Not more than $₹ 3000$ is to be spent daily on the job and the number of large vans cannot exceed the number of small vans. Formulate this problem as a $\mathrm{LPP}$ given that the objective is to minimize cost

A firm manufactures two products, each of which must be processed through two departments, $1$ and $2$. The hourly requirements per unit for each product in each department, the weekly capacities in each department, selling price per unit, labour cost per unit, and raw material cost per unit are summarized as follows:

The problem is to determine the number of units to produce each product so as to maximize total contribution to profit. Formulate this as a $\mathrm{LPP}$.