Parthasarathi Mukhopadhyay and Manabendra Nath Mukherjee Solutions for Exercise 2: EXERCISE

A fertiliser $A$ consists of $10\%$ nitrogen and $6\%$ phosphoric acid. Another fertiliser $B$ consists of $5\%$ nitrogen and $10\%$ phosphoric acid. A farmer needs at least $14 \mathrm{kg}$ of nitrogen and $14 \mathrm{kg}$ of phosphoric acid for his crop. The costs per $\mathrm{kg}$ of $A$ and $B$ are respectively $₹ 6$ and $₹ 5$. Determine how much of each type of fertilisers should be used so that the nutrient requirements are met at a minimum cost. Find also the minimum cost.

Food $X$ contains $6$ units of vitamin $A$ per gram and $7$ units of vitamin $B$ per gram and costs $12$ paise per gram. Food $Y$ contains $8$ units of vitamin $A$ per gram and $12$ units of vitamin $B$ per gram and costs $20$ paise per gram. The daily minimum requirements of vitamin $A$ and $B$ are $100$ units and $120$ units respectively. Formulate the problem as a linear programming in which the cost of the product mixture is to be minimized.

Consider two different types of food stuff, say $F_1$ and $F_2$. Assume that these foodstuffs contain vitamins $v_1, v_2, v_3$. For a human body, minimum daily requirements of these vitamins are $1 \mathrm{mg}$. of $v_1, 50 \mathrm{mg}$ of $v_2$ and $10 \mathrm{mg}$ of $v_3$. Suppose that one unit of foodstuff $F_1$ contains $1 \mathrm{mg}$ of ${\mathrm{v}}_1, 100 \mathrm{mg}$ of $v_2$ and $10 \mathrm{mg}$ of $v_3$ whereas one unit of foodstuff $F_2$ contains $1 \mathrm{mg}$ of $v_1$, $10 \mathrm{mg}$ of $v_2$ and $100 \mathrm{mg}$ of $v_3$. Cost of one unit of $F_1$ is $\mathrm{Rs}. 1$ and that of $F_2$ is $\mathrm{Rs}. 1.5$. Formulate the problem as a linear programming model in which cost of diet that would supply the body at least minimum requirements of each vitamin is to be minimised. Then solve it graphically.

$K$ is a new cereal formed of a mixture of bran and rice that contains at least $88$ grams of protein and at least $36 \mathrm{mg}$ of iron. Knowing that bran contains $80 \mathrm{gm}$ of protein and $40 \mathrm{mg}$ of iron per $\mathrm{kg}$. and that rice contains $100 \mathrm{gm}$ of protein and $30 \mathrm{mg}$. of iron per $\mathrm{kg}$. Find the minimum cost of producing this new cereal if bran costs $\mathrm{Rs}. 5$ per $\mathrm{kg}$. and rice costs $\mathrm{Rs}. 4$ per $\mathrm{kg}$.

A diet which fulfill certain minimum daily requirements for three nutrients calcium, protein and calories, consists of two foods $X$ and $Y$ whose price and nutrient contents are shown below:

Find the combination of food so that cost is minimum.

A sand dealer has two depots $A$ and $B$ with stocks of $3000$ and $2000$ bags of sand respectively, each of same volume. He receives orders from three builders $P, Q$ and $R$ for $1500, 2000$ and $1500$ bags respectively. The costs of transportation of each lot of $100$ bags from $A$ to $P, Q$ and $R$ are $\mathrm{Rs}. 20, \mathrm{Rs}. 60$ and $\mathrm{Rs}. 40$ respectively. The cost of transportation of each lot of $100$ bags to $P, Q$ and $R$ from $B$ are $\mathrm{Rs}. 40, \mathrm{Rs}. 20$ and $\mathrm{Rs}. 30$ respectively. How should the dealer fulfill the orders so as to keep the cost of transportation minimum? Formulate the problem as an LPP and then solve it graphically.

A transport company has offices in five localities $A, B, C, D, E$. On some day the offices located at $A$ and $B$ had $8$ and $10$ spare trucks whereas offices $C, D, E$ required $6, 8, 4$ trucks respectively. The distance in kilometer between the five localities are given below:

How should the trucks from $A$ and $B$ be sent to $C, D$ and $E$ so that the total distance between covered by the trucks is minimum. Formulate the problem as a linear programming problem and solve it graphically.

A refrigerator manufacturing company has its stores at three places $A, B$ and $C$. From these stores, supply of refrigerators is made to three shops located at $P, Q$ and $R$. Company decides that refrigerators from $A$ will be sent only to the shops $P$ and $Q$ and those from $B$ to $Q$ and $R$ only. However, from the store $C$ refrigerators will be sent to each of the three shops. The monthly requirements of the shops $P, Q$ and $R$ are $17, 18$ and $8$ refrigerators, while the storage capacity of the stores $A, B$ and $C$ are $16, 12$ and $15$ refrigerators respectively. The costs of transportation of each refrigerator from the stores to the shops are given in the following table.

How many refrigerators should be sent to the shops from the stores so as to make the cost of transportation minimum? Formulate the problem mathematically and then solve graphically. Find also the minimum cost of transportation.