question_answer

The objective of A diet problem is to ascertain the quantities of certain foods that should be eaten to meet certain nutritional requirement at minimum cost. The consideration is limited to milk, beaf' and eggs, and to vitamins A, B, C. The number of milligrams of each of these vitamins contained within A unit of each food is given below

vitamin	Litre of milk	Kg of beaf	Doze of eggs	Minimum daily requirements
A	1	1	10	1 mg
B	100	10	10	50 mg
C	10	100	10	10 mg
Cost	Rs. 1.00	Rs. 1.10	Rs. 0.50

What is the linear programming formulation for this problem?

Answer:

Let the daily diet consists of x litres of milk, y kgs of beaf and z dozens of eggs. Then, Total cost per day \[=\text{ }Rs.\text{ (}x+1.10y+0.50z\text{)}\] Let Z denotes the total cost in Rs. Then, \[Z=x+1.10y+0.50z\] Total amount of vitamin A in the daily diet is \[(x+y+10z)mg.\] But the minimum requirement is 1 mg of vitamin A. \[\therefore \]      \[x+y+10z\ge 1\] Similarly, total amounts of vitamins B and C in the daily diet are \[(100x+10y+10z)\] mg and \[(10x+100y+10z)\] mg respectively and their minimum requirements are of 50 mg and 10 mg respectively \[\therefore \]      \[100x+10y+10z\ge 50\] and       \[10x+100y+10z\ge 10\] Finally, the quantity of milk, kgs of beaf and dozens of eggs cannot assume negative values. \[\therefore \]      \[x\ge 0,\] \[y\ge 0,\] \[z\ge 0\] Hence, the mathematical formulation of the given LPP is Minimize \[Z=x+1.10y+0.50z\] Subject to constraints \[x+y+10z\ge 1\]             \[100x+10y+10z\ge 50\]             \[10x+100y+10z\ge 10\] and       \[x\ge 0,\] \[y\ge 0,\] \[z\ge 0\]