A dietician wishes to mix together two kinds of food X and Y in such a way that the mixture contains at least 10 units of vitamin A, 12 units of vitamin B, and 8 units of vitamin C. The vitamin content of 1kg of food is given below:
Food Vitamin A Vitamin B Vitamin C X 1 2 3 Y 2 2 1
One kg of food X costs Rs16 and one kg of food Y costs Rs20. Find the least cost of the mixture that will produce the required diet.
A dietician wishes to mix together two kinds of food X and Y in such a way that the mixture contains at least 10 units of vitamin A, 12 units of vitamin B, and 8 units of vitamin C. The vitamin content of 1kg of food is given below:
| Food | Vitamin A | Vitamin B | Vitamin C |
| X | 1 | 2 | 3 |
| Y | 2 | 2 | 1 |
One kg of food X costs Rs16 and one kg of food Y costs Rs20. Find the least cost of the mixture that will produce the required diet.
Solution and Explanation
Let the mixture contain x kg of food X and y kg of food Y.
The mathematical formulation of the given problem is as follows.
Minimize Z=16x+20y...(1)
Subject to the constraints,
x+2y≥10....(2)
x+y≥6....(3)
3x+y≥8.....(4)
x,y≥0....(5)
The feasible region determined by the system of constraints is as follows.
| Corner point | Z=16x+20y | |
| A(10,0) | 160 | |
| B(2,4) | 112 | \(\rightarrow\)Minimum |
| C(1,5) | 116 | |
| D(0,8) | 160 |
As the feasible region is unbounded, therefore,112 may or may not be the minimum value of Z.
For this, we draw a graph for the inequality,16x+20y<112 or 4x+5y<28, and check whether the resulting half plane has points in common with the feasible region or not. It can be seen that the feasible region has no common point with 4x+5y<28
Therefore, the minimum value of Z is 112 at(2,4).
Thus, the mixture should contain 2kg of food X and 4kg of food Y. The minimum cost of the food is Rs112.
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Notes on Linear Programming Problem
Concepts Used:
Linear Programming Problems
The Linear Programming Problems (LPP) is a problem that is concerned with finding the optimal value of the given linear function. The optimal value can be either maximum value or minimum value. Here, the given linear function is considered an objective function. The objective function can contain several variables, which are subjected to the conditions and it has to satisfy the set of linear inequalities called linear constraints.
Linear Programming Simplex Method
Step 1: Establish a given problem. (i.e.,) write the inequality constraints and objective function.
Step 2: Convert the given inequalities to equations by adding the slack variable to each inequality expression.
Step 3: Create the initial simplex tableau. Write the objective function at the bottom row. Here, each inequality constraint appears in its own row. Now, we can represent the problem in the form of an augmented matrix, which is called the initial simplex tableau.
Step 4: Identify the greatest negative entry in the bottom row, which helps to identify the pivot column. The greatest negative entry in the bottom row defines the largest coefficient in the objective function, which will help us to increase the value of the objective function as fastest as possible.
Step 5: Compute the quotients. To calculate the quotient, we need to divide the entries in the far right column by the entries in the first column, excluding the bottom row. The smallest quotient identifies the row. The row identified in this step and the element identified in the step will be taken as the pivot element.
Step 6: Carry out pivoting to make all other entries in column is zero.
Step 7: If there are no negative entries in the bottom row, end the process. Otherwise, start from step 4.
Step 8: Finally, determine the solution associated with the final simplex tableau.