Question

This time, our immune system is the best defense. With this, a Melagail wishes to mix two types of foods in such a way that vitamin contents of the mixture contain at least 8 units of vitamin A and 10 units of vitamin C. Food A contains 2 units/kg of vitamin A and 1 unit/kg of vitamin C. Food B contains 1 unit/kg of vitamin A and 2 units/kg of vitamin C. It costs ₱50 per kg to purchase Food A and ₱70 per kg to purchase Food B. Formulate this problem as a linear programming problem to minimize the cost of such a mixture.

A. Define the decision variables

B. Write the LP Model

C. Determine the feasible region

D. Determine the optimal solution

E. Final Answer: (interpret the result)

Answer 1

(A) Let the mixture contains x kg of food A and y kg of Food B.

B) For vitamin A-

      Food A contains 2 unit and food B contains 1 units and total unit is at least 8.

      $\Rightarrow 2x+y\ge 8$

  For Vitamin C-

        Food A contains 1 unit and food B contains 2 units and total unit is at least 10.

        $\Rightarrow x+2y\ge 10$

 Also cost of food A is 50 and cost of food B is 70 per kg

  Total cost is 50x+70y

     $\Rightarrow Z=50x+70y$

  LP model is-

    minimize $Z=50x+70y$

  Subject to-

   $2x+y\ge 8\\x+2y\ge10\\ \text{and } x,y\ge 0.$

(c)The feasible region is shown in figure below-

 (D) On solving the above equation 

    we get the corner points (0,8),(2,4) and (10,0) 

  At $(0,8)\Rightarrow Z=0+70(8)=560$

  At $(2,4)\Rightarrow Z=50(2)+70(4)=100+280=380(min)$

  At $(10,0)\Rightarrow Z=50(10)+0=500$

Optimal solution is $x=2,y=4$ and minimum $Z=380$

(E) Hence Minimum cost is 380. and mixture has 2 kg of food A and 4kg of food B.