Answer to Question #193502 in Operations Research for Eleine

Question #193502

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)


1
Expert's answer
2021-05-17T08:15:39-0400

(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.


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