Question

Pedro is a primary school teacher who raises pigs for additional income. He must decide how to feed his pigs and is considering combining two brands, A and B. Both brands are available from local suppliers. Pedro wants to feed his pigs at minimal cost, but ensuring that they eat the minimum calorie and vitamin requirements. The cost, calorie and vitamin content for the two brands are in the following table.

contenido (libra)	Marca A	Marca B
calorías	800	1000
vitaminas	140 UNIDADES	70 UNIDADES
costo(por libra)	$0.40	$0.80

Each pig requires, per day, at least 8000 calories and at least 700 units of vitamin. Also, since brand A has an ingredient that is toxic if ingested in large quantities, no more than a third of the total weight of daily intake can be from brand A.
a) (15 points) Clearly define the decision variables and formulate a PL model for this problem.
b) (10 points) Using some linear programming software, solve your model formulated in the subsection
previous. Clearly specify the optimal values of the decision variables as well as that of the function
objective. Include a screenshot of the software output.

Answer 1

Answer a:

Based on the given information in the question, we will formulate an LP model as mentioned below:

Decision Variables:

Let,

x1 = # Pounds of diet to be feed from the brand A

x2 = # Pounds of diet to be feed from the brand B

Objective Function:

As an objective is to minimize the total cost, the objective function would be:

Min Z = 0.40 x1 + 0.80 x2

Subject to Constraints:

C1 = 800 x1 + 1000 x2 ≥ 8000 (Min. requirement of calories)

C2 = 140 x1 + 70 x2 ≥ 700 (Min. requirement of vitamins)

C3 = x1 ≤ 3 (x1 + x2) (The quantity of brand A should be less than or equal to three times the total weight)

∴ C3 = x1 - 3 x1 - 3 x2 ≤ 0

∴ C3 = - 2 x1 - 3 x2 ≤ 0

Non-Negativity Condition:

x1, x2 ≥ 0

Answer b:

As no specific information is mentioned in the question, we will solve the given LP model by using MS-Excel Solver data tool as mentioned in the below steps:

Step 1: Prepare the following table in the excel

(Note: Make Sure to prepare this table in the exact Rows and Columns as mentioned in the below screenshot)

Step 2: Now, type the following formulas in the column of 'Total' as mentioned in the below table. Then, press enter.

Hence, we get the following table:

Step 3: Now, open Solver (Data > Analyze > Solver), and fill the exact data in Solver Pop-Up Window as shown in the below screenshot:

Step 4: Once, you press the solve button as mentioned in the above step, you will get the following table. Here, the highlighted values are the optimal values for the decision variables:

Thus, the optimal solution:

Brand A = 10 Pounds

Brand B = 0 Pounds

Min Cost = 4 $

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