Give the objective function. Give all constraints, and be sure to write them in

Question

Give the objective function.

Give all constraints, and be sure to write them in standard form with only variables (with coefficients) on the left side of the equation or inequality and a constant (number) on the right side.

Be sure to define variables used.  

Note that we are NOT solving these problems using Excel.

A company makes two products, ketchup and tomato sauce. Both come in one size, 1 pint. They have committed to make 800 pints of ketchup and 600 pints of sauce. The products are made with two ingredients, premium tomato paste and cheap tomato paste. Tomato paste is made tomatoes and corn syrup. Table 4.1 give the mixture of the pastes and their cost.

Table 4.1

Tomato Paste tomatoes (%) corn syrup (%) Cost/Pint ($)

Premium 60 25 $2.00

Cheap 40 35 $1.20

Each pint of tomato sauce must be at most 32% of corn syrup and each pint of ketchup must contain at least 55% tomatoes. (The rest of the makeup of the products are made of ingredients of insignificant cost.)

With these demands and constraints, minimize cost.

Explanation / Answer

Linear Programming Problem – Formulation only.

Let x = quantity (pints) of premium tomato paste and

      y = quantity (pints) of cheap tomato paste.

Then, the total cost = z = 2x + 1.2y. ………………………(1) This is the objective function.

Constraints:

Premium Tomato Paste contains 60% tomatoes and 25% corn syrup and

Cheap Tomato Paste contains 40% tomatoes and 35% corn syrup

=>x of premium and y of cheap tomato pastes will have

Tomato: 0.6x + 0.4y ………………………………………………(2) and

Corn: 0.25x + 0.35y ……………………………………………….(3)

They have committed to make 800 pints of ketchup and 600 pints of sauce and each pint of tomato sauce must be at most 32% of corn syrup and each pint of ketchup must contain at least 55% tomatoes =>

Maximum corn syrup = 600 x 0.32 = 192 ……………………………….(4) and

Minimum tomato = 800 x 0.55 = 440 ……………………………………..(5)

Combining (2) and (5): 0.6x + 0.4y 440 ……………………………………(6)

Combining (3) and (4): 0.25x + 0.35y 192 ……………………………………(7)

(6) and (7) form the constraints.

Thus, the LPP is:

Minimize Z = 2x + 1.2y subject to

0.6x + 0.4y 440

0.25x + 0.35y 192 DONE.

[The above LPP can be comfortably solved by graphical method.]

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