Question

A home industry produces two kinds of products consisting of two types of products namely 'Pastries' and 'Bread Lovers'. Each product requires 3 components of different raw materials, namely flour, butter and eggs. The 'Pastries' product requires 3 flour, 2 butter and 2 eggs. While the product 'Bread Lovers' require 2 flour, 3 butter and 4 eggs. The available resources for the three raw materials for Pastries and Bread Lovers are 15 flour, 8 butter and 10 eggs, respectively. The cost required to make 'Pastries' is $12,000, and 'Bread Lovers' are $15,000. What is the number of each product that must be produced in order to minimize the operating costs of the home industry?

Answer 1

THE PARTICULAR PROBLEM CAN BE SOLVED USING LINEAR PROGRAMMING METHOD. THE PROBLEM IS STRUCTURED AS FOLLOWS:

GIVEN TWO GOODS PASTRIES AND BREAD

TWO GOODS ARE PRODUCED USING 3 RAW MATERIALS NAMELY FLOUR, BUTTER, EGG.

TAKING QUANTITY OF PASTRY AS X AND QUANTITY OF BREAD AS Y.

GIVEN COST OF PRODUCING PASTRY IS $12000 AND COST OF PRODUCING BREAD IS $15000

THUS THE COST FUNCTION IS : C = 12000X + 15000Y

NOW TO PRODUCE I UNIT PASTRY 3UNITS OF FLOUR IS NEEDED AND TO PRODUCE I UNIT BREAD 2UNITS OF FLOUR IS NEEDED. ALSO GIVEN AVAILABLE FLOUR IS 15UNITS. THUS WE CAN WRITE
3X+2Y 15.

TO PRODUCE I UNIT PASTRY 2UNITS OF BUTTER IS NEEDED AND TO PRODUCE I UNIT BREAD 3UNITS OF BUTTER IS NEEDED. ALSO GIVEN AVAILABLE BUTTER IS 8 UNITS. THUS WE CAN WRITE
2X+3Y 8.

TO PRODUCE I UNIT PASTRY 2UNITS OF EGG IS NEEDED AND TO PRODUCE I UNIT BREAD 4 UNITS OF EGG IS NEEDED. ALSO GIVEN AVAILABLE EGG IS 10 UNITS. THUS WE CAN WRITE
2X+4Y 10.

THUS THE PROBLEM IS TO, MINIMISE C = 12000X + 15000Y

   SUBJECT TO, 3X+2Y 15.
2X+3Y 8.
2X+4Y 10.

**REST CALCULATION PART IS ATTACHED IN THE IMAGE BELOW**

IMAGE :1

IMAGE :2

IMAGE :3

THUS COST MINIMISING QUANTITIES OF PASTRY IS I UNIT AND BREAD IS 2 UNITS.

MINIMUN COST = 12000*1 + 15000*2 = 12000+ 30000 = $42000