Linear Programming solution to question below

Question

A Pharmaceutical company has developed a new pill to be taken by smokers that will make them nauseous if they smoke. this new oill is a combinations of three ingredients that are costly and in linited supply. the supply availablities and costs are as follows:

Ingredient	Supply (kg)	Cost per kg
A	10	140
B	8	175
c	9	260

Production requirements stipulate that atleast 12kg of the pill must be produced.

Blending specifications for this new pill is as follows:

1. ingredient A must be at least 45 % of the total mixture but cannot exceed 60 % of the total mixture.

2. ingrdient B must comprise at least 10 % of the total mixture.

3. ingdient C must not be more than 50 % of the total mixture.

Formulate a linear programming model fro this company to determine the quantity of each ingredient to use with the objective of minimizing cost.

Answer 1

The first thing to note is that no minimum percentage has been given for the most expensive ingredient C, which implies that the minimum is 0%.

The minimum quantity of mixture is 12kg. The maximum quantity M is calculated from the limited supply of ingredient A, which must be at least 45% of the mixture, that is, 0.45M=10kg, and M=10/0.45=200/9kg. So we can write: 12≤A+B+C≤200/9 kg. In the graph below, X=A+B+C, the sum of the quantities of the ingredients.

When the mixture weighs 12kg, 5.4kg (45%) of it will be A. That leaves 6.6kg for the remaining ingredients: 1.2≤B≤6.6 and 5.4≥C≥0 because B+C=6.6kg. The cost of 12kg can be calculated: (A,B,C)={(5.4,1.2,5.4)...(5.4,6.6,0)} giving us a range for cost={ 2370...1911 }. Therefore, the range of costs for a 12kg mixture is [1911,2370]. The unit cost has a range of [159.12,197.5] per kg.

When the mixture weighs 200/9kg, the whole 10kg supply of A is used up. This time B+C=200/9-10=110/9kg. The picture below illustrates how the feasibility region (thick black line segment) is found.

Note that maxC (50% of the mixture, C=100/9kg), indicated by the broken blue vertical line, is actually greater than the 9kg supply for C, so it is superseded by the constraint.

The range for (A,B,C) involves the complete 8kg supply of B, leaving 38/9kg for C; and the complete 9kg supply of C, leaving 29/9kg for B: (A,B,C)={(10,8,38/9)...(10,29/9,9)} giving a cost range [3897.8,4303.9]. The unit cost range is [175.4,198.675].

So it’s clear that the 12kg mixture gives the minimum unit rate and production cost: 159.12/kg and 1911, made from 5.4kg of A and 6.6kg of B, no C.