solve the following LP problem graphically:

Question

a housewife wishes to feed her children a breakfast menu which contains a specified level of nourishment.After consulting her dietician, she decides that her children should get at least one milligram of thiamine, 5 mgs of niacin and 400 calories a day at breakfast, The children have a choice of eating the latest cereals-Noisies, the old standby Crispies, ot as children often prefer, a mixture if the two. The small print on the side of each cereal boxcontains, apart from knitting patterns and assorted recipes, the fact that one spoonful of Noisies contains 0.1 mg of thiamine, 1 mg of niacin and 110 calories, while one spoonful of Crispies contains 0.25 mg of thiamine, 0.25 mg of niacin and 120 calories.

a.formulate the nutrient constraints.

b.If Noisies cost 3.6 cents per spoonful and Crispies 4.2 cents per spoonful determine the cheapest mixture of Noisies and Crispies which satified the nutrient constraints.

Answer 1

Let N be the number of spoonfuls of Noisies and C the number of spoonfuls of Crispies. The nutrients are represented by t=thiamine, n=niacin and calorific content by c. So in N spoonfuls we have the quantities of the nutrients: 0.1Nt, Nn, 110Nc; in C spoonfuls we have 0.25Ct, 0.25Cn, 120Cc. We need at least 5n, 400c, and 1t in the combined spoonfuls.

a. 5n=Nn+0.25Cn; so N+0.25C>5; similarly, 110N+120C>400 and 0.1N+0.25C>1.

N>5-0.25C. Or, 0.1N>1-0.25C; N>10-2.5C. Or, N>(400-120C)/110; N>(40-12C)/11.

These requirements can be displayed graphically, using N as a vertical axis and C as a horizontal axis. The inequalities appear as regions above the lines: N=5-0.25C; N=10-2.5C; N=40/11-12C/11. Label the lines t, n and c as appropriate. The intersection of n and t lies above c, so the open region above n and t bounded by the intersection of n and t meets all the minimum nutritional requirements, therefore the intersection point gives us the minimum values of N and C, allowing us to calculate the minimum cost. The equations we need are N=5-0.25C and N=10-2.5C. So, 5-0.25C=10-2.5C, 2.25C=5, C=20/9, from which N=40/9 or 2C. The intersection point is (20/9,40/9). If we work in whole spoonfuls, we can make C=3, and N>5-0.75; N>4.25, so N=5.

b. Cost=3.6N+4.2C cents, so putting in C=3 and N=5 we get cost=5*3.6+3*4.2=30.6 cents or approx 31 cents.

The nutritional content is therefore 1.25mg thiamine, 5.75mg niacin and 910 calories.

[At C=20/9 (2.22) and N=40/9 (4.44) spoonfuls, the cost comes to 25.3 cents and the nutritional content would be 1mg thiamine, 5mg niacin and 755.6 calories. At C=2.5 and N=4.5 spoonfuls, the cost is 26.7 cents and the nutritional content 1.075mg thiamine, 5.125mg niacin and 795 calories.]