Diet Problem with Application of WinQSB for LP

Source of the problem: page 68, W. L. Winston, "Operations Research, Application and Algorithms", 4th Edition, Thomson Learning, 2004, ISBN: 0-534-38058-1Software used: WinQSB by Yih-Long Chang
Prepared by: Dr. Parisay

Problemstatement:

My diet requires that all the food I eat come from one of the four "basic food groups" (chocolate cake, ice cream, soda, and cheesecake). At present, the following four foods are available for consumption: brownies, chocolate ice cream, cola, and pineapple cheesecake. Each brownie costs 50 ¢, each scoop of chocolate ice cream costs 20 ¢, each bottle of cola costs 30 ¢, and each piece of pineapple cheesecake costs 80 ¢. Each day, I must ingest at least 500 calories, 6 oz of chocolate, 10 oz of sugar, and 8 oz of fat. The nutritional content per unit of each food is shown in the following table. Formulate a linear programming model that can be used to satisfy my daily nutritional requirements at minimum cost.

Summary of the problem as a table:

O.F.: Minimize total cost

Type of food / Calories / Chocolate (ounces) / Sugar (Ounces) / Fat (Ounces / Variables
Brownie / 400 / 3 / 2 / 2 / BRO
Choclate ice cream (1 scoop) / 200 / 2 / 2 / 4 / ICE
Cola (1 bottle) / 150 / 0 / 4 / 1 / COL
Pineapple cheesecake (1 piece) / 500 / 0 / 4 / 5 / PIN
at least / 500 / 6 / 10 / 8

Problem formulation:

O.F.: Min Z =50*BRO +20*ICE+30*COL +80*PIN

S.T.:

CALORY: 400*BRO +200*ICE +150*COL +500*PIN >= 500

CHOCOL: 3*BRO +2*ICE +0*COL +0*PIN >= 6

SUGAR: 2*BRO +2*ICE + 4*COL +4*PIN >= 10

FAT: 2*BRO +4*ICE +1*COL +5*PIN >= 8

All variables are positive.

Input to WinQSB:

Output from WinQSB:

Summary of final solution:

Z = 90 cents

Basic variables: ICE = 3 COL = 1

Nonbasic variables: BRO = 0 PIN = 0

Constraints CHOCOL and SUGAR are binding.

Constraints CALORY and FAT are nonbinding.

Perform Sensitivity Analysis Using WinQSB:

Enter the problem and solve it. Select “results” from top menu, select “perform parametric analysis” from pull down menu, on left hand side box select radio button for O.F. or RHS, on right hand side box select one of variables listed or constraints listed, click OK, you will get a table as summary of sensitivity analysis

Select “results” from top menu, select “graphic parametric analysis”, you will get the graph related to the sensitivity table.

Sensitivity analysis:

WinQSB sensitivity table and graph for the cost of one scoop of chocolate “ICE” cream (objective function coefficient of a basic variable) is bleow:

WinQSB sensitivity table and graph for the minimum requirements of “CHOCOL”ate (right hand side of a binding constraint) is below.

WinQSB sensitivity graph for the minimum requirements of FAT (right hand side of a nonbinding constraint) is below.

WinQSB sensitivity graph for the cost of one unit of BROwnie (objective function coefficient of a non-basic variable):

Prepared: Oct 2005

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