Question

John Smith operates a sandwich truck with which he visits construction projects and industrial areas. His sandwiches are very popular for their freshness and quality because John uses the best quality ingredients that he buys daily, very early in the morning, from his supplier. The ingredients he buys every day are:

Ingredients	Cost
24 pounds of turkey breast prepared in the oven	$170
30 pounds of honey-cured ham	135
24 pounds of roast beef	189
24 pounds of swiss cheese	81
300 French bread rolls	90
Assorted condiments	45

In addition to the cost of $710 of the foods listed above, John incurs other expenses that add up to $420 daily. These correspond to gasoline, payment of the truck, insurance, paper napkins, plastic wrap for sandwiches and the wages of an assistant. John slices the meats and cheese in one ounce portions, proceeds to prepare no more than 300 sandwiches, which is the maximum he can accommodate in his truck. He has no problem selling the sandwiches he prepares every day. He discards the ingredients that he does not use to guarantee the freshness of his sandwiches. The following table presents the data corresponding to the ingredients of each of the five sandwiches that he makes as well as their sale prices:

Sandwich	Turkey	Roast Beef	Ham	Cheese	Price
A	4 ounces	--	--	1 ounce	$3.90
B	--	4 ounces	--	1 ounce	$5.50
C	--	--	4 ounces	2 ounces	$4.50
D	2 ounces	2 ounces	2 ounces	2 ounces	$6.25
E	3 ounces	3 ounces	3 ounces	--	$6.75

Formulate a model to find the number of sandwiches of each type that John must prepare daily in order to maximize his profit. Do not forget to define with absolute clarity each of the variables you use in your model as well as its meaning. Be careful when formulating your objective function without ignoring anything. Just formulate the model, no need to solve it.

Answer 1

Linear programming model is formulated as below:

Decision variables: Let

A be the number of type A sandwiches to be prepared daily

B be the number of type B sandwiches to be prepared daily

C be the number of type C sandwiches to be prepared daily

D be the number of type D sandwiches to be prepared daily

E be the number of type E sandwiches to be prepared daily

Objective function: Maximize Z = 3.9A+5.5B+4.5C+6.25D+675E-(710+420) (this is the total daily profit)

Constraints:

4A+2D+3E <= 384 (24 pounds of turkey = 24*16 = 384 ounces)

4B+2D+3E <= 384 (24 pounds of roast bee = 24*16 = 384 ounces)

4C+2D+3E <= 480 (30 pounds of ham = 30*16 = 480 ounces)

1A+1B+2C+2D <= 384 (24 pounds of cheese = 24*16 = 384 ounces)

A+B+C+D+E <= 300   (maximum total number of sandwiches to be prepared daily)

A, B, C, D, E >= 0