Question

The HeadStart runs a day camp for 6- to 10- year olds during the summer. Its manager is trying to reduce the center's operating costs to avoid having to raise the tuition fee. The manager is currently planning what to feed the children for lunch. She would like to keep costs to a minimum, but also wants to make sure she is meeting the nutritional requirements of the children. She has already decided to go with peanut butter and jelly sandwiches, and some combination of apples, milk, and/or cranberry juice. The nutritional content of each food choice and its cost are given in the table below:

	Bread	Peanut Butter	Jelly		Milk	Juice
	(slice)	(tbsp)	(tbsp)	Apples	(cup)	(cup)
Unit Cost	$0.06	$0.05	$0.08	$0.35	$0.20	$0.40
			Nutritional Data
Calories from Fat	15	80	0	0	60	0
Calories	80	100	70	90	120	110
Vitamin C (mg)	0	0	4	6	2	80
Fiber (g)	4	0	3	10	0	1

The nutritional requirements are as follow. Each child should receive between 300 and 500 calories, but no more than 30% of these calories should come from fat. Each child should receive at least 60 milligrams (mg) of vitamin C and at least 10 grams (g) of fiber.

To ensure tasty sandwiches, the manager wants each child to have a minimum of 2 slices of bread, 1 tablespoon (tbsp) of peanut butter, and 1 tbsp of jelly, along with at least 1 cup of liquid (milk and/or cranberry juice).

The manager would like to select the food choices that would minimize cost while meeting all these requirements.

Formulate and solve a linear programming model for this problem on a spreadsheet.

a) How many ounces of each type of food are included in one serving of the lunch? (Round to three decimals)

	Bread	Peanut Butter	Jelly	Apples	Milk	Juice
	(slice)	(tbsp)	(tbsp)		(cup)	(cup)
Diet (ounces)

b) How many calories from fat are in one serving of the lunch? (Round to three decimals)

c) What is the total cost per serving of the lunch? (Round to two decimals) $

Answer 1

Let A be the number of Bread slice served, B be the quantity of Peanut butter in tablespoons, C be the quantity of Jelly in tablespoons, D be the number of Apples served, E be the number of cups of Milk served and F be the number of cups of Juice served at a time to each child.

Our objective is to minimize the cost of single serving. Thus our objective function is-

Min(0.06*A+0.05*B+0.08*C+0.35*D+0.20*E+0.40*F)

Our constraints are-

From calorie requirement-

80*A+100*B+70*C+90*D+120*E+110*F >= 300

80*A+100*B+70*C+90*D+120*E+110*F <= 500

From Calorie from Fat constraint-

15*A+80*B+60*E <= 0.30*(80*A+100*B+70*C+90*D+120*E+110*F)

From Vitamin C constraint

4*C+6*D+2*E+80F >= 60

From Fibre Constraint-

4*A+3*C+10*D+1*F >= 10

From individual item conditions-

A >= 2

B>= 1

C>= 1

E+F>=1

The Solver screenshots are as shown-

A) Quantity of each type of food-

Bread 2 slices, Peanut Butter 1 tablespoon, Jelly 1 tablespoon, no apples, 0.308 cup of milk and 0.692 cup of juice.

B) From the LHS of the Calories from Fat constraint in the solver equation, 128.462 calories are obtained from fats.

C) From the figure in the Objective cell of solver equation, 0.59 $ shall be spent on each serving.