Question

Alison is making a large fruit salad for a party. She has everything she needs at home, except for watermelons and grapefruit. She needs 12 watermelons, and 31 grapefruit. She goes to a nearby fruit stand, where she finds two vendors selling bags of mixed fruit. Vendor 1 is selling bags containing four watermelons, and five grapefruit, for $18 per bag. Vendor 2 is selling bags containing two watermelons and ten grapefruits for $13.50 per bag. Vendor 1 bags have a mass of 2 kg; Vendor 2 bags have a mass of 3.2 kg. Alison cannot carry more than 16 kg.

(a) Formulate a cost minimization integer model for this problem.

B) On the graph below, draw the constraints and show the (linear) feasible region. On this region show all the dots representing the feasible integer solutions. Determine the optimal values for the two variables and the OFV, and give the recommendation.

Answer 1

a)

Minimization integer model is as follows:

Let X be the number of bags to be purchased from vendor 1

Y be the number of bags to be purchased from vendor 2

Min 18X+13.5Y

s.t.

2X+3.2Y <= 16

4X+2Y >= 12

5X+10Y >= 31

X, Y >= 0

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b)

Constraints and objective function are plotted on the graph as shown below:

The corner points of the feasible region are highlighted on the graph.

Corner points and the value of objective function at each of the corner points is tabulated below:

X	Y	Objective value
0.727	4.545	74.4435
8	0	144
6.2	0	111.6
1.933	2.133	63.6

Minimum objective value is 63.6 at point (1.933, 2.133)

Therefore, optimal values of the two variables are:

X = 1.933

Y = 2.133

However, these values are not integers.

Rounding-up these values to integers, the resulting optimal values are:

X = 2

Y = 3

All the constraints are satisfied with these optimal values.

Minimum total cost = 18*2+13.5*3 = $ 76.5