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Question: We wish to select a cheapest diet of various fruits (apples, oranges, kiwis and lychees) subject …

by | Nov 30, 2023 | Posted Questions



We wish to select a cheapest diet of various fruits (apples,
oranges, kiwis and lychees) subject to various requirements
including minimum Vitamin C content, minimum fibre content, maximum
calories and maximum acidity. We are not concerned with integrality
in this question. The dietary numbers are made up.

vitamin C units fibre calories acidity cost per fruit

apple 80 10 100 0 $1.20

orange 120 15 80 1 $2.00

kiwi 40 10 30 1 $0.125

lychee 30 20 40 0 $1.00

We require at least 600 units of Vitamin C and at least 100
units of fibre. Our diet should have at most 800 calories and at
most 9 units of acidity (certain fruits count

to the total acidity). We formulate the problem as a Maximum
problem by taking the negative of the cost. Our LINDO input is:

max -1.2apple-2orange-0.125kiwi-lychee

st

vitaminc)80apple+120orange+40kiwi+30lychee>600

fibre)10apple+15orange+10kiwi+20lychee>100

calories)100apple+80orange+30kiwi+40lychee<800

acidity)orange+kiwi<9

end

Each of the following questions are independent of each other.
When giving costs, you can report the cost to the nearest cent.

a) How much does the diet cost? How much more would a new
optimal diet cost if the minimum requirement of Vitamin C is
increased by 100.

b) If I allowed up to 15 units of acidity, what would a new
optimal diet cost?

c) If I need to restrict my calorie intake (from fruits) to have
at most 600 calories, what would a new optimal diet cost?

d) If I raise the minimum Vitamin C requirement from 600 to 690
and lower the maximum acidity from 9 to 7, what would a new optimal
diet cost? Note that we are proposing two changes that are at most
half of the `allowable’ changes. Explain why the marginal values
are still useful.

e) Consider adding a new fruit to the diet. A single rambutan
costs $.50 and has 50 units of Vitamin C, has 10 units of bre, has
25 calories and has half the acidity of an orange or kiwi. Would a
new optimal diet would contain this fruit?

The LINDO output:

LP OPTIMUM FOUND AT STEP 0

OBJECTIVE FUNCTION VALUE

1) -4.725000

VARIABLE VALUE REDUCED COST

APPLE 3.000000 0.000000

ORANGE 0.000000 0.675000

KIWI 9.000000 0.000000

LYCHEE 0.000000 0.550000

ROW SLACK OR SURPLUS DUAL PRICES

VITAMINC) 0.000000 -0.015000

FIBRE) 20.000000 0.000000

CALORIES) 230.000000 0.000000

ACIDITY) 0.000000 0.475000

NO. ITERATIONS= 0

RANGES IN WHICH THE BASIS IS UNCHANGED:

OBJ COEFFICIENT RANGES

VARIABLE CURRENT ALLOWABLE ALLOWABLE

COEF INCREASE DECREASE

APPLE -1.200000 0.950000 0.675000

ORANGE -2.000000 0.675000 INFINITY

KIWI -0.125000 INFINITY 0.475000

LYCHEE -1.000000 0.550000 INFINITY

RIGHTHAND SIDE RANGES

ROW CURRENT ALLOWABLE ALLOWABLE

RHS INCREASE DECREASE

VITAMINC 600.000000 184.000000 160.000000

FIBRE 100.000000 20.000000 INFINITY

CALORIES 800.000000 INFINITY 230.000000

ACIDITY 9.000000 6.000000 4.000000

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