Problem
Thin and Trim Diet Centers is developing a new dietary drink. Three ingredients are used and the objective is to minimize cost.
Cost of ingredient 1 is $3.50 per unit
Cost of ingredient 2 is $6.60 per unit
Cost of ingredient 3 is $6.50 per unit
The model will tell Thin and Trim how much of each ingredient to use to meet minimum requirements for Vitamin A and iron, as well as the maximum limit on sodium
The decision variables for the problem are:
x1 = Units of Ingredient 1
x2 = Units of Ingredient 2
x3 = Units of Ingredient 3
MIN 3.5 X1 + 6 X2 + 6.5 X3
SUBJECT TO
2) 3 X1 + 8 X2 + 10 X3 >= 280 Vitamin A Minimum
3) 5 X1 + 6 X2 + 6 X3 >= 220 Iron Minimum
4) 10 X1 + 30 X2 + 40 X3 <= 1050 Sodium Maximum
END
LP OPTIMUM FOUND AT STEP 3
OBJECTIVE FUNCTION VALUE
1) 213.750000
VARIABLE VALUE REDUCED COST
X1 5.000000 .000000
X2 30.000000 .000000
X3 2.500000 .000000
ROW SLACK OR SURPLUS DUAL PRICES
2) .000000 -1.125001
3) .000000 -.375000
4) .000000 .175000
RANGES IN WHICH THE BASIS IS UNCHANGED:
OBJ COEFFICIENT RANGES
VARIABLE CURRENT ALLOWABLE ALLOWABLE
COEF INCREASE DECREASE
X1 3.500000 .583334 .750000
X2 6.000000 .750000 .218750
X3 6.500000 .318182 1.500000
RIGHTHAND SIDE RANGES
ROW CURRENT ALLOWABLE ALLOWABLE
RHS INCREASE DECREASE
2 280.000000 1.111111 8.571428
3 220.000000 60.000000 10.000000
4 1050.000000 37.500000 4.545455
a How many units of each ingredient should be used?
b How much will the mixture cost?
c Will an excess amount of either Vitamin A or iron be supplied?
d Will the maximum sodium level be reached?
e What would happen if the cost of ingredient 1 dropped to $3.00 (i.e., would the same solution be optimal)?
f What would happen to the value of the objective function if the Vitamin A requirement were 281?
g What would happen to the value of the objective function if the sodium restriction were 1051?
h What would happen to the solution if 250 units of iron were required?