Operations Research by H.a TAHA Solution Manual (8th Edition)

SOLUTION MANUAL

SOLUTION MANUAL

Set 9.1d

7

1. Straightforward formulation: Let xit = 1 if load i is assigned to trailer t, o otherwise Li = linear feet of load i ri = revenue from load i 10

2

M axim ize z = ∑ ∑ ri x it subject to i =1 t =1 10

∑ L x ≤ 36, t = 1, 2 i

it

i =1 2

∑ x ≤ 1, i = 1, ...,10, x = (0,1), i = 1, 2, ...10 it

it

t =1

2. Formulation using if-then: Let xit = feet in trailer t assigned to load i yi = (0, 1), i = 1, 2, …, 10, wit = (0, 1), i = 1, 2, ..., 10, t = 1, 2 10

2

Maximize z = ∑∑ ri xit subject to i =1 t =1 10

∑ x ≤ 36, t = 1, 2 it

i =1

xi1 ≤ Li yi , xi 2 ≤ Li (1 − yi ), i = 1, 2,...,10 (above constraint is not as efficient as xi1 + xi 2 ≤ 1, i = 1, 2,...,10 in formulation 1) (if xit > 0 then xit = Li ) translates to xit ≤ M (1 − wit ), Li − xit ≤ Mwit , − Li + xit ≤ Mwit , i = 1, 2,...,10, t = 1, 2 xit , wit , yi = (0,1), i = 1, 2,...,10, t = 1, 2 Solution: z = $7929. Problem has several alternative optima. (See file ampl9.1d-7.txt.)

Solution 1 Solution 2 Trailer Load Feet Load Feet 1 1 5 1 5 5 7 2 11 6 9 6 9 8 14 9 10 Total 35 ft Total 35 ft 2 2 11 4 15 4 15 5 7 9 10 8 14 Total 36 ft Total 36 ft

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