Operations Research by H.A TAHA Solution Manual (8th Edition) [PDF]

Citation preview

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



M axim ize z =



2



∑∑rx i



it



subject to



i =1 t =1 10



∑Lx



≤ 36, t = 1, 2



i



it



it



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



i =1 2



∑x 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



it



≤ 36, t = 1, 2



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



9-14