CAPITULO-9- Solucionario de Taha

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 r i = revenue from load i M aximize  z =

10

2

i =1

t =1

∑ ∑ ri x it 

subject to

10

∑  Li xit  ≤ 36, t  = 1, 2 i =1

2

∑  x it

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

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

∑ xit  ≤ 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 Trailer 1

Load 1 5 6 8 Total

2

2 4 9 Total

Solution 2 Feet 5 7 9 14 35 ft 11 15 10 36 ft 9-14

Load 1 2 6 9 Total 4 5 8 Total

Feet 5 11 9 10 35 ft 15 7 14 36 ft