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

Set 9.1d

1. Straightforward formulation: 7777 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

1 1

10

1

2

1

Maximize subject to

36, 1, 2

1, 1, ...,10, (0,1), 1, 2, ...10

i it

i t

i it

i

it it

t

z r x

L x t

x i x i

= =

=

=

=

≤ =

≤ = = =

∑∑

∑

∑

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

1 1

10

1

1 2

1 2

Maximize subject to

36, 1, 2

, (1 ), 1, 2,...,10

(above constraint is not as efficient as 1, 1, 2,...,10 in formulation 1)

(if 0 then ) translat

i it

i t

it

i

i i i i i i

i i

it it i

z r x

x t

x L y x L y i

x x i

x x L

= =

=

=

≤ =

≤ ≤ − =

+ ≤ =

> =

∑∑

∑

es to

(1 ), , , 1, 2,...,10, 1, 2

, , (0,1), 1, 2,...,10, 1, 2

it it i it it i it it

it it i

x M w L x Mw L x Mw i t

x w y i t

≤ − − ≤ − + ≤ = =

= = =

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