Research on Zero-Wait Scheduling Problems in Multiproduct Processes with Due Dates - Intelligent Computing for Sustainable Energy and Environment - page 324

~

iM ~

( )

starting (completion) time of the last operation of product i .

iM

( )

σ

processing sequence under the minimum of makespan .
~ penalty of products between the due-data window.

σ , σ ,  σ

1 2 N

~

the delay between products i and j on the first unit.

ij

A product batch must be scheduled without any wait as soon as it finishes processing
on one unit under ZW policy. We used the procedure of calculating delays presented
by Reddi and Ramamurthy [8]. Let

~ define the delay between products i and j ,
when product j follows product i in the sequence. It is given by

ij

~

  

m

~

m − 1

~

  

 

= max 0 , −

(1)

ij ij ij

m = 2 , M

k = 2 k = 1

Objective Function
Using these donations, the fuzzy ZW scheduling model is formulated as following,

Minimize:

[ ]

~

N

(

~

) (

~

)


=

= h max 0 , e − + w max 0 , − t

(2)

i i iM i iM i

i 1

Subject to:

~ ~ ~

= + , ∀ j > 1

( )

i j i j − 1 i j

k k k

~ ~ ~ ~

≥ + + , i ∈ N , j ∈ M

( ) ( )

i j i − 1 j i − 1 i i j k

k k k k k

~ ~ ~

= +

i j i j i j

k k k

~ ≥ 0

ij

2.2 Solution Description

We used the Triangular Fuzzy Numbers (TFN) to represent the fuzzy processing time
of operations, which is determined by a triplet

~

( )

L M U

= O , O , O

.

A fuzzy set ~

of the universe X can be specified by a membership

()

μ x

function , which takes its value in the interval [0, 1]. It is completely

characterized by the set of ordered pairs:

~

~

{ } ( )

()

= x , μ x x ∈ X

(3)

~