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In this paper, we consider the no-wait two-machine scheduling problem with convex resource allocation and learning effect under the condition of common due date assignment. We take the total earliness, tardiness and common due date cost as the objective function, and find the optimal common due date, the resource allocation and the schedule of jobs to make the objective function minimum under the constraint condition that the total resource is limited. The corresponding algorithm is given and proved that the problem can be solved in polynomial time.

In the scheduling problem of industrial production, the processing of the job is complicated. The processing time of the job in the classic scheduling problem is a constant, but in the actual production, the actual processing time of the job is often related to the normal processing time of the job, allocation of resources, learning effects, deteriorating effects of the machine and other factors. In view of the complexity of the current scheduling problem, a large number of scholars have given their own research. In 1980, Vickson [

In this paper, we consider the two-machine no-wait flow shop scheduling problem with learning effects and convex resource allocation. Under limited resource availability, some results are given.

There are n independent jobs { J 1 , J 2 , ... , J n } to be processed on a two-machine flow shop setting. Each job is required to be processed on machine M 1 and then on machine M 2 , and between the two machines, the jobs are not allowed to wait, the operation O j i ( j = 1 , 2 , ... , n ; i = 1 , 2 ). The processing time of each job is give as follows:

P j i = ( p ¯ j i r a u j i ) m (1)

where P j i is the actual processing time of operation O j i , p ¯ j i is the normal processing time of operation O j i , u j i represents the amount of a non-renewable resource allocated to operation O j i , r is the actual scheduling position of the job J j , a is the learning index( a ≤ 0 ), m is a positive constant.

In this model, d is the common due date, C j is the completion time of job J j , E j = max { 0 , d − C j } is the earliness of job J j , T j = max { 0 , C j − d } is the tardiness of job J j . The purpose is to determine the optimal schedule π = ( J [ 1 ] , J [ 2 ] , J [ 3 ] , ... , J [ n ] ) , the optimal resource allocation strategy u [ j ] 1 * , u [ j ] 2 * , and

the common due date d of jobs under the condition ∑ i = 1 2 ∑ j = 1 n u j i ≤ G so that the following objective function is minimized:

f ( π , u , d ) = ∑ j = 1 n ( α E j + β T j + γ d ) (2)

where weights α , β , γ are given constants ( α ≥ 0 , β ≥ 0 , γ ≥ 0 ). In what follows, the problem studied will be denoted by using the extended three-field notation scheme (Graham et al. [

Lemma 1. (Gao et al. [

Lemma 2. (Gao et al. [

F 2 | n o − w a i t , P j i = ( p ¯ j i r a u j i ) m , ∑ i = 1 2 ∑ j = 1 n u j i ≤ G | ∑ j = 1 n ( α E j + β T j + γ d )

problem, an optimal sequence exists such that d = C [ k ] , where

k = min { max { ⌈ ( β − γ ) n α + β ⌉ , 0 } , n } (3)

As in Gao et al. [

f ( π , u , d ) = ∑ j = 1 n ( α E j + β T j + γ d ) = α ∑ j = 1 k-1 ( d − C [ j ] ) + β ∑ j = k + 1 n ( C [ j ] − d ) + γ n d = α ( ∑ j = 2 k ( j − 1 ) p [ j ] 1 + ( k − 1 ) p [ k ] 2 − ∑ j = 1 k − 1 p [ j ] 2 )

+ β ( ∑ j = k + 1 n ( n − j + 1 ) p [ j ] 1 − ( n − k ) p [ k ] 2 + ∑ j = k + 1 n p [ j ] 2 ) + γ ( n ∑ j = 1 k p [ j ] 1 + n p [ k ] 2 ) = α ∑ j = 2 k ( j − 1 ) p [ j ] 1 + β ∑ j = k + 1 n ( n − j + 1 ) p [ j ] 1 + γ n ∑ j = 1 k p [ j ] 1 + ( α ( k − 1 ) p [ k ] 2 − β ( n − k ) p [ k ] 2 + γ n p [ k ] 2 ) + ( β ∑ j = k + 1 n p [ j ] 2 − α ∑ j = 1 k − 1 p [ j ] 2 ) = ∑ j = 1 n ω j ( p ¯ [ j ] 1 j a u [ j ] 1 ) m + ∑ j = 1 n υ j ( p ¯ [ j ] 2 j a u [ j ] 2 ) m (4)

where

ω j = { n γ , j = 1 α ( j − 1 ) + γ n , j = 2 , 3 , ... , k β ( n − j + 1 ) , j = k + 1 , k + 2 , ... , n (5)

and

υ j = { − α , j = 1 , 2 , ... , k − 1 α ( k − 1 ) − β ( n − k ) + γ n , j = k β , j = k + 1 , k + 2 , ... , n (6)

Lemma 3. For a given sequence π = ( J [ 1 ] , J [ 2 ] , J [ 3 ] , ... , J [ n ] ) , the optimal resource allocation u * ( π ) for the problem

F 2 | n o − w a i t , p j i = ( p ¯ j i r a u j i ) m , ∑ i = 1 2 ∑ j = 1 n u j i ≤ G | ∑ j = 1 n ( α E j + β T j + γ d )

is:

u [ j ] 1 * = G ( ω j ) 1 m + 1 ( p ¯ [ j ] 1 j a ) m m + 1 ∑ j = 1 n ( ω j ) 1 m + 1 ( p ¯ [ j ] 1 j a ) m m + 1 + ∑ j = 1 n ( υ j ) 1 m + 1 ( p ¯ [ j ] 2 j a ) m m + 1 ， j = 1 , 2 , ... , n (7)

u [ j ] 2 * = G ( υ j ) 1 m + 1 ( p ¯ [ j ] 2 j a ) m m + 1 ∑ j = 1 n ( ω j ) 1 m + 1 ( p ¯ [ j ] 1 j a ) m m + 1 + ∑ j = 1 n ( υ j ) 1 m + 1 ( p ¯ [ j ] 2 j a ) m m + 1 , j = 1 , 2 , ... , n (8)

Proof. Obviously under the condition ∑ i = 1 2 ∑ j = 1 n u j i = G , we can get the minimum value of the objective function.

Min f ( π , u , d ) = ∑ j = 1 n ( α E j + β T j + γ d )

S.t. ∑ i = 1 2 ∑ j = 1 n u j i − G = 0 (9)

According to Lagrangian multiplier method, from (9) and (4), we have:

L ( π , u , d , λ ) = f ( π , u , d ) + λ ( ∑ i = 1 2 ∑ j = 1 n u j i − G ) = ∑ j = 1 n ( α E j + β T j + γ d ) + λ ( ∑ i = 1 2 ∑ j = 1 n u j i − G ) = ∑ j = 1 n ω j ( p ¯ [ j ] 1 j a u [ j ] 1 ) m + ∑ j = 1 n υ j ( p ¯ [ j ] 2 j a u [ j ] 2 ) m + λ ( ∑ j = 1 n u [ j ] 1 + ∑ j = 1 n u [ j ] 2 − G ) (10)

where λ is the Lagrangian multiplier.

Since each of the objectives is a convex function, according to Lagrangian multiplier method.

Differentiating (10) with respect to u [ j ] 1 , we have

∂ L ( π , u , d , λ ) ∂ u [ j ] 1 = λ − m ω j × ( p ¯ [ j ] 1 j a ) m ( u [ j ] 1 ) m + 1 = 0 , j = 1 , 2 , ... , n (11)

then

u [ j ] 1 * = ( m ω j λ ) 1 m + 1 ( p ¯ [ j ] 1 j a ) m m + 1 , j = 1 , 2 , ... , n (12)

Differentiating (10) with respect to u [ j ] 2 , we have

∂ L ( π , u , d , λ ) ∂ u [ j ] 2 = λ − m υ j × ( p ¯ [ j ] 2 j a ) m ( u [ j ] 2 ) m + 1 = 0 , j = 1 , 2 , ... , n (13)

then

u [ j ] 2 * = ( m υ j λ ) 1 m + 1 ( p ¯ [ j ] 2 j a ) m m + 1 , j = 1 , 2 , ... , n (14)

Differentiating (10) with respect to λ , we have

∂ L ( π , u , d , λ ) ∂ λ = ∑ j = 1 n u [ j ] 1 + ∑ j = 1 n u [ j ] 2 − G = 0 (15)

Substituting (12) and (14) into (15), we have

λ 1 m + 1 = ∑ j = 1 n ( m ω j ) 1 m + 1 ( p ¯ [ j ] 1 j a ) m m + 1 G + ∑ j = 1 n ( m υ j ) 1 m + 1 ( p ¯ [ j ] 2 j a ) m m + 1 G (16)

From (16) and (12), we have

u [ j ] 1 * = G ( ω j ) 1 m + 1 ( p ¯ [ j ] 1 j a ) m m + 1 ∑ j = 1 n ( ω j ) 1 m + 1 ( p ¯ [ j ] 1 j a ) m m + 1 + ∑ j = 1 n ( υ j ) 1 m + 1 ( p ¯ [ j ] 2 j a ) m m + 1 ， j = 1 , 2 , ... , n

From (16) and (14), we have

u [ j ] 2 * = G ( υ j ) 1 m + 1 ( p ¯ [ j ] 2 j a ) m m + 1 ∑ j = 1 n ( ω j ) 1 m + 1 ( p ¯ [ j ] 1 j a ) m m + 1 + ∑ j = 1 n ( υ j ) 1 m + 1 ( p ¯ [ j ] 2 j a ) m m + 1 , j = 1 , 2 , ... , n

Substituting (7) and (8) into (4), under optimal resource allocation u [ j ] 1 * , u [ j ] 2 * and π = ( J [ 1 ] , J [ 2 ] , J [ 3 ] , ... , J [ n ] ) , we obtain that a new unified expression for f ( π , u , d ) = ∑ j = 1 n ( α E j + β T j + γ d ) .

f ( π , u ∗ ( π ) , d ) = ∑ j = 1 n ( α E j + β T j + γ d ) = ∑ j = 1 n ω j ( p ¯ [ j ] 1 j a u [ j ] 1 ∗ ) m + ∑ j = 1 n υ j ( p ¯ [ j ] 2 j a u [ j ] 2 ∗ ) m = ∑ j = 1 n ω j ( p ¯ [ j ] 1 j a u [ j ] 1 ) m + ∑ j = 1 n υ j ( p ¯ [ j ] 2 j a u [ j ] 2 ) m = G − m ( ∑ j = 1 n ( ω j ) 1 m + 1 ( p ¯ [ j ] 1 j a ) m m + 1 × ( ∑ j = 1 n ( ω j ) 1 m + 1 ( p ¯ [ j ] 1 j a ) m m + 1 + ∑ j = 1 n ( υ j ) 1 m + 1 ( p ¯ [ j ] 2 j a ) m m + 1 ) m ) + ∑ j = 1 n ( υ j ) 1 m + 1 ( p ¯ [ j ] 2 j a ) m m + 1 × ( ∑ j = 1 n ( ω j ) 1 m + 1 ( p ¯ [ j ] 1 j a ) m m + 1 + ∑ j = 1 n ( υ j ) 1 m + 1 ( p ¯ [ j ] 2 j a ) m m + 1 ) m ) ) m + 1 = G − m ( ∑ j = 1 n ( ω j ) 1 m + 1 ( p ¯ [ j ] 1 j a ) m m + 1 + ∑ j = 1 n ( υ j ) 1 m + 1 ( p ¯ [ j ] 2 j a ) m m + 1 ) ) m + 1 (17)

In what follows, we derive the optimal schedule for

F 2 | n o − w a i t , p j i = ( p ¯ j i r a u j i ) m , ∑ i = 1 2 ∑ j = 1 n u j i ≤ G | ∑ j = 1 n ( α E j + β T j + γ d )

It is clear that the minimum value of (17) is equal to minimizing

∑ j = 1 n ( ω j ) 1 m + 1 ( p ¯ [ j ] 1 j a ) m m + 1 + ∑ j = 1 n ( υ j ) 1 m + 1 ( p ¯ [ j ] 2 j a ) m m + 1

Let us define binary variables x j r , such that x j r = 1 , if job J j ( j = 1 , 2 , ... , n ) is scheduled at position r ( r = 1 , 2 , ... , n ) , otherwise x j r = 0 . Then, the problem

F 2 | n o − w a i t , p j i = ( p ¯ j i r a u j i ) m , ∑ i = 1 2 ∑ j = 1 n u j i ≤ G | ∑ j = 1 n ( α E j + β T j + γ d )

can be solved by the following linear assignment problem:

Min ∑ r = 1 n ∑ j = 1 n θ j r x j i (18)

S.t. ∑ r = 1 n x j r = 1 , j = 1 , 2 , ... , n (19)

∑ j = 1 n x j r = 1 , r = 1 , 2 , ... , n (20)

x j r = 0 o r 1 , j = 1 , 2 , .. , n , r = 1 , 2 , ... , n (21)

where

θ j r = ( ω r ) 1 m + 1 ( p ¯ j 1 r a ) m m + 1 + ( υ r ) 1 m + 1 ( p ¯ j 2 r a ) m m + 1 (22)

ω r = { n γ , r = 1 α ( r − 1 ) + γ n , r = 2 , 3 , ... , k β ( n − r + 1 ) , r = k + 1 , k + 2 , ... , n (23)

and

υ r = { − α , r = 1 , 2 , ... , k − 1 α ( k − 1 ) − β ( n − k ) + γ n , r = k β , r = k + 1 , k + 2 , ... , n (24)

By solving this linear assignment problem, we can get the optimal job sequence π = ( J [ 1 ] , J [ 2 ] , J [ 3 ] , ... , J [ n ] ) of the problem

Based on the above analysis, our algorithm for

can be described as follows.

Algorithm 1

Step 1. According to (3), calculate k = min { max { ⌈ ( β − γ ) n α + β ⌉ , 0 } , n } .

Step 2. According to (22), calculate the θ j r .

Step 3. Solve the linear assignment problem (18)-(21) to determine the optimal job sequence π = ( J [ 1 ] , J [ 2 ] , J [ 3 ] , ... , J [ n ] ) .

Step 4. According to (7) and (8), calculate the optimal resource allocation u [ j ] 1 * , u [ j ] 2 * .

Step 5. According to (1), calculate the actual processing times p j i .

Step 6. Calculate the common due date d = C [ k ] .

Theorem 1. The problem

can be solved in O ( n 3 ) time by Algorithm 1.

Proof. According to the lemmas 1, 2, 3 and the assignment problem (18)-(21), we can get the correctness of Algorithm 1, the complexity of step 2 in Algorithm 1 is O ( n 2 ) , step 3 is O ( n 3 ) , the complexity of steps 1,4,5,6 is O ( n ) . So the complexity of Algorithm 1 is O ( n 3 ) .

Hsu was supported by the Ministry Science and Technology of Taiwan under Grant MOST 106-2221-E-252-003.

Geng, X.N., Wang, J.B. and Hsu, C.-J. (2018) Flow Shop Scheduling Problem with Convex Resource Allocation and Learning Effect. Journal of Computer and Communications, 6, 239-246. https://doi.org/10.4236/jcc.2018.61024