Single machine scheduling problems with general position-dependent processing times and past-sequence-dependent delivery times

JAMCJ Appl Math ComputDOI 10.1007/s12190-013-0722-9

O R I G I NA L R E S E A R C H

Single machine scheduling problems with generalposition-dependent processing times andpast-sequence-dependent delivery times

Chuanli Zhao · Hengyong Tang

Received: 3 July 2013© Korean Society for Computational and Applied Mathematics 2013

Abstract This paper considers single machine scheduling with past-sequence-dependent (psd) delivery times, in which the processing time of a job depends onits position in a sequence. We provide a unified model for solving single machinescheduling problems with psd delivery times. We first show how this unified modelcan be useful in solving scheduling problems with due date assignment considera-tions. We analyze the problem with four different due date assignment methods, theobjective function includes costs for earliness, tardiness and due date assignment.We then consider scheduling problems which do not involve due date assignmentdecisions. The objective function is to minimize makespan, total completion timeand total absolute variation in completion times. We show that each of the problemscan be reduced to a special case of our unified model and solved in O(n3) time. Inaddition, we also show that each of the problems can be solved in O(n logn) time forthe spacial case with job-independent positional function.

Keywords Scheduling · Single machine · Position-dependent processing times ·Past-sequence-dependent delivery times

Mathematics Subject Classification 90B35 · 68M20

1 Introduction

In most classical scheduling problems, it is assumed that job processing times arefixed. However, in many realistic settings, it appears that the processing time of jobsdepending on their position in the sequence [2]. Two well known special cases are

C. Zhao (B) · H. TangSchool of Mathematics and Systems Science, Shenyang Normal University, Shenyang, Liaoning,110034, People’s Republic of Chinae-mail: [email protected]

mailto:[email protected]

C. Zhao, H. Tang

learning effect and aging effect (positional deterioration). In a learning environment,the processing times decrease as a function of the position; while in an aging envi-ronment, the processing times increase as a function of the position. The learningeffect on scheduling problem was first introduced by Biskup [3]. Cheng et al. [5, 6]introduced two new scheduling models with learning effects. For the single-machinecase, they derived polynomial-time optimal solutions for the problems to minimizemakespan and total completion time. For the flow shop, they presented polynomial-time optimal solutions for some special cases of the problems to minimize makespanand total completion time. Cheng et al. [9] considered a two-machine flowshopscheduling problem with a truncated learning function. The objective is to minimizethe makespan. They proposed a branch-and-bound and three crossover-based geneticalgorithms to find the optimal and approximate solutions. Yang et al. [31] introduceda general learning effect model in which the learning effect from job processing de-pends not only on the sum of the processing times, but also on the processing com-plexity of the jobs already processed. They provided some properties that are helpfulfor finding the optimal solutions for some single-machine and flowshop schedulingproblems under the general effect model. Other studies include Cheng et al. [7, 8];Lee et al. [16]; Mosheiov [19–21]; and Wu et al. [28, 30]. An updated survey of theresults on scheduling problems with learning effect was provided by Biskup [4]. Theaging effect was mentioned for the first time by Mosheiov [20]. Other studies includeMosheiov [22]; Kuo and Yang [15]; and Janiak and Rudek [11]. Rustogi and Stru-sevich [26] presented a critical review of scheduling models with job-independentpositional effects. They showed that a linear assignment algorithm can be replacedby a faster matching algorithm and delivers the required solution much faster. Only afew papers consider general job-dependent position-dependent processing times. Oz-turkoglu and Bulfin [24] investigated a scheduling problem with position dependentprocessing times and rate-modifying activity simultaneously. They proposed heuris-tics for both makespan and total completion time. Mosheiov and Sidney [21] consid-ered scheduling with general job-dependent learning function. They introduced poly-nomial time solutions for several classical objective functions. Mosheiov [23] studieda scheduling problem with general position dependent processing times. They derivedpolynomial algorithm for makespan minimization on an m-machine proportionateflow shop. Wu and Lee [29] considered both the machine and human learning effectssimultaneously. They presented the solution procedures for some single machine andsome flow shop problems. Zhao and Tang [33] considered scheduling with generaljob-dependent aging effect. The objective is to make a decision on the maintenanceand the sequence of jobs to minimize the makespan. They showed that the prob-lem can be solved by polynomial-time algorithms. Kim and Ozturkoglu [13] studiedscheduling jobs with job dependent position dependent processing times. The objec-tive is to make a decision on the maintenance and the sequence of jobs to minimizethe makespan and completion time objective. They proposed a number of heuristicsand genetic algorithms for the problems.

In many industries, the job’s waiting time may have an adverse effect on the to-tal processing time of a job before delivery to the customer. Such an adverse effectsis viewed as a post-sequence-dependent (psd) delivery time, which is proposed byKoulamas and Kyparisis [14]. Koulamas and Kyparisis [14] assumed that the psd

Single machine scheduling problems

delivery time is an extra time which needed to remove waiting time-induced adverseeffects on the job’s condition prior to delivering it to the customer. For analytical con-venience, it is generally assumed that the psd delivery time of a job is proportionalto the job’s waiting time. They show that single machine scheduling problems withpsd delivery times and with either completion time-related criteria or due date relatedcriteria can be reduced to the corresponding problems without psd delivery times andsolved by simple polynomial-time algorithms. Liu et al. [18] further studied somesingle machine scheduling problems with psd delivery times and introduced poly-nomial algorithms for several objective functions, includes the minimization of thetotal weighted completion time, the total weighted discounted completion time, thetotal absolute variation in completion times and the sum of earliness, tardiness andcommon due date penalty.

The concepts of psd delivery time and position-dependent job processing timeshave been studied in the literature. However, to the best of our knowledge, apart fromYang and Yang [32], there are no research results on scheduling models concerningthe psd delivery time and position-dependent job processing times at the same time.Yang and Yang [32] analysed single-machine scheduling problems with simultaneousconsiderations of position-dependent job processing times and job delivery times. Intheir model, there is a particular rule that explains how exactly the processing timechanges (pj,r = pj r

aj ). They investigated the minimization problems of the sumof earliness, tardiness, and due-window-related cost, the total absolute differences incompletion times, and the total absolute differences in waiting times. The polynomialtime algorithms are proposed.

In this paper, we extend the model of Yang and Yang [32] to the general setting.We consider single machine scheduling with psd delivery times and general job-dependent position-dependent processing times. Our model is not restricted to anyspecific job-position function.

2 Problem formulation and unified model

The problem under investigation can be described as follows.We consider a single machine and a set N = {J1, J2, . . . , Jn} of n jobs. All jobs

are available for processing at time zero and preemption is not permitted. Each job Jj

has a normal processing time pj . The actual processing time of job Jj , if scheduledin position r of a sequence, is given by

pj,r = gj (r)pj ,

where gj (r) is a function that specifies a job-dependent positional effect.For a given schedule π = [J[1], J[2], . . . , J[n]], we use J[j ] to indicate the job

occupying the j -th position in π . Denote the starting time of job J[j ] by s[j ],which satisfies s[1] = 0 and s[j ] = ∑j−1

i=1 g[i](i)p[i], j = 2, . . . , n. The processingof job J[j ] must be followed immediately by its psd delivery time q[j ]. It is as-sumed that q[j ] is proportional to the starting time (or waiting time) of job J[j ], i.e.,

q[j ] = δ∑j−1

i=1 g[i](i)p[i], j = 1, . . . , n.

C. Zhao, H. Tang

Let C[j ] denote the completion time of job J[j ] in π (i.e., the completion time ofthe processing of J[j ] on the machine plus the job’s psd delivery time). Therefore,

C[1] = g[1](1)p[1] + q[1] = g[1](1)p[1],C[j ] = s[j ] + g[j ](j)p[j ] + q[j ]

= (1 + δ)

j−1∑

i=1

g[i](i)p[i] + g[j ](j)p[j ], j = 2, . . . , n.

We first provide a unified model. We consider the general problem with psd de-livery time and position-dependent processing times, denoted as P : 1|qpsd,pj,r =gj (r)pj |f .

Theorem 1 For a given schedule π = [J[1], J[2], . . . , J[n]], if the objective value ofproblem P is given by f = ∑n

j=1 w[j ],jp[j ], where w[j ],j is a job-dependent position

weight, then it can be formulated as an assignment problem and solved in O(n3) time.

Proof We will show that the problem P can be reduced to an assignment problem.Since the objective value of problem P is given by f = ∑n

j=1 w[j ],jp[j ], we cancreate an assignment matrix, where the rows represent the jobs and the columns rep-resent their potential positions. �

We introduce a binary variable xj,r such that xj,r = 1 if job Jj is in the r-thposition, and xj,r = 0 otherwise, j, r = 1,2, . . . , n. Then the problem P can be for-mulated as the following assignment problem AP and solved in O(n3) time:

Minn∑

j=1

n∑

r=1

cj,rxj,r

s.t.n∑

r=1

xj,r = 1, j = 1, . . . , n,

n∑

j=1

xj,r = 1, r = 1, . . . , n,

xj,r ∈ {0,1}, j = 1, . . . , n, r = 1, . . . , n,

where cj,r = wj,rpj .Now we consider the spacial case gj (r) = g(r), i.e.,the actual processing time of

job j , if scheduled in position r of a sequence, is given by

pj,r = g(r)pj ,

where g(r) is a function that specifies a job-independent positional effect.We introduce a useful lemma.

Lemma 1 [10] Let there be two sequences of numbers xi and yi (i = 1, . . . , n). Thesum

∑ni=1 xiyi of products of the corresponding elements is the least if the sequences

are monotonic in the opposite sense.


By Lemma 1, for problem P , if f = ∑nj=1 wjp[j ] and wj is job-independent

position weight, the optimal solution for problem P can be obtained by arranging theelements of the wj and p[j ] vectors in opposite orders.

Theorem 2 For a given schedule π = [J[1], J[2], . . . , J[n]], if the objective value ofproblem P is given by f = ∑n

j=1 wjp[j ], where wj is a job-independent positionweight, then it can be solved in O(n logn) time.

3 Applications to solve various due date assignment problems

In this section, we show how our unified method can be used to solve a large setof scheduling problems involving both due date assignment and position-dependentprocessing times.

Many different due date assignment methods have been studied in the literature. Inthis section, we study our problem with the four most frequent due date assignmentmethods:

• The common due window assignment method (Liman et al. [17], referred asCONW), where the scheduler can assign a single desired time window, [d, d +D],for the completion time of each job, and the objective includes a linear penalty forboth d and D.

• The common due date assignment method (Panwalkar et al. [25], referred as CON),in which all jobs are assigned the same due date, that is dj = d for j = 1, . . . , n,and d ≥ 0 is a decision variable.

• The slack due date assignment method (Adamopoulos and Pappis [1], referred toas SLK), in which all jobs are given a flow allowance that reflects equal waitingtimes (equal slacks), that is, dj = pj +q for j = 1, . . . , n, where q ≥ 0 is a decisionvariable.

• The unrestricted due date assignment method (Seidmann et al. [27], referred to asDIF), in which each job can be assigned a different due date with no restrictions.

Based on the solutions of Theorem 1 and Theorem 2, we only need to showthat for each problem we considered, the objective can be formulation as: f =∑n

j=1 w[j ],jp[j ] for general case pj,r = gj (r)pj , and f = ∑nj=1 wjp[j ] for spacial

case pj,r = g(r)pj .Firstly, we consider the CONW due date assignment method. The objective is to

minimize a cost function given by:

f =n∑

j=1

(αEj + βTj + γ1d + γ2D). (1)

Liman et al. [17] presented the following result for the CONW due date assignmentproblem with fixed processing times.

Lemma 2 For the 1|CONW|∑(αEj + βTj + γ1d + γ2D) problem,

(i) If γ1 > γ2, an optimal schedule exists in which the due-window starts at timezero.

C. Zhao, H. Tang

(ii) If β < min{γ1, γ2}, an optimal schedule exists in which the due-window is re-duced to a due-date that starts (and is completed) at time zero.

(iii) There exists an optimal schedule in which both the due-window starting timed and the due-window completion time d + D coincide with job completiontimes, and d = C[k] and d + D = C[k+m], where k = �n(γ2−γ1)

α� and k + m =

�n(β−γ2)β

�.

It can be easily verified that the above lemma holds for our problem 1|qpsd,pj,r =gj (r)pj , CONW|∑(αEj + βTj + γ1d + γ2D). Therefore,

d = C[k], (2)

d + D = C[k+m], (3)

E[j ] ={

C[k] − C[j ] for j = 1, . . . , k − 1,

0 for j = k, . . . , n,(4)

T[j ] ={

0 for j = 1, . . . , k + m,

C[j ] − C[k+m] for j = k + m + 1, . . . , n.(5)

Let π = [J[1], J[2], . . . , J[n]], then C[j ] = (1 + δ)∑j−1

i=1 g[i](i)p[i] + g[j ](j)p[j ].By substituting Eqs. (2)–(5) into Eq. (1), we get a new expressed for objective func-tion:

f =n∑

j=1

(αEj + βTj + γ1d + γ2D)

=k∑

j=1

αE[j ] +n∑

j=k+m+1

βT[j ] +n∑

j=1

γ1C[k] +n∑

j=1

γ2(C[k+m] − C[k])

= α

k∑

j=1

(C[k] − C[j ]) + β

n∑

j=k+m+1

(C[j ] − C[k+m])

+n∑

j=1

γ1C[k] +n∑

j=1

γ2(C[k+m] − C[k])

=k∑

j=1

(α + γ1 − γ2)C[k] −k∑

j=1

αC[j ] +k∑

j=1

γ2C[k+m]

+k+m∑

j=k+1

(γ1 − γ2)C[k] +k+m∑

j=k+1

γ2C[k+m]

+n∑

j=k+m+1

βC[j ] +n∑

j=k+m+1

(γ2 − β)C[k+m] +n∑

j=k+m+1

(γ1 − γ2)C[k]

=k∑

j=1

((α + γ1 − γ2)k(1 + δ) − α

) k−1∑

j=1

g[j ](j)p[j ] + (α + γ1 − γ2)kg[k](k)p[k]


− α

k−1∑

j=1

[(1 + δ)(k − j) + 1

]g[j ](j)p[j ] − αg[k](k)p[k]

+k+m+1∑

j=1

γ2k(1 + δ)g[j ](j)p[j ] + γ2kg[k+m](k + m)p[k+m]

+k−1∑

j=1

(γ1 − γ2)m(1 + δ)

k−1∑

j=1

g[j ](j)p[j ] + (γ1 − γ2)mg[k](k)p[k]

+k+m−1∑

j=1

γ2m(1 + δ)g[j ](j)p[j ] + γ2mg[k+m](k + m)p[k+m]

+k+m−1∑

j=1

β(n − k − m)(1 + δ)g[j ](j)p[j ]

+ β(n − k − m)(1 + δ)g[k+m](k + m)p[k+m]

+n∑

j=k+m+1

[(1 + δ)(n − j) + 1

]g[j ](j)p[j ]

+k+m−1∑

j=1

(γ2 − β)(n − k − m)(1 + δ)g[j ](j)p[j ]

+ (γ2 − β)(n − k − m)g[k+m](k + m)p[k+m]

+k−1∑

j=1

(γ1 − γ2)(n − k − m)(1 + δ)g[j ](j)p[j ]

+ (γ1 − γ2)(n − k − m)g[k](k)p[k]

=k−1∑

j=1

((αj + γ1n)(1 + δ) − α

)g[j ](j)p[j ] + [

(γ1 + γ2δ)n + α(k − 1)]g[k](k)p[k]

+k+m−1∑

j=k+1

γ2n(1 + δ)g[j ](j)p[j ] + [βδ(n − k − m) + γ2n

]g[k+m](k + m)p[k+m]

+n∑

j=k+m+1

β[(n − j)(1 + δ) + 1

]g[j ](j)p[j ]

=n∑

j=1

w[j ],j p[j ],

C. Zhao, H. Tang

where

w[j ],j =

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

((αj + γ1n)(1 + δ) − α)g[j ](j) for j = 1, . . . , k − 1,

((γ1 + γ2δ)n + α(j − 1))g[j ](j) for j = k,

γ2n(1 + δ)g[j ](j) for j = k + 1, . . . , k + m − 1,

(βδ(n − j) + γ2n)g[j ](j) for j = k + m,

β[(n − j)(1 + δ) + 1]g[j ](j) for j = k + m + 1, . . . , n.

Therefore, 1|qpsd,pj,r = gj (r)pj , CONW|∑(αEj + βTj + γ1d + γ2D) can beformulated as the following assignment problem AP :

Minn∑

j=1

n∑

r=1

cj,rxj,r

s.t.n∑

r=1

xj,r = 1, j = 1, . . . , n,

n∑

j=1

xj,r = 1, r = 1, . . . , n,

xj,r ∈ {0,1}, j = 1, . . . , n, r = 1, . . . , n,

where

cj,r =

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(αr + γ1n)(1 + δ)gj (r)pj for r = 1, . . . , k − 1,

((γ1 + γ2δ)n + α(r − 1))gj (r)pj for r = k,

γ2n(1 + δ)gj (r)pj for r = k + 1, . . . , k + m − 1,

(βδ(n − r) + γ2n)gj (r)pj for r = k + m,

β((n − r)(1 + δ) + 1)gj (r)pj for r = k + m + 1, . . . , n.

If gj (r) = g(r), then wj,r = wr and 1|qpsd,pj,r = gj (r)pj , CONW|∑(αEj +βTj + γ1d + γ2D) can be solved in O(n logn) time: arranging the elements of thewr and pj vectors in opposite orders.

Theorem 3 The 1|qpsd,pj,r = gj (r)pj , CONW|∑(αEj +βTj +γ1d +γ2D) prob-

lem can be solved in O(n3) time and the 1|qpsd,pj,r = g(r)pj , CONW|∑(αEj +βTj + γ1d + γ2D) problem can be solved in O(n logn) time.

We then consider the problem with CON method. The objective is to minimize acost function that includes the costs of earliness, tardiness and due date assignmentgiven by:

f =n∑

j=1

(αEj + βTj + γ d).

We first present some earlier results obtained by Panwalkar et al. [25] for the CONdue date assignment problem with fixed processing times.


Lemma 3 For the 1|CON|∑(αEj + βTj + γ d) problem, there exists an optimal

due date d equal to C[k], where k = �n(β−γ )α+β

�.

It can be easily verified that the above lemma holds for our problem 1|qpsd,pj,r =gj (r)pj , CON|(αEj + βTj + γ d). Therefore

d = C[k],

E[j ] ={

C[k] − C[j ] for j = 1, . . . , k − 1,

0 for j = k, . . . , n,

T[j ] ={

0 for j = 1, . . . , k,

C[j ] − C[k] for j = k + 1, . . . , n.

We get a new expressed for objective function:

f =n∑

j=1

(αEj + βTj + γ d)

= α

k∑

j=1

(C[k] − C[j ]) + β

n∑

j=k+1

(C[j ] − C[k]) + γ

n∑

j=1

C[k]

= (α + γ )

k∑

j=1

C[k] − α

k−1∑

j=1

C[j ] + β

n∑

j=k+1

C[j ] + (γ − β)

n∑

j=k+1

C[k]

= (α + γ )(1 + δ)k

k∑

j=1

g[j ](j)p[j ] + (α + γ )kg[k](k)p[k]

− α

k−1∑

j=1

[(1 + δ)(k − j) + 1

]g[j ](j)p[j ] − αg[k](k)p[k]

+ β(1 + δ)(n − k)

k−1∑

j=1

g[j ](j)p[j ] + β(1 + δ)(n − k)g[k](k)p[k]

+ β

n∑

j=k+1

[(1 + δ)(n − j) + 1

]g[j ](j)p[j ]

+ (γ − β)(n − k)(1 + δ)

k−1∑

j=1

g[j ](j)p[j ] + (γ − β)(n − k)g[k](k)p[k]

=k−1∑

j=1

{γ (1 + δ)n + α

[(1 + δ)j − 1

]}g[j ](j)p[j ]

+ [nγ + (k − 1)α + βδ(n − k)

]g[k](k)p[k]

+n∑

j=k+1

β[1 + (1 + δ)(n − j)

]g[j ](j)p[j ].

C. Zhao, H. Tang

Therefore, 1|qpsd,pj,r = gj (r)pj , CON|∑(αEj + βTj + γ d) can be formulatedas an assignment problem AP , where

cj,r =

⎧⎪⎨

⎪⎩

γ (1 + δ)n + α[(1 + δ)r − 1]gj (r)pj for r = 1, . . . , k − 1,

nγ + (r − 1)α + βδ(n − r)gj (r)pj for r = k,

β[1 + (1 + δ)(n − r)gj (r)pj for r = k + 1, . . . , n.

Theorem 4 The 1|qpsd,pj,r = gj (r)pj , CON|∑(αEj + βTj + γ d) problem canbe solved in O(n3) time and the 1|qpsd,pj,r = g(r)pj , CON|∑(αEj + βTj + γ d)

problem can be solved in O(n logn) time.

We now consider the SLK due date assignment problem. The objective is to mini-mize a cost function given by:

f =n∑

j=1

(αEj + βTj + γ dj ).

Adamopoulos and Pappis [1] presented the following result for the SLK due dateassignment problem with fixed processing times.

Lemma 4 For the 1|SLK|∑(αEj + βTj + γ dj ) problem, there exists an optimal

slack allowance q equal to C[k−1], where k = �n(β−γ )α+β

�.

It can be easily verified that the above lemma holds for our problem 1|qpsd,pj,r =gj (r)pj , SLK|∑(αEj + βTj + γ dj ), we have

dj = C[j ] − C[j−1] + C[k−1] for j = 1, . . . , n,

E[j ] ={

C[k−1] − C[j−1] for j = 1, . . . , k − 1,

0 for j = k + 1, . . . , n,

T[j ] ={

0 for j = 1, . . . , k,

C[j−1] − C[k−1] for j = k + 1, . . . , n,

f =n∑

j=1

(αEj + βTj + γ dj )

=k−1∑

j=1

αE[j ] +n∑

j=k+1

βT[j ] +n∑

j=1

γ dj

= α

k−1∑

j=1

(C[k−1] − C[j−1]) + β

n∑

j=k+1

(C[j−1] − C[k−1])

+ γ

n∑

j=1

(C[j ] − C[j−1] + C[k−1])

= (α + γ )

k−1∑

j=1

C[k−1] − (α + γ )

k−1∑

j=1

C[j−1] + γ

n∑

j=1

C[j ]


+ (β − γ )

n∑

j=k+1

C[j−1] − (β − γ )

n∑

j=k+1

C[k−1]

= [(α + γ )(k − 1) − (n − k)(β − γ )

]

×[

(1 + δ)

k−2∑

j=1

g[j ](j)p[j ] + g[k−1](k − 1)p[k−1]

]

− (α + γ )

[k−3∑

j=1

[(1 + δ)(k − j − 2)

]g[j ](j)p[j ] +

k−2∑

j=1

g[j ](j)p[j ]

]

+ γ

n−1∑

j=1

[(1 + δ)(n − j) + 1

]g[j ](j)p[j ] + γg[n](n)p[n]

+ (β − γ )(n − k)(1 + δ)

k−1∑

j=1

g[j ](j)p[j ]

+ (β − γ )(1 + δ)

n−2∑

j=k

(n − j − 1)g[j ](j)p[j ]

+ (β − γ )

n−1∑

j=k

g[j ](j)p[j ]

=k−2∑

j=1

{(1 + δ)

[α(j + 1) + γ (n + 1)

] − α}g[j ](j)p[j ]

+ {(1 + δ)

[α(k − 1) + γ n

] + γ}g[k−1](k − 1)p[k−1]

+n−2∑

j=k

{(1 + δ)

[β(n − j + 1) + γ

] + β}g[j ](j)p[j ]

+ [β + γ (1 + δ)

]g[n−1](n − 1)p[n−1] + γg[n](n)p[n].

Therefore, 1|qpsd,pj,r = gj (r)pj , SLK|∑(αEj + βTj + γ d) can be formulatedas an assignment problem AP , where

cj,r =

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

{(1 + δ)[α(r + 1) + γ (n + 1)] − α}gj (r)pj for r = 1, . . . , k − 2,

((1 + δ)(αr + γ n) + γ )gj (r)pj for r = k − 1,

{(1 + δ)[β(n − r + 1) + γ ] + β}gj (r)pj for r = k, . . . , n − 2,

(β + γ (1 + δ))gj (r)pj for r = n − 1,

γgj (r)pj for r = n.

Theorem 5 The 1|qpsd,pj,r = gj (r)pj , SLK|∑(αEj + βTj + γ dj ) problem canbe solved in O(n3) time and the 1|qpsd,pj,r = g(r)pj , SLK|∑(αEj + βTj + γ dj )


C. Zhao, H. Tang

For the problem with the DIF due date assignment method, the objective is tominimize a cost function given by:

f =n∑

j=1

(αEj + βTj + γ dj ).

Seidmann, Panwalkar and Smith [27] presented the following result for the DIFdue date assignment problem with fixed processing times.

Lemma 5 For the problem 1|DIF|∑(αEj + βTj + γ dj ), if γ ≥ β then d[j ] = 0;otherwise, d[j ] = C[j ]; for j = 1, . . . , n.

It can be easily verified that the above lemma holds for our problem 1|qpsd,pj,r =gj (r)pj , DIF|∑(αEj + βTj + γ dj ), and we have

f =n∑

j=1

εC[j ]

= ε

n∑

j=1

[

(1 + δ)

j−1∑

i=1

g[i](i)p[i] + g[j ](j)p[j ]

]

= ε

n∑

j=1

[(1 + δ)(n − j) + 1

]g[j ](j)p[j ],

where ε = min{β,γ }.Therefore, 1|qpsd,pj,r = gj (r)pj , DIF|∑(αEj + βTj + γ dj ) can be formulated

as an assignment problem AP , where cj,r = ε[(1 + δ)(n − r) + 1]gj (r)pj .

Theorem 6 The 1|qpsd,pj,r = gj (r)pj , DIF|∑(αEj + βTj + γ dj ) problem canbe solved in O(n3) time and the 1|qpsd,pj,r = g(r)pj , DIF|∑(αEj + βTj + γ dj )


4 Applications to solve scheduling problems without due dates

We show in this section that a large collection of other scheduling problems, whichdo not involve due dates, can also be solved by our unified model.

We will consider the makespan, the total completion time and the total absolutevariation of completion times [12] minimization problems.

Let π = [J[1], J[2], . . . , J[n]], then the makespan is:

f = Cmax =n−1∑

j=1

(1 + δ)g[j ](j)p[j ] + g[n](n)p[n].

The total completion time is:


f =n∑

j=1

C[j ]

=n∑

j=1

[

(1 + δ)

j−1∑

i=1

g[i](i)p[i] + g[j ](j)p[j ]

]

=n∑

j=1

[(1 + δ)(n − j) + 1

]g[j ](j)p[j ].

The total absolute variation of completion times is:

f =n∑

j=1

n∑

i=j

|Ci − Cj |

=n∑

j=1

(2j − 1 − n)C[j ]

=n∑

j=1

(2j − 1 − n)

[

(1 + δ)

j−1∑

i=1

g[i](i)p[i] + g[j ](j)p[j ]

]

=n∑

j=1

[(j − 1)(n − j + 1) + j (n − j)δ

]g[j ](j)p[j ].

Therefore, 1|qpsd,pj,r = gj (r)pj |Cmax can be formulated as an assignment prob-lem where

cj,r ={

(1 + δ)gj (r)pj for r = 1, . . . , n − 1,

gj (r)pj for r = n.

1|qpsd,pj,r = gj (r)pj | = ∑Cj can be formulated as an assignment problem

where cj,r = [(1 + δ)(n − r) + 1]gj (r)pj .1|qpsd,pj,r = gj (r)pj |∑∑ |Ci − Cj | can be formulated as an assignment prob-

lem where cj,r = [(r − 1)(n − r + 1) + r(n − r)δ]gj (r)pj .

Theorem 7 The 1|qpsd,pj,r = gj (r)pj |f , f ∈ {Cmax,∑

Cj ,∑∑ |Ci − Cj |}

problems can be solved in O(n3) time and the 1|qpsd,pj,r = g(r)pj |f , f ∈{Cmax,

∑Cj ,

∑∑ |Ci − Cj |} problems can be solved in O(n logn) time.

We conclude this section with Table 1 and Table 2. Table 1 summarizes most ofresults in the literature. Table 2 summarizes most of our results.

5 Conclusions

Scheduling problems with position-dependent processing times and past-sequence-dependent delivery times have received increasing attention. We have identified aunified structure for a large number of scheduling problems with position-dependent

C. Zhao, H. Tang

Table 1 Previous results for problems with position-dependent processing times and (or) past-sequence-dependent delivery times

Problem Complexity Ref.

1|qpsd,pj,r = pj raj |∑∑ |Ci − Cj | O(n3) [32]

1|qpsd,pj,r = pj raj , CONW|∑(αEj + βTj + γ d + δD) O(n3) [32]

1|pj,r = g(r)pj |Cmax O(n logn) [26]

1|pj,r = g(r)pj |∑Cj O(n logn) [26]

1|pj,r = g(r)pj |∑∑ |Ci − Cj | O(n logn) [26]

1|pj,r = g(r)pj ,CON|∑(αEj + βTj + γ d) O(n logn) [26]

1|qpsd |Cmax O(n logn) [14]

1|qpsd |∑Cj O(n logn) [14]

1|qpsd,CON|∑(αEj + βTj + γ d) O(n logn) [18]

1|qpsd,pj,r = pj ra |∑∑ |Ci − Cj | O(n logn) [32]

1|qpsd,pj,r = pj ra,CONW|∑(αEj + βTj + γ d + δD) O(n logn) [32]

Table 2 Summary of our results

Problem Complexity Ref.

1|qpsd,pj,r = gj (r)pj |Cmax O(n3) Theorem 7

1|qpsd,pj,r = gj (r)pj |∑Cj O(n3) Theorem 7

1|qpsd,pj,r = gj (r)pj |∑∑ |Ci − Cj | O(n3) Theorem 7

1|qpsd,pj,r = gj (r)pj ,CON|∑(αEj + βTj + γ d) O(n3) Theorem 4

1|qpsd,pj,r = gj (r)pj ,SLK|∑(αEj + βTj + γ dj ) O(n3) Theorem 5

1|qpsd,pj,r = gj (r)pj ,DIF|∑(αEj + βTj + γ dj ) O(n3) Theorem 6

1|qpsd,pj,r = gj (r)pj ,CONW|∑(αEj + βTj + γ d + δD) O(n3) Theorem 3

1|qpsd,pj,r = g(r)pj |Cmax O(n logn) Theorem 7

1|qpsd,pj,r = g(r)pj |∑Cj O(n logn) Theorem 7

1|qpsd,pj,r = g(r)pj |∑∑ |Ci − Cj | O(n logn) Theorem 7

1|qpsd,pj,r = g(r)pj ,CON|∑(αEj + βTj + γ d) O(n logn) Theorem 4

1|qpsd,pj,r = g(r)pj ,SLK|∑(αEj + βTj + γ dj ) O(n logn) Theorem 5

1|qpsd,pj,r = g(r)pj ,DIF|∑(αEj + βTj + γ dj ) O(n logn) Theorem 6

1|qpsd,pj,r = g(r)pj ,CONW|∑(αEj + βTj + γ d + δD) O(n logn) Theorem 3

processing times and past-sequence-dependent delivery times. By convert it to an as-signment problem, We provided a polynomial time solution for the unified problem.We then show how this unified model can be useful in solving scheduling prob-lems with or without due date assignment considerations. We also show that theproblems can be solved by simple polynomial-time algorithms for the spacial casewith job-independent positional function. Scheduling with position-dependent pro-cessing times and past-sequence-dependent delivery times in other machine settingsare clearly interesting and significant topics for future research.


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Single machine scheduling problems with general position-dependent processing times and past-sequence-dependent delivery times