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A note on the scheduling problem with revised deliverytimes and job-dependent tardiness penaltiesChristos Koulamasa

a Department of Decision Sciences and Information SystemsFlorida International University,Miami, FL, USAAccepted author version posted online: 23 Oct 2013.Published online: 11 Mar 2014.

To cite this article: Christos Koulamas (2014) A note on the scheduling problem with revised delivery times and job-dependent tardiness penalties, IIE Transactions, 46:6, 619-622, DOI: 10.1080/0740817X.2013.851435

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IIE Transactions (2014) 46, 619–622Copyright C© “IIE”ISSN: 0740-817X print / 1545-8830 onlineDOI: 10.1080/0740817X.2013.851435

A note on the scheduling problem with revised delivery timesand job-dependent tardiness penalties

CHRISTOS KOULAMAS

Department of Decision Sciences and Information Systems, Florida International University, Miami, FL 33199, USAE-mail: koulamas@fiu.edu

Received October 2012 and accepted July 2013

This article considers the single-machine scheduling problem with revised delivery times and job-dependent tardiness penalties. Itis shown that the exact dynamic programming algorithm for the simpler problem with job-independent tardiness penalties can beextended to the problem with job-dependent tardiness penalties. When the job-dependent tardiness penalties are bounded by apolynomial function of the number of jobs, the dynamic programming algorithm can be converted to a Fully Polynomial TimeApproximation Scheme by implementing the “scaling the input” technique.

Keywords: single machine, tardiness, FPTAS

1. Introduction

In numerous scheduling problems, the delivery times (duedates) are negotiated between the shop manager andthe customers. Subsequently, it is often realized that thepromised delivery times cannot be met due to unforeseendelays in the delivery of parts and/or subassemblies. Inthose cases, the shop manager negotiates revised deliverytimes with the customers. These negotiations usually resultin penalties proportional to the delivery delays and also inthe outsourcing of certain jobs or in additional tardinesspenalties for these jobs if they are late even with respect tothe revised delivery times.

Steiner and Zhang (2011) defined this problem asthe revised delivery-time quotation problem with tardi-ness penalties and showed that it is equivalent to the1/rej/

∑j∈S Tj + ∑

j∈R e j problem; that is, to the single-machine total tardiness problem with job rejection (out-sourcing). In the 1/rej/

∑j∈S Tj + ∑

j∈R e j problem, thereare n jobs available at time 0 to be processed non-preemptively on a single machine; job j has a processingtime p j , a due date d j , and a rejection (outsourcing) cost e j .We can assume, without loss of generality, that all data arenon-negative integers. Any job can be either scheduled orrejected. If job j is scheduled, then its tardiness is definedas Tj = max(0, Cj − d j ), where Cj is the completion timeof job j . If job j is rejected, then its rejection (outsourcing)cost e j must be paid. The objective is to determine a job se-quence so that the

∑j∈S Tj + ∑

j∈R e j value is minimizedwhere S, R denote the subsets of the scheduled and therejected jobs, respectively.

The equivalence of the 1/rej/∑

j∈S Tj + ∑j∈R e j prob-

lem to the revised delivery-time quotation problem followsfrom the observation that d j corresponds to the originallyquoted delivery time for job j , Cj corresponds to the revised(actual) delivery time of job j , Tj corresponds to the lin-ear penalty due to the delayed delivery, and e j correspondsto either the outsourcing cost (if job j is outsourced) orto an additional (higher) penalty cost if job j is processedin-house and delivered late even with respect to its reviseddelivery time.

Steiner and Zhang (2011) proposed an O(n4 P3)Dynamic Programming (DP) algorithm for the1/rej/

∑j∈S Tj + ∑

j∈R e j problem (where P = ∑nj=1 p j )

by generalizing Lawler’s O(n4 P) DP algorithm (Lawler,1977) for the 1//

∑Tj problem (without the job rejection

option).Steiner and Zhang (2011) converted their DP algorithm

to a Fully Polynomial Time Approximation Scheme (FP-TAS) by emulating Lawler’s conversion of his DP algorithmto a FPTAS (Lawler, 1982) and also by implementing Ko-valyov’s bound improvement procedure (Kovalyov, 1995).The Steiner–Zhang FPTAS runs in

O(

n10 log n + n10

ε3+ n2 log n log max

j=1,...,n{e j }

)time,

because the bound improvement procedure takesO(n10 log n) time, the initial bounds are obtained inO(n2 log n log max j=1,...,n{e j }) time, and the final imple-mentation of the scaling scheme runs in O(n10/ε3) time,where ε denotes the desired level of approximation.

0740-817X C© 2014 “IIE”

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620 Koulamas

The objective of this note is to extend the above resultsto the more general 1/rej/

∑j∈S w j Tj + ∑

j∈R e j weightedtardiness problem with job rejection (outsourcing) in whichw j denotes the weight (importance) of job j . In the1/rej/

∑j∈S w j Tj + ∑

j∈R e j problem, the delayed deliverypenalties are job-dependent, enabling the shop manager totake into account the relative importance of each job whenpreparing the delivery schedules.

It is well known that the weighted tardiness 1//∑

w j Tjproblem (without job rejection and with arbitrary jobweights) is strongly NP-hard (Lawler, 1977). Therefore, itis unlikely that a pseudo-polynomial algorithm can be de-veloped for the 1//

∑w j Tj problem (and therefore for the

more general 1/rej/∑

j∈S w j Tj + ∑j∈R e j problem) unless,

of course, P = NP.Following Lawler (1977), we focus on the agreeable case

in which the inequality pi < p j implies that wi ≥ w j for anytwo jobs i , j . In that case, the job weights (and thereforethe actual tardiness penalties) are disproportionate to thejob sizes. Kanet (2007) stated that this may be the casein assembly operations in which the opportunity cost ofa tardy component might be the lost revenue or the delaycost of the entire assembly.

We close this section by providing an outline on howthe DP algorithm of Steiner and Zhang (2011) can be ex-tended to solve the 1/agr, rej/

∑j∈S w j Tj + ∑

j∈R e j prob-lem (with agreeable job processing times and weights).

According to Lawler (1977), let S(i, j, k) denote the jobsubset in the interval i, i + 1, ..., j with processing times lessthan pk; formally, S(i, j, k) = { j ′ : i ≤ j ′ ≤ j, p′

j < pk}.Steiner and Zhang (2011) defined the state v(S(i, j, k), s, t)where s, t denote the starting time and the completiontime, respectively, of the non-rejected jobs in the S(i, j, k)subset. In order to determine the minimum cost value forv(S(i, j, k), s, t), the cost function v̄(S(i, j, k), s, t, δ) wascomputed as

v̄(S(i, j, k), s, t, δ)= min

τ≥s[v(S(i, k′ + δ, k′), s, τ − pk′) + max(0, τ − dk′)

+ v(S(k′ + δ + 1, j, k′), τ, t)],

where job k′ with p′k = max{p′

j : j ′ ∈ S(i, j, k)} is sched-uled in the δth position in the non-rejected job subset fromS(i, j, k) and τ is the completion time of job k′.

The cost function v̄(S(i, j, k), s, t, k′) = v(S(i, j, k) −{k′}, s, t) + e′

k was also computed as the cost of rejectingjob k′. Therefore,

v(S(i, j, k), s, t) = min[v̄(S(i, j, k), s, t, k′),min

δ[v̄(S(i, j, k), s, t, δ)]. (1)

There are no more than n3 P2 states and the evaluationof Equation (1) for all possible δ and τ values results in anoverall O(n4 P3) running time. It should be mentioned thatwhen S(k′ + δ + 1, j, k′) = ∅, there is no need to know the

value of τ and in that case Steiner and Zhang (2011) useda more efficient recursive equation.

The above recursive equation can be extended to solvethe 1/agr, rej/

∑j∈S w j Tj + ∑

j∈R e j problem by replacingthe max(0, τ − dk′) term with the w′

k max(0, τ − dk′) termin the computation of the v̄(S(i, j, k), s, t, δ) value; there-fore, the 1/agr, rej/

∑j∈S w j Tj + ∑

j∈R e j problem can besolved in O(n4 P3) time by the above DP algorithm.

In the next section, we focus on how to convert this DPalgorithm into a FPTAS for the 1/agr, rej/

∑j∈S w j Tj +∑

j∈R e j problem.

2. A FPTAS for the 1/agr, rej/∑

j∈S w j Tj + ∑j∈R e j

problem

We begin our presentation with an outline of the stepsof the FPTAS of Steiner and Zhang (2011) for the1/rej/

∑j∈S Tj + ∑

j∈R e j problem. These steps can besummarized as follows:

1. Determine a pair of initial bounds B ≤ v∗ ≤ nB wherev∗ denotes the optimal (minimal) cost value forthe 1/rej/

∑j∈S Tj + ∑

j∈R e j problem. This is accom-plished by constructing and solving a series of minmaxtardiness problems in O(n2 log n log max j=1,...,n{e j })time.

2. Using the bounds determined in Step 1, construct aK-scaled problem and solve it using the optimal DPalgorithm in order to determine a pair of tight boundsX ≤ v∗ ≤ 3X. This step takes O(n10 log n) time.

3. Set K = εX/n(n + 1) and run the optimal DP algorithmfor the K-scaled problem. This step takes O(n4 X3/K3) ≈O(n10/ε3) time and yields a solution to the problem witha cost value less than or equal to (1 + ε)v∗.

In the above procedure, the K-scaled problem is definedwith p′

j = p j/K�, d ′j = d j/K , and e′

j = e j/K , where �denotes the floor function; this function returns the largestinteger that is not greater than its argument.

Let s ′ be the optimal sequence for the K-scaled problemwith cost v′ and let v′′ be the cost of the s ′ sequence withthe original p j , d j , w j , e j values. Clearly,

v′′ =∑j∈S

max(0, C′j − d j ) +

∑j∈R

e j

≤ K

⎧⎨⎩∑j∈S

max

⎛⎝0,

∑l≤ j

(⌊ pl

K

⌋+ 1

)− d j

K

⎞⎠ +

∑j∈R

(e j

K

)⎫⎬⎭

= K

⎧⎨⎩

∑j∈S

max

⎛⎝0,

∑l≤ j

p′j − d

′j

⎞⎠ +

∑j∈R

e′j

⎫⎬⎭

+ K |S| (|S| + 1)= Kv

′ + K |S| (|S| + 1) ≤ Kv′ + Kn(n + 1), (2)

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FPTAS 621

where C′j denotes the completion time of job j in its actual

position in s ′, the l ≤ j summation range implies that thesummation includes job j and all non-rejected jobs sched-uled before j in s ′ and |S| denotes the cardinality of thesubset S; observe that the first inequality in Equation (2)holds because of the inequality p j ≤ K{p j/K� + 1}.

The inequality Kp′j ≤ p j yields

Kv′ ≤ v∗, (3)

because v′, v∗ are both optimal for their respective prob-lems. By combining Equations (2) and (3):

v′′ ≤ v∗ + Kn(n + 1). (4)

It is implicitly assumed that |S| > 0; if the problem datastructure implies that |S| = 0 (e.g., when

∑nj=1 e j ≤ B),

then any selection of K > 0 induces no error term. Inthat case, the optimal solution is S = ∅, R = {1, . . . , n} andthere is no reason to run the FPTAS. We will add this stepto the proposed FPTAS for the 1/agr, rej/

∑j∈S w j Tj +∑

j∈R e j problem.In the case of the 1/agr, rej/

∑j∈S w j Tj + ∑

j∈R e j prob-lem, the implementation of the above scaling schememodifies the

∑j∈S max(0,

∑l≤ j (pl/K� + 1) − (d j/K))

term in the first inequality of Equation (2) to∑j∈S w j max(0,

∑l≤ j (pl/K� + 1) − (d j/K)), which con-

verts the Kn(n + 1) error term to K∑n

i=1 iwi ≈O(nK

∑ni=1 wi ).

Also, the computation of the initial bounds according toStep 1 will necessitate the solution of a series of minmaxweighted tardiness problems.

It is well known that there are two standard approachesto convert a DP algorithm to an FPTAS.

The approach of “thinning out” the state space ofthe DP algorithm in order to convert it to an FPTASis not directly applicable to the DP algorithm for the1/agr, rej/

∑j∈S w j Tj + ∑

j∈R e j problem because this DPalgorithm uses a state update scheme similar to the oneused in the DP algorithm for the 1//

∑Tj problem and,

according to Woeginger (2000), this scheme inhibits theimplementation of the “thinning out” the state spaceapproach.

The “scaling the input” approach should round the jobprocessing times because, as pointed out by a referee, theDP algorithm of Lawler (1977) assumes that all job pro-cessing times are non-negative integers.

The above observations imply that in order to develop anFPTAS for the 1/agr, rej/

∑j∈S w j Tj + ∑

j∈R e j problemwe should first derive a pair of initial bounds B ≤ v∗ ≤ nBfor the 1/agr, rej/

∑j∈S w j Tj + ∑

j∈R e j problem (wherev∗ now denotes the optimal (minimal) cost value for the1/agr, rej/

∑j∈S w j Tj + ∑

j∈R e j problem) and then roundthe job processing times. In order to end up with an FPTAS,we need to impose the additional restriction of w j ≤ g(n)for all j = 1, . . . , n, where g(n) is a polynomial function of

n. In that case, the K∑n

i=1 iwi error term is bounded byKn2g(n), which is polynomial in n.

2.1. A pair of initial bounds for the1/agr, rej/

∑j∈S w j Tj + ∑

j∈R e j problem

Let us consider the 1/agr, rej/ max j∈S{w j Tj } + max j∈R{e j }problem in which the objective is to minimize the sum ofthe maximum weighted tardiness of the scheduled jobsand the maximum rejection cost of the rejected jobs ona single machine. To the best of our knowledge, the1/agr, rej/ max j∈S{w j Tj } + max j∈R{e j } problem has notbeen considered in the literature.

Clearly, B ≤ v∗, where B denotes the optimal(minimal) max j∈S{w j Tj } + max j∈R{e j } value for the1/agr, rej/ max j∈S{w j Tj } + max j∈R{e j } problem.

It is easy to show that in an optimal1/agr, rej/ max j∈S{w j Tj } + max j∈R{e j } sequence, thejobs j ∈ S are sequenced first according to Lawler’salgorithm (Lawler, 1973), which solves the 1// max{w j Tj }problem optimally in O(n2) time, followed by the jobsj ∈ R sequenced in any order. Let us renumber the jobsaccording to the non-increasing order of their e j valueswith ties broken in favor of the maximum w j value so thate1 ≥ e2 ≥ · · · ≥ en.

Lemma 1. In an optimal 1/agr, rej/ max j∈S{w j Tj } +max j∈R{e j } sequence, either S = ∅ or S = {1, . . . , k} with|S| = k for some k, 1 ≤ k ≤ n.

Proof. Let us assume that |S| < k and that job l >

1 is the lowest numbered job such that l /∈ S; thatis, S = {1, 2, . . . , l − 1} ∪ S′; in that case, max j∈R{e j } =el = max j=l,...,n{e j }. If S′ = ∅, then all jobs in S′ canbe transferred to R without increasing the incumbentmax j∈S{w j Tj } and max j∈R{e j } values; subsequently, S ={1, . . . , l − 1} and the lemma is proved with k = l − 1. �

Lemma 1 leads to the following AlgorithmA1 to determine an optimal sequence for the1/agr, rej/ max j∈S{w j Tj } + max j∈R{e j } problem andits cost B. Recall that the jobs have been renumbered sothat e1 ≥ e2 ≥ · · · ≥ en. Also, define en+1 = 0.

Algorithm A1.

Step 0. Let S = ∅, R = {1, 2, . . . ,n}, B = e1, k = 1.Step 1. Let Stemp = S ∪ {k}, Rtemp = R − {k}, Btemp =

max j∈Sj∈Stemp{w j Tj } + ek+1 where the jobs in Stemp

are sequenced according to Lawler’s algorithm forthe 1// max{w j Tj } problem.

Step 2. If Btemp ≥ B, terminate.Step 3. S = Stemp, R = Rtemp, B = Btemp.Step 4. If k = n, terminate.Step 5. Set k = k + 1 and go to Step 1.

End.

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Lawler’s algorithm for the 1// max{w j Tj } problem runsin O(n2) time; therefore, each step of Algorithm A1 runs inO(n2) time, which implies that Algorithm A1 runs in O(n3)time.

Consequently, we can derive an initial lower bound B ≤v∗ for the 1/agr, rej/ max j∈S{w j Tj } + max j∈R{e j } problemin O(n3) time.

Let |S| = k, |R| = n − k in the optimal 1/agr, rej/max j∈S{w j Tj } + max j∈R{e j } sequence determined byAlgorithm A1, where 1 ≤ k ≤ n. This sequence is also afeasible sequence for the 1/agr, rej/

∑j∈S w j Tj + ∑

j∈R e j

problem with∑j∈S

w j Tj +∑j∈R

e j ≤ k maxj∈S

{w j Tj } + (n − k) maxj∈R

{e j }

≤ n(

maxj∈S

{w j Tj } + maxj∈R

{e j })

= nB.

Therefore, a pair of initial bounds B ≤ v∗ ≤ nB is derivedfor the 1/agr, rej/

∑j∈S w j Tj + ∑

j∈R e j problem in O(n3)time.

We conclude this section by summarizing the steps todevelop a FPTAS for the 1/agr, rej/

∑j∈S w j Tj + ∑

j∈R e j

problem.

1. Apply Algorithm A1 to obtain a pair of initial boundsB ≤ v∗ ≤ nB in O(n3) time.

2. If∑n

j=1 e j ≤ B, stop; the optimal solution is S = ∅, R ={1, . . . , n}.

3. Run a bound improvement procedure to replace the ini-tial B ≤ v∗ ≤ nB bounds with the tighter X ≤ v∗ ≤ 3Xbounds. This step takes O(n10g(n) log n) time.

4. Set K = εX/n2g(n) and run the DP algorithm for the K-scaled problem in O(n4 X3/K3) ≈ O(n10g(n)/ε3) time.

The resulting solution has a cost of v′′, takes a total of

O(

n10g(n)ε3

+ n10g(n) log n + n3)

≈ O(

n10g(n)ε3

+ n10g(n) log n)

time,

and is an FPTAS because v′′ ≤ v∗ + Kn2g(n) ≤ v∗ + εX ≤(1 + ε)v∗.

3. Conclusions

We considered the 1/agr, rej/∑

j∈S w j Tj + ∑j∈R e j prob-

lem. We showed that the exact DP algorithm for the simpler1/rej/

∑j∈S Tj + ∑

j∈R e j problem can be extended to the

1/agr, rej/∑

j∈S w j Tj + ∑j∈R e j problem. We also showed

that when w j ≤ g(n) for all j = 1, . . . , n (where g(n) is poly-nomial function of n), the DP algorithm can be convertedto an FPTAS by implementing the “scaling the input”technique.

Our main objective was to develop an FPTAS for the1/agr, rej/

∑j∈S w j Tj + ∑

j∈R e j problem. It remains achallenging question to determine whether an alternativeapproach can improve the running time of the FPTAS.

Acknowledgement

The author would like to thank the referees for their in-sightful comments about the structure of the FPTAS, whichhelped improve earlier versions of this note.

References

Kanet, J.J. (2007) New precedence theorems for one-machine weightedtardiness. Mathematics of Operations Research, 32(3), 579–588.

Kovalyov, M.Y. (1995) Improving the complexities of approximationalgorithms for optimization problems. Operations Research Letters,17(2), 85–87.

Lawler, E.L. (1973) Optimal sequencing of a single processor subject toprecedence constraints. Management Science, 19(5), 544–546.

Lawler, E.L. (1977) A “pseudopolynomial” algorithm for sequencingjobs to minimize total tardiness. Annals of Discrete Mathematics, 1,331–342.

Lawler, E.L. (1982) A fully polynomial approximation scheme for thetotal tardiness problem. Operations Research Letters, 1(6), 207–208.

Steiner, G. and Zhang, R. (2011) Revised delivery-time quotation inscheduling with tardiness penalties. Operations Research, 59(6),1504–1511.

Woeginger, G.J. (2000) When does a dynamic programming formulationguarantee the existence of a fully polynomial time approximationscheme (FPTAS)? INFORMS Journal on Computing, 12(1), 57–74.

Biography

Christos Koulamas is Ryder Eminent Scholar, Senior Associate Dean,and Chairperson of the Department of Decision Sciences and Informa-tion Systems at Florida International University. He received his Ph.D.in 1985 from Texas Tech University. His research interests include op-erations management and supply chain management, among others.He has published over 90 refereed journal articles in journals such asOperations Research, INFORMS Journal on Computing, Production andOperations Management, Operations Research Letters, IIE Transactions,Naval Research Logistics, Decision Sciences, Discrete Applied Mathe-matics, ASME Transactions, European Journal of Operational Research,Computers & Operations Research, Information Processing Letters, Inter-national Journal of Production Economics, International Journal of Pro-duction Research, and the Journal of the Operational Research Society,among others.

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