Research Article Parallel-Machine Scheduling with Time ... · bound of any optimal schedule. For...

Research ArticleParallel-Machine Scheduling with Time-Dependent andMachine Availability Constraints

Cuixia Miao and Juan Zou

School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong 273165, China

Correspondence should be addressed to Cuixia Miao; [email protected]

Received 9 February 2015; Revised 11 April 2015; Accepted 12 April 2015

Academic Editor: Chin-Chia Wu

Copyright © 2015 C. Miao and J. Zou. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We consider the parallel-machine scheduling problem in which the machines have availability constraints and the processing timeof each job is simple linear increasing function of its starting times. For the makespan minimization problem, which is NP-hardin the strong sense, we discuss the Longest Deteriorating Rate algorithm and List Scheduling algorithm; we also provide a lowerbound of any optimal schedule. For the total completion time minimization problem, we analyze the strong NP-hardness, andwe present a dynamic programming algorithm and a fully polynomial time approximation scheme for the two-machine problem.Furthermore, we extended the dynamic programming algorithm to the total weighted completion time minimization problem.

1. Introduction

Consider the following scheduling problem with deteriorat-ing job and machine availability constrains.There are 𝑛 inde-pendent deteriorating jobs 𝐽 = {𝐽

1, . . . , 𝐽

𝑛} to be processed on

𝑚 identical parallel machines. The actual processing time ofjob 𝐽𝑗(𝑗 = 1, . . . , 𝑛) is 𝑝

𝑗= 𝛼𝑗𝑡, where 𝛼

𝑗(>0) and 𝑡 are the

deteriorating rate and the starting time of job 𝐽𝑗, respectively.

Each job 𝐽𝑗has a weight 𝑤

𝑗. All jobs are released at time

𝑡0(>0). The case with 𝑡

0= 0 is not considered because the

job would have its processing rimes equal to zero if 𝑡0= 0.

We assume that the jobs are nonresumable in our problems.Machine𝑀

𝑖(1 ≤ 𝑖 ≤ 𝑚) is not continuously available and

unavailable during the period 𝐼𝑖= [𝐻(𝑖)

1, 𝐻(𝑖)

2). In addition, we

assume that 𝑡0(1+𝛼max) < 𝐻

𝑖

1< 𝐻𝑖

2and𝐻(𝑖)

1< 𝑡0∏𝑛

𝑗=1(1+𝛼𝑗),

where 𝛼max = max𝑗=1,...,𝑛

{𝛼𝑗}. Otherwise, all the jobs can

be finished before the nonavailable period and the problembecomes trivial. Without loss of generality, we assume that allparameters are integral unless stated otherwise.

Our objective is to minimize the makespan and the total(weighted) completion time. Following Gawiejnowicz [1], wedenote our problems as 𝑃

𝑚|𝑛𝑟 − 𝑎, 𝑝

𝑗= 𝛼𝑗𝑡|𝐶max and 𝑃𝑚|𝑛𝑟 −

𝑎, 𝑝𝑗= 𝛼𝑗𝑡| ∑(𝑤

𝑗)𝐶𝑗, where 𝑛𝑟 − 𝑎means nonresumable.

The model described above falls into the categoriesof the scheduling with deteriorating job and the machine

scheduling with availability constraints. The scheduling withdeteriorating job was first considered by J. N. D. Gupta andS. K. Gupta [2] and Browne and Yechiali [3]. Cheng et al.[4] gave a survey and the monograph by Gawiejnowicz [1]presented this scheduling from different perspectives andcovers results and examples. Graves and Lee [5] pointed outthatmachine schedulingwith an availability constraint is veryimportant and is still relatively unexplored.They studied thisproblem whose maintenance needs to be performed within afixed period. Lee [6] presented extensive study of single andparallel machine scheduling problems with an availabilityconstraint, with respect to various performancemeasures andtwo cases are considered: resumable and nonresumable. Ajob is said to be resumable if it cannot be finished before thenonavailable interval of a machine and can continue after themachine is available again. On the other hand, a job is said tobe nonresumable if it has to restart rather than continue. Maet al. [7] gave a survey of this scheduling.

In this paper, we consider the deteriorating job schedulingwith machine availability constraints on 𝑚 identical parallelmachines. The jobs are nonresumable and our objective is tominimize the makespan and the total (weighted) completiontime.

Relevant PreviousWork. Wu and Lee [8] initiated the deterio-rating job scheduling with machine availability constraints;

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015, Article ID 956158, 6 pageshttp://dx.doi.org/10.1155/2015/956158

2 Mathematical Problems in Engineering

they showed that minimizing the makespan of schedulingdeteriorating jobs on a single machine with an availabilityconstraint can be transformed into 0-1 integer programming.Ji et al. [9] gave some results for the linear deterioratingjobs with an availability constraint on a single machine.Gawiejnowicz and Kononov [10] considered the complexityand approximability of scheduling resumable proportionallydeteriorating jobs. Fan et al. [11] considered the schedul-ing resumable deteriorating jobs on a single machine withnonavailability constraints. Li and Fan [12] addressed thenonresumable scheduling problem 1|𝑛𝑟−𝑎, 𝑝

𝑗= 𝛼𝑗𝑡| ∑𝑤

𝑗𝐶𝑗.

The problems they considered are on the single machine.In this paper, we consider the parallel-machine schedulingproblem with deteriorating jobs and machine availabilityconstraints, and we show that the problems are strongly NP-hard and present some algorithms.

2. Minimizing the Makespan

In this section, we first show that problem 𝑃𝑚|𝑛𝑟 − 𝑎, 𝑝

𝑗=

𝛼𝑗𝑡|𝐶max is NP-hard in the strong sense. Ji and Cheng [13]

showed that problem 𝑃𝑚|𝑝𝑗= 𝛼𝑗𝑡|𝐶max is NP-hard in the

strong sense when 𝑚 is arbitrary. In their problem, all themachines are available all the time.Thus, our problem𝑃

𝑚|𝑛𝑟−

𝑎, 𝑝𝑗= 𝛼𝑗𝑡|𝐶max is NP-hard in the strong sense when 𝑚 is

arbitrary.In the following, we discuss the Longest Deteriorating

Rate (LDR for short) algorithm and List Scheduling (LS forshort) algorithm and analyze a lower bound of any optimalschedule.

2.1. LDR and LS Algorithms

LS Algorithm. Given a sequence of jobs 𝐽1, . . . , 𝐽

𝑛, assign the

jobs one by one according to the list. Each job is assigned tothemachine where the job can be finished as early as possible.

LDR Algorithm. Sort the jobs in the nonincreasing orderof their deteriorating rates, and then assign the jobs by LSalgorithm.𝐶LDRmax , 𝐶

LSmax, and 𝐶

∗

max denote the makespan correspond-ing LDR, LS, and the optimal solution, respectively.

Theorem 1. 𝐶𝐿𝐷𝑅max /𝐶∗

max can be arbitrarily large even for thetwo-machine problem 𝑃

2|𝑛𝑟 − 𝑎, 𝑝

𝑗= 𝛼𝑗𝑡|𝐶max.

Proof. Consider a problem with the following instance: 𝑡0=

1, 𝛼1= 𝛼2= 7, 𝛼

3= 𝛼4= 𝛼5= 3, 𝐻(1)

1= 𝐻(2)

1= 64, 𝐻(1)

2=

𝐻(2)

2= 𝑛, and𝑚 = 2. In the optimal schedule with makespan

𝐶∗

max = 64, jobs 𝐽1 and 𝐽2 are scheduled before𝐻(1)

1on the first

machine, and jobs 𝐽3, 𝐽4, and 𝐽

5are scheduled before𝐻(2)

1on

the secondmachine. However, 𝐶LDRmax = 4𝑛. As a consequence,

𝐶LDRmax /𝐶

∗

max = 4𝑛/64 → ∞ when 𝑛 → ∞.

Theorem 2. If min𝑖=1,...,𝑚

{𝐻(𝑖)

1} ≥ 𝑡

0∏𝑛

𝑗=1(1 + 𝛼

𝑗), then

𝐶𝐿𝑆

max/𝐶∗

max ≤ (1 + 𝛼max)(𝑚−1)/𝑚 and 𝐶𝐿𝐷𝑅max /𝐶

∗

max ≤ (1 +

𝛼min)(𝑚−1)/𝑚, where 𝛼max = max

𝑗=1,...,𝑛{𝛼𝑗} and 𝛼min =

min𝑗=1,...,𝑛

{𝛼𝑗}.

The proof of this theorem is similar to the proofs ofTheorems 1 and 3 in Liu et al. [14].

2.2. Lower Bound of Any Optimal Schedule. Without loss ofgenerality, let 𝑡

0= 1, 𝐻

1= max

𝑖=1,...,𝑚{𝐻(𝑖)

1}, and 𝐻

2=

min𝑖=1,...,𝑚

{𝐻(𝑖)

2}.

Theorem 3.

𝐶∗

max ≥𝐻2(1 + 𝛼min)

𝑛/𝑚

𝐻1

. (1)

Proof. In any optimal schedule, let 𝐽𝑆𝑖

and 𝐽𝐵𝑆𝑖

denote theset of jobs scheduled on machine 𝑀

𝑖and before 𝐻𝑖

1on

𝑀𝑖(𝑖 = 1, . . . , 𝑚), respectively. Then, the set of jobs sched-

uled after 𝐻𝑖2is 𝐽𝑆𝑖/𝐽𝐵

𝑆𝑖, and ∏

𝐽𝑗∈𝐽𝐵

𝑆𝑖

(1 + 𝛼𝑗) ≤ 𝐻

𝑖

1. We

have the load of machine 𝑀𝑖denoted by 𝐿

𝑖holding 𝐿

𝑖=

𝐻𝑖

2∏𝐽𝑗∈𝐽𝑆𝑖/𝐽𝐵

𝑆𝑖

(1 + 𝛼𝑗) ≥ 𝐻

𝑖

2∏𝐽𝑗∈𝐽𝑆𝑖

(1 + 𝛼𝑗)/𝐻𝑖

1for 𝑖 =

1, . . . , 𝑚. Note that 𝐶∗max = max𝑖=1,...,𝑚

{𝐿𝑖}. Thus, 𝐶∗max ≥

𝑚√∏𝑚

𝑖=1𝐿𝑖≥𝑚√𝐻1

2⋅ ⋅ ⋅ 𝐻𝑚

2∏𝑛

𝑗=1(1 + 𝛼

𝑗)/𝐻1

1⋅ ⋅ ⋅ 𝐻𝑚

1≥ 𝐻2(1 +

𝛼min)𝑛/𝑚/𝐻1.

3. Minimizing the Total Completion Time

In this section, we discuss the total completion time mini-mization problem.

Ji and Cheng [13] showed that problem 𝑃𝑚|𝑝𝑗= 𝛼𝑗𝑡| ∑𝐶

𝑗

is NP-hard in the strong sense when 𝑚 is arbitrary, whichimplies that our problem 𝑃

𝑚|𝑛𝑟−𝑎, 𝑝

𝑗= 𝛼𝑗𝑡| ∑𝐶

𝑗is NP-hard

in the strong sense when𝑚 is arbitrary.

3.1. Dynamic Programming Algorithm. In this subsection,we present a dynamic programming algorithm for 𝑃

2|𝑛𝑟 −

𝑎, 𝑝𝑗= 𝛼𝑗𝑡| ∑𝐶

𝑗when machine𝑀

2is always available. For

convenience, let 𝑡0= 1, and let the only nonavailable interval

on machine𝑀1be [𝐻(1)

1, 𝐻(1)

2), 𝛼 = ∏𝑛

𝑗=1(1 + 𝛼

𝑗), and 𝛼𝑗 =

∏𝑗

𝑖=1(1 + 𝛼

𝑖).

Smallest Deteriorating Rate (SDR for Short). Sort the jobs inthe nondecreasing order of their deteriorating rates such that𝛼1≤ 𝛼2≤ ⋅ ⋅ ⋅ ≤ 𝛼

𝑛.

Lemma 4. In any optimal solution to problem 𝑃2|𝑛𝑟 − 𝑎, 𝑝

𝑗=

𝛼𝑗𝑡| ∑𝐶

𝑗, the jobs scheduled before the nonavailable interval

are processed by the SDR order and so are the jobs scheduledafter the nonavailable interval and on machine𝑀

2.

We can proof this lemma by the interchanging argument.We assume that the jobs are reindexed in the SDR order.

Let𝑓𝑗(V, 𝑢) denote the optimal value of the objective function

Mathematical Problems in Engineering 3

satisfying the following conditions:

(i) The jobs in consideration are 𝐽1, . . . , 𝐽

𝑗.

(ii) The total processing time of𝑀1before𝐻(1)

1is V.

(iii) The total processing time of𝑀2is 𝑢.

To get 𝑓𝑗(V, 𝑢), we distinguish three cases as follows.

Case 1. Job 𝐽𝑗is scheduled before𝐻(1)

1.

In this case, 𝑢 does not change. The starting time of 𝐽𝑗is

(V + 1)/(1 + 𝛼𝑗), the total processing time of𝑀

1before 𝐻(1)

1

is (V + 1)/(1 + 𝛼𝑗) − 1 = (V − 𝛼

𝑗)/(1 + 𝛼

𝑗) before inserting job

𝐽𝑗, and the completion of 𝐽

𝑗is V+1. Thus, 𝑓

𝑗(V, 𝑢) = 𝑓

𝑗−1((V−

𝛼𝑗)/(1 + 𝛼

𝑗), 𝑢) + V + 1.

Case 2. Job 𝐽𝑗is scheduled on machine𝑀

2.

In this case, V does not change. The starting time of 𝐽𝑗is

(𝑢+1)/(1+𝛼𝑗), the total processing time of𝑀

2is (𝑢+1)/(1+

𝛼𝑗) − 1 = (𝑢 − 𝛼

𝑗)/(1 + 𝛼

𝑗) before inserting job 𝐽

𝑗, and the

completion of 𝐽𝑗is 𝑢+1. Thus, 𝑓

𝑗(V, 𝑢) = 𝑓

𝑗−1(V, (𝑢−𝛼

𝑗)/(1+

𝛼𝑗)) + 𝑢 + 1.

Case 3. Job 𝐽𝑗is scheduled after𝐻(1)

2.

In this case, both 𝑢 and V do not change. The completionof 𝐽𝑗is𝐻(1)2𝛼𝑗/(V+1)(𝑢+1) since𝛼𝑗 = ∏𝑗

𝑖=1(1+𝛼𝑖) = (V+1)(𝑢+

1)∏𝐽𝑙∈𝐽𝑗

𝐻(1)

2

(1+𝛼𝑙), where 𝐽𝑗

𝐻(1)

2

denotes the set of jobs scheduled

after𝐻(1)2. Thus, 𝑓

𝑗(V, 𝑢) = 𝑓

𝑗−1(V, 𝑢) + 𝐻(1)

2𝛼𝑗/(V + 1)(𝑢 + 1).

Combining the above cases, we design a dynamic pro-gramming algorithm as follows.

Algorithm DP1.

Step 1. Reindex the jobs in nondecreasing order of theirdeteriorating rates such that 𝛼

1≤ 𝛼2≤ ⋅ ⋅ ⋅ ≤ 𝛼

𝑛.

Step 2 (Initialization).

𝑓1(V, 𝑢) =

{{{{{{{

{{{{{{{

{

V + 1 V = 𝛼1and 𝑢 = 0,

𝑢 + 1 V = 0 and 𝑢 = 𝛼1,

𝐻(1)

2(1 + 𝛼

1) V = 0 and 𝑢 = 0,

∞ otherwise.

(2)

Step 3 (Iteration).

𝑓𝑗(V, 𝑢)

= min

{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{

{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{

{

min{𝑓𝑗−1(

V − 𝛼𝑗

1 + 𝛼𝑗

, 𝑢) + V + 1, 𝑓𝑗−1(V,𝑢 − 𝛼𝑗

1 + 𝛼𝑗

) + 𝑢 + 1, 𝑓𝑗−1(V, 𝑢) +

𝐻(1)

2𝛼𝑗

(V + 1) (𝑢 + 1)} if

V − 𝛼𝑗

1 + 𝛼𝑗

,

𝑢 − 𝛼𝑗

1 + 𝛼𝑗

,

𝐻(1)

2𝛼𝑗

(V + 1) (𝑢 + 1)∈ 𝑍,

min{𝑓𝑗−1(

V − 𝛼𝑗

1 + 𝛼𝑗

, 𝑢) + V + 1, 𝑓𝑗−1(V,𝑢 − 𝛼𝑗

1 + 𝛼𝑗

) + 𝑢 + 1} ifV − 𝛼𝑗

1 + 𝛼𝑗

,

𝑢 − 𝛼𝑗

1 + 𝛼𝑗

∈ 𝑍,

𝐻(1)

2𝛼𝑗

(V + 1) (𝑢 + 1)∈𝑍,

min{𝑓𝑗−1(

V − 𝛼𝑗

1 + 𝛼𝑗

, 𝑢) + V + 1, 𝑓𝑗−1(V, 𝑢) +

𝐻(1)

2𝛼𝑗

(V + 1) (𝑢 + 1)} if

V − 𝛼𝑗

1 + 𝛼𝑗

,

𝐻(1)

2𝛼𝑗

(V + 1) (𝑢 + 1)∈ 𝑍,

𝑢 − 𝛼𝑗

1 + 𝛼𝑗

∈𝑍,

min{𝑓𝑗−1(V,𝑢 − 𝛼𝑗

1 + 𝛼𝑗

) + 𝑢 + 1, 𝑓𝑗−1(V, 𝑢) +

𝐻(1)

2𝛼𝑗

(V + 1) (𝑢 + 1)} if

𝑢 − 𝛼𝑗

1 + 𝛼𝑗

,

𝐻(1)

2𝛼𝑗

(V + 1) (𝑢 + 1)∈ 𝑍,

V − 𝛼𝑗

1 + 𝛼𝑗

∈𝑍,

𝑓𝑗−1(

V − 𝛼𝑗

1 + 𝛼𝑗

, 𝑢) + V + 1 ifV − 𝛼𝑗

1 + 𝛼𝑗

∈ 𝑍,

𝑢 − 𝛼𝑗

1 + 𝛼𝑗

,

𝐻(1)

2𝛼𝑗

(V + 1) (𝑢 + 1)∈𝑍,

𝑓𝑗−1(V,𝑢 − 𝛼𝑗

1 + 𝛼𝑗

) + 𝑢 + 1 if𝑢 − 𝛼𝑗

1 + 𝛼𝑗

∈ 𝑍,

V − 𝛼𝑗

1 + 𝛼𝑗

,

𝐻(1)

2𝛼𝑗

(V + 1) (𝑢 + 1)∈𝑍,

𝑓𝑗−1(V, 𝑢) +

𝐻(1)

2𝛼𝑗

(V + 1) (𝑢 + 1)if

𝐻(1)

2𝛼𝑗

(V + 1) (𝑢 + 1)∈ 𝑍,

V − 𝛼𝑗

1 + 𝛼𝑗

,

𝑢 − 𝛼𝑗

1 + 𝛼𝑗

∈𝑍,

∞ otherwise.

(3)

Step 4 (Solution). Theoptimal value ismin{𝑓𝑛(V, 𝑢) | V = 0, 1,

. . . , 𝐻(1)

1− 1; 𝑢 = 0, 1, . . . , 𝛼 − 1}.

Theorem 5. The problem 𝑃2|𝑛𝑟 − 𝑎, 𝑝


𝑗is solvable

in 𝑂(𝑛𝛼𝐻(1)1) time by Algorithm DP1.

Proof. The correctness of AlgorithmDP1 is guaranteed by theabove discussion. Note that V = 0, 1, . . . , 𝐻(1)

1− 1 and 𝑢 =

0, 1, . . . , 𝛼−1.Thus, the recursive function has at most𝑂((𝛼−1)(𝐻(1)

1−1)) = 𝑂(𝛼𝐻

(1)

1) states. Each iteration takes𝑂(𝑛) time

to execute. Hence, the running time is 𝑂(𝑛𝛼𝐻(1)1).


3.2. A Fully Polynomial Time Approximation Scheme. In thissubsection,we present a fully polynomial time approximationscheme for problem 𝑃

2|𝑛𝑟 − 𝑎, 𝑝


𝑗when machine

𝑀2is always available. Following Woeginger [15], we show

that 𝑃2|𝑛𝑟−𝑎, 𝑝


𝑗is DP-benevolent, which follows

that there exists a fully polynomial time approximationscheme for our problem. For convenience, let 𝑡

0= 1.

The fully polynomial time approximation scheme is basedon Lemma 4 stated in Section 3.2. Thus, we first sort the jobsin the nondecreasing order of their deteriorating rates suchthat 𝛼

1≤ 𝛼2≤ ⋅ ⋅ ⋅ ≤ 𝛼

𝑛. The dynamic programming

algorithm proposed in the following goes through 𝑛 phases.In the 𝑘th (𝑘 = 1, 2 . . . , 𝑛) phase, we input the vector 𝑋

𝑘=

[𝛼𝑘]; meanwhile, a state set 𝑆

𝑘is generated. Any state in 𝑆

𝑘is

a vector [𝑙1, 𝑙2, 𝑙3, 𝑧] which encodes a partial schedule for the

first 𝑘 jobs 𝐽1, . . . , 𝐽

𝑘. The component 𝑙

1represents the total

processing time before𝐻(1)1

onmachine𝑀1, 𝑙2represents the

total processing time after𝐻(1)2

on machine𝑀1, 𝑙3represents

the total processing time onmachine𝑀2, and the component

𝑧 represents the objective value of the current schedule. Theinitial set 𝑆

0contains the only state [1,𝐻(1)

2, 1, 0]. The state 𝑆

𝑘

is generated from the state 𝑆𝑘−1

by three mappings 𝐹1, 𝐹2, and

𝐹3which are defined as follows:

𝐹1(𝛼𝑘, 𝑙1, 𝑙2, 𝑙3, 𝑧) = [𝑙

1(1 + 𝛼

𝑘) , 𝑙2, 𝑙3, 𝑧 + 𝑙1(1 + 𝛼

𝑘)] ,

𝐹2(𝛼𝑘, 𝑙1, 𝑙2, 𝑙3, 𝑧) = [𝑙

1, 𝑙2(1 + 𝛼

𝑘) , 𝑙3, 𝑧 + 𝑙2(1 + 𝛼

𝑘)] ,

𝐹3(𝛼𝑘, 𝑙1, 𝑙2, 𝑙3, 𝑧) = [𝑙

1, 𝑙2, 𝑙3(1 + 𝛼

𝑘) , 𝑧 + 𝑙

3(1 + 𝛼

𝑘)] .

(4)

Intuitively, function 𝐹1puts job 𝐽

𝑘at the end before 𝐻(1)

1on

machine𝑀1if it is possible for the given state; and function

𝐹2puts job 𝐽

𝑘at the end after 𝐻(1)

2on machine 𝑀

1and

function 𝐹3puts job 𝐽

𝑘at the end on machine 𝑀

2. Finally,

set 𝐺(𝑙1, 𝑙2, 𝑙3, 𝑧) = 𝑧.

Combining the above discussion, we design a dynamicprogramming algorithm as follows.

Algorithm DP2.

Initialize 𝑠0:= {[1,𝐻

(1)

2, 1, 0]}

For 𝑘 = 1 to 𝑛 do

Let 𝑠𝑘:= 0

For every [𝑙1, 𝑙2, 𝑙3, 𝑧] ∈ 𝑠

𝑘−1do

[𝑙1, 𝑙2(1 + 𝛼

𝑘), 𝑙3, 𝑧 + 𝑙2(1 + 𝛼

𝑘)] assignments 𝑠

𝑘

and

[𝑙1, 𝑙2, 𝑙3(1 + 𝛼

𝑘), 𝑧 + 𝑙

3(1 + 𝛼

𝑘)] assignments 𝑠

𝑘

If 𝑙1(1 + 𝛼

𝑘) ≤ 𝐻

(1)

1

Then [𝑙1(1 +𝛼

𝑘), 𝑙2, 𝑙3, 𝑧 + 𝑙1(1 +𝛼

𝑘)] assign-

ments 𝑠𝑘

Endfor

Endfor

Output min{𝐺(𝑙1, 𝑙2, 𝑙3, 𝑧) : [𝑙

1, 𝑙2, 𝑙3, 𝑧] assignments

𝑠𝑛}.

Note that the number of states in the above dynamicprogramming is bounded by𝐻(1)

2𝑛∏𝑛

𝑗=1(1 + 𝛼

𝑗). There holds

the following result.

Theorem 6. There exists a fully polynomial time approxima-tion scheme for problem 𝑃

2|𝑛𝑟 − 𝑎, 𝑝


𝑗when one

machine is always available.

Proof. The functions𝐹1,𝐹2, and𝐹

3are vectors of polynomials

with nonnegative coefficients and the polynomial functionsin 𝐹1, 𝐹2, and 𝐹

3that yield the components are polynomials.

Moreover, all polynomials linearly depend on 𝑙1, 𝑙2, 𝑙3, and 𝑧.

The inequality inside operator “if ” can be checked in poly-nomial time. The objective function 𝐺 is a polynomial withnonnegative coefficients. Therefore, similar to the examplein Section 5.3 of Woeginger [15], it is not hard to verifythat the above dynamic programming satisfies the conditionsof Lemma 6.1 and Theorem 2.5 from [15]. Thus, problem𝑃2|𝑛𝑟 − 𝑎, 𝑝


𝑗is DP-benevolent. As a result,

there exists a fully polynomial time approximation schemefor problem 𝑃

2|𝑛𝑟 − 𝑎, 𝑝


𝑗when one machine is

always available.

4. Minimizing the Total WeightedCompletion Time

In this section, we extend the dynamic programming algo-rithm to the total weighted completion time minimizationproblem, that is, 𝑃

2|𝑛𝑟 − 𝑎, 𝑝


𝑗𝐶𝑗. We also assume

that machine𝑀2is always available.

We assume that the jobs are reindexed such that 𝑤1(1 +

𝛼1)/𝛼1≥ 𝑤2(1 + 𝛼

2)/𝛼2≥ ⋅ ⋅ ⋅ ≥ 𝑤

𝑛(1 + 𝛼

𝑛)/𝛼𝑛. For

convenience, denot this order by Weight Deteriorating Rate(WDR for short).

Lemma 7. In any optimal solution to problem 𝑃2|𝑛𝑟 − 𝑎, 𝑝

𝑗=

𝛼𝑗𝑡| ∑𝑤

𝑗𝐶𝑗, the jobs scheduled before the nonavailable interval

are processed by the WDR order and so are the jobs scheduledafter the nonavailable interval and on machine𝑀

2.

We can proof this lemma by the interchanging argument.We assume that the jobs are reindexed in the WDR

order. Similar to the context of Section 3.1, let 𝑓𝑗(V, 𝑢) denote

the optimal value of the objective function satisfying thefollowing conditions:

(i) The jobs in consideration are 𝐽1, . . . , 𝐽

𝑗.

Mathematical Problems in Engineering 5

(ii) The total processing time of𝑀1before𝐻(1)

1is V.

(iii) The total processing time of𝑀2is 𝑢.

We design a dynamic programming algorithm as follows.

Algorithm DP3.

Step 1. Reindex the jobs such that 𝑤1(1 + 𝛼

1)/𝛼1≥ 𝑤2(1 +

𝛼2)/𝛼2≥ ⋅ ⋅ ⋅ ≥ 𝑤

𝑛(1 + 𝛼

𝑛)/𝛼𝑛.

Step 2 (Initialization).

𝑓1(V, 𝑢) =

{{{{{{{

{{{{{{{

{

𝑤1(V + 1) V = 𝛼

1and 𝑢 = 0,

𝑤1(𝑢 + 1) V = 0 and 𝑢 = 𝛼

1,

𝑤1𝐻(1)

2(1 + 𝛼

1) V = 0 and 𝑢 = 0,

∞ otherwise.

(5)

Step 3 (Iteration).

𝑓𝑗(V, 𝑢)

= min

{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{

{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{

{

min{𝑓𝑗−1(

V − 𝛼𝑗

1 + 𝛼𝑗

, 𝑢) + 𝑤𝑗(V + 1) , 𝑓

𝑗−1(V,𝑢 − 𝛼𝑗

1 + 𝛼𝑗

) + 𝑤𝑗(𝑢 + 1) , 𝑓

𝑗−1(V, 𝑢) +

𝑤𝑗𝐻(1)

2𝛼𝑗

(V + 1) (𝑢 + 1)} if

V − 𝛼𝑗

1 + 𝛼𝑗

,

𝑢 − 𝛼𝑗

1 + 𝛼𝑗

,

𝐻(1)

2𝛼𝑗

(V + 1) (𝑢 + 1)∈ 𝑍,

min{𝑓𝑗−1(

V − 𝛼𝑗

1 + 𝛼𝑗

, 𝑢) + 𝑤𝑗(V + 1) , 𝑓

𝑗−1(V,𝑢 − 𝛼𝑗

1 + 𝛼𝑗

) + 𝑤𝑗(𝑢 + 1)} if

V − 𝛼𝑗

1 + 𝛼𝑗

,

𝑢 − 𝛼𝑗

1 + 𝛼𝑗

∈ 𝑍,

𝐻(1)

2𝛼𝑗

(V + 1) (𝑢 + 1)∈𝑍,

min{𝑓𝑗−1(

V − 𝛼𝑗

1 + 𝛼𝑗

, 𝑢) + 𝑤𝑗(V + 1) , 𝑓

𝑗−1(V, 𝑢) +

𝑤𝑗𝐻(1)

2𝛼𝑗

(V + 1) (𝑢 + 1)} if

V − 𝛼𝑗

1 + 𝛼𝑗

,

𝐻(1)

2𝛼𝑗

(V + 1) (𝑢 + 1)∈ 𝑍,

𝑢 − 𝛼𝑗

1 + 𝛼𝑗

∈𝑍,

min{𝑓𝑗−1(V,𝑢 − 𝛼𝑗

1 + 𝛼𝑗

) + 𝑤𝑗(𝑢 + 1) , 𝑓

𝑗−1(V, 𝑢) +

𝑤𝑗𝐻(1)

2𝛼𝑗

(V + 1) (𝑢 + 1)} if

𝑢 − 𝛼𝑗

1 + 𝛼𝑗

,

𝐻(1)

2𝛼𝑗

(V + 1) (𝑢 + 1)∈ 𝑍,

V − 𝛼𝑗

1 + 𝛼𝑗

∈𝑍,

𝑓𝑗−1(

V − 𝛼𝑗

1 + 𝛼𝑗

, 𝑢) + 𝑤𝑗(V + 1) if

V − 𝛼𝑗

1 + 𝛼𝑗

∈ 𝑍,

𝑢 − 𝛼𝑗

1 + 𝛼𝑗

,

𝐻(1)

2𝛼𝑗

(V + 1) (𝑢 + 1)∈𝑍,

𝑓𝑗−1(V,𝑢 − 𝛼𝑗

1 + 𝛼𝑗

) + 𝑤𝑗(𝑢 + 1) if

𝑢 − 𝛼𝑗

1 + 𝛼𝑗

∈ 𝑍,

V − 𝛼𝑗

1 + 𝛼𝑗

,

𝐻(1)

2𝛼𝑗

(V + 1) (𝑢 + 1)∈𝑍,

𝑓𝑗−1(V, 𝑢) +

𝑤𝑗𝐻(1)

2𝛼𝑗

(V + 1) (𝑢 + 1)if

𝐻(1)

2𝛼𝑗

(V + 1) (𝑢 + 1)∈ 𝑍,

V − 𝛼𝑗

1 + 𝛼𝑗

,

𝑢 − 𝛼𝑗

1 + 𝛼𝑗

∈𝑍,

∞ otherwise.

(6)

Step 4 (Solution). The optimal value is min{𝑓𝑛(V, 𝑢) | V =

0, 1, . . . , 𝐻(1)

1− 1; 𝑢 = 0, 1, . . . , 𝛼 − 1}.

Theorem 8. The problem 𝑃2|𝑛𝑟 − 𝑎, 𝑝


𝑗𝐶𝑗is

solvable in 𝑂(𝑛𝛼𝐻(1)1) time by Algorithm DP3.

5. Conclusions

In this paper, we considered the parallel-machine schedulingwith time-dependent and machine availability constraints.We showed that our two problems 𝑃

𝑚|𝑛𝑟 − 𝑎, 𝑝

𝑗=

𝛼𝑗𝑡|𝐶max(∑𝐶𝑗) are NP-hard in the strong sense. We ana-

lyzed the LDR and LS algorithms and the lower bound ofany optimal schedule and presented dynamic programmingalgorithm and fully polynomial time approximation schemefor the problem 𝑃

2|𝑛𝑟 − 𝑎, 𝑝


𝑗. Furthermore, we

extended the dynamic programming algorithm to problem𝑃2|𝑛𝑟 − 𝑎, 𝑝


𝑗𝐶𝑗.

For future research, the other objectives areworth consid-ering.The design of PTAS for our problems is another worthytopic.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

The authors thank the editor and the anonymous reviewersfor their helpful and detailed comments on an earlier versionof their paper. This work was supported by The NationalNatural Science Foundation of China (11201259), the Doc-toral Fund of the Ministry of Education (20123705120001,20123705110003), Domestic Visiting Scholar Program forOutstanding Teachers of Higher Education in ShandongProvince, and the Natural Science Foundation of ShandongProvince (ZR2014AM012, BS2013SF016, J13LI09).

References

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