Results of Parallel-Machine Scheduling Model with Maintenance Activity considering Time-Dependent Deterioration, Delivery Times, and Resource Allocation

,is paper investigates parallel-machine scheduling models with maintenance activity, delivery times, time-dependent deterioration, and resource allocation. We consider two forms of the problem: the first is to minimize the sum of total completion times, total machine loads, the total absolute deviation of job completion times, and the total resource allocation; the second is to minimize the sum of total waiting times, total machine loads, the total absolute deviation of job waiting times, and the total resource allocation. ,e problems are proved to be solvable in polynomial time.


Introduction
Initiated by Vickson [1], scheduling problems with controllable processing times through resource allocation have been studied extensively by researchers since 1980. Wang et al. [2] introduce resource allocation scheduling with learning effect. ey analyse the problem with two different processing time functions p j � p j r a − a j u j and p j � (p j r a /u j ) k , u j > 0. ey provide a polynomial time algorithm to the problem. Zhu et al. [3] investigate scheduling problems and show that all the presented problems are polynomially solvable. Liu and Feng [4] show two-machine no-wait flow shop scheduling problems. Liu et al. [5] consider a parallel-machine scheduling problem to minimize the sum of resource consumption and outsourcing cost given. Liu et al. [6] study scheduling problems on single machine to determine the optimal job schedule, the optimal due window location, and the optimal resource allocation.
In scheduling, Lee and Leon [23] introduce a new class of scheduling situations with rate-modifying activity in which the machines need to be maintained and become unavailable. Recently, scheduling problem with a rate-modifying activity becomes a research hotspot. Yin et al. [24] consider a single-machine batch delivery scheduling and common due date assignment problem. ey provide some properties of the optimal schedule for the problem. Wang and Wei [25] introduce identical parallel-machine scheduling problems and show that the problems remain polynomially solvable under the proposed model. Wang and Wang [26] address SLK due date assignment scheduling problem with a ratemodifying activity. ey present a polynomial time solution for this problem. Zhu et al. [27] address a single machine scheduling problem and show that the problems are solvable in o(n 4 ) time for a linear resource allocation function and are solvable in o(n 2 log n) time for a convex resource allocation function. Ji et al. [28] consider single-machine common due-window and deteriorating rate-modifying activity scheduling problem. ey provide polynomial solution algorithms for the corresponding problems. Yang and Yang [29] introduce unrelated parallel-machine scheduling problems with multiple rate-modifying activities. Hsu et al. [30] extend an unrelated parallel-machine scheduling environment and present a more efficient algorithm to solve the extended problem.
Jobs' processing time is impacted greatly by the environment in modern manufacture. ere is some extra time to be needed to deal with this thing. is treatment can be performed after the component has been processed by the machine but before it is delivered to the customer so it can be delivered with a guarantee of safety. e extra time to eliminate the adverse effects between the main processing and the delivery of a job is called as a past-sequence-dependent delivery time. Delivery time is firstly introduced into scheduling problem by Koulamas and Kyparisis. Liu et al. [31] consider the single-machine delivery time scheduling problem, which was introduced in Koulamas and Kyparisis [32]. Liu [33] introduce parallel-machine learning effect scheduling with p-s-d delivery time. Shen and Wu [34] study single-machine scheduling with delivery time and general learning effects. Sun et al. [35] consider scheduling models with DeJong's learning effect, delivery times, ratemodifying activity, and resource allocation, respectively. e problems are proved to be solvable in polynomial time.
In this paper, we investigate parallel-machine scheduling problem with time-dependent deterioration delivery time maintenance activity and resource allocation, and some new results are given. e remaining part of this paper is formed as follows: we take shape the problem and present some notations and one lemma in Section 2. In Section 3, we introduce the version of the problem with linear resource consumption function. We analyse the problem with convex resource allocation and present an example to illustrate the application in Section 4. In the last section, we conclude the paper.

Problem Formulation
ere are n independent and nonpreemptive jobs, which are simultaneously available for processing on m identical parallel machines. We suppose that m < n throughout the paper. Let a ij be the normal processing time of job J ij and p ij be the actual processing time of job J ij . Let a i [k] and p i [k] be the normal processing time and the actual processing time of the kth job J i[k] on machine M i . C i [j] and W i [j] are the completion time and the waiting time of the jth job on machine M i in a sequence.
One deteriorating maintenance activity (denoted by DMA) is allowed on each machine throughout the scheduling to improve machine production efficiency. e deteriorating maintenance duration is represented by f(t) � β + σt, which is longer if it is started later, where β > 0 is the basic maintenance activity time, σ > 0 is the deterioration rate of maintenance activity, and t is the starting time of the deteriorating maintenance activity. e deteriorating maintenance activity can be scheduled immediately after any job's processing completion and its starting time is not known in advance. e position of deteriorating maintenance activity on machine M i is denoted by k i , if the deteriorating maintenance activity is scheduled immediately after completing the processing of the k i th job J [k i ] . e machine will revert to its initial condition, and the aging effect will start anew after the maintenance activity.
Let n i denote the number of jobs located on machine M i and k i denote the position of deteriorating maintenance activity on machine M i . Hence, the deteriorating maintenance activity duration is f(t) � β + σ( k i l�1 p i [l] ). If a job is scheduled on the jth position of machine M i in a sequence, its actual processing time with resource consumption is as follows.
A linear resource consumption function is given by where λ i [j] is the weight of the resource which is allocated to job J i [j] . u i [j] is the amount of the resource allocated to job , and a i � min a ij , j � 1, 2, . . . , n i if j ≥ k i . Jobs' deterioration rate is b. A convex resource consumption function is given by 2 Mathematical Problems in Engineering where v is a positive constant, u i [j] is the amount of the resource allocated to job J i [j] with 0 ≤ u i[j] ≤ u i [j] , and u i [j] is the maximum resource allocation of job J i [j] . As in the study of Koulamas and Kyparisis [32], the processing of job J i[r] must be followed by the p-s-d delivery time q i [r] , which can be calculated as q i [1] � 0 and , before maintenance activity, and , after maintenance activity, where c ≥ 0 is a normalizing constant.
As usual, we assume that the postprocessing operation of any job J i[l] modelled by its delivery time q i[l] is performed off-line. erefore, it is not affected by the availability of the machine and it can be implemented immediately upon completion of the main operation resulting in C i [1] � p i [1] and Let denote the p-s-d delivery time by q ps d . In addition, we denote TADC the total absolute deviation of job completion times and TADW the total absolute deviation of job waiting times, i.e., TADC � m i�1 n i l�1 Let TC indicate the job's total processing times and TW indicate the job's total waiting times, i.e., TC � m i�1 n i r�l C i [r] and TW � m i�1 n i r�l W i [r] . We denote the load of machine M i by L i , and the total machine load (TML) is m i�l L i , i.e., TML � m i�1 L i . As in the study of Liu and Feng [4], we will try to find the optimal job sequence, the optimal DMA, and the optimal resource consumption on parallel-machine schedule such that the following cost functions are minimized: where δ 1 , δ 2 , δ 3 , δ 4 represent the per unit time contribution for the total machine load, the total processing (waiting) time, and the total absolute deviation of job completion (waiting) times with δ 1 > 0, δ 2 > 0, δ 3 > 0, and δ 4 > 0. G ij is the per unit time cost associated with resource allocation. Let LRA denote linear resource allocation and CRA denote convex resource allocation. Using the three-field notation introduced by Graham et al., for scheduling problems, we denote the two versions of the problems as P m |q psd , LRA, DMA|z, P m |q psd , CRA, DMA|z, We first present some notations and one lemma before the main results. If the number of jobs on machine M i is n i and the position of the job preceding the deterioration maintenance activity is k i , the jobs' completion times and waiting times on machine M i are as follows.
For the linear case, For the convex case, Let P(n, m, k) � (n 1 , n 2 , . . . , n m ; k 1 , k 2 , . . . , k m ) denote an allocation vector. We provide a lemma concerning an upper bound on the number of P(n, m, k) vectors.

Cases with Linear Resource Consumption Function
In this section, we introduce the problem to minimize the sum of total machine load, total completion times, and total absolute deviation of job completion times with resource consumption on all the machines. From the above analysis, for machine M i , we calculate the machine load, the total completion times, and the total absolute deviation of job completion times as follows: Hence, the sum of total machine load, total completion times, and total absolute deviation of job completion times with resource consumption on all the machines is Mathematical Problems in Engineering 5 Let i � 1, 2, . . . , m, l � k i + 1, k i + 2, . . . , n i − 1, us, From the above equation, for any job sequence, the optimal resource allocation for a job depends on the sign of [l] is negative, the maximum feasible amount of the resource should be allocated to job is positive, no resource should be allocated to job J i [l] , and if G i[l] − λ i[l] w i [l] is equal to zero, any of value of resource consumption will not affect the total cost. Let u * i[l] denote the optimal resource allocation for job J i [l] , where From equation (12), we can obtain the optimal resource allocation for any given optimal sequence.
Since A 1 is a constant, when n i and k i are given, we can express the problem as the following assignment problem: 6 Mathematical Problems in Engineering where Consequently, when the P(n, m, k) vector is given, optimal job scheduling and optimal resource allocation are given in Algorithm 1.
Since the P(n, m, k) vector is given, we know that the problem can be solved in O(n 3 ) time. Together with Lemma 1, it is easy to obtain the following theorem.

3.2.
e Problem P m |q ps d , LRA, DMA|δ 1 TML+ δ 2 TW+ δ 3 TADW + δ 4 m i�1 n i j�1 G ij u ij . In this section, we study the problem to minimize the sum of the total machine load, the total waiting times, and the total absolute deviation of job waiting times with resource consumption. For machine M i , we compute the machine load, the total waiting times, and the total absolute deviation of job waiting times as follows: Hence, the sum of total machine load, total waiting times, and total absolute deviation of job waiting times with resource consumption on all the machines is Mathematical Problems in Engineering 7 Let Step 1: jobs are scheduled by (AP 1 ).
Step 2: optimal job resource allocation is calculated by formula (12).
ALGORITHM 1: Algorithm for the problem to minimize the sum of total completion times, total machine loads, the total absolute deviation of job completion times, and the total resource allocation under linear resource consumption. 8 Mathematical Problems in Engineering us, For any job sequence, the optimal resource allocation for a job depends on the sign of denote the optimal resource allocation for job J i [l] , where From equation (19), we can get the optimal resource allocation for any given optimal sequence. Accordingly, when n i and k i are given, we can indicate the problem as the following assignment problem: where Hence, when the P(n, m, k) vector is given, optimal job scheduling and optimal resource allocation are given in Algorithm 2.
us, when the P(n, m, k) vector is given, the problem can be solved in O(n 3 ) time. Together with Lemma 1, we have the following theorem.

Cases with Convex Resource Consumption Function
In this section, we will consider the problems under convex resource consumption function, i.e. P m |q ps d , CRA, DMA|z, Similar to the analysis of problem P m |q ps d , LRA, DMA|δ 1 TML+ δ 2 TC + δ 3 TADC + δ 4 m i�1 n i j�1 G ij u ij , if n i and k i are given, we calculate the problem to minimize the sum of the total machine load, the total completion times and the total absolute deviation of job completion times with convex resource consumption as follows:

Mathematical Problems in Engineering
where By taking the first derivative of H 1 with respect to u i[l] , i � 1, 2, . . . , m, l � 1, 2, . . . , n i , equating the result to zero, and solving it for u i [l] , we can obtain the optimal resource allocation (denoted by u * i [l] ).
By substituting u * i[l] into the objective function H 1 , we obtain a new unified expression as follows: (v/v+1) .
erefore, we can formulate the minimum problem as the following assignment problem: Step 1: jobs are scheduled by (AP 2 ).
Step 2: optimal job resource allocation is calculated by formula (19).
ALGORITHM 2: Algorithm for the problem to minimize the sum of total waiting times, total machine loads, the total absolute deviation of job waiting times, and the total resource allocation under linear resource consumption. where G ij a ij (v/v+1) , i � 1, 2, . . . , m, j � 1, 2, . . . , n, l � 1, 2, . . . , n i . (28) Consequently, when the P(n, m, k) vector is given, optimal job scheduling and optimal resource allocation are given Algorithm 3.
Together with Lemma 1, we have the following theorem. Similar to the analysis of problem P m |q ps d , LRA, DMA|δ 1 TML + δ 2 TW+ δ 3 TADW + δ 4 m i�1 n i j�1 G ij u ij , if n i and k i are given, we calculate the problem to minimize the sum of total waiting times and total absolute deviation of job waiting times with convex resource consumption as follows: where i � 1, 2, . . . , m, l � 1, 2, . . . , k i , i � 1, 2, . . . , m, l � k i + 1, k i + 2, . . . , n i − 1, Hence, taking the first derivative of H 2 with respect to u i [l] , i � 1, 2, . . . , m, l � 1, 2, . . . , n i , equating the result to zero, and solving it for u i [l] , we can obtain the optimal resource allocation (denoted by u * i[l] ).
By substituting u * i[l] into the objective function H 2 , we obtain a new unified expression as follows: erefore, we can formulate the minimum problem as the following assignment problem: (G ij a ij ) (v/v+1) . erefore, when the P(n, m, k) vector is given, optimal job scheduling and optimal resource allocation are given Algorithm 4.
From the above analysis and Lemma 1, we have the following theorem.

Theorem 4.
e problem P m |q ps d , CRA, DMA|δ 1 TML+ Example. We consider three jobs assigned to two parallel machines to minimize the sum of the total machine loads, the total completion times, the total absolute deviation of job completion times, and the total resource allocation under linear resource allocation. Using the three-field notation, we denote the problem as P m |q ps d , LRA, DMA|δ 1 TML+ δ 2 TC+ δ 3 Step 1: jobs are scheduled by (AP 3 ).
Step 2: optimal job resource allocation is calculated by formula (25).
ALGORITHM 3: Algorithm for the problem to minimize the sum of total completion times, total machine loads, the total absolute deviation of job completion times, and the total resource allocation under the convex resource consumption. erefore, the optimal resource allocation is u 1 � 0, u 2 � 4, and u 3 � 4, and the objective function value is δ 1 TML π 1 + δ 2 TC π 1 + δ 3 TADC π 1 + δ 4 By the same method, we have the minimization objective function value and the optimal resource allocation of other schedule as Table 1.
From Table 1, it is easy to obtain that the optimal schedule is (J  [2] ). e optimal resource allocation is u 1 � 4, u 2 � 0, and u 3 � 4.

Conclusions
In this paper, parallel-machine scheduling problems with past-sequence-dependent delivery times, time-dependent deterioration, maintenance activity, and resource consumption are considered. We present two versions of the scheduling problems can be solved polynomially. Future research will be worth extending to multiple maintenance activities or other objective scheduling problems.

Data Availability
All data generated or analysed during this study are included in this article.