Abstract

This paper considers the combination of the general sum-of-processing-time effect and position-dependent effect on a single machine. The actual processing time of a job is defined by functions of the sum of the normal processing times of the jobs processed and its position and control parameter in the sequence. We consider two monotonic effect functions: the nondecreasing function and the nonincreasing function. Our focus is the following objective functions, including the makespan, the sum of the completion time, the sum of the weighted completion time, and the maximum lateness. For the nonincreasing effect function, polynomial algorithm is presented for the makespan problem and the sum of completion time problem, respectively. The latter two objective functions can also be solved in polynomial time if the weight or due date and the normal processing time satisfy some agreeable relations. For the nondecreasing effect function, assume that the given parameter is zero. We also show that the makespan problem can remain polynomially solvable. For the sum of the total completion time problem and is the deteriorating rate of the jobs, there exists an optimal solution for ; a V-shaped property with respect to the normal processing times is obtained for . Finally, we show that the sum of the weighted completion problem and the maximum lateness problem have polynomial-time solutions for under some agreeable conditions, respectively.

1. Introduction

Recent years, position-effect and processing-time-dependent scheduling problems have been paid more attention. Significant contributions also are presented to solve these problems, including the following. Browne and Yechiali [1] gave some applications to concern the control of queues and communication systems, where there exists deterioration phenomenon in the process of awaiting processing. Kunnathur and Gupta [2] and Mosheiov [3] presented several real-life situations of deteriorating jobs, including the search for an object under worsening weather and performance of medical treatments under some deteriorated health conditions. We refer to the surveys [4] for detailed state-of-the-art reviews in this time-dependent scheduling, as well as for references to practical applications. Among most common rationales for deterioration, the authors often mention the loss of the processing quality of machinery over time and/or the decrease in the productivity of a human operator who gets tired. Cheng et al. [5] considered deteriorated-effect scheduling problems, where the actual processing time of a job means a function of the logarithm of the sum of the normal processing time of the jobs processed and the setup times are proportional to the actual processing times of the jobs processed. Yin et al. [6] addressed another deterioration model to minimize the makespan and the total completion time, where the actual processing time of a job depends on its starting time and its position. They showed that there exists optimal sequence based on the relationships between problem parameters, including the shortest processing time, longest processing time, or V-shaped with respect to the normal processing times. Rudek [7, 8] considered the general sum-of-processing time-based learning or aging effects and showed that the total weighted completion times’ problem is strongly NP-hard, respectively. Gawiejnowicz [9] gave a detail review for four decades of time-dependent scheduling, including main results, new topics, etc. Jiang et al. [10] studied general truncated sum-of-actual processing-time-based effect on the single machine. The actual processing time of a job is affected by the sum-of-actual processing times of previous jobs and by a job-dependent truncation parameter. More recent papers considered deteriorating jobs: Li et al. [11], Liang et al. [12], Gawiejnowicz and Kurc [13], and Wang et al. [14].

Learning effects are divided into the following two types. (1) Position dependent: the actual processing time of job depends on and on its position in the sequence. (2) Cumulative: the actual processing time of job depends on and on the sum of normal processing times of jobs sequenced earlier. Biskup [15] and Cheng and Wang [16] were one of the pioneers who brought the concept of learning into the field of scheduling. Biskup [17] presented a detailed review for learning effect in 2008. Wang and Wang [18] investigated a general model with the agreeable position weight. The general models can cover the majority of existent sum-of-processing-time-based scheduling models. Luo [19] presented more general sum-of-processing-time-based scheduling models, which cover the normal processing time or the actual processing times. The distinctive proof technique is developed based on the adding-term operation, the subtracting-term operation, and the Lagrange mean value theorem. Lin [20] studied job-dependent learning effect and controllable processing time on the unrelated parallel machine. The three objective functions are considered, including the weighted sum of total completion time, total load, and total compression cost. Extensive surveys of different scheduling models can be found in Azadeh et al. [21], Pei et al. [22], and Tai [23]. More application of scheduling models, especially, many real-world problems have been explained by using mathematical models such as higher-order spectral analysis of stray flux signals for faults’ detection in induction motors, vortex theory for two-dimensional Boussinesq equations, normal complex contact metric manifolds admitting a semisymmetric metric connection, urea injection and uniformity of ammonia distribution in the SCR system of diesel engine, and new complex and hyperbolic forms for Ablowitz-Kaup-Newell-Segur wave equation with fourth order can be found in the following papers: Iglesias Mart et al. [24], Sharifi and Reasi [25], Jiao and Zheng [26], and Eskita et al. [27].

Motivated on the above discussion, the general sum-of-processing-time-based effect and position-dependent effect are provided. The job processing times are defined by functions of the sum of the normal processing times of jobs processed, its position and and control parameter in the sequence. Two monotonic effect functions are studied: nondecreasing function and nonincreasing function. Our four objective functions is the makespan, the sum of completion time, the sum of the weighted completion time, and the maximum lateness. Our contribution in this paper is listed as follows:(i)The nonincreasing effect function.(ii)The makespan problem and the sum of the completion time problem can be solved in polynomial time.(iii)The total weighted completion times’ problem and the lateness problem can also be solved in polynomial time if the weight or due date and the normal processing time satisfy some agreeable relations.(iv)The nondecreasing effect function and the given parameter is zero.(v)The makespan problem can remain polynomially solvable.(vi)The sum of completion time problem for can be optimally solved, where is the deteriorating rate of the jobs. Moreover, for the sum of the completion time problem with , a V-shaped property based on the normal processing times is obtained in an optimal sequence which satisfies some agreeable relations.(vii)The sum of the weighted completion times’ problem and the maximum lateness problem for have polynomial-time solutions under some agreeable conditions.

The rest of the paper is organized as follows. In Section 2, we give the problem description. In Sections 3 and 4, we consider two different actual processing times. Our conclusion will be given in Section 5.

2. Problem Description

Single-machine scheduling problems can be normally narrated as follows: jobs’ set.(i)(ii): the normal processing time of job , (iii): the weight of job , (iv): the due date of job , (v): the actual processing time of the job scheduled in the position in the sequence(vi): a bivariate continuous convex function on and a continuous function on with the normal processing time of a job scheduled in the position in a job sequence(vii): the completion time of job in job sequence (viii): the makespan(ix): the total completion(x): the total weighted completion time(xi): the maximum lateness

The proposed scheduling model is considered as follows:where is a given control parameter. Moreover, assume that for , , and .

Note that the bivariate function is only continuous convex function. Next, we will consider two monotonic function on : nondecreasing function and nonincreasing function. For the former, assume that , and . However, for the latter, we only consider the special case of the continuous convex function , and a given parameter .

3. The Nonincreasing Function on

This section will consider the nonincreasing function on and the nonincreasing function on . Four objective functions will be studied, including the makespan, the sum of the completion times, the sum of the weighted completion times, and the lateness.

Theorem 1. Problem can be obtained as an optimal schedule by nondecreasing normal processing times (the shortest processing time (SPT) rule).

Proof. The properties of the optimal solutions for some single-machine problems are proved by the pairwise job interchange technique. Let and be two job schedules where the difference between and is a pairwise interchange of two adjacent jobs and , i.e., and , where and are partial sequences. Assume that denotes the completion time of the last job scheduled in position of . Under , the completion times of jobs and areUnder , the completion times of jobs and areNote that p_i ≤ p_j. Next, we will show that σ dominates σ^ʹ. Taking the difference between (4) and (14), it is obtained thatBased on the monotonicity of function and , we haveNext, the parameter will be discussed by four cases as follows:(1). Then, we have(2). Then, we have(3). Then, we have(4). Then, we haveLet , . Then, we can obtainLet , and we haveBased on , , and , we can obtain that , i.e., the function is an nondecreasing function for . Thus, for . Moreover, we have .
Note that the completion times of the job if scheduled in in the job sequence and , respectively, are denoted as follows:From , we have , i.e, the starting time of the first job in partial job sequence of job sequence is earlier than job sequence . Therefore, we have for job in the partial sequence . Hence, the optimality of the SPT rule can be showed by repeating this argument for the proposed scheduling problem.
Note that by the equations (2) and (4). Then, the following theorem will be presented.

Theorem 2. Problem can be solved by the SPT rule.

For the total weighted completion time and the maximum lateness, we only show that this two problems can be solved in polynomial time under some special agreeable relations, respectively.

Theorem 3. For the problem , an optimal schedule can be obtained by the weighted smallest processing time, i.e., the WSPT rule, if the jobs have agreeable weights, i.e., implies for all jobs and .

Proof. Similar to the same notations in the proof of Theorem 1. Let the jobs and satisfy the agreeable relation, i.e., which implies and . Next, we will show that .
Since partial job sequence in job sequence and has the same job position, then the completion time of job of partial job sequence is the equal, i.e., , .
From Theorems 1 and 2, we haveAdditionally, we have for by Theorem 1.

Theorem 4. For the problem , an optimal schedule can be obtained by the earliest due date, i.e., the EDD rule, if the jobs have agreeable weights, i.e., implies for all jobs and .

Proof. Using the same notations of Theorem 1, we will show that based on the agreeable relation, i.e., and . From Theorems 1 and 2, we haveMoreover, we can obtain . Hence, interchanging the job position will not increase the value of .

4. The Nondecreasing Function on

In this section, the special case of the nondecreasing function on will be given, and . Firstly, some notations are defined as follows: and denote the deteriorating or learning rate and the learning rate, respectively. are three given positive numbers, where , , and .

The special sum-of-processing-time model can be described as follows:where . , , and .

Next, some useful lemmas will be given. Based on and , the proofs of some lemmas can be obtained by differentiation.

Lemma 1. , for , , , , , and .

Lemma 2. , for , , , , , and .

Lemma 3. , for , , , , , , and .

Similar to the notations of Theorem 1, we will give the following results. Under , the completion times of jobs and are

Under , the completion times of jobs and are