Scheduling with Learning Effects and/or Time-Dependent Processing Times to Minimize the Weighted Number of Tardy Jobs on a Single Machine

Abstract

We consider single machine scheduling problems with learning/deterioration effects and time-dependent processing times, with due date assignment consideration, and our objective is to minimize the weighted number of tardy jobs. By reducing all versions of the problem to an assignment problem, we solve them in O(n⁴) time. For some important special cases, the time complexity can be improved to be O(n²) using dynamic programming techniques.

1. Introduction

Traditionally, the processing times of jobs in scheduling problems were fixed, regardless of the jobs' positions in the schedule or their starting times. In recent years, however, more and more researchers are investigating scheduling problems with variable processing times of two types: learning/deterioration effects and time-dependent processing times.

Learning Effect. As early as in the 1930s, Wright [1] noticed that in the aircraft industry the working costs per unit decreased with an increasing production output. He formulated the so-called 80% hypothesis, stating that with every redoubling of output the unit processing time decreases by 20%. This learning effect has a large impact on mass production. For example, it was reported that, in the semiconductor industry, the efficiency gains cause the price to drop by 10–30% [2]. Recently, new proof of the impact of learning effects from a new car-assembly plant was presented in [3].

Learning effects are important for production problems involving significant level of human activities, such as machine setup, machine cleaning, machine maintenance, machine failure removal, machine operating and controlling, machine data reading/understanding/interpretation, and all kinds of handwork. They are especially important when the production environment changes. Such changes may include, among others, investments in new machines or replacement of equipment, new workers, workflow changes, and the acceptance of new jobs. Even small changes to the production environment like a software update, a new design of the format of important documents, and a new organization of the spare parts depot cause learning effects. Every time a worker needs to get used to a new circumstance, a learning experience will occur. Since there are usually high levels of such human involvement in scheduling environments which change frequently, and these activities are subject to learning, it is important to consider learning effects on scheduling problems [4].

Biskup [5] proposed the following model: 𝑝_[𝑗]=𝑎_[𝑗]𝑗^𝑐, (1) where 𝑝_[𝑗] and 𝑎_[𝑗], respectively, represent the actual processing time and “normal’’ processing time (i.e., without learning effect) of the job at the 𝑗th position in the scheduling sequence and 𝑐≤0 represents the rate of learning. It is easy to verify that if 𝑐=log₂0.8=−0.322, it corresponds to the aforementioned 80% hypothesis.

If 𝑐>0, then the model captures the effect of process deterioration [6]. Many researchers used an equivalent model 𝑝_𝑖𝑟=𝑝_𝑖𝑟^𝑐, where 𝑟 represents the position of job 𝑖 in the schedule. Mosheiov [7] investigated scheduling problems with learning effects described by (1) on parallel machines and Mosheiov and Sidney [8] studied more general learning functions. Scheduling on parallel machines with a more general form of learning effect was studied in [9]. A comprehensive survey about the learning effect in scheduling problems can be found in [10]. Further reviews of scheduling models with positional effects and learning were presented in [11–13].

Time-Dependent Processing Time. In some other practical situations, the actual processing time of a job depends on its starting time. For example, Gupta et al. [14] modeled a multiple loan repayment problem in financial management as a scheduling problem with time-dependent processing times. In steel production, steel ingots are to be heated to the required temperature before rolling can begin. The time taken to heat depends on the size and current temperature of the ingot, which further depends on the amount of time the ingot has been waiting to be processed in. Another example is in forest-fire fighting, where the time required to control a fire is longer if the start of the fire fighting effort is delayed [15]. Some other empirical applications exist in hospital emergency room scheduling and crime scene response scheduling for police [16, 17].

The concept of time-dependent processing times was introduced by Browne and Yechiali [18]. One of the most commonly used models is 𝑝_[𝑗]=𝑎_[𝑗]+𝑏𝑆_[𝑗], (2) where 𝑎_[𝑗] also denotes the “normal’’ processing time, 𝑏 is a constant, and 𝑆_[𝑗] denotes the starting time of the job in the 𝑗th position. If 𝑏>0, then it reflects deteriorating processing time. Mosheiov presented a V-shape policy and a ∧-shape policy, respectively, in [17, 19] for scheduling deteriorating jobs. Some researchers used position-dependent 𝑏_[𝑗] [20]. However, the problem usually becomes significantly harder in most of those cases. Gawiejnowicz gives a comprehensive exposition of time-dependent models [21].

The two types of variable processing times defined above were usually considered separately. Lee [22] was the first to consider them simultaneously. Wang [23] proposed the following combined model: 𝑝_[𝑗]=𝑎_[𝑗]+𝑏𝑆_[𝑗]𝑗^𝑐, (3) where 𝑝_[𝑗], 𝑆_[𝑗], 𝑏, and 𝑐≤0 all have the same meaning as before. This was followed by studies by Wang [24], Wang and Cheng [25], Wang et al. [26], Wang and Cheng [27], and Koulamas et al. [28].

This model reflects, for example, the following practical scenario: an operator obtains additional skills by learning from experience; at the same time, the machine that he/she operates may be subjected to wear and tear, that is, deteriorates with time [13]. A recent review by Janiak et al. focused on a variety of models with these types of combined effects [12].

We can see that each of the above discussed cases of variable processing times is a special case of this general model. If 𝑏=0, we have the learning effect problem; if 𝑐=0, we have the time-dependent processing time problem.

The general problem we study in this paper may be stated as follows: 𝑛 independent, nonpreemptive jobs, 𝐽={1,2,…,𝑛}, are available for processing at time zero and are to be processed on a single machine. A schedule is defined by a job sequence 𝜋=([1],[2],…,[𝑛]), where [𝑗] represents the job that is in the 𝑗th position in 𝜋 for 𝑗=1,2,…,𝑛. The processing time of the job at position 𝑗 is defined by (3). Our objective is to determine a schedule which minimizes a general unified cost function that is the sum of scheduling costs, expressed by using positional penalties. This unified cost function can be formulated as follows: 𝑔(𝜋)=^𝑛_𝑗=1𝜉_𝑗𝜂_[𝑗]+^𝑛_𝑗=𝑙+1𝜓_[𝑗], (4) where 𝜉_𝑗 is a positional, job-independent penalty for any job scheduled in the 𝑗th position for 𝑗=1,…,𝑛, 𝜂_[𝑗] represents the normal processing time of the job in position 𝑗, 𝑙 is a variable index to be determined, and 𝜓_[𝑗] is the scheduling penalty, for example, for tardiness, incurred by jobs scheduled in the last 𝑛−𝑙 positions. Our model is adapted from [29], where Leyvand et al. proposed a unified framework for scheduling problems with controllable processing time and solved it using an algorithm for general linear assignment in 𝑂(𝑛³) time. The unified objective function above is also a generalization of scheduling functions studied in [30].

We will use and extend the standard three-field notation 𝐴|𝐵|𝐶 introduced by Graham et al. [31] for scheduling problems. The 𝐴 field describes the machine environment. For example, if 1 appears in the 𝐴 field, it means that we deal with a single-machine scheduling problem. The 𝐵 field defines the job-processing characteristics and constraints. The 𝐶 field contains the optimizing criteria (for ease of presentation, shown in a symbolic fashion without including the summation indices).

The paper is organized as follows. In Section 2, we present an 𝑂(𝑛⁴) optimization algorithm to determine the optimal schedule for the unified-problem. In Section 3, we present a group of scheduling problems, where the objective function includes costs for the total weighted number of tardy jobs, makespan, and due date assignment. We show that, for three different due date assignment methods, the objective can be reformulated as a special case of (4), which enables us to solve all these problems in 𝑂(𝑛⁴) time. Using dynamic programming techniques, we improve this time complexity to be 𝑂(𝑛²) for some important special cases in Section 4. The last section contains a summary and our concluding remarks.

2. The Unified Problem

We show in this section that our problem can be solved in 𝑂(𝑛⁴) time by solving a linear assignment problem. First we consider the case when 𝑙 is given. For 1≤𝑖, 𝑗≤𝑛, let us define 𝑐_𝑖𝑗 as 𝑐_𝑖𝑗=𝜉_𝑖𝜂_𝑗𝜉for𝑖≤𝑙_𝑖𝜂_𝑗+𝜓_𝑗for𝑙+1≤𝑖≤𝑛. (5)

Our sequencing problem then reduces to the classical linear assignment problem of finding 𝜋 which minimizes ∑^𝑛_𝑖=1𝑐_𝑖[𝑖]. Since the calculation of the assignment costs depends on 𝑙, we denote this assignment problem by P1(𝑙). It is well known that a linear assignment problem can be solved in 𝑂(𝑛³) time (see [32]).

Therefore, we have the following result.

Lemma 1. The optimal sequence, denoted by 𝜋^∗, can be obtained by solving a linear assignment problem requiring 𝑂(𝑛³) time for fixed 𝑙.

The results of our analysis are summarized as shown in Algorithm 1.

Step  1. Calculate the 𝑐𝑖𝑗 values by (5). Step  2. Solve the assignment problem (𝑃1(𝑙)) to determine the optimal job sequence, and denote the resulting optimal sequence       by 𝜋∗=([1],[2],…,[𝑛]).

Algorithm 1 

The optimization algorithm for the solution of the 1||∑𝜉_𝑗𝜂_[𝑗]+∑𝜓_[𝑗] problem.

If 𝑙 is not known, then we have to enumerate all 𝑙 and run Algorithm 1 repeatedly as a subroutine (see Algorithm 2).

Initialization: 𝑍∗=∞, 𝑙=0.    while  𝑙≤𝑛  do      Step  1. Apply Algorithm 1. Denote the optimal job sequence by 𝜋∗(𝑙)=([1],[2],…,[𝑛]) and the minimum cost by 𝑍∗(𝑙).      Step  2. If 𝑍∗(𝑙)≤𝑍∗, then set 𝑍∗=𝑍∗(𝑙), 𝑙∗=𝑙, 𝜋∗=𝜋∗(𝑙).      Step  3. 𝑙=𝑙+1.    end

Algorithm 2 

From Lemma 1, we can get the following result easily.

Theorem 2. Algorithm 2 solves the 1‖∑𝜉_𝑗𝜂_[𝑗]+∑𝜓_[𝑗] problem in 𝑂(𝑛⁴) time.

In the next section, we present various applications of our general approach to solve a large set of important scheduling problems with variable processing times.

3. Applications to Solve Scheduling Problems

Let 𝐶_[𝑘] be the completion time of the 𝑘th job in the processing sequence whose processing time varies according to (3). The following crucial equality is given in [23].

Lemma 3. Assuming that the jobs are processed from time zero and there is no idle time between them, then 𝐶_[𝑘]=^𝑘_𝑗=1𝑗^𝑐𝑘_𝑖=𝑗+11+𝑏𝑖^𝑐𝑎_[𝑗], (6) where ∏^𝑘_𝑖=𝑘+1(1+𝑏(𝑘+1)^𝑐)=1 by definition.

Define 𝑓_𝑗𝑘=𝑗^𝑐𝑘_𝑖=𝑗+11+𝑏𝑖^𝑐 (7) for 𝑗=1,…,𝑛 and 𝑘=𝑗+1,…,𝑛. Then (6) above can be rewritten as 𝐶_[𝑘]=^𝑘_𝑗=1𝑓_𝑗𝑘𝑎_[𝑗]. (8)

3.1. Minimizing the Total Weighted Number of Tardy Jobs with Due Date Assignment

In this section, we show how our unified method can be used to minimize the total weighted number of tardy jobs with due date assignment decisions and variable processing times.

Since the solution of due date assignment problems also includes the determination of the due dates, a schedule for these problems is defined by a job sequence 𝜋=([1],[2],…,[𝑛]) and a due date assignment vector 𝐝=(𝑑₁,𝑑₂,…,𝑑_𝑛), and our objective is to determine the schedule which minimizes various objective functions defined in the subsections below. Therefore, to also show their dependence on the due dates, we redefine 𝑔(𝜋) from (4) as 𝑔(𝜋,𝐝). Cases will be considered by providing special forms of 𝑔(𝜋,𝐝).

Our objective is to find a job sequence 𝜋^∗=([1],[2],…,[𝑛]) and a set of due dates 𝐝^∗=(𝑑^∗₁,𝑑^∗₂,…,𝑑^∗_𝑛) to minimize a cost function that includes the costs of tardiness, due date assignment, and makespan. For the CON, SLK, and DIF due date assignment methods, it is given by 𝑔(𝜋,𝐝)=^𝑛_𝑗=1𝜓_[𝑗]𝑈_[𝑗]+𝛾^𝑛_𝑗=1𝑑_[𝑗]+𝛿𝐶_max, (9) where 𝛾 is a nonnegative parameter representing the cost of one unit of due date, 𝛿 is a nonnegative parameter representing the cost of one unit of makespan, 𝜓_[𝑗] is the cost of job [𝑗] being tardy, and 𝑈_[𝑗] is the tardiness indicator variable; that is, 𝑈_[𝑗]=1 if 𝐶_[𝑗]>𝑑_[𝑗] and 0 otherwise.

It is easy to see that the jobs are processed from time zero and there is no idle time between them; that is, the condition of Lemma 3 is satisfied for all the due date assignment methods. Therefore, (6) holds in all cases. Next we show that, under an optimal due date assignment strategy, the scheduling cost can be reduced to the format of (4) for all of them. We present results with different due date assignment methods, including CON, DIF, and SLK, to minimize total weighted number of tardy jobs.

In the following subsections, we will reduce the scheduling problem for each due date assignment method to the format of (4), which means that they can all be solved in 𝑂(𝑛⁴) time. For any sequence 𝜋, let us define set 𝐸(𝜋) as the set of early (or on-time) jobs under an optimal due date assignment strategy and define the set of tardy jobs by 𝑇(𝜋)=𝐽⧵𝐸(𝜋). Then the following lemma from [33] is applicable to our problems (we omit the proof since it is rather straightforward).

Lemma 4. There is an optimal schedule in which the corresponding 𝜋 sequences 𝐸(𝜋) before set 𝑇(𝜋).

Let 𝑙=|𝐸(𝜋)| denote the number of early jobs in the optimal schedule, and then we have 𝐸(𝜋)={[1],[2],…,[𝑙]}.

DIF Due Date Assignment Method. By Lemma 4, we can schedule the 𝑙 early jobs before all the tardy jobs. For job 𝑗, if it is early, its due date is at least as large as its completion time. Therefore, the optimal assignment strategy is to let 𝑑^∗_[𝑗]=𝐶_[𝑗], for 𝑗=1,…,𝑙. On the other hand, if job 𝑗 is tardy, then we can set its due date as early as possible; that is, let 𝑑^∗_[𝑗]=0 for 𝑗=𝑙+1,…,𝑛, since the job tardiness cost does not depend on the amount of tardiness. From (8) and (9), we have 𝑔(𝜋)=^𝑛_𝑗=𝑙+1𝜓_[𝑗]+𝛾^𝑙_𝑘=1𝐶_[𝑘]+𝛿𝐶_[𝑛]=^𝑛_𝑗=𝑙+1𝜓_[𝑗]+𝛾^𝑙𝑘_𝑘=1𝑗=1𝑓_𝑗𝑘𝑎_[𝑗]+𝛿^𝑛_𝑗=1𝑓_𝑗𝑛𝑎_[𝑗]=^𝑙_𝑗=1𝛾^𝑙_𝑘=𝑗𝑓_𝑗𝑘+𝛿𝑓_𝑗𝑛𝑎_[𝑗]+𝛿^𝑛_𝑗=𝑙+1𝑓_𝑗𝑛𝑎_[𝑗]+^𝑛_𝑗=𝑙+1𝜓_[𝑗]. (10)

We can see that this is indeed in the format of (4), where 𝜉_𝑗=𝛾^𝑙_𝑘=𝑗𝑓_𝑗𝑘+𝛿𝑓_𝑗𝑛for𝑗≤𝑙𝛿𝑓_𝑗𝑛for𝑙+1≤𝑗≤𝑛. (11)

This equation becomes simpler if there is only learning effect or only time-dependent processing times. If there is only learning effect, we have 𝑏=0; therefore 𝑓_𝑗𝑘 in (7) is reduced to 𝑗^𝑐 for every 𝑘. Thus the equation above becomes 𝜉_𝑗=𝛾𝑙−𝑗+1𝑗^𝑐+𝛿𝑗^𝑐for𝑗≤𝑙𝛿𝑗^𝑐for𝑙+1≤𝑗≤𝑛. (12)

On the other hand, if there is only time-dependent processing time effect, we have 𝑐=0; therefore, 𝑓_𝑗𝑘 becomes (1+𝑏)^𝑘−𝑗. Thus the equation above becomes 𝜉_𝑗=𝛾𝑙−𝑗+1^𝑙_𝑘=𝑗(1+𝑏)^𝑘−𝑗+𝛿(1+𝑏)^𝑛−𝑗𝛿()for𝑗≤𝑙1+𝑏^𝑛−𝑗for𝑙+1≤𝑗≤𝑛. (13)

CON Due Date Assignment Method. It can be easily seen that the optimal strategy is to assign 𝐶_[𝑙] as the common due date 𝑑^∗_[𝑗]=𝐶_[𝑙] for all 𝑗. From (8) and (9), we have 𝑔(𝜋)=^𝑛_𝑗=𝑙+1𝜓_[𝑗]+𝑛𝛾𝐶_[𝑙]+𝛿𝐶_[𝑛]=^𝑛_𝑗=𝑙+1𝜓_[𝑗]+𝑛𝛾^𝑙_𝑗=1𝑓_𝑗𝑙𝑎_[𝑗]+𝛿^𝑛_𝑗=1𝑓_𝑗𝑛𝑎_[𝑗]=^𝑙_𝑗=1𝑛𝛾𝑓_𝑗𝑙+𝛿𝑓_𝑗𝑛𝑎_[𝑗]+^𝑛_𝑗=𝑙+1𝜓_[𝑗]+𝛿^𝑛_𝑗=𝑙+1𝑓_𝑗𝑛𝑎_[𝑗]. (14)

We can see that this is indeed the format of (4), where 𝜉_𝑗=𝑛𝛾𝑓_𝑗𝑙+𝛿𝑓_𝑗𝑛for𝑗≤𝑙𝛿𝑓_𝑗𝑛for𝑙+1≤𝑗≤𝑛. (15)

If there is only learning effect, we have 𝑏=0; therefore 𝑓_𝑗𝑘 is reduced to 𝑗^𝑐 for every 𝑘. Thus the equation above becomes 𝜉_𝑗=𝑛𝛾𝑗^𝑐+𝛿𝑗^𝑐for𝑗≤𝑙𝛿𝑗^𝑐for𝑙+1≤𝑗≤𝑛. (16)

On the other hand, if there is only time-dependent processing time effect, we have 𝑐=0; therefore, 𝑓_𝑗𝑘 becomes (1+𝑏)^𝑘−𝑗. Thus the equation above becomes 𝜉_𝑗=𝑛𝛾(1+𝑏)^𝑙−𝑗+𝛿(1+𝑏)^𝑛−𝑗𝛿()for𝑗≤𝑙1+𝑏^𝑛−𝑗for𝑙+1≤𝑗≤𝑛. (17)

SLK Due Date Assignment Method. Let ℎ=𝑙−1. We know from [34] that the optimal common slack value is 𝑠^∗=𝐶_[ℎ]. From (8) and (9), we have 𝑔(𝜋)=^𝑛_𝑗=𝑙+1𝜓_[𝑗]+𝛾^𝑛_𝑗=1𝑠^∗+𝑝_[𝑗]+𝛿𝐶_[𝑛]=^𝑛_𝑗=𝑙+1𝜓_[𝑗]+𝑛𝛾𝐶_[ℎ]+𝛾𝐶_[𝑛]+𝛿𝐶_[𝑛]=^𝑛_𝑗=𝑙+1𝜓_[𝑗]+𝑛𝛾^ℎ_𝑗=1𝑓_𝑗ℎ𝑎_[𝑗]+𝛾+𝛿^𝑛_𝑗=1𝑓_𝑗𝑛𝑎_[𝑗]=^ℎ_𝑗=1𝑛𝛾𝑓_𝑗ℎ+𝛾+𝛿𝑓_𝑗𝑛𝑎_[𝑗]+𝛾+𝛿^𝑛_𝑗=ℎ+1𝑓_𝑗𝑛𝑎_[𝑗]+^𝑛_𝑗=𝑙+1𝜓_[𝑗]. (18)

We can see that this is indeed in the format of (4), where 𝜉_𝑗=𝑛𝛾𝑓_𝑗ℎ+𝛾+𝛿𝑓_𝑗𝑛𝑓for𝑗≤ℎ𝛾+𝛿_𝑗𝑛forℎ+1≤𝑗≤𝑛. (19)

If there is only learning effect, we have 𝑏=0; therefore 𝑓_𝑗𝑘 is reduced to 𝑗^𝑐. Thus the equation above becomes 𝜉_𝑗=𝑛𝛾𝑗^𝑐+𝛾+𝛿𝑗^𝑐𝑗for𝑗≤ℎ𝛾+𝛿^𝑐forℎ+1≤𝑗≤𝑛. (20)

On the other hand, if there is only time-dependent processing time effect, we have 𝑐=0; therefore, 𝑓_𝑗𝑘 becomes (1+𝑏)^𝑘−𝑗. Thus the equation above becomes 𝜉_𝑗=𝑛𝛾(1+𝑏)^ℎ−𝑗+𝛾+𝛿(1+𝑏)^𝑛−𝑗()for𝑗≤ℎ𝛾+𝛿1+𝑏^𝑛−𝑗forℎ+1≤𝑗≤𝑛. (21)

Application of the Unified Optimization Algorithm. According to the analysis above, the objective function (9) can be written in the format of (4) for all three due date assignment methods, and thus we can present the following optimization algorithm to solve the problem for the weighted number of tardy jobs objective (see Algorithm 3).

Step  1. Apply Algorithm 2 where 𝜉𝑗 is calculated by (15) for the CON method, by (19) for the SLK method, and by (11) for the    DIF method. Step  2. For the CON method, assign 𝐶[𝑙∗] to all 𝑑∗[𝑗]. For the SLK method, assign slack value 𝑠∗=𝐶[𝑙∗−1] to all jobs. For the DIF    method, let 𝑑∗[𝑗]=𝐶[𝑗] for 𝑗=1,2,…,𝑙∗, and 𝑑∗[𝑗]=0 for 𝑗=𝑙∗+1,…,𝑛.

Algorithm 3 

A unified optimization algorithm for solving the 1∣𝑝_[𝑗]=(𝑎_[𝑗]+𝑏𝑆_[𝑗])𝑗^𝑐, 𝑋∣∑^𝑛_𝑗=1𝜓_𝑗𝑈_𝑗+𝛾∑^𝑛_𝑗=1𝑑_𝑗+𝛿𝐶_max problems for 𝑋∈{CON, SLK, DIF}.

By Theorem 2, we have the following result for Algorithm 3.

Theorem 5. Algorithm 3 solves the 1∣𝑝_[𝑗]=(𝑎_[𝑗]+𝑏𝑆_[𝑗])𝑗^𝑐, 𝑋∣∑^𝑛_𝑗=1𝜓_𝑗𝑈_𝑗+𝛾∑^𝑛_𝑗=1𝑑_𝑗+𝛿𝐶_max problems for 𝑋∈{𝐶𝑂𝑁,𝑆𝐿𝐾,𝐷𝐼𝐹} in 𝑂(𝑛⁴) time.

It is easy to verify that if, instead of the makespan, we consider the total completion time in the objective of Theorem 5, that is, we replace 𝛿𝐶_max with 𝜃∑𝐶_𝑗, the same arguments hold. The only difference is that, in the definition of 𝜉_𝑗, we use 𝜃∑^𝑛_𝑘=𝑗𝑓_𝑗𝑘 instead of 𝛿𝑓_𝑗𝑛. Therefore, we also state the following result.

Theorem 6. The 1∣𝑝_[𝑗]=(𝑎_[𝑗]+𝑏𝑆_[𝑗])𝑗^𝑐, 𝑋∣∑^𝑛_𝑗=1𝜓_𝑗𝑈_𝑗+𝛾∑^𝑛_𝑗=1𝑑_𝑗+𝜃∑𝐶_𝑗 problems for 𝑋∈{𝐶𝑂𝑁,𝑆𝐿𝐾,𝐷𝐼𝐹} are solvable in 𝑂(𝑛⁴) time.

In the next two sections, we omit the makespan considerations; that is, 𝛿=0. The objective function (9) now becomes 𝑔(𝜋)=^𝑛_𝑗=1𝜓_𝑗𝑈_𝑗+𝛾^𝑛_𝑗=1𝑑_𝑗. (22)

3.2. Faster Algorithms for Cases with Time-Dependent Processing Times Only

In this section, we present quadratic-time algorithms for cases without learning effect (𝑐=0 in (3)) or makespan considerations (𝛿=0) using dynamic programming techniques. In this case (7) becomes 𝑓_𝑗𝑘=(1+𝑏)^𝑘−𝑗. (23)

Our method is to give the recursive relation for the optimal scheduling cost, which contains a term 𝑎_[𝑗] times, some coefficient. To this end, we need to rewrite 𝐶_[𝑘] in (8) and ∑^𝑘_𝑗=1𝐶_[𝑗] in terms of the 𝑎_[𝑗]'s as follows: 𝐶_[𝑘]=^𝑘_𝑗=1(1+𝑏)^𝑘−𝑗𝑎_[𝑗]. (24)

For the ease of the following summation calculation, we change it to the following form: 𝐶_[𝑗]=^𝑗_𝑖=1(1+𝑏)^𝑗−𝑖𝑎_[𝑖],_𝑘𝑗=1𝐶_[𝑗]=^𝑘𝑗_𝑗=1𝑖=1𝑓_𝑖𝑗𝑎_[𝑖]=^𝑘𝑘_{𝑗=1𝑖=𝑗}𝑓_𝑗𝑖𝑎_[𝑗]=^𝑘𝑘_{𝑗=1𝑖=𝑗}(1+𝑏)^𝑖−𝑗𝑎_[𝑗]=^𝑘_𝑗=1(1+𝑏)^{𝑘−𝑗+1}−1𝑏𝑎_[𝑗], (25)(26) where we changed the order of summation for the second equality.

Since the per-unit due date assignment cost 𝛾 is the same for each job 𝑗 in (22) and the order of tardy jobs is immaterial, we can assume that the early jobs are always sequenced in SPT order. Thus, we sort the jobs into SPT order and decide which job is to be early or tardy in a backward fashion. For the sake of simplicity, suppose they are already sorted in this way.

Notice that in (24) and (26), the coefficient of 𝑎_[𝑗] is dependent only on the number of early jobs after job [𝑗]∶𝑘−𝑗. In fact, 𝑎_𝑗 has a “weight’’ of (1+𝑏)^𝑘−𝑗 (in other words, it contributes (1+𝑏)^𝑘−𝑗 “times’’) in the calculation of 𝐶_[𝑘], and it has a “weight’’ of ((1+𝑏)^{𝑘−𝑗+1}−1)/𝑏 (in other words, it contributes ((1+𝑏)^{𝑘−𝑗+1}−1)/𝑏 “times’’) in the summation ∑^𝑘_𝑗=1𝐶_[𝑗].

This enables us to design the following backward dynamic programming. Define 𝑤(𝑗,𝑘) to be the minimum cost of scheduling jobs 𝑗,𝑗+1,…,𝑛 s.t. 𝑘 of them are early, for 1≤𝑗≤𝑛 and 0≤𝑘≤𝑛−𝑗+1. We have the following set of boundary conditions for all cases when all jobs are tardy: 𝑤𝑗,0=^𝑛_𝑖=𝑗𝜓_[𝑖], (27) and another boundary condition when 𝑗=𝑛: 𝑤(𝑛,1)=𝛾𝑎_𝑛. (28)

Now we describe the recursive functions to calculate 𝑤(𝑗,𝑘) for the three due date assignment methods.

DIF. If job 𝑗 is early, then there will be 𝑘−1 early jobs among 𝑗+1,…,𝑛. The optimal due date for job 𝑗 is 𝐶_𝑗. By (22) and (26), 𝑎_𝑗 is counted 𝛾((1+𝑏)^𝑘−1)/𝑏 “times.’’ Therefore 𝑤()𝑗,𝑘=𝑤𝑗+1,𝑘−1+𝛾1+𝑏^𝑘−1𝑏𝑎_𝑗; (29) otherwise, when job 𝑗 is tardy, the optimal due date for job 𝑗 is 0; thus 𝑤𝑗,𝑘=𝑤𝑗+1,𝑘+𝜓_𝑗. (30) Therefore, we have the following recursive equation: 𝑤𝑤()𝑗,𝑘=min𝑗+1,𝑘−1+𝛾1+𝑏^𝑘−1𝑏𝑎_𝑗,𝑤𝑗+1,𝑘+𝜓_𝑗. (31)

CON. By the due date assignment rule, each job is assigned the same due date, which is the sum of the processing times of all the early jobs. Therefore, if job 𝑗 is early, then it will be counted 𝑛 times. Thus 𝑤𝑗,𝑘=𝑤𝑗+1,𝑘−1+𝑛𝛾(1+𝑏)^𝑘−1𝑎_𝑗 (32) by (22) and (24) and the fact that there are 𝑘−1 early jobs after job 𝑗; otherwise, when job 𝑗 is tardy, 𝑤𝑗,𝑘=𝑤𝑗+1,𝑘+𝜓_𝑗. (33) Therefore, we have the following recursive equation: 𝑤𝑤()𝑗,𝑘=min𝑗+1,𝑘−1+𝑛𝛾1+𝑏^𝑘−1𝑎_𝑗,𝑤𝑗+1,𝑘+𝜓_𝑗. (34)

SLK. From [34], we know that the optimal common slack value is 𝑠^∗=𝐶_[𝑙−1], where 𝑙 is the number of early jobs. In other words, the last early job does not contribute to the due date assignment cost, but the other ones (the first 𝑙−1) do. If job 𝑗 is early and 𝑘≥2, then 𝑤𝑗,𝑘=𝑤𝑗+1,𝑘−1+𝑛𝛾(1+𝑏)^𝑘−1𝑎_𝑗 (35) by (22) and (24) and the fact that there are 𝑘−1>0 early jobs after job 𝑗; if job 𝑗 is early and 𝑘=1, then it is the last early job and thus 𝑤𝑗,𝑘=𝑤𝑗+1,𝑘−1; (36) that is, 𝑤𝑗,1=𝑤𝑗+1,0=^𝑛_𝑖=𝑗+1𝜓_[𝑖]; (37) if job 𝑗 is tardy, we have 𝑤𝑗,𝑘=𝑤𝑗+1,𝑘+𝜓_𝑗. (38) Therefore, we have the following recursive equation: 𝑤𝑤()𝑗,𝑘=min𝑗+1,𝑘−1+𝑛𝛾1+𝑏^𝑘−1𝑎_𝑗,𝑤𝑗+1,𝑘+𝜓_𝑗, (39) for 𝑘≥2, and (for 𝑘=1) 𝑤𝑗,1=min𝑤𝑗+1,0,𝑤𝑗+1,1+𝜓_𝑗=min^𝑛_𝑖=𝑗+1𝜓_[𝑖],𝑤𝑗+1,1+𝜓_𝑗. (40) From the analysis above, we can present Algorithms 4 and 5.

Initialization:                   𝑤(𝑛,1)∶=𝛾𝑎𝑛  For 𝑗:=1To 𝑛Do                  𝑤(𝑗,0)∶=∑𝑛𝑖=𝑗𝜓[𝑖] Recursion:  For 𝑗:=𝑛−1Down To 1Do   For 𝑘:= 1To 𝑛−𝑗+1 Do    for CON:           𝑤(𝑗,𝑘)=min𝑤𝑗+1,𝑘−1+𝑛𝛾(1+𝑏)𝑘−1𝑎𝑗,𝑤𝑗+1,𝑘+𝜓𝑗    for DIF:         𝑤(𝑗,𝑘)=min𝑤𝑗+1,𝑘−1+𝛾(1+𝑏)𝑘−1𝑏𝑎𝑗,𝑤𝑗+1,𝑘+𝜓𝑗 Optimal solution value:                min{𝑤(1,𝑘)∣0≤𝑘≤𝑛}. End

Algorithm 4 

The optimization algorithm for the solution of 1∣𝑝_[𝑗]=𝑎_[𝑗]+𝑏𝑆_[𝑗], 𝑋∣∑^𝑛_𝑗=1𝜓_𝑗𝑈_𝑗+𝛾∑^𝑛_𝑗=1𝑑_𝑗 problems for 𝑋∈{CON, DIF}.

Initialization:                 𝑤(𝑛,1)∶=𝛾𝑎𝑛  For 𝑗∶=1To 𝑛Do                𝑤(𝑗,0)∶=∑𝑛𝑖=𝑗𝜓[𝑖] Recursion:  For 𝑗∶=𝑛−1Down To 1Do      𝑤(𝑗,1)=min𝑤𝑗+1,0,𝑤𝑗+1,1+𝜓𝑗=min∑𝑛𝑖=𝑗+1𝜓[𝑖],𝑤𝑗+1,1+𝜓𝑗 For 𝑗:=𝑛−1Down To 1Do    For 𝑘:= 2 To 𝑛−𝑗+1Do        𝑤(𝑗,𝑘)=min𝑤(𝑗+1,𝑘−1)+𝑛𝛾(1+𝑏)𝑘−1𝑎𝑗,𝑤(𝑗+1,𝑘)+𝜓𝑗 Optimal solution value:           min{𝑤(1,𝑘)∣0≤𝑘≤𝑛}.  End

Algorithm 5 

The optimization algorithm for the solution of 1∣𝑝_[𝑗]=𝑎_[𝑗]+𝑏𝑆_[𝑗],SLK∣∑^𝑛_𝑗=1𝜓_𝑗𝑈_𝑗+𝛾∑^𝑛_𝑗=1𝑑_𝑗 problem.

In summary, we have the following results.

Theorem 7. Algorithm 4 solves the 1∣𝑝_[𝑗]=𝑎_[𝑗]+𝑏𝑆_[𝑗],𝑋∣∑^𝑛_𝑗=1𝜓_𝑗𝑈_𝑗+𝛾∑^𝑛_𝑗=1𝑑_𝑗 problems for 𝑋∈{𝐶𝑂𝑁,𝐷𝐼𝐹} in 𝑂(𝑛²) time. Algorithm 5 solves the 1∣𝑝_[𝑗]=𝑎_[𝑗]+𝑏𝑆_[𝑗],𝑆𝐿𝐾∣∑^𝑛_𝑗=1𝜓_𝑗𝑈_𝑗+𝛾∑^𝑛_𝑗=1𝑑_𝑗 problem in 𝑂(𝑛²) time.

Proof. The correctness of the algorithms follows from (24)–(40) and the preceding discussion. For the time complexity, we note that, in both algorithms, the outer recursion 𝑗 takes 𝑂(𝑛) time, and the inner recursion on 𝑘 also takes 𝑂(𝑛) time. Therefore, the total time complexity is 𝑂(𝑛²).

3.3. Faster Algorithms for Cases with Learning Effect Only

In this section, we present quadratic-time algorithms for cases without time-dependent effect (𝑏=0 in (3)), or makespan consideration (𝛿=0), using forward dynamic programming techniques. In this case, we have from (7) 𝑓_𝑗𝑘=𝑗^𝑐. (41)

As in the previous section, our method is to give the recursive relation for the optimal scheduling costs, which contain the term 𝑎_[𝑗] times, some coefficient. To this end, we need to rewrite 𝐶_[𝑘] in (8) in terms of 𝑎_[𝑗] as follows: 𝐶_[𝑘]=^𝑘_𝑗=1𝑗^𝑐𝑎_[𝑗]. (42)

Similarly to the discussion in the previous section, we can assume that the early jobs are already sequenced in SPT order. Our goal is to decide which job is to be early or tardy in a forward fashion this time.

Notice that, in (42), the coefficient of 𝑎_[𝑗] is dependent only on the number of early jobs before job [𝑗]. This enables us to design the following forward dynamic programming algorithms.

CON. Define 𝑤(𝑗,𝑘) to be the minimum cost of scheduling jobs 1,2,…,𝑗 s.t. 𝑘 of them are early, for 1≤𝑘≤𝑗≤𝑛. We have the following set of boundary conditions when jobs 1,2,…,𝑗 are all tardy: 𝑤𝑗,0=^𝑗_𝑖=1𝜓_[𝑖], (43) for 1≤𝑗≤𝑛, and 𝑤(1,1)=𝑎₁. (44)

If job 𝑗 is early, then its processing time gets counted in the (common) due date of every job, that is, 𝑛 times, and thus 𝑤𝑗,𝑘=𝑤𝑗−1,𝑘−1+𝑛𝑗^𝑐𝑎_𝑗 (45) by (22) and (42) and the fact that there are 𝑘−1 early jobs before job 𝑗; otherwise, job 𝑗 is tardy, and we have 𝑤𝑗,𝑘=𝑤𝑗−1,𝑘+𝜓_𝑗. (46) Therefore, we have the following recursive equation: 𝑤𝑗,𝑘=min𝑤𝑗−1,𝑘−1+𝑛𝑗^𝑐𝑎_𝑗,𝑤𝑗−1,𝑘+𝜓_𝑗. (47)

SLK. As in the previous section, we know from [34] that the optimal common slack value is 𝑠^∗=𝐶_[𝑙−1], where 𝑙 is the number of early jobs. In other words, the last early job does not contribute to the due date assignment cost, but the other ones (the first 𝑙−1) do. Due to this special property, we do not have the same direct recursive equations as in the CON case. However, we can still utilize those recursive equations to get a set of recursive equations for the SLK case.

Let 𝑊(𝑟) be the minimum cost to schedule jobs of 1,2,…,𝑛 s.t. job 𝑟 is the last early one. By the above discussion and definition of 𝑤(𝑗,𝑘) in the CON part (in which the due date assignment cost for the 𝑘 early jobs is calculated recursively), we have 𝑊(𝑟)=min_{0≤𝑘≤𝑟−1}𝑤(𝑟−1,𝑘)+^𝑛_𝑖=𝑟+1𝜓_𝑖 (48) for 1≤𝑟≤𝑛. We define 𝑤(0,0)=0 and the minimum cost to schedule jobs 1,2,…,𝑛 is given by 𝑊=min_{1≤𝑟≤𝑛}𝑊(𝑟). (49)

From the analysis above, we are ready to present Algorithm 6.

Initialization:                  𝑤(1,1)∶=𝛾𝑎1  For 𝑗∶=1To 𝑛Do                 𝑤(𝑗,0)∶=∑𝑖𝑖=𝑖𝜓[𝑖] Recursion:  For 𝑗∶=1To 𝑛Do   For 𝑘∶=0To 𝑗Do          𝑤𝑗,𝑘=min𝑤𝑗−1,𝑘−1+𝑛𝑗𝑐𝑎𝑗,𝑤𝑗−1,𝑘+𝜓𝑗               𝑊(𝑗)=min0≤𝑘≤𝑗−1𝑤(𝑗−1,𝑘)+∑𝑛𝑖=𝑗+1𝜓𝑖 Optimal solution value:  For CON:                 min{𝑤(1,𝑘)∣0≤𝑘≤𝑛}.  For SLK:                  𝑊=min1≤𝑗≤𝑛𝑊(𝑗).  End

Algorithm 6 

The optimization algorithm for the solution of 1∣𝑝_[𝑗]=𝑎_[𝑗]𝑗^𝑐,𝑋∣∑^𝑛_𝑗=1𝜓_𝑗𝑈_𝑗+𝛾∑^𝑛_𝑗=1𝑑_𝑗 problems for 𝑋∈{CON, SLK}.

In summary, we have the following result.

Theorem 8. Algorithm 6 solves the 1∣𝑝_[𝑗]=𝑎_[𝑗]𝑗^𝑐,𝑋∣∑^𝑛_𝑗=1𝜓_𝑗𝑈_𝑗+𝛾∑^𝑛_𝑗=1𝑑_𝑗 problems for 𝑋∈{𝐶𝑂𝑁,𝑆𝐿𝐾} in 𝑂(𝑛²) time.

Proof. The correctness of the algorithms follows from the analysis before them. For the time complexity, we note that, in both algorithms, the outer recursion 𝑗 takes 𝑂(𝑛) time, and the inner recursion 𝑘 also takes 𝑂(𝑛) time. Therefore, the total time complexity is 𝑂(𝑛²).

4. Summary and Concluding Remarks

We considered single machine scheduling problems with learning/deterioration effects and time-dependent processing times, with due date assignment considerations and with the objective to minimize the weighted number of tardy jobs. We gave polynomial-time solutions for all problems considered. It remains an open question whether there exists a 𝑂(𝑛²) time algorithm for the DIF due date assignment method with learning effect only. Another challenge is to design a 𝑂(𝑛²) time algorithm for the case of 𝛿>0, that is, to consider the makespan.

Conflict of Interests

The authors declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This research was supported in part by Discovery Grant 1798-08 by the Natural Sciences and Engineering Research Council of Canada.