Due-Window Assignment and Resource Allocation Scheduling with Truncated Learning Effect and Position-Dependent Weights

This paper studies single-machine due-window assignment scheduling problems with truncated learning effect and resource allocation simultaneously. Linear and convex resource allocation functions under common due-window (CONW) assignment are considered. The goal is to find the optimal due-window starting (finishing) time, resource allocations and job sequence that minimize a weighted sum function of earliness and tardiness, due window starting time, due window size, and total resource consumption cost, where the weight is position-dependent weight. Optimality properties and polynomial time algorithms are proposed to solve these problems.


Introduction
Scheduling models and problems with learning effects (see Biskup [1]; Lu et al. [2]; Azzouz et al. [3]; Wang et al. [4]) and/or resource allocations (see Shabtay and Steiner [5]; Yang et al. [6]) have become popular topics for scheduling researchers in recent years. Scheduling with learning effects and resource allocations simultaneously was introduced by Wang et al. [7], who focused on single-machine scheduling problems. Lu et al. [8] studied single-machine due-date assignment scheduling with learning effects and resource allocations. ey proved that several problems can be solved in polynomial time. Wang and Wang [9] and Li et al. [10] considered common and slack due-window assignment problems with learning effects and resource allocations. Wang and Wang [11] considered single-machine scheduling problems with learning effects and convex resource allocation function. For the scheduling criterion (the total resource compression criterion) minimization subject to the constraint that the total resource compression criterion (the scheduling criterion) is less than or equal to a fixed constant, they proved that the problems can be solved in polynomial time. Wang et al. [12] and Liu and Jiang [13] considered due-date assignment scheduling with job-dependent learning effects and resource allocation. Liu and Jiang [14] considered flow shop due-date assignment scheduling with resource allocation and learning effect. Shi and Wang [15] considered flow shop due-window assignment scheduling with resource allocation and learning effect.
In recent years, many researchers focused on the study of scheduling with due-window, where a time interval is assumed, such that jobs completed within this interval are not penalized (Janiak et al. [16] and Wang et al. [17]). Wang et al. [18] considered the single-machine due-window scheduling problems with position-dependent weights. For the weighted sum of earliness and tardiness, due window starting time, and due window size, where the weight only dependent on its position in a sequence (i.e., a positiondependent weight), they proved that the problems can be solved in polynomial time. In this study, we continue the work of Wang et al. [18], i.e., we consider the due-window assignment scheduling problems with learning effect and resource allocation in the single-machine environment. e goal is to find the optimal due-window starting (finishing) time, resource allocations, and job sequence such that a sum of scheduling cost (including weighted sum function of earliness and tardiness, due window starting time, due window size, where the weight is position-dependent weight) and total resource consumption cost is minimized. e contributions of this paper are given as follows.
(1) e structural properties of single-machine scheduling problems are derived. (2) For the linear resource allocation, we proved that the sum of scheduling cost and total resource consumption cost can be solved in polynomial time. For the convex resource allocation, three versions of scheduling cost and total resource consumption cost can be solved in polynomial time respectively. (3) It is further extended the model to the case with slack due-window (SLKW) assignment model. e rest of the article is organized as follows: In Section 2, we introduce the problem. In Sections 3 and 4, we provide some properties to optimally solve these problems under linear and convex resource allocation. In Section 5, we conclude the paper.

Problem Formulation
We study a scheduling problem consisting of a set of n independent jobs N � J 1 , J 2 , . . . , J n that need to be processed on a single machine. For the linear resource allocation, the actual processing time of job J j is where p j is the basic processing time of job J j (i.e., the processing time without any resource allocation and truncated learning effect), α j ≤ 0 is the job-dependent learning rate (Mosheiov and Sidney [19]) of job J j , 0 < δ < 1 is a truncation parameter (Wang et al. [20]), β j is the compression rate of job J j , and u j is the amount of resource allocated to job J j and satisfies 0 ≤ u j ≤ u j ≤ ((p j max r α j , δ { })/β j ). For the convex resource allocation, the actual processing time of job J j is where η > 0 is a constant, i.e., P A j is a convex decreasing function of resource u j .
Let [d 1 , d 2 ]be the common due-window for all jobs, where d 1 ≥ 0(d 2 , d 1 ≤ d 2 ) denotes the starting (finishing) time of the common due window. e length of the duewindow is D � d 2 − d 1 . Both d 1 and d 2 are decision variables in this paper. e goal of this paper is to find jointly the optimal due-window location, the optimal resource allocation and sequence π such that the following objective function is minimized: where [j] denotes the job scheduled in jth position, w j (j � 0, 1, 2, . . . , n, n + 1) denotes a position-dependent weight, L [j] is the earliness-tardiness of job J [j] , and Using the three-field classification, the problem can be denoted as 1|CONW, [21]), where CONW denotes the common due-window assignment. Wang et al. [18] considered single-machine scheduling problems with CONW and slack due-window (SLKW) assignments problems , q ′ and q ″ are decision variables and D � q ″ − q ′ . Wang et al. [18] proved that these both problems can be solved in O(n log n) time, respectively.

Linear Resource Allocation
Lemma 1 [Wang et al. [18]]. For any given sequence π, there exists an optimal sequence in which d 1 � C [k] for some k and where Proof. From Lemma 1, we have where ξ j (j � 1, 2, . . . , n) are given by (6).
□ Algorithm 1 Step 1. Calculate the indices k and l according to Lemma 1.
Step 4. Calculate the optimal resource allocation by (7). Step can be solved by Algorithm
By taking the first derivative of the objective given by (15) with respect to u [j] , equating it to zero and solving it for J [j] , we have (16).

Lemma 3. For a given sequence, the optimal resource allocation of the problem 1|CONW, P
By substituting (16) into (15), we have Let where ξ r (r � 1, 2, . . . , n) are given by (6). As in Section 3, for the problem 1|CONW, , we can propose the following algorithm: Step 1. Calculate the indices k and l according to Lemma 1.
Similarly to Section 4.

Conclusions
is paper considered the single-machine due-window assignment scheduling problems with learning effect and resource allocation. For the linear and convex resource allocations, we showed that some different models are polynomially solvable, respectively. Future research may focus on the flow shop scheduling problems with learning effect and resource allocation or study the Pareto-optimal solutions with respect to the criterion n j�1 w j L [j] + w 0 d 1 + w n+1 D and the resource compression cost n j�1 v [j] u [j] .

Data Availability
No data were used in the study.

Conflicts of Interest
e author declares that there are no conflicts of interest regarding the publication of this paper.