Parallel-machine scheduling in shared manufacturing

We consider parallel-machine scheduling in the context of shared manufacturing where each job has a machine set to which it can be assigned for processing. Such a set is called the processing set. In the shared manufacturing setting, a job can be assigned not only to certain machines for processing, but can also be processed on the remaining machines at a certain cost. Compared with traditional scheduling with job rejection, the scheduling model under study embraces the notion of sustainable manufacturing. Showing that the problem is NP-hard, we develop a fully polynomial-time approximation scheme to solve the problem when the number of machines is fixed.

1. Introduction. In many applications of the parallel-machine scheduling model, the machines may have the same speed of processing jobs but different functions. Consequently, in such a manufacturing environment, each job has a restricted set of machines on which it can be processed, called its processing set. The general scheduling model with the processing set is as follows: There are n jobs J = {J 1 , J 2 , ..., J n } and m machines M= {M 1 , M 2 , ..., M m }, whereby each job J j is associated with a machine set M j to which it can be assigned for processing, where M j ⊆ M. There are several significant special forms of the processing set, namely the nested processing set, the inclusive processing set, the interval processing set and the treehierarchical processing set. In the case of the nested processing set, for each machine pair M j and M k , we have M j ∩ M k = ∅, M j ⊆ M k , or M k ⊆ M j . In the case of the inclusive processing set, for each machine pair M j and M k , we have either M j ⊆ M k or M k ⊆ M j . The special property of the interval processing set is that for any job J j , M j = M aj , M aj+1 , . . . , M bj for some 1 ≤ a j ≤ b j ≤ m. And for the tree-hierarchical processing set, associated with each machine set M j is a node of a tree.
In the last decade, scheduling with processing set restrictions has received considerable research attention. Leung and Li [29] presented a survey of research on traditional parallel-machine scheduling with processing set restrictions. Since then, many researchers have pursued this line of research [11,16]. Leung and Li [30] provided an expository literature review of this field. Li [33] considered parallel batching machine scheduling with the inclusive processing set restriction and non-identical machine capacity. Leung and Ng [31] provided fast approximation algorithms for uniform-machine scheduling with two kinds of processing sets.
Over the past 20 years, researchers have devoted much effort to study machine scheduling with rejection (MSR). Slotnick [41] and Shabtay et al. [40] provided comprehensive literature reviews on this topic. Ou et al. [35] studied parallel-machine scheduling that combines the inclusive processing set restriction and job rejection. They developed a 5 3 + ε -approximation algorithm that runs in O nm ε 2 log P time to minimize the makespan of all the accepted jobs plus the total rejection cost.
However, rejection may render resources under-utilized [26], which will lead to less social benefit. Therefore, we consider scheduling in the sharing environment. In the academic literature, there is no exact definition of the sharing notion. Price [38] suggested that sharing is the distribution of economic goods and services without considering the return in an intimate social group. Benkler [8] regarded sharing as a non-reciprocal pro-social behaviour. Belk [6] considered sharing as the act and process of distributing personal property to others for their use. Chasin et al. [10] held that the definition of sharing varies in different contexts where sharing takes place.
Applications of sharing can be found in many aspects of the modern life. Barnes and Mattsson [4] indicated that the sharing phenomenon and practice are expanding rapidly with the development of information and communications technology (ICT). Sharing knowledge [42], resources [43], experience [34], services [36], and products [22] are significant forms of the sharing practice. Many firms that practise sharing have achieved commercial success, see, e.g., Uber [27] and Airbnb [3].
Sharing has been a popular buzz word in academic circles. Belk [7] systematically analyzed the differences between sharing, gift giving, and commodity exchange. Lamberton and Rose [28] researched commercial sharing systems. Porter and Kramer [37] emphasized that sharing not only creates a bigger individual value but also leads to greater economic and social values. Albinsson and Perera [2] examined sharing events, and explored the connection between sharing, community building, and alternative consumption. Heinrichs [15] and Hamari et al. [13] revealed that sharing provides new possibilities for sustainable development. Irvine et al. [17] proposed a new interpretation of shared value and illustrated the relationship between individual and shared values. Ryu et al. [39] conducted a survey of the sharing paradigm.
Sharing has gradually penetrated into manufacturing. Brandt [9] first put forward the notion of shared manufacturing. Adhau et al. [1] revealed that manufacturing enterprises attempt to share manufacturing resources with other enterprises, in order to make better use of the resources and obtain greater benefits. Becker and Stern [5] studied resource sharing in transport logistics. They suggested that resource sharing in the manufacturing industry can contribute to the productivity and profitability of the collaborating enterprises. Li et al. [32] studied scheduling of shared distributed manufacturing resources. He and Zhang [14] systematically reviewed the advantages and research status of shared manufacturing. Jiang and Li [23] regarded shared manufacturing as a realized form of social manufacturing. They examined various kinds of shared factories that share production orders, processing resources, and manufacturing capability.
To the best of our knowledge, there is no research on scheduling in the context of shared manufacturing discussed above. Besides, as sharing contributes to sustainable development, it is reasonable to believe that scheduling in shared manufacturing can lead to enhancement of social benefits. Specifically, we consider parallel-machine scheduling that takes processing set restrictions into consideration in the shared manufacturing setting, seeking to develop an algorithm that can handle all the possibilities of processing set restrictions.
We organize the rest of the paper as follows: In Section 2 we introduce the problem under study. In Section 3 we provide a fully polynomial-time approximation scheme (FPTAS) for the problem. Finally, in Section 4, we conclude the paper and suggest topics for future research.
2. Problem description. We formally introduce the problem under study as follows: There are a set of n independent jobs J = {J 1 , J 2 , ..., J n } to be processed on a set of m identical parallel machines M= {M 1 , M 2 , ..., M m }, whereby each job J j . Each machine can process at most one job at a time and job preemption is not allowed. All the jobs are available for processing at time 0.Each job J j has a machine set M j to which J j can be assigned for processing without incurring any cost except the time cost, where M j is an arbitrary subset of M. If job J j is assigned to one of machines in M j , the processing time p j is independent of the machines. However, if job J j is assigned to machine M i for processing, where M i ∈ M\M j , job J j has the processing time p ij and incurs an additional shared service cost w ij . Our goal is to find an optimal schedule to minimize the sum of the makespan and total shared service cost. We use the three-field notation [12] to denote the problem under study as P m |M j , SM | C max + S, where SM and S denote shared manufacturing and the total shared service cost, respectively.
We first show that the problem is NP -hard. It is easy to see that P m |M j | C max is a special case of our problem when w ij → ∞, ∀i, j . In addition, our scheduling problem reduces to problem P m C max when M j = M, for j = 1, 2, . . . , n. Since both problems P m |M j | C max and P m C max are NP -hard, P m |M j , SM | C max + S is NP -hard, too. In the following we provide an FPTAS to deal with this intractable problem.

3.
FPTAS. An algorithm A is called a (1+ε)-approximation algorithm for a minimization problem if it produces a solution that is at most (1+ε) times as big as the optimal value, running in time that is polynomial in the input size. A family of approximation algorithms {A ε } is a fully polynomial-time approximation scheme if, for each ε > 0, algorithm A ε is a (1+ε)-approximation algorithm that is polynomial in the input size and in 1/ε. From now on we assume, without loss of generality, that 0 < ε ≤ 1. If ε > 1, then a 2-approximation algorithm can be taken as a (1+ε)approximation algorithm. In this section we propose an FPTAS for the problem under study based on the technique of trimming the state space (see, e.g., Kovalyov and Kubiak [24,25]).
First, we introduce the variables x j , j = 1, 2, . . . , n, where x j =k if job J j is assigned to machine M k , k ∈ {1, 2, . . . , m}, for processing. Let X denote the set of all the vectors x = (x 1 , x 2 , . . . , x n ) with x j =k, j = 1, 2, . . . , n, k = 1, 2, . . . , m. We define the following initial and recursive functions on X: In addition, Consequently, problem P m |M j , SM | C max + S reduces to the following problem: Minimize Q (x) for x ∈ X. To solve the above problem, we introduce the procedure Partition(A, e, δ) proposed by Kovalyov and Kubiak [24], where A ⊆ X, e is a non-negative integer function on X, and 0 < δ ≤ 1. The procedure partitions A into disjoint subsets A e 1 , A e 2 , . . . , A e ke such that |e (x) − e (x )| ≤ δ min {e (x) , e (x )} for any x, x ∈ A e h , h = 1, 2, . . . , k e . We provide the details of Partition(A, e, δ) in the following.
Procedure Partition(A, e, δ) requires O (|A| log |A|) operations to arrange the vectors of A in non-decreasing order of e (x) and O (|A|) operations to provide a partition. We present two main properties of Partition that we will use in the development of our FPTAS A ε . Kovalyov and Kubiak [24,25] first provided the properties, which Ji and Cheng [18,19], and Ji et al. [20,21] extended subsequently.
We provide a formal description of the FPTAS A ε for problem P m |M j , SM | C max + S as follows: Step 2. (Generation of Y 1 , Y 2 , . . . , Y n ) For set Y j−1 , generate Y j by adding k, k = 1, 2, . . . , m, in position j of each vector from Y j−1 . Calculate the following for any x ∈ Y j , assuming x j = k: If j = n, then set Y n = Y n and go to Step 3. If j < n, then set δ=ε/2 (n + 1) and perform the following computation.
As a consequence, the time complexity of algorithm A ε is O n m+2 L m+2 ε m+1 .

Conclusions.
We consider parallel-machine scheduling with processing set restrictions in the shared manufacturing setting. Different from traditional scheduling with job rejection, the jobs in our problem cannot be rejected. Under the scheduling model in shared manufacturing, a job can be assigned to certain machines for processing, or it can be processed on the remaining machines at a certain cost. Since resource sharing is conductive to sustainable development, it is reasonable to believe that the scheduling model under study embraces the notion of sustainable manufacturing. We give an FPTAS to solve the problem in which the number of machines is fixed. Future research may explore other scheduling models with various objectives in the context of shared manufacturing.