Engineering Applications of Artificial Intelligence
Multi-product sequencing and lot-sizing under uncertainties: A memetic algorithm
Highlights
► We study a stochastic multi-product sequencing and lot-sizing problem. ► Two types of uncertainties are considered: random lead time and random yield. ► A decomposition approach is used to separate sequencing and lot-sizing algorithms. ► A new memetic algorithm is proposed for lot-sizing sub-problem. ► Results show that the algorithms developed can be efficiently used for large scale instances.
Introduction
In this paper, a multi-product lot-sizing and sequencing problem under uncertainties is studied. The source of the problem is derived from an automated semiconductor manufacturing plant where there are non-negligible percentages of rejects and breakdowns. However, this situation can concern any automatic production line working under uncertainties and the proposed approach could be easily extended to different types of production lines.
The example given here is from our experience of designing a paced line to produce several types of conductor patterns. These parts are used to obtain electronic modules (printed circuits). Since the considered semi-conductor factory is greatly automated, there is no staff other than maintenance for almost the whole day. The facility functions with three shifts. Specifically, the main task for the day shift is to define the production plan for the next 24 h and start the manufacturing process. The evening and night shifts, which consist of maintenance personnel only, insure the production line continue to function, but cannot change the production plan. As a consequence, the production plan is set for 24 h and will not be adjusted to take into account rejects or breakdowns that may occur in the later shifts.
After processing, parts (conductor patterns) are placed in an automatic storage system, and they are used for the assembly of electronic modules. The automatic storage system is expensive and restricted in volume. Consequently, it should work with one day stock limit, if possible. So, this line and storage system should be able supply the next assembly line just-in-time. In other words, the following policy is applied: all the items used for assembly in period r+1 must be in the storage system by the end of planning period r. At the beginning of the period r, the demand for all items types for assembly in the planning period r+1 is known (taking into account the current stocks in the automatic storage system, if they exist, and the production plans of the assembly line for period r and r+1). Thus, for each period r, the following question has to be answered: how many items of each type must be released to production in the beginning of the period r to obtain the necessary quantity for all components in the next production run r+1 of the assembly line? With this sort of production, there is a non-negligible percentage of rejects, because some finished components are produced with unacceptable quality. Quality control is made at the end of the line with no intermediate quality control. In addition, the machines of the line are often stopped briefly because of breakdowns.
This leads to a new and very interesting production control problem dealing with optimal lot-sizing and scheduling under uncertainties. There are two types of possible policies which are in conflict. To diminish the influence of random breakdowns, the safety time (the difference between the duration of the planning horizon and the time necessary to produce all lots) can be increased, but in this case, it is necessary to reduce the sizes of lots, so the production plan can be more easily perturbed by rejects. On the other hand, the size of lots can be increased to diminish the impact of random rejects, but the line will be more sensitive to random breakdowns, because of insufficient safety time. There would be not enough time to repair of all these breakdowns and the line cannot produce the necessary quantity of items. Moreover, a set-up time is necessary between processing of two different products for reconfiguration of manufacturing facility. The set-up time depends on type of already manufactured product and new items to process. Thus, for the both policies mentioned above, there is an additional level of action affecting the safety time.
In this paper, we will reexamine the probabilistic formulation of this lot-sizing and scheduling problem, initially evoked in Dolgui (2002). For this problem, a first approach was suggested in Dolgui et al. (2005): authors have shown that this problem can be reduced to a single machine problem with sequence-dependent set-up times and that its optimal solution can be obtained using a decomposition into several optimization sub-problems: enumerating, sequencing and lot-sizing. Note that the latter two problems are NP-hard but the sequencing sub-problem can be transformed in a well-known Traveling Salesman Problem, for which there exist a large number of effective algorithms. For lot-sizing sub-problem, in Dolgui et al. (2005), the authors presented an idea how dynamic programming (DP) approach could be used to solve it. Nevertheless, only the decomposition framework and a recursive DP expression for lot-sizing were provided. No evaluating tests were performed. Thus, the question on the effectiveness of this approach for small, medium and large size cases is still open. This needed to be explored further which is the motivation of the current paper.
We present a global approach intended to treat actual problems of industrial sizes. This study will employ the earlier proposed overall decomposition approach. DP will be tested on several numerical examples and a memetic algorithm (MA) based on a local search (LS) and a genetic algorithm (GA) will be suggested for large scale cases.
The rest of the paper is organized as follows. Overall assumptions and problem statement are presented in Section 2. A review of related literature is given in Section 3. Section 4 introduces the solution framework: how the uncertainties are modeled and decomposition is accomplished. In Section 5, a Memetic Algorithm (MA) with its elements is presented. In Section 6, several experimental results comparing the algorithms (DP, LS, MA) are reported. Section 7 contains some concluding remarks and further perspectives.
Section snippets
Problem formulation
In this paper, a paced flow line that produces items of several types in lots is considered. This consists of several machines located sequentially according to a manufacturing process. One lot is a set of items of the same type that pass through the line sequentially without any other types inserted. A machine can produce no more than one item at the same time. There are no buffers between machines. The processing times are known and transfer times between machines can be integrated in
Survey of literature
This is a recently stated problem, examined only in two previous publications (Dolgui, 2002) and (Dolgui et al., 2005). Nevertheless, there is an extensive literature on related problems of operations management (see for example, Dolgui and Proth, 2010). Thus, the objective of this section is to try to position this new problem on corresponding research domains and show problems containing certain similar elements. This can be useful to better understand the problem considered.
Our problem
Calculation of the overall service level for a feasible solution
For the problem considered, the criterion is the overall service level, i.e. the probability of obtaining the required quantities of items for each type at the end of period. This criterion takes into account both rejects and breakdowns, i.e. random number of good quality items and random total working time of the line. Note that the line considered is a paced line where if a machine breakdowns then the entire production line is halted. Also, the setups are made for all machines simultaneously.
Lot-sizing subproblem
The sub-problem P1 is a simple enumeration, and sub-problem P2 can be solved efficiently using known methods from literature. Therefore, hereafter, we deal with only the sub-problem P3. It is supposed that the sequence of lots π⁎ was set, all lots were renumbered accordingly and total set-up time S(π⁎) was calculated.
While DP from Dolgui et al. (2005) provides an optimal solution, it might be inefficient for large scale instances because of the rapidly growing computing time and computer memory
Computer experiments
This section is divided into five parts. Section 6.1 presents how the problem instances were generated and the main parameters of the MA proposed. Section 6.2 announces the methods that will be tested and compared. Section 6.3 demonstrates the limitations of DP method. Section 6.4 provides an analysis of quality for MA solutions for the instances of small size. Finally, in Section 6.5, the MA is compared with the LS for large scale problems.
Conclusions
A real-life problem of optimal lot-sizing and sequencing for a production line with random breakdowns and rejects was studied. The line considered manufactures intermediate components of different types for subsequent assembly into modules. The problem is to choose sequences and lot sizes. The objective is to maximize the probability to have a sufficient number of components by the end of a given planning horizon, i.e. maximize the overall service level. The benefit of this optimization is
Acknowledgments
The authors thank Chris Yukna for his help with English.
References (49)
- et al.
Lot sizing with random yields and tardiness costs
Comput. Oper. Res.
(2000) - et al.
A survey of scheduling problems with setup times and costs
Eur. J. Oper. Res.
(2008) - et al.
Inventory control in a multi-supplier system
Int. J. Prod. Econ.
(2006) - et al.
Production lot sizing with process deterioration and machine breakdown under inspection schedule
Omega
(2009) - et al.
A hybrid genetic algorithm approach on multi-objective of assembly planning problem
Eng. Appl. Artif. Intell.
(2002) - et al.
Machine scheduling with job class setup and delivery considerations
Comput. Oper. Res.
(2010) Robust planning in optimization for production system subject to random machine breakdown and failure in rework
Comput. Oper. Res.
(2010)- et al.
A genetic algorithm with a mixed region search for the asymmetric traveling salesman problem
Comput. Oper. Res.
(2003) - et al.
A genetic algorithm to solve the general multi-level lot-sizing problem with time-varying costs
Int. J. Prod. Econ.
(2000) - et al.
Supply planning under uncertainties in MRP environments: a state of the art
Annu. Rev. Control
(2007)
Lot sizing in a no-wait flow shop
Oper. Res. Lett.
Applying genetic algorithms to dynamic lot sizing with batch ordering
Comput. Ind. Eng.
Optimal lot sizing for an unreliable production system based on net present value approach
Int. J. Prod. Econ.
Optimal lot sizing for an unreliable production system under partial backlogging and at most two failures in a production cycle
Int. J. Prod. Econ.
An algorithm for the multiple lot sizing problem with rigid demand and interrupted geometric yield
J. Math. Anal. Appl.
The finite multiple lot sizing problem with interrupted geometric yield and holding costs
Eur. J. Oper. Res.
Genetic algorithm for supply planning in two-level assembly systems with random lead times
Eng. Appl. Artif. Intell.
How to protect against demand and yield risks in MRP systems
Int. J. Prod. Econ.
Managing yield by lot splitting in a serial production line with learning, rework and scrap
Int. J. Prod. Econ.
On a production-inventory system of deteriorating items subject to random machine breakdowns with a fixed repair time
Math. Comput. Model.
Lot sizing with a Markov production process and imperfect items scrapped
Int. J. Prod. Econ.
Simultaneous lotsizing and scheduling by combining local search with dual reoptimization
Eur. J. Oper. Res.
A comparative analysis of several asymmetric traveling salesman problem formulations
Comput. Oper. Res.
Evolutionary approaches to the design and organization of manufacturing systems
Comput. Ind. Eng.
Cited by (16)
Integrated lot-sizing and scheduling: Mitigation of uncertainty in demand and processing time by machine learning
2023, Engineering Applications of Artificial IntelligenceCitation Excerpt :For a detailed review of these models, see Drexl and Kimms (1997). Several studies consider different types of uncertainty in the lot-sizing and scheduling problem, such as uncertainty of demand quantity (Wan and Zhan, 2021) or demand timing (Akartunalı and Dauzère-Pérès, 2022), lead time uncertainty (Slama et al., 2021), processing costs (Hu and Hu, 2020), and setup time uncertainty (Ramezanian and Saidi-Mehrabad, 2013), capacity constraint uncertainty (Zhu et al., 2022), random machine breakdown (Dolgui et al., 2010), random product rejection (Schemeleva et al., 2012), random workforce efficiency (Li and Hu, 2017), etc. However, most of them are based on stochastic or fuzzy programming in a predictive manner without real-time response.
Robust machine layout design under dynamic environment: Dynamic customer demand and machine maintenance
2019, Expert Systems with Applications: XCitation Excerpt :Machine breakdown has been one of the most studied disruptions in flexible job shop scheduling (Nouiri et al., 2017). The machine failure rate has been represented by the Poisson distribution (Schemeleva, Delorme, Dolgui & Grimaud, 2012) or generated randomly (Nodem, Kenne & Gharbi, 2011). Machine lifetime is commonly modelled using the Weibull distribution (Fitouhi & Nourelfath, 2012).
Modeling and optimization of a road–rail intermodal transport system under uncertain information
2018, Engineering Applications of Artificial IntelligenceCitation Excerpt :To be specific, this algorithm uses an effective genetic search method to explore the search space and an efficient local search method to exploit information in the search region. MA proposed by Moscato (1989), is similar to the process of natural evolution, but in which adds a selective mutation, through this process to solve the complicated problems (Tang and Yao, 2007; Schemeleva et al., 2012; Wang et al., 2014; Qu et al., 2017). MA is different from genetic algorithm (GA), the latter is a simple process of simulating biological evolution, while MA adds local search into the process of biological evolution to determine how to change.
Evaluation of solution approaches for a stochastic lot-sizing and sequencing problem
2018, International Journal of Production EconomicsCitation Excerpt :We decided to use a linear model from Sherali and Driscoll (2002) to obtain an optimal solution (in subsection 4.2.1) and a genetic algorithm of Nagata and Soler (2012) to solve the problem approximately (subsection 4.2.2). Further, subsection 4.3 contains the methods proposed earlier to solve the lot-sizing part of the problem: exact algorithm for the lot-sizing part (DP procedure by Dolgui et al. (2005)) described in subsection 4.3.1 and a meta-heuristic approach (MA) from Schemeleva et al. (2012) recapitulated in subsection 4.3.2. A review of the literature has shown that the ATSP formulation proposed by Desrochers and Laporte in 1991 is still very effective.
Artificial intelligence in engineering risk analytics
2017, Engineering Applications of Artificial IntelligenceA memetic algorithm for a stochastic lot-sizing and sequencing problem
2015, IFAC-PapersOnLine