1 Introduction

Lot-sizing and Scheduling (LSS) problem determines the production quantity, machine assignment, and production sequence in order to respond to the customers’ demand with the minimum cost (Rehman et al. 2019). This problem stands on the edge of tactical and operational planning levels in Production Planning and Control (PPC). Although some researchers classify the LSS into the operational level, it is practically linked to the Master Production Schedule (MPS) from the tactical level, and the detailed schedule plan for the operational level can be derived from it.

Integrating the decisions of the LSS is essential when the setup times/costs are sequence-dependent (Fandel and Stammen-Hegene 2006). Neglecting these sequence-dependent setups leads to the loss of available production capacity, weak performance, and finally, results in the increased amount of unfulfilled demand, customer dissatisfaction, and the loss of competitiveness in the industry (Toso et al. 2009). Furthermore, there are two important motivations for this integration: achieving a higher level of feasibility for the production plans (the feasibility point-of-view) and reducing the actual cost of the production plans (the optimality point-of-view) (Almada-Lobo et al. 2015).

Maintenance planning is a crucial matter in manufacturing systems. According to the Total Productive Maintenance (TPM) method, maintenance and production plans should be simultaneously determined. Generally, maintenance planning can be carried out by planning the Preventive Maintenance (PM) activities in order to reduce the component’s failure probability and item degradation (Wu and Zuo 2010). An unsynchronized PM plan without the consideration of the production requirements losses both its effectiveness and applicability and also reduces the productivity of the manufacturing system (Feng et al. 2018).

Addressing the feasibility of the integrated LSS problems, combining the maintenance planning with the integrated LSS problem will lead to a higher level of feasibility for both the maintenance and production problems, a more realistic and practical maintenance schedule, and an enhanced level of the system’s availability for the production plan.

This study integrates the age-based maintenance modelling with one of the variants of the integrated LSS problems, the General Lot-sizing and Scheduling Problem (GLSP). The contributions of the paper can be described in four points:

  • Integrating availability and PM planning with the GLSP

  • Using product-oriented formulation for the sequence-dependent setup in the integrated production and PM planning models at the tactical level

  • Proposing a five-step heuristic approach based on the Lagrangian relaxation algorithm, decomposition, and valid equalities

  • Developing the bi-objective formulation to extend the application of the integrated model

The manuscript is arranged as follows. Section 2 provides a brief literature review on the GLSP and the integrated production and PM planning problem. Section 3 presents the mathematical model for the developed GLSP-PM. Section 4 describes the proposed solution approach. Section 5 addresses the internal analysis of the proposed methodology along with the extended GLSP-PM model. Section 6 points out the conclusion and the suggestions for future studies.

2 Literature review

2.1 General lot-sizing and scheduling problem

Drexl and Kimms (1997) reviewed the mathematical models of the integrated LSS problem ranging from small time-bucket to big time-bucket models. A detailed literature review on each of the existing models can be found in Copil et al. (2017). GLSP, which was introduced by Fleischmann and Meyr (1997), combines the features of the small-bucket and big-bucket models and therefore, it is a hybrid model. It is based on the two-level time structure. The planning horizon is divided into a predefined number of macroperiods with a fixed length, and each of these macroperiods contains a set of non-overlapping microperiods with a variable length. The length of the microperiods is the variable of the model, which is expressed by the quantity of a particular product (lot-size) manufactured during the associated miscroperiod. The system is set up to produce only one type of item in each microperiod, and the sequence of the changeovers between the microperiods defines the sequence of the products in the macroperiods (small-bucket feature). The assessment of the inventory and possible shortage levels along with the final demand delivery are accomplished at the end of the macroperiods (big-bucket feature). According to Fleischmann and Meyr (1997), GLSP can be viewed as a general form of the integrated LSS problem since with some modifications, it is converted to other related problems (e.g., CLSP, CSLP, DLSP, PLSP, and CLSD). It is shown that the GLSP has greater efficiency in modelling complex real-world cases while the two-level time structure increases the complexity of the model in comparison with other integrated LSS models. In this paper, a literature review on the GLSP is provided based on the publications in the recent decade (2013–2023). Subsequently, for a comprehensive literature review on the GLSP from 1997 to 2017, we refer to the work of Copil et al. (2017). Table 1 reviews the selected recent articles based on the structure of the model, manufacturing system, and the solution approach.

Table 1 Classification of the recent articles for the GLSP
Full size table

Focusing on the different approaches of mathematical modelling, Araujo and Clark (2013) introduced a new reformulation of the GLSP using the facility location concept and setup strengthening constraints. Meyr and Mann (2013) proposed new decomposition heuristics for the GLSP with non-identical parallel lines (GLSPPL). The multi-line GLSPPL was iteratively decomposed into a series of single-line GLSPs, and each problem was solved by the heuristic approach which embedded dual network flow re-optimization into Threshold Accepting (TA) and Simulated Annealing (SA) algorithms. Seeanner and Meyr (2013) introduced a mathematical model for GLSP with multiple production stages (GLSPMS). The main idea of the GLSPMS was to use a common time grid for all of the parallel production lines in each stage. They provided some reformulations, valid inequalities, and construction heuristics (such as relax-and-fix (RF)) for the solution method. Seeanner (2013) comprehensively studied the multi-stage LSS problem (including GLSPMS) and developed an improved version of the GLSPMS. The possible reformulations, heuristics, and meta-heuristics methods were also provided for the improved model. Guimarâes et al. (2014) reviewed the big-bucket LSS problems with sequence-dependent setups and categorized them into sequence-oriented and product-oriented models. Mohammadi and Poursabzi (2014) extended the model for the GLSP in the job shop manufacturing system with multi-level planning (MLGLSP_MM), previously presented by Fandel and Stammen-Hegene (2006), and applied the rolling horizon (RH) heuristics to solve it. Toledo et al. (2015) developed the generalized model of the two-level GLSP. Wolter and Helber (2016) addressed the hybrid GLSP and PLSP model combined with predictive and use-based maintenance (GLMSP). Some critical resources of the system were subject to wear and tear, and partial or complete maintenance actions could be performed prior to the production. Finally, an adapted fix-and-optimize (FO) heuristic was used to solve the model. Alem et al. (2018) investigated the GLSP under the demand uncertainty. Two different approaches of budget-uncertainty set robust optimization and two-stage stochastic programming were designed and compared via Monte Carlo simulation. With relatively the same concept, Curcio et al. (2018) modelled the GLSP with uncertain demand by the exact and approximate multi-stage stochastic programming methods. Rehman et al. (2019) featured the concept of updating the demand information in the multi-level, multi-stage GLSP and proposed three heuristic algorithms based on the RF approach within the RH framework. Worbelauer et al. (2019) studied a structural overview of the literature on the simultaneous LSS models involving secondary resources and proposed a set of models covering the mathematical formulation of the GLSPPL with different types of secondary resources. Wichmann et al. (2019) presented a novel model for the energy-oriented GLSP (EOGLSP), which parallels the energy-oriented microperiod (with a fixed length) and production-oriented microperiod (with a variable length). Alipour et al. (2020) designed the GLSP for perishable food products with the consideration of quality control and production wastes. The RF and hybrid FO and RF heuristics were used to solve the model. Soler et al. (2021) proposed nine MIP models for the GLSP and CLSD with parallel production lines that share scarce resources. Lee and Lee (2022) applied the concept of time flow for modelling the GLSP and used valid inequalities and approximate dynamic programming algorithm to solve the model.

Apart from the theoretical studies, which focus on the different mathematical models, a considerable amount of studies has been devoted to optimizing the GLSP in numerous manufacturing plants. Martinez et al. (2016) studied the application of the GLSP in the production planning and process selection of the molded pulp industry. Alves et al. (2016) addressed three mathematical formulations for the GLSP in the refractory cement industry. Pagliarussi et al. (2017) applied the GLSPPL model to the fruit juice beverage industry. Martinez et al. (2018) introduced the integrated process configuration, lot sizing, and scheduling model for the molded pulp industry and used MIP heuristics, based on the RF and FO, to solve the model. Martinez et al. (2019) proposed exact approaches, such as valid inequalities (symmetry breaking constraints) and branch-and-check algorithm, based on the logic-based Benders decomposition, with a three-phase heuristic to solve the integrated model introduced later by Martinez et al. (2018). Boas et al. (2021) used the RF and FO heuristics for the GLSP with process configuration selection. The proposed model can be applied to the molded pulp, paper, furniture, and electrofused grain industries. Schimidt et al. (2022) used a novel two-level GLSP model in the cosmetic industry. Mediouni et al. (2022) studied the multi-level GLSP in a soft drink company and used exact and RF heuristics methods for the solution approach. Araujo et al. (2023) applied RF heuristics to the GLSP in the personal care consumer goods industry. Wallrath et al. (2023) studied the application of the GLSP in a real-world chemical batch plant. Milenković et al. (2023) used the simultaneous LSS problem in an animal feed industry.

Although many researchers have decided to use heuristic approaches to solve the GLSP, meta-heuristic algorithms have also been applied in the literature. Figueira et al. (2013) proposed the multi-stage GLSP for the pulp and paper industry and used a hybrid VNS approach to solve it. The solution approach combined approximate (GVNS and heuristic) methods with an exact method. Seeanner et al. (2013) embedded the principles of the Variable Neighborhood Decomposition Search (VNDS) and FO heuristic to solve the GLSPMS. Toledo et al. (2014) suggested the Genetic Algorithm (GA) embedded with the mathematical programming techniques as a new solution procedure for the GLSP in a two-staged soft drinking company with multiple parallel lines. Figueira et al. (2015) designed an optimization-based DSS for the problem studied in Figueira et al. (2013). They provided valid inequalities and a matheuristic approach to solve the model. Rohaninejad et al. (2015) modeled the GLSP in the flexible job shop system and provided the hybrid GA and PSO with a local search heuristic for the solution approach. Furlan et al. (2015) featured a customized GA embedded with a residual LP model to solve the multi-stage GLSP in the pulp and paper mills. Goerler et al. (2020) added the rework and lifetime constraints to the GLSP with defective items and used the Late Acceptance Hill-Climbing Matheuristic (LAHCM) algorithm to solve the model. Popovi’c et al. (2023) used the hybrid VNS/LP approach to solve a two-stage multi-machine GLSP in a fruit juice production industry.

2.2 Integrated production and preventive maintenance planning problem

According to Wu and Zuo (2010), the PM planning models can be classified into three different groups: age reduction models, hazard rate reduction models, and hybrid models. In the age reduction models, the component’s age is considered in the hazard function and a PM action will reduce the virtual or effective age of the component. Depending on whether a certain PM is perfect or imperfect, the range of age reduction varies from a younger age to zero age. This concept is closely related to age-based maintenance, which is used in the Time-Based Maintenance (TBM) technique. Age-based maintenance modelling has been widely studied in the literature. To name a few recent articles, Sheu et al. (2018) proposed a general framework for age maintenance policies in a system with random working times. Feng et al. (2018) developed the combined SA and GA (SAGA) algorithm for the integrated imperfect PM optimization and sequence-dependent group scheduling problem in the flexible flow-shop manufacturing cells. Eryilmaz (2020) featured the age-based PM policy in an arbitrary coherent system. Ait El Cadi et al. (2021) addressed a stochastic analytical model for the integrated production and PM control problem. Jomaa et al. (2021) used two VNS algorithms to solve the permutation flow-shop scheduling problem with the PM. Razavi Al-e-hashem et al. (2022) developed a bi-objective optimization model for the robust maintenance planning and scheduling problem with disruption cost. Salmasnia and Talesh-Kazemi (2022) studied an integrated pricing, inventory, and maintenance planning problem with common cause failures for perishable products in a two-component parallel manufacturing system. An et al. (2022) proposed the joint PM and Flexible Job shop Rescheduling Problem (FJRP) optimization model with new machine insertion and processing speed selection. An et al. (2023a) developed a multi-objective optimization model for the FJRP with new job insertion and machine PM. An et al. (2023b) considered a multi-level imperfect PM in the FJRP with real-time order acceptance and FJRP. An et al. (2023c) applied a hybrid multi-objective evolutionary algorithm to solve the adaptive FJRP with condition-based PM.

The age-based maintenance is also used in the integrated Production and Maintenance Planning Problem (PMPP). With respect to the simultaneous LSS problem, the most related class of the PMPP is the integrated Multi-Item Capacitated Lot-Sizing Problem (MICLSP) and PM planning problem. Alimian et al. (2019) provided a comprehensive literature review on the models of this class. Focusing on age-based maintenance in this category, Fitouhi and Nourelfath (2012) were the first to combine the average availability and age-based PM planning in the integrated model. Fitouhi and Nourelfath (2014) extended their late study for multi-state systems. Yildirim and Ghazi Nezami (2014) studied energy consumption in the combined model. Nourelfath et al. (2016) added a quality inspection aspect to the integrated model for the imperfect processes. Aghezzaf et al. (2016) suggested hybrid failure rates for modeling the imperfect PM in failure-prone manufacturing systems. Saidi-Mehrabad et al. (2017) designed an integrated model with perfect and imperfect PM options. Alimian et al. (2019) suggested a robust optimization approach for the integrated model with uncertain demand. Alimian et al. (2020) combined production and PM planning with cell formation and group scheduling decisions in the dynamic cellular manufacturing system. Alimian et al. (2022) applied the RH heuristic to solve the capacitated lot-sizing and scheduling problem with due dates and PM planning. Avilés et al. (2022) proposed an integrated MIP model for the pulp and paper industry. Merghem et al. (2023) considered the outsourcing opportunity and defective products in the integrated model for a hybrid manufacturing-remanufacturing system. DerakhshanHoreh and Bijari (2023) developed FO heuristic approaches to solve the joint optimization model in the flow-shop environment with a limited buffer.

Summarizing the featured literature review in Sect. 2, the differences between this study and the two relevant literatures of GLSP and integrated production and PM planning problem can be addressed in the following four points:

  1. (1)

    Regarding the GLSP models, only Wolter and Helber (2016) studied the combination of the GLSP and use-based maintenance. Also, no research has been devoted to modelling the combination of availability and PM planning in the GLSP. Therefore, this paper is the first study to integrate the GLSP with the age-based maintenance problem, which is called “General Lot-sizing and Scheduling Problem with PM Planning (GLSP-PM)”.

  2. (2)

    Focusing on the heuristic solution methods that have been used for the GLSP, the proposed five-step heuristic approach is different from the featured heuristic methods in the literature of the GLSP due to the unique structure of the GLSP-PM model. More precisely, this paper is the first study to use the Lagrangian relaxation algorithm for the GLSP.

  3. (3)

    Regarding the integrated MICLSP with PM planning models, Liu et al. (2017) have addressed the sequence-dependent setup in the integrated production and PM planning models with periodic PM and sequence-oriented formulation. The present study uses the product-oriented formulation with age-based noncyclical PM planning to model the integrated problem with different changeover cases.

  4. (4)

    Finally, this paper is the first study to use the bi-objective formulation for simultaneous focus on the total cost of the manufacturing system and its average availability in the integrated PMPP models.

3 Mathematical model

3.1 Problem description

Let us consider a manufacturing system that utilizes batch production as its method of manufacturing process. This system is viewed as a whole; in other words, the studied system is viewed as a single entity or a system with only one component (station or machine). The task of this system is to produce a set of P products in a given planning horizon H. The planning horizon consists of T macroperiods in which each macroperiod is divided into a fixed number of M non-overlapping microperiods. The number and duration of both macroperiods and microperiods are known and fixed in advance.

The maintenance planning is based on the age-based maintenance policy. It is developed as the implementation of PM actions against corrective maintenance (CM) actions in the age reduction model. The PM is perfect, while the CM is minimal. Therefore, performing a noncyclical PM will reduce the system’s age to zero (As Good As New (AGAN)), and the CM action that follows a sudden failure does not alter the effective age of the system (As Bad As Old (ABAO)).

The other main assumptions of the model are listed as follows:

  • All of the products are available from the beginning of the planning horizon.

  • The setup times and costs are sequence-dependent, and the triangular inequality is applied to them.

  • The system has a fixed failure rate due to its failure probability distribution.

  • The expected number of failures in a particular microperiod follows the nonhomogeneous Poisson process.

  • At any time, the system is either in the production, setup, maintenance, or idle state.

  • No preemption is allowed for the PM or the lots unless a CM is required immediately after a sudden failure occurrence during the production state.

  • The PM action has the 1st priority over the production and setup activities, and so, it is implemented at the beginning of the microperiods.

  • At the beginning of the planning horizon, the manufacturing system is completely new.

  • The shortage is permitted and considered as a backorder.

  • No initial inventory or backorder is available at the beginning of the planning horizon.

All of the PM actions are similar and have identical durations, but lot i may contain a different product with a different production time from lot j in the previous microperiod. Focusing on the permutation of the products and the PM actions between two consecutive microperiods, four changeover cases can happen. Figure 1 depicts the different changeover cases. Case 1 shows the changeover from lot i to the different lot j with an in-between PM action. On the contrary, case 2 shows the changeover to a new product without an intermediary PM. In case 3, the same product is chosen to be produced in the next microperiod, and a PM is implemented at the beginning of the next microperiod. Finally, the same product is planned to be manufactured in two successive microperiods without an in-between PM. This case is also known as “setup carry-over” in the literature.

Fig. 1
figure 1

Possible cases for the changeover

Full size image

3.2 Nomenclature

Table 2 represents the symbols and their associated definition.

Table 2 Notation
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3.3 GLSP-PM

The proposed mathematical model for the GLSP-PM is developed as Eqs. (1)–(26):

$$\begin{gathered} Min\,\,\,f = \,\,\sum\limits_{i = 1}^{P} {\sum\limits_{t = 1}^{T} {h_{it} I_{it} } } + \sum\limits_{i = 1}^{P} {\sum\limits_{t = 1}^{T} {b_{it} B_{it} } } + \sum\limits_{i = 1}^{P} {\sum\limits_{m = 1}^{M} {\sum\limits_{t = 1}^{T} {c_{it} x_{imt} } } } + \sum\limits_{i = 1}^{P} {\sum\limits_{j \ne i}^{P} {\sum\limits_{m = 1}^{M} {\sum\limits_{t = 1}^{T} {(lz_{jit} z_{jimt} + } } } } lv_{jit} v_{jimt} ) \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \sum\limits_{i = 1}^{P} {\sum\limits_{m = 1}^{M} {\sum\limits_{t = 1}^{T} {lu_{it} u_{imt} } } } + \sum\limits_{m = 1}^{M} {\sum\limits_{t = 1}^{T} {(CMC \times N_{mt} + } PMC \times w_{mt} )} \hfill \\ \end{gathered}$$
(1)

s.t.

$$a_{mt} = \left\{ {\begin{array}{*{20}c} \begin{gathered} 0 \hfill \\ (1 - w_{mt} )(a_{(m - 1)t} + \pi ) \hfill \\ (1 - w_{mt} )(a_{M(t - 1)} + \pi ) \hfill \\ \end{gathered} & \begin{gathered} \quad \quad \quad \quad for\,\,m = 1\,,\,\,t = 1\,\,\, \hfill \\ \quad \quad \quad \quad for\,\,\,m > 1,\,\,t \in C \hfill \\ \quad \quad \quad \quad for\,\,\,m = 1\,,\,\,t > 1 \hfill \\ \end{gathered} \\ \end{array} } \right.$$
(2)
$$N_{mt} = \int_{{a_{mt} }}^{{a_{mt} + \pi }} {r(x)dx\qquad \qquad \qquad \forall m \in G,t \in C} {\mkern 1mu}$$
(3)
$$A_{mt} = \frac{{\pi - \left( {CMT \times N_{mt} + PMT \times w_{mt} + \sum\nolimits_{i = 1}^{P} {\sum\nolimits_{j \ne i}^{P} {(sz_{jit} z_{jimt} + sv_{jit} v_{jimt} )} } + \sum\nolimits_{i = 1}^{P} {su_{it} u_{imt} } } \right)}}{\pi }\,\,\,\,\,\,\,\,\,\,\forall m \in G,t \in C\,$$
(4)
$$\sum\limits_{i = 1}^{P} {y_{imt} } = 1\qquad \qquad \qquad \qquad \forall m \in G,t \in C$$
(5)
$$x_{imt} \le \left( {\sum\limits_{t = 1}^{T} {d_{it} } } \right)y_{imt} \qquad \qquad \qquad \qquad \forall i \in Q,m \in G,t \in C$$
(6)
$$\sum\limits_{m = 1}^{M} {x_{imt} } - I_{it} + I_{i(t - 1)} + B_{it} - B_{i(t - 1)} = d_{it} \qquad \qquad \qquad \forall i \in Q,t \in C$$
(7)
$$\sum\limits_{i = 1}^{P} {p_{it} x_{imt} } \le \pi A_{mt} \qquad \qquad \qquad \qquad \forall m \in G,t \in C\,$$
(8)
$$z_{jimt} \ge y_{j(m - 1)t} + y_{imt} - 1\,\, - w_{mt} \qquad \qquad \qquad \qquad \forall i,j \in Q\,(i \ne j),m > 1,t \in C$$
(9)
$$v_{jimt} \ge y_{j(m - 1)t} + y_{imt} + w_{mt} - 2\qquad \qquad \qquad \qquad \forall i,j \in Q\,(i \ne j),m > 1,t \in C$$
(10)
$$u_{imt} \ge y_{i(m - 1)t} + y_{imt} + w_{mt} - 2\qquad \qquad \qquad \qquad \forall i \in Q\,,m > 1,t \in C$$
(11)
$$q_{imt} \ge y_{i(m - 1)t} + y_{imt} - 1\, - w_{mt} \qquad \qquad \qquad \qquad \forall i \in Q\,,m > 1,t \in C\,$$
(12)
$$q_{imt} \le \,y_{imt} \qquad \qquad \qquad \qquad \forall i \in Q\,,m > 1,t \in C$$
(13)
$$q_{imt} \le \,y_{i(m - 1)t} \qquad \qquad \qquad \qquad \forall i \in Q\,,m > 1,t \in C\,\,$$
(14)
$$q_{imt} \le \,1 - w_{mt} \qquad \qquad \qquad \qquad \forall i \in Q,m > 1,t \in C\,\,$$
(15)
$$z_{jimt} \ge y_{jM(t - 1)} + y_{imt} - 1 - w_{mt} \qquad \qquad \qquad \qquad \forall i,j \in Q\,(i \ne j),m = 1,t > 1\,\,\,\,\,$$
(16)
$$v_{jimt} \ge y_{jM(t - 1)} + y_{imt} + w_{mt} - 2\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\forall i,j \in Q\,(i \ne j),m = 1,t > 1\,$$
(17)
$$u_{imt} \ge y_{iM(t - 1)} + y_{imt} + w_{mt} - 2\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\forall i \in Q\,,m = 1,t > 1\,$$
(18)
$$q_{imt} \ge y_{iM(t - 1)} + y_{imt} - 1 - w_{mt} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\forall i \in Q\,,m = 1,t > 1$$
(19)
$$q_{imt} \le \,y_{imt} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\forall i \in Q\,,m = 1,t > 1\,\,$$
(20)
$$q_{imt} \le \,y_{iM(t - 1)} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\forall i \in Q\,,m = 1,t > 1\,\,\,$$
(21)
$$q_{imt} \le \,1 - w_{mt} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\forall i \in Q\,,m = 1,t > 1\,\,$$
(22)
$$B_{i0} = 0\,\,\,\& \,\,\,I_{i0} = 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\forall i \in Q\,\,\,\,\,\,\,\,\,\,\,$$
(23)
$$x_{imt} ,I_{it} ,B_{it} \ge 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\forall i \in Q,m \in G,t \in C\,\,\,\,\,\,\,\,\,$$
(24)
$$y_{imt} ,z_{jimt} ,v_{jimt} ,u_{imt} ,q_{imt} ,w_{mt} = 0\,\,or\,1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\forall i,j \in Q\,(i \ne j),m \in G,t \in C\,\,$$
(25)
$$a_{mt} ,N_{mt} ,A_{mt} \ge 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\forall m \in G,t \in C$$
(26)

The objective function of the mathematical model minimizes the sum of inventory holding (1st part), shortage (2nd part), production (3rd part), sequence-dependent setup and changeovers (4 and 5th part), and maintenance (6th part) costs. Equation (2) calculates the effective age of the system in the entire planning horizon. The first sub-equation describes the assumption that the system is new at the beginning of the planning process. The second sub-equation computes the effective age of the system during a given microperiod of a specific macorperiod (except the 1st microperiod) with respect to the age of the previous microperiod and the decision on the PM implementation. The third sub-equation is used to relate the age of the 1st microperiod (in a specific macroperiod) to the last microperiod of the previous macroperiod. It must be noted that since the PM is perfect, carrying out a PM will reduce the effective age of the system to zero; otherwise, a fixed amount of π is added to the age of the system. Equation (3) estimates the expected number of sudden failures during a single micropeiod. This estimation is done with respect to the Poisson process, the effective age of the system, and the hazard rate function of the system (which can be calculated via formula \(r(x) = \frac{f(x)}{{1 - F(x)}}\)). As described earlier, each sudden failure will be followed by a minimal CM action. Equation (4) computes the availability of the system in a specific microperiod (in percentages). Each changeover (except for the setup carry-over case), PM, and CM activity will consume a specific amount of time from the total available time of the microperiod. Therefore, the available production time during a microperiod is equal to the duration of that particular microperiod minus the time that the system is being setup or repaired. Equation (5) assures that exactly one product must be produced in a particular microperiod. Constraint (6) guarantees that a product is manufactured if the system is setup for it in a microperiod. Constraint (7) describes the inventory balance equation for the system in each macroperiod. Constraint (8) shows that the production level of the products must be planned according to the available production time of the microperiod. Constraint (9) models the 2nd case of the changeovers in Fig. 1, while Constraint (10) models the 1st case of the possible changeovers. As seen, the either-or constraining approach is used to model these two cases since the only difference between them is in the presence (or absence) of a PM action between the two different products. In the same way, Constraints (11) and (12) model the 3rd and 4th cases of the changeovers to a similar product. Since the setup carry-over has zero length and cost, the model may generate an excessive number of changeover variables for this particular case. Constraints (13)–(15) prevent the model from generating these excessive changeover variables. Constraints (16)–(22) use the same approach, that is used in constraints (9)–(15), to model the changeovers by relating the 1st microperiod of a particular macroperiod to the last microperiod of the previous macroperiod. Constraint (23) shows the assumption that no initial inventory or shortage is available at the beginning of the planning horizon. Constraints (24)–(26) address the non-negativity, integer, and binary aspects of the decision variables.

As discussed in the previous section, the length of the microperiod is the main variable of the GLSP since it is completely based on the lot-size. However, there exist some studies that fix the length of the microperiod as a parameter (e.g., Toledo et al. 2009, 2014). In the proposed study, the overall length of the microperiods is fixed to fully synchronize the production and maintenance aspects and to pave the way for an accurate calculation of the system’s age. In fact, it must be noted that the total production time of a microperiod (\({p}_{it}{x}_{imt}\)) is a variable since the lot-size is still the main variable in all of the microperiods. Also, the total maintenance time of a microperiod (PMT × \({w}_{mt}\)+CMT × \({N}_{mt}\)) is a variable too. Thus, the nature of the GLSP is preserved in the proposed mathematical model.

The proposed model is a mixed-integer nonlinear programming problem in its primal form. In order to apply the proposed solution method, the model must be linearized. The linearization approach is provided in the 1st part of the appendix. Furthermore, for a better insight into the performance of the integrated model, a small-sized example is discussed. The data of the numerical example and the full description of the associated results are featured in the 2nd part of the appendix.

4 Solution approach

The simultaneous LSS problem (and expectedly GLSP) is NP-hard (Fleischmann and Meyr (1997), Lee and Lee (2023)), and integrating it with the maintenance problem (GLSP-PM) makes it strongly NP-hard. In this case, a heuristic or meta-heuristic approach should be implemented to find a satisfying feasible solution in a reasonable amount of computational time (Ramezanian and Saidi-Mehrabad 2013). Therefore, a heuristic method, based on the Lagrangian relaxation algorithm, decomposition, and valid equalities, is developed to tackle the solution process. The basic concepts and steps of the proposed heuristic approach are described in this section.

In the context of the Lot-Sizing (LS) problem, the capacity is a complicated constraint that increases the complexity of the problem. By removing this constraint from the constraints list, the process of finding a feasible solution for the LS model may become tractable in a polynomial time. There are many ways to perform the mentioned task, such as forcing the model to generate feasible solutions in the available time of the system, penalizing the capacity constraint in the objective function of the model, or using the relaxation methods (Rostami and Bagherpour 2020). The Lagrangian relaxation (LR) algorithm is a well-known relaxation method that has been widely used for MILP problems (including LS problems). Regarding the two integrated problems on the background of the GLSP-PM, Aghezzaf and Najid (2008) applied LR to the integrated MICLSP and PM planning model. Xiao et al. (2015) developed a hybrid LR-SA heuristic approach to solve the CLSD problem. For a comprehensive survey on the LR algorithm, we refer to the recent article of Bragin (2023).

In the proposed model, the right-hand side of Constraint (4) calculates the available production time of the manufacturing system. From another point of view, this constraint is the key point of integration in the GLSP-PM. By relaxing this constraint via the LR algorithm, the integrated model can be decomposed into two sub-problems: the maintenance sub-problem and the general LSS sub-problem with relaxed capacity constraints. The maintenance sub-problem is written as follows:

$$Min\,\,\,f_{1} = \,\,\sum\limits_{m = 1}^{M} {\sum\limits_{t = 1}^{T} {(CMC \times N_{mt} + } PMC \times w_{mt} )}$$
(27)

s.t.

Eqs. (2)–(3)

$$A_{mt} = \frac{{\pi - \left( {CMT \times N_{mt} + PMT \times w_{mt} } \right)}}{\pi }\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\forall m \in G,t \in C$$
(28)

Eq. (26)

$$w_{mt} = 0\,\,or\,1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\forall m \in G,t \in C\,\,\,$$
(29)

In which Eq. (27) is the objective function of the maintenance sub-problem, and Eq. (28) computes the total availability of the system by only using the data of the maintenance aspect. It is worth mentioning that the decomposed maintenance sub-problem is the genuine age reduction model for the PM planning problem under maintenance cost minimization.

The outcomes of the maintenance sub-problem are the optimal PM plan and the total availability of the system. This data can be entered as a parameter in the relaxed general LSS sub-problem. Moreover, focusing on the four cases of the changeover and their associated equations, it can be realized that by knowing the exact PM time, some of the setup and changeover cases are canceled. This fact can be used to effectively cut the solution space of the second sub-problem and decrease its complexity. Consider Ω as the set of microperiods in macroperiod η, in which the PM actions are implemented according to the optimal PM plan of the maintenance sub-problem and, Φ as the set of microperiods for η in which the PM actions are not planned to be performed (ΩUΦ = G). With respect to Sect. 3.3, the so-called “valid equalities” can be written as Eqs. (30)–(33):

$$\sum\limits_{i = 1}^{P} {\sum\limits_{j \ne i}^{P} {v_{jim\eta } } } + \sum\limits_{i = 1}^{P} {u_{im\eta } } = 1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\forall i,j \in Q\,(i \ne j),m \in \Omega$$
(30)
$$\sum\limits_{i = 1}^{P} {\sum\limits_{j \ne i}^{P} {z_{jim\eta } } } + \sum\limits_{i = 1}^{P} {q_{im\eta } } = 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\forall i,j \in Q\,(i \ne j),m \in \Omega$$
(31)
$$\sum\limits_{i = 1}^{P} {\sum\limits_{j \ne i}^{P} {v_{jim\eta } } } + \sum\limits_{i = 1}^{P} {u_{im\eta } } = 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\forall i,j \in Q\,(i \ne j),m \in \Phi$$
(32)
$$\sum\limits_{i = 1}^{P} {\sum\limits_{j \ne i}^{P} {z_{jim\eta } } } + \sum\limits_{i = 1}^{P} {q_{im\eta } } = 1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\forall i,j \in Q\,(i \ne j),m \in \Phi \,$$
(33)

According to Eq. (30), only changeover cases with an intermediary PM occur in the microperiods with planned PM action, while with respect to Eq. (31), the other cases could not occur in the mentioned microperiods. Equations (32) and (33) describe the opposite fact for the microperiods without any pre-planned PM actions. These valid equalities must be written separately for each macroperiod of the problem.

Now, by adding the valid equalities as the new constraints and, the optimal PM plan along with the total availability of the system as parameters to the second sub-problem, the relaxed general LSS sub-problem is developed as below:

$$\begin{gathered} Min\,\,\,f_{2} = \,\,\sum\limits_{i = 1}^{P} {\sum\limits_{t = 1}^{T} {h_{it} I_{it} } } + \sum\limits_{i = 1}^{P} {\sum\limits_{t = 1}^{T} {b_{it} B_{it} } } + \sum\limits_{i = 1}^{P} {\sum\limits_{m = 1}^{M} {\sum\limits_{t = 1}^{T} {c_{it} x_{imt} } } } + \sum\limits_{i = 1}^{P} {\sum\limits_{j \ne i}^{P} {\sum\limits_{m = 1}^{M} {\sum\limits_{t = 1}^{T} {(lz_{jit} z_{jimt} + lv_{jit} v_{jimt} )} } } } + \sum\limits_{i = 1}^{P} {\sum\limits_{m = 1}^{M} {\sum\limits_{t = 1}^{T} {lu_{it} u_{imt} } } } \hfill \\ \qquad\qquad\,\,\,\,\, + \sum\limits_{m = 1}^{M} {\sum\limits_{t = 1}^{T} {\lambda_{mt} } \left( {\sum\limits_{i = 1}^{P} {p_{it} x_{imt} } + \sum\limits_{i = 1}^{P} {\sum\limits_{j \ne i}^{P} {(sz_{jit} z_{jimt} + sv_{jit} v_{jimt} )} } + \sum\limits_{i = 1}^{P} {su_{it} u_{imt} } - \pi A_{mt} } \right)} \hfill \\ \end{gathered}$$
(34)

s.t.

Eqs. (5)–(7)

Eqs. (9)–(25)

Eqs. (30)–(33)

In which Eq. (34) minimizes the total cost of the relaxed general LSS sub-problem (including the infeasibility penalty cost), and \({\lambda }_{mt}\) is the LR coefficient for the availability constraint in each microperiod.

The relaxed general LSS sub-problem is solved by the LR algorithm. The steps of the LR algorithm as well as the pseudo-code are provided in the following:

  • Step 1: Set the LR coefficients to zero (\({\lambda }_{mt}\)= 0), best bound to zero, and the step size to value μ.

  • Step 2: Solve the relaxed problem with a MIP solver.

  • Step 3: Save \({{\text{f}}}_{2}^{*}\) as the new bound, and \({{\text{x}}}^{*}\), \({{\text{z}}}^{*}\), \({{\text{v}}}^{*}\), \({{\text{u}}}^{*}\), and \({{\text{q}}}^{*}\) as the optimal values for the lot-size and changeover variables.

  • Step 4: If \((\sum\limits_{i = 1}^{P} {p_{it} x^{*}_{imt} } + \sum\nolimits_{i = 1}^{P} {\sum\nolimits_{j \ne i}^{P} {(sz_{jit} z^{*}_{jimt} + sv_{jit} v^{*}_{jimt} )} } + \sum\nolimits_{i = 1}^{P} {su_{it} u^{*}_{imt} } ) > \pi A_{mt}\), add μ to the mtth LR coefficient (\({\lambda }_{mt}\)= \({\lambda }_{mt}\)+ μ), otherwise \({\lambda }_{mt}\)= max (0, \({\lambda }_{mt}\)- μ).

  • Step 5: If \({{\text{f}}}_{2}^{*}\) is better than the best bound (\({{\text{f}}}_{2}^{*}\)>best bound), save \({{\text{f}}}_{2}^{*}\) as the new best bound.

  • Step 6: If there is no improvement in the best bound after σ iterations, halve the step size (μ = μ/2).

  • Step 7: Repeat steps 2–6 until the stopping criterion is met.

  • Stopping criterion: the maximum iteration number (θ) is reached, or the step size becomes smaller than ε (μ < ε).

figure a

The outcomes of the second sub-problem are the optimal lot-size, inventory, and shortage levels of the products plus the optimal sequence of the lots in each microperiod. The total integrated cost of the system is calculated by summing the optimal cost of the maintenance sub-problem (\({f}_{1}^{*}\)) and the optimal cost of the relaxed general LSS problem (\({f}_{2}^{*}\)). Finally, the proposed heuristic approach can be summarized in the following five steps:

  • Step 1: Relax the availability constraint in the GLSP-PM and decompose the integrated model

  • Step 2: Solve the maintenance sub-problem and obtain the optimal PM plan, the expected number of sudden failures, total availability, and the effective age function of the system.

  • Step 3: Write the valid equalities according to the optimal PM plan.

  • Step 4: Add the valid equalities as the new constraints and the optimal PM plan along with the total availability as the new parameters to the second sub-problem.

  • Step 5: Solve the relaxed general LSS sub-problem using the LR algorithm and obtain the optimal sequence of the products with their optimal lot-size, inventory, and shortage levels.

5 Internal analysis of the proposed methodology

This section investigates the effectiveness of the proposed solution method and the advantages of the GLSP-PM. Additionally, an extended version of the proposed mathematical model is also developed to increase the performance of the GLSP-PM in real-world conditions.

5.1 Effectiveness analysis of the proposed solution method

To test the efficiency of the proposed solution method, 26 instances in the range of small to large scale are generated. Table 3 presents the characteristics of the generated instances, and Table 4 shows the parameter setting, which is used in instances No. 1 and 2. To apply the mentioned parameter setting to other instances with different number of microperiods, the value of the parameters which reflects the durations (the 4th column in Table 4 except the lifetime distribution) must be multiplied by ratio 4/M in order to convert the weekly microperiod structure to the instance-specific structure. For example, multiplying 4/30 by the value of the durations converts the weekly structure to the daily microperiod structure, which is used in instances No. 25 and 26. Finally, the parameter setting for parameters θ, μ, σ, and ε of the LR algorithm is respectively 10, 1, 1, and 0.001.

Table 3 Characteristics of the instances
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Table 4 Parameter setting for instance No. 1 and 2
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Similar to the numerical example, all of the generated instances are solved using GAMS (ver. 26.1) software with CPLEX 12.8 solver on a Windows 7 Ultimate SP1 (32-bit) with Intel Core i5-2500 CPU at 3.30 GHz processor and 8.00 GB RAM. The computational results are shown in Tables 5 and 6. Table 5 reports the optimal integrated cost, relative gap, and the solution time of the simple CPLEX run and the two sub-problems of the proposed heuristic approach. Before interpreting the results in Table 5, some necessary points must be explained:

  1. (1)

    The maximum solution time for the simple CPLEX run or each iteration of the LR algorithm is 6000 s.

  2. (2)

    The “relative gap” measures the relative gap between the best found and the best possible solution that the software finds during a single run.

  3. (3)

    If the software reports an “out of memory” error, the particular run is scheduled to be terminated earlier to record the total cost, gap, and solution time of the run.

  4. (4)

    The LR algorithm may have different gap and solution time per iteration in a single generated instance. Thus, the relative gap and the solution time of the LR algorithm are reported from the iteration that finds the best bound.

Table 5 Computational results of the instances
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Table 6 Comparison of the different cost components of the two solution methods
Full size table

Focusing on Table 5, the simple CPLEX run manages to solve the small-scale instances in an acceptable amount of computational time (except instance No. 6) with a maximum of 8.67% error (the relative gap of instance No. 8). Meanwhile, it can be seen that the proposed solution approach performs even better regarding the solution time and computational error of the decomposed sub-problems. Considering the fact that the simple CPLEX run acts as an exact method to find the optimal solution of the integrated model, credit can be given to the simple CPLEX method for the small instances. However, the heuristic approach explicitly highlights itself in the medium and large-scale instances as the simple method either reaches the time limit (in five instances) or reports a solution with a higher relative gap in comparison with the heuristic method (in all of the eighteen instances). For a detailed analysis of Table 5, the average solution time and relative gap (average computational error) of the simple CPLEX method in the medium and large instances are 3456.8 s and 19.91%, while the same performance metrics for the proposed heuristic approach (more precisely, the optimal iteration of the relaxed general LSS sub-problem) are respectively 2090.2 s and 6.11% in the mentioned instances. As presented, the decomposed maintenance sub-problem of the heuristic method requires less than a second to be optimally solved (by using GAMS/CPLEX as the exact solution method with the featured linearization techniques). Therefore, it can be concluded that the main reason behind the superiority of the heuristic approach is the combination of the valid equalities and the LR algorithm that is applied in the second sub-problem. Evaluating the computational results of the CPLEX and relaxed general LSS sub-problem, it can be derived that the heuristic approach (with the hybrid LR/valid equalities technique) decreases the error of the solution process (up to half or even third in the worst-case performance).

Table 6 compares the cost components of the GLSP-PM obtained from the simple CPLEX run and the proposed heuristic approach. It must be noted that when the GAMS reports an “out of memory” error, the detailed cost components may not be reported in the GAMS log. To fix this issue, the software is scheduled to be terminated a few seconds earlier in order to obtain the full results of the solution process. By comparing the maintenance cost of the two solution methods in Table 6, it is observed that a little difference exists between the two maintenance costs. Running a 1-sample sign test reports (0,18.40)$ confidence interval and 3.60$ median for the mean differences with the 95% confidence level. This negligible mean difference could justify the decomposition aspect of the heuristic approach. Ideally, the proposed model must be solved in its integrated form. However, the scale of the GLSP-PM rises when the number of microperiods is increased. In this case, the implementation of a proper solution approach is essential. With respect to Table 6, it can be seen that using the proposed heuristic approach (with the decomposition technique) does not highly aggravate the total maintenance cost or the optimal PM plan of the integrated problem. The remaining main difference in the cost components of the production aspect can be summarized in the ability of the solution method to balance the production and shortage levels in the presence of the sequencing process. Since the reported solution of the heuristic method has higher production cost (especially in the medium and large-scale instances) and lower shortage cost (in all of the instances except instances No. 1 and 3) than the solution of the simple CPLEX run, it can be claimed that the proposed heuristic approach successfully achieves the proper balance of lot-sizing and scheduling in the GLSP-PM. The column “Gap” in Table 6 measures the relative gap between the total integrated cost of the simple CPLEX and the total cost of the heuristic method (the best bound of the relaxed general LSS sub-problem plus the optimal cost of the maintenance sub-problem in Table 5) via Eq. (35).

$${\text{Gap = }}\frac{{\left( {{\text{Simple}}\,{\text{CPLEX}}\,{\text{total}}\,{\text{cost}} - {\text{Heuristic}}\,{\text{approach}}\,{\text{total}}\,{\text{cost}}} \right)}}{{{\text{Simple}}\,{\text{CPLEX}}\,{\text{total}}\,{\text{cost}}}} \times 100\%$$
(35)

Reviewing Tables 5 and 6, the proposed heuristic approach generally finds a feasible solution that provides a lower integrated cost (with an average gap of 17.11%) and a realistic production plan (with an effective sequence, lot-size, and lower shortage level) either in a reasonable amount of computational time (with the average of 2090.2 s for the best iteration of LR) or with the lower computational error (with the average of 4.23% in all of the instances and the maximum of 8.7% excluding instances No. 23–26).

5.2 Advantages of GLSP-PM

Compared to the original form of the GLSP with the additional presence of sudden failures, the proposed GLSP-PM has three advantages in the following criteria: cost, availability, and feasibility. To better explain these advantages, the GLSP-PM is compared with a similar integrated model without any allowed option for PM actions. The new model, called “No-PM GLSP”, is developed by adding a new constraint to the GLSP-PM in which the PM variables are fixed to zero in all of the microperiods. Therefore, the only existing tool to maintain the system is CM actions.

Table 7 compares the average availability and the cost components of the GLSP-PM and No-PM GLSP models. The column “RDP” calculates the relative difference percentage between the cost component of the GLSP-PM and No-PM GLSP models via Eq. (36), while Eq. (37) computes the gap between the average availability of the models. The absence of the PM actions in the No-PM GLSP raises the quantity of sudden failures and subsequently, the available production time of the system is occupied excessively by the following CM actions. Inevitably, the trend decreases the production rate of the system (with an average RDP of 8.66% reduction in the production cost in comparison with the GLSP-PM model). This process also affects the sequencing aspect of the model, as the No-PM GLSP has a lower setup and changeover cost (with an average RDP of 15.46%). This subject is the consequence of the model’s decision to employ more unnecessary setup carry-overs. Finally, the unmet demand leads to a higher shortage cost (with an average RDP of 33,864.9%), and the minimally repaired system incurs higher maintenance cost (with an average RDP of 173.44%) to the manufacturer.

$${\text{RDP}} = \frac{{\left( {{\text{cost}}\,{\text{component}}\,{\text{of}}\,{\text{No-PM}}\,{\text{GLSP}} - {\text{cost}}\,{\text{component}}\,{\text{of}}\,{\text{GLSP-PM}}} \right)}}{{{\text{cost}}\,{\text{component}}\,{\text{of}}\,{\text{GLSP-PM}}}} \times 100\%$$
(36)
$${\text{Gap}} = {\text{average}}\,{\text{availability}}\,{\text{of}}\,{\text{No-PM}}\,{\text{GLSP}} - {\text{average}}\,{\text{availability}}\,{\text{of}}\,{\text{GLSP-PM}}$$
(37)
Table 7 Comparison of the average availability and different cost components of the GLSP-PM and No-PM GLSP models
Full size table

As presented in Table 7, the second advantage of the GLSP-PM is the model’s ability to boost the average availability of the system. This advantage has a strong relationship with the main advantage (cost-effectiveness) since the amount of 11.8% reduction in the total availability of the system leads to the 43.15% growth in the total integrated cost of the model. On the other hand, it is obvious that the GLSP-PM generates infeasible solutions with the reported average availability in the optimal solution of the No-PM GLSP (input as the parameter for the GLSP-PM). Thus, the improved feasibility of the production plan can be pointed out as the third major advantage of the GLSP-PM. Furthermore, by observing the huge RDP in the instances with two macroperiods, it can be concluded that production and maintenance integration is critical in the long-term and extended planning horizons of the GLSP.

5.3 Extended GLSP-PM

Switching from the theoretical environment to the real-world condition, the GLSP-PM may have two disadvantages. First, the PMs are not practically perfect in real-world cases, which in fact return the machine’s/system’s state to the Partially As Good As New (PAGAN) state. Second, there have always been some conflicts between the production and maintenance departments of a manufacturing system. Alimian et al. (2020) showed this fact through a sensitivity analysis on one of the maintenance parameters of the integrated PMPP model and concluded that a bi-objective model could tackle this issue. Therefore, the extended version of the GLSP-PM, called “Extended GLSP-PM”, is proposed. The extended model includes the imperfect PM and bi-objective concepts. The Extended GLSP-PM can be written as follows:

$$\begin{gathered} Min\,\,\,f_{3} = \,\,\sum\limits_{i = 1}^{P} {\sum\limits_{t = 1}^{T} {h_{it} I_{it} } } + \sum\limits_{i = 1}^{P} {\sum\limits_{t = 1}^{T} {b_{it} B_{it} } } + \sum\limits_{i = 1}^{P} {\sum\limits_{m = 1}^{M} {\sum\limits_{t = 1}^{T} {c_{it} x_{imt} } } } + \sum\limits_{i = 1}^{P} {\sum\limits_{j \ne i}^{P} {\sum\limits_{m = 1}^{M} {\sum\limits_{t = 1}^{T} {(lz_{jit} z_{jimt} + } } } } lv_{jit} v_{jimt} ) \hfill \\ \qquad\qquad\,\,\,\,\,\, + \sum\limits_{i = 1}^{P} {\sum\limits_{m = 1}^{M} {\sum\limits_{t = 1}^{T} {lu_{it} u_{imt} } } } + \sum\limits_{m = 1}^{M} {\sum\limits_{t = 1}^{T} {(CMC \times N_{mt} + } PMC \times w_{mt} )} \hfill \\ \end{gathered}$$
(38)
$$Max\,f_{4} = \frac{{\sum\limits_{m = 1}^{M} {\sum\limits_{t = 1}^{T} {A_{mt} } } }}{MT}$$
(39)

s.t.

$$a_{mt} = \left\{ {\begin{array}{*{20}c} \begin{gathered} 0 \hfill \\ (1 - \beta w_{mt} )(a_{(m - 1)t} + \pi ) \hfill \\ (1 - \beta w_{mt} )(a_{M(t - 1)} + \pi ) \hfill \\ \end{gathered} & \begin{gathered} \qquad \qquad \quad \quad for\,\,m = 1\,,\,\,t = 1 \hfill \\ \qquad \qquad \quad \quad for\,\,\,m > 1,\,\,t \in C \hfill \\ \qquad \qquad \quad \quad for\,\,\,m = 1\,,\,\,t > 1 \hfill \\ \end{gathered} \\ \end{array} } \right.$$
(40)

Eqs. (3)–(26)

In which the first objective function of the model still aims to minimize the total cost of the integrated model, while the second objective function maximizes the average availability of the system during the entire planning horizon. The parameter β shows the imperfectness ratio of the PM actions for the calculation of the effective age function in revised Eq. (39). Again, the nonlinear model can be linearized by the conventional linearization approaches or the same approach that is used in Sect. 3.

For further studying the properties of the extended model, the numerical example of the appendix is solved by using the ε-constraint method through the same software and system configuration that is utilized in this Section. Table 8 shows the payoff matrix of the Extended GLSP-PM. As presented, the lowest total cost is achieved with the lowest level of average availability, while vice versa, the highest level of system availability is followed by the maximum integrated cost. Thus, the approach of the bi-objective formulation is justified as the conflict between the two featured objectives exists. Figure 2 depicts the Pareto front of the numerical example when the PM’s imperfectness ratio is set to 50%. In this example, the Pareto front ranges from (90.4%, 6780.1$) to (87.3%, 4016.7$). Table 9 compares the different cost components and average availability of the GLSP-PM, No-PM GLSP, and one of the Pareto solutions of the Extended GLSP-PM in the numerical example. As shown, the Extended GLSP-PM always has higher maintenance and total integrated cost than the original GLSP-PM. The average availability of the GLSP-PM is the highest value, while the average availability of the No-PM GLSP is the lowest among the three models. The imperfect PM practically highlights the existing conflict between the two objectives, which links to the contrast between the production and maintenance aspects. In the absence of the PM’s perfect property, the model decides to implement more imperfect PM activities (than the original plan of the perfect PM) to boost the average availability of the system. Also, the aforementioned decision is accompanied by another choice of increasing the lot-sizes for further response to the demand. This process incurs more amount of maintenance costs to the manufacturing system, which is not desirable for the first objective of the extended model. Hence, the model must find a way (new solutions) to balance the goals of the two sub-problems (the two objectives). Reviewing Table 9, the Extended GLSP-PM stands in the middle of the line that starts from GLSP-PM and ends with No-PM GLSP. Moving from GLSP-PM to No-PM GLSP increases the shortage, maintenance, and total integrated costs while the trend reduces the optimal level of average availability and production cost.

Table 8 Payoff matrix of the Extended GLSP-PM
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Fig. 2
figure 2

Pareto front of the numerical example with 50% imperfectness ratio of the PM

Full size image
Table 9 Comparison of the three featured models
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6 Conclusion

This paper addresses the new concept of integrating age-based maintenance modelling with the General Lot-sizing and Scheduling Problem (GLSP). The proposed mathematical model (GLSP-PM) has the ability to specify the inventory and shortage levels, while the lot-sizes and the optimal sequence of the lots are determined in close relation to the optimal availability and PM plan in the medium term. Considering the special structure of the GLSP-PM, a heuristic solution approach, based on the Lagrangian relaxation algorithm, decomposition, and valid equalities, is developed. The computational result approves the efficiency of the solution method and its capability to present the daily production and PM plan of a manufacturing system in a reasonable amount of computational effort. Compared to the original GLSP with the presence of sudden failures and no option for the PM actions, it is shown that not only does the GLSP-PM decrease the total cost of the manufacturing system, but it also increases the availability of the system and improves the feasibility of the production plan. The Extended GLSP-PM model features the bi-objective formulation of the problem, which tackles the imperfect PM and the conflicts of the production and maintenance aspects. The proposed model can be used by any manufacturing system that utilizes a batch production process, such as computer-hardware manufacturing industries, dairy and food industries, molded pulp and paper industries, blistering and packaging departments of the pharmaceutical industries, and etc.

Since this is the first article to combine age-based maintenance modelling with the GLSP, many suggestions can be addressed for further studies. With respect to the concept, the idea can be applied to small time-bucket models. Focusing on the maintenance aspect, the hazard rate reduction models can be used, and the performance of the models is compared. Regarding the manufacturing aspect, the model can be extended for different system configurations (such as parallel lines, flow-shop, job shop, multi-state, and etc.). The study of possible bi-objective formulation for the production and maintenance costs is another interesting idea. Addressing the solution method, the LR phase of the solution approach can be upgraded (using self-adaptive parameters) or improved by combining it with other heuristics (e.g., RH, RF, FO) or meta-heuristic methods. Finally, a novel contribution can be made by modelling the GLSP-PM in a stochastic, fuzzy, or uncertain environment.