On the availability and changeover cases of the general lot-sizing and scheduling problem with maintenance modelling: a Lagrangian-based heuristic approach

Abstract

The present paper proposes a novel concept to integrate maintenance modelling with an integrated lot-sizing and scheduling problem. The maintenance aspect of the problem is studied as age-based maintenance, while the production section is modeled as the General Lot-sizing and Scheduling Problem. The mathematical model aims to minimize the total integrated cost of the manufacturing system by determining the sequence of the products with their optimal lot-size, inventory, and shortage levels in close relation to the specified preventive maintenance plan and the availability of the system. Based on the unique structure of the proposed model, a heuristic solution approach is developed, which includes the Lagrangian relaxation algorithm, decomposition, and valid equalities. The computational result justifies the procedure of the proposed solution method and approves its efficiency in terms of cost and solution time for the range of small to large-scale instances. Furthermore, it is discussed that not only does the integrated model decrease the total cost of the manufacturing system, but it also increases the average availability of the system and improves the feasibility of the production plan. Finally, an extended model is developed to tackle the conflicts of the production and maintenance sub-problems via the bi-objective formulation.

Appendix

1.1 A. Linearization

Equation (2) of the GLSP-PM has nonlinear terms. Additionally, Eq. (3) may generate nonlinear terms. Alimian et al. (2019) studied the linearization techniques and conditions for the integrated model, which has a similar maintenance aspect to the proposed mathematical model. Thus, the same techniques and conditions can be applied to the GLSP-PM. According to the mentioned article, if the lifetime distribution of the system conforms to the Weibull (α,2) distribution, Eq. (3) results in the linear Eq. (40):

$$N_{mt} = \int_{{a_{mt} }}^{{a_{mt} + \pi }} {{\raise0.7ex\hbox{${2x}$} \!\mathord{\left/ {\vphantom {{2x} {\alpha^{2} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\alpha^{2} }$}}dx = \frac{{(a_{mt} + \pi )^{2} - a_{mt}^{2} }}{{\alpha^{2} }}\, = \,\frac{{2\pi a_{mt} + \pi^{2} }}{{\alpha^{2} }}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\forall m \in G,t \in C}$$

(41)

As for Eq. (2), it is replaced by the following linear equations:

$$a_{mt} = 0\quad \quad \quad \quad \quad for\,\,m = 1\,\,,\,\,t = 1\,$$

(42)

$$\left\{ \begin{gathered} a_{mt} \ge - w_{mt} \overline{M} + (a_{(m - 1)t} + \pi ) \hfill \\ a_{mt} \le (1 - w_{mt} )\overline{M} \hfill \\ \end{gathered} \right.\quad \quad \quad \quad \,\,\,\,\,for\,\,m = 2,3,...,M,\,\,t = 1,2,...,T$$

(43)

$$\left\{ \begin{gathered} a_{mt} \ge - w_{mt} \overline{M} + (a_{M(t - 1)} + \pi ) \hfill \\ a_{mt} \le (1 - w_{mt} )\overline{M} \hfill \\ \end{gathered} \right.\quad \quad \quad \quad \quad \,\,\,\,\,\,\,for\,\,m = 1\,\,,\,\,t = 2,3,...,T$$

(44)

1.2 B. Numerical example

The planning horizon of the example contains three months (T = 3), and each month is planned weekly (M = 4). Therefore, each month is a macroperiod (with a value of 1 for the duration), which has four weeks as the microperiods. The total number of four products must be produced monthly, leaving the system to assign only one product to each week. Table 10 shows the characteristics and maintenance parameters of the example. Table 11 shows the demand of the products. The production and setup parameters are shown in Tables 12, 13, 14 and 15.

Table 10 Characteristic and maintenance parameters of the system

Table 11 Demand of the products

Table 12 Production parameters of the products

Table 13 Parameters of the 1st changeover case for the products

Table 14 Parameters of the 2nd changeover case for the products

Table 15 Parameters of the 3rd changeover case for the products

Assuming that a month contains 30 days and each day has an amount of 16 h for the manufacturing process, the duration of a month and week can be respectively converted to 480 and 120 h. Hence, all of the durations that are shown in the mentioned tables can be multiplied by 480 in order to change the time scale of the example from month to hour. This approach has the advantage of generating real numbers for the number of sudden failures in comparison to the studies of Alimian et al. (2019, 2020), which featured monthly time-scale and decimal values for the expected number of sudden failures.

The numerical example is 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 results are featured in Tables 16, 17, 18, 19, 20 and 21. Focusing on Table 16, the lot-sizes are mainly determined according to the external demand of the products. Addressing Table 18, no inventory stocking is decided by the model, but four major (product i3 in t1 and t2 and t3 and product i2 in t3) and five minor (products i1 and i2 in t1 and t2 and product i3 in t3) cases of backlogging happen. The demand of product i4 is completely answered in each macroperiod while most of the demand of products i1 and i2 is met in the first two macroperiods. The binary setup variable and the changeover cases can be derived from Table 17. For instance, products i4, i1, i2, and i3 are planned to be manufactured in the corresponding microperiod m1 to m4 in macroperiod t2. Only one PM action is decided to be implemented at the beginning of microperiod m3 of t2, and the other microperiods of macroperiod t2 are without any planned PM. Focusing on the changeovers in macroperiod t2, the 2nd changeover case is applied in microperiods m2 and m4, while the 1st case is carried out in microperiod m3. Also, a setup carry-over is performed for product i4 in the last microperiod of t1 to microperiod m1 of t2. Generally, apart from the only setup carry-over case in macroperiod t3, the model decides to apply this changeover case in the transition from a macroperiod to a new one. Table 19 shows the optimal solution for the maintenance aspect. It can be interpreted that the model decides to plan a PM when the system’s age reaches the value of 360 h. This is a response to the constant length of the microperiods, maintenance characteristics, and the lifetime distribution of the system. As presented, if a perfect PM is implemented, the effective age of the system and the total number of sudden failures are minimized; otherwise, a constant value (the duration of a microperiod) is added to the system’s age and the quantity of the sudden failures increases in the corresponding microperiod. Reviewing Table 19, the average availability of the system during macroperiod t1 to t3 is respectively 91.28, 90.20, and 90.93%. Finally, the optimal integrated cost, along with the optimal cost components of the model, are featured in Table 20.

Table 16 Production level of the products

Table 17 Sequence of the lots in the macroperiods

Table 18 Inventory and backorder level of the products

Table 19 PM plan, effective age function, and number of sudden failures in the microperiods

Table 20 Optimal cost components of the model

As stated in Sect. 3, although the overall length of the microperiods is fixed, the duration of the production and maintenance parts in the microperiods are still variable in the GLSP-PM. In other words, an upper bound is set for the length of the microperiods while the length of production (lot-size), setup and changeover, idleness, and maintenance (PM and CM actions) parts change from microperiod to microperiod. Table 21 shows this fact by reporting the length of the production and maintenance parts for each microperiod in the numerical example.

Table 21 Length of production and maintenance parts in the microperiods