Abstract
In the competitive business environment, manufacturers are seeking strategies to improve the product quality and the system reliability while reducing the costs. This paper addresses the problem of finding the optimal production and maintenance schedules for a deteriorating manufacturing system with the objective of minimizing the expected cost per unit time. The system consists of one machine which deteriorates with time and it may shift from an in-control state to an out-of-control state with a larger proportion of imperfect products. In addition, the hedging point policy is applied as the production-inventory control policy. The predictive maintenance is performed based on process inspections, whose sampling intervals are variable. To deal with the proposed problem, we build a joint model that coordinates production, inventory, maintenance, and quality control with 16 scenarios. Then we propose a novel approach speeding up the Monte Carlo simulation to calculate the objective function. Thus it becomes feasible to optimize the objective function by metaheuristic algorithms. Then we use the genetic algorithm to illustrate its feasibility. Next, the advantage of the proposed approach is verified by comparing with the traditional integral method. Finally, a sensitivity analysis with an orthogonal experiment is conducted to help managers find the factors with the most significant effect on the cost.
1. Introduction
In the competitive business environment, manufacturers are seeking strategies to improve the product quality and the system reliability while reducing the costs. Performance of a system is highly related to the coordination and cooperation of the subsystems. Production-inventory systems (PIS), maintenance systems (MS), and quality systems (QS) are three main subsystems of manufacturing systems with different goals.
For the PIS, an important issue is how to control the production rate to keep suitable inventory level. In order to solve this issue, researchers have proposed two famous models: the economic production quantity (EPQ) [1] and the hedging point policy (HPP) [2]. The HPP is more suitable for systems subject to random failures and repairs. Because the HPP entails the build-up and preservation of a final product safety stock while the machine is operational in order to hedge against future shortages caused by machine failures, the optimality of the HPP has been demonstrated for failure and repair times described by homogeneous Markov processes in the case of constant demand rate [3] and stochastic demand rate [4]. For general failure and repair time distributions, the optimal control policy cannot be solved analytically [5, 6], but the structure of the optimal inventory policy can be approximated by the HPP [7–9]. Later, Rezg, Dellagi, and Chelbi [10] added one decision variable (when to start building up a buffer stock) to the traditional HPP, which can further decrease the total cost. Rather than HPP or EPQ, Souheil, Dellagi, and Rezg [11] used subcontracting to solve the issue and to satisfy a required service level.
For the MS, there are enormous maintenance policies in the literature for various situations. The most common three types are corrective maintenance (CM), preventive maintenance (PM), and predictive maintenance (PRDM). CM is performed after failure to make the equipment continue to run. It is often a minimal repair [12], which just makes the equipment as good as old. PM is performed before failure to prevent the loss caused by sudden failure. It is usually performed based on time or age of the equipment [13]. PRDM is maintenance carried out when necessary, based on process inspections, diagnostic tests, or other means of condition monitoring [14–16]. The most widely used tool of process inspections is the X-bar control chart, whose sampling interval can be fixed or variable. In addition, Zied, Dellagi, and Rezg [17] showed the effect of the production rate variation on the optimal maintenance strategy. Later, Lv and Liu [18] proposed a maintenance policy for multicomponent systems with economic dependence and structural dependence.
For the QS, manufacturers can either rework the imperfect products before sale with additional rework cost [19, 20] or sell the whole production quantity with warranty, where additional warranty cost is incurred [21]. In this paper, the manufacturer chooses the latter policy.
Despite the conventional researches dealing separately with these subsystems, the integrated strategy is an appropriate approach to handle the interactions between them. For the integration of the three systems, relevant papers are summarized in Table 1. In this table, “Monte Carlo” means that the Monte Carlo method [22] is used to calculate the objective function in a paper. “Metaheuristic method” means that the metaheuristic method [23], which includes genetic algorithms, particle swarm algorithms, etc., is used to optimize the objective function in a paper. “Orthogonal experiment” means that the orthogonal experiment is used to conduct the sensitivity analysis in a paper. It is noticeable that Mifdal, Hajej, and Dellagi [24] considered the integration of the PIS and the MS with a machine producing several products and satisfying some random demands, which is very complicated but meaningful.
The remainder of this paper is organized as follows. In the next section, we specify targeted contributions of this paper. Section 3 describes the notations and the problem. In Section 4, a model with 16 scenarios is developed. In Section 5, the developed model is solved by the Monte Carlo method and the genetic algorithm. In Section 6, an illustrative numerical example is presented. In addition, comparative and sensitivity analyses are also conducted in this section. Finally, Section 7 concludes the paper.
2. Targeted Contributions
By integrating all the three systems, this paper aims at minimizing the expected cost per unit time by determining 5 decision variables concerning the three systems. Comparing with the existing literature, the main contributions of this paper are the following:
The process inspection is often integrated with the economic production quantity model as the production-inventory policy in the previous integration models. However, the hedging point policy is also a very important production-inventory policy and it often interacts with the process inspection in practice. To solve the manufacturing problem with the process inspection and the hedging point policy, this paper builds an optimization model with 16 scenarios.
The Monte Carlo method is directly used to calculate the objective function in the process of a metaheuristic algorithm. Although some researchers have managed to use the Monte Carlo method to calculate objective functions in the process of genetic algorithms [25, 26], the process of genetic algorithms has to be modified because of the low speed of the Monte Carlo simulation. However, this paper proposes an approach speeding up the Monte Carlo simulation dramatically. This makes the direct application of the Monte Carlo method in the process of metaheuristic algorithms practical. As an experiment, this paper uses the Monte Carlo method to calculate the objective function in the process of the genetic algorithm, which is one of the most important metaheuristic algorithms.
To assess the accuracy of the Monte Carlo method, this paper also calculates the expected cost per unit time by the integral method. Because 16 scenarios may occur, this calculation is very difficult. However, this paper solves it successfully by calculating conditional expectations by conditioning. This calculation method can be used in other manufacturing problems involving expectations of complex random variables.
In the majority of previous papers, the sensitivity analysis was conducted by single factor experiments. In this paper, an orthogonal experiment is used to conduct the sensitivity analysis, which can make the sensitivity analysis more comprehensive with fewer trials.
3. Problem Statement
3.1. Notation
The notations used in this paper are defined as follows:
Decision Variables : Hedging level (safety stock) : Duration of the first sampling interval : Coefficient of the control limit : Maximum number of samples in a production cycle : Sample size
Model Parameters : Random variable denoting the duration of the in-control state in a production cycle : Probability density function of : Hazard rate function of : Random variable denoting the time to failure of the machine : Hazard rate function of : Random variable denoting the preventive maintenance duration : Probability density function of : Random variable denoting the predictive maintenance duration : Probability density function of : Number of samples needed to detect the state shift after it occurs in a production cycle, considering the type II error : Setup cost : Demand rate : Maximum production rate : Proportion of imperfect products when the production process is in the in-control state : Proportion of imperfect products when the production process is in the out-of-control state : Warranty period : Hazard rate function of perfect products : Hazard rate function of imperfect products : Cost per product at each minimum repair for the warranty : Index of sampling intervals : Mean value of the quality characteristic in the in-control state : Standard deviation of the quality characteristic : Magnitude of the shift in the mean of quality characteristic when the assignable cause occurs : Fixed cost per sample : Variable cost per sample : Cost of one corrective maintenance action : Cost of performing preventive maintenance per unit time : Cost of performing predictive maintenance per unit time : Cost of inspecting an alert signal : Holding cost of a product unit during a unit of time : Shortage cost of a product unit : Length of the th sampling interval : Time at the end of the th sampling interval : Probability of type I error : Probability of type II error : Duration of the build-up of the buffer stock : Duration of the decrease of the buffer stock : Number of imperfect products produced in a production cycle : Number of perfect products produced in a production cycle : Corrective maintenance cost in a production cycle : Preventive maintenance cost in a production cycle : Predictive maintenance cost in a production cycle : Sampling cost in a production cycle : Cost of inspecting alert signals in a production cycle : Inventory cost in a production cycle : Cost for the free minimal repair warranty incurred in a production cycle : Shortage cost in a production cycle : Setup cost in a production cycle : Total cost of a production cycle : Length of a production cycle
Other notations will be introduced where they are needed.
3.2. Problem Description
In the following, the policies for the production-inventory system, the maintenance system, and the quality system are explained.
3.2.1. Production-Inventory System
The so-called hedging point policy is used to instantly control the production rate of the machine as follows:where is the inventory level at time and denotes the state of the machine at time . if the machine is available for production. if the machine is under maintenance.
In addition, if the inventory level is greater than zero when the maintenance is complete, the machine will not be started until the inventory level is reduced to zero. If the inventory is not available to adjust demand during the interruption due to maintenance, customer demand is lost at a relevant cost.
3.2.2. Maintenance System
This paper investigates an imperfect manufacturing process, which includes two states: the in-control state and the out-of-control state. The manufacturing process starts operating in the in-control state and one assignable cause may occur and lead to shifting process to the out-of-control state. The quality characteristic of products follows the normal distribution with mean value and standard deviation in the in-control state. When the process shifts from the in-control state to the out-of-control state, the mean value changes from to , where is the magnitude of quality shift, and the standard deviation is assumed to remain the same.
A Shewhart X-bar control chart is used to monitor the quality characteristic with an alert signal to inform operators when the process shifts to the out-of-control state. When the process mean falls inside the control limits [LCL, UCL], the process is considered to be in the in-control state. Once it falls outside the control limits, an alert signal is generated since the assignable cause probably has affected the process. Here, and , where is the coefficient of the control limit and is the sample size.
The alert signal of the control chart triggers an error-free inspection to ascertain its authenticity. If the assignable cause indeed happens, the predictive maintenance (PRDM) is immediately carried out to restore the process to the in-control state [27]. If the alert signal is false (type I error), the process will continue operating without any maintenance action. Once the equipment fails, the corrective maintenance (CM) has to be performed at once to restore the equipment and the process. The CM is minimal repair and the repair time is negligible. If no PRDM is conducted after samplings, there may be two possible conditions: (i) the process is in the in-control state and (ii) the process is in the out-of-control state but is undetected (type II error). Therefore, it is necessary to stop the production process to implement the preventive maintenance (PM) at the end of the (+1)th sampling interval (without sampling). If the process is in the out-of-control state, the assignable cause will be detected during the PM. Thus the PM has to be replaced by the PRDM. The PM and the PRDM are considered to restore the equipment to the good-as-new condition.
The lengths of sampling intervals are determined such that the integrated hazard over them is constant. That is,where . is assumed to follow the Weibull distribution. Thus, its hazard rate function can be assumed to be Therefore, the can be obtained:
3.2.3. Quality System
When the system is in the in-control state, it produces some imperfect products. In the out-of-control state, the system produces more imperfect products than in the in-control state. Both the perfect and imperfect products will be sold with warranty. The free minimal repair warranty (FRW) policy is adopted. That is, all products sold can obtain free minimal repair when failing and the repair time is negligible. In addition, the manufacturer will pay for the repair.
4. Model Development
First of all some key variables need to be calculated. The duration of the build-up of the buffer stock is Under the FRW, a perfect product fails according to a nonhomogeneous Poisson process (NHPP) with rate , while an imperfect product fails according to a NHPP with rate . Therefore, given a perfect product, the mean number of failures within the warranty period is . On the other hand, given an imperfect product, the mean number of failures within the warranty period is . Consequently, the mean warranty costs of a perfect product and an imperfect product are, respectively,The whole cost of each sampling is The duration of the decrease of the buffer stock is The probability of type I error is where represents the cdf of standard normal distribution. The probability of type II error is is a geometric random variable with parameter . Thus,
4.1. The 16 Scenarios
The period between the two successive starts of the good-as-new equipment is described as a production cycle. Thus, according to the policies proposed in Section 3.2, 16 scenarios may occur in a production cycle. This section contributes to describing these scenarios in detail and formulating the conditional expectations of and for different scenarios.
4.1.1. Scenario 1: , ,
This scenario can be rewritten as and , , . Then the behavior of inventory in this scenario can be described as Figure 1. In this scenario, the process shifts to out-of-control state before the hedging safety stock is built. Due to the type II error, the alert signal is not sent until the th sampling. Then the PRDM is conducted for . Because , shortage does not occur. The conditional expectations of and given that , that is, and , when , , will be formulated.
Because , can be obtained through calculating the conditional expectations of , and . The CM is minimal repair and the repair time is negligible; hence the machine fails according to an NHPP with rate . Therefore, the mean number of failures within the production period is . Consequently, the conditional expectations of areThe conditional expectations of areThe PM is not conducted; henceThe conditional expectations of areThe conditional expectations of areThe conditional expectations of areThe conditional expectations of areThe conditional expectations of areThe conditional expectations of areHence,Shortage does not occur; henceAbove all, where denotes when , , .
From Figure 1, it can be obtained thatwhere denotes when , , .
4.1.2. Scenario 2: , ,
This scenario can be rewritten as and , , . Thus the behavior of inventory in this scenario can be described as Figure 2. In contrast to Scenario 1, shortage occurs since . HenceOther costs are the same as Scenario 1. Thereforewhere denotes when , , . From Figure 2, it can be obtained thatwhere denotes when , , .
4.1.3. Scenario 3: , ,
This scenario can be rewritten as and , , . Thus the behavior of inventory in this scenario can be described as Figure 3. The alert signal is not sent in the first samples. Hence the preventive maintenance (PM) is implemented at the end of the (+1)th sampling interval (without sampling). Then the assignable cause is detected during the PM. Thus, the PM is replaced by the PRDM. Because , shortage does not occur.
The conditional expectations of all kinds of costs given that can be calculated using the same analogy as Scenario 1. And they are displayed in Tables 1 and 2 of the Annex (Supplementary 1). Hencewhere denotes when , , .
From Figure 3, it can be obtained thatwhere denotes when , , .
4.1.4. Scenario 4: , ,
This scenario can be rewritten as and , , . Thus the behavior of inventory in this scenario can be described as Figure 4. In contrast to Scenario 3, shortage occurs since . HenceOther costs are the same as Scenario 3. Thereforewhere denotes when , , . From Figure 4, it can be obtained thatwhere denotes when , , .
4.1.5. Scenario 5: , ,
This scenario can be rewritten as and , , . Thus the behavior of inventory can be described as Figure 5. In this scenario, the process shifts to out-of-control state after the hedging safety stock is built and before the first sampling. Due to the type II error, the alert signal is not sent until the th sampling. Then the PRDM is conducted for . Because , shortage does not occur.
The conditional expectations of all kinds of costs given that can be calculated using the same analogy as Scenario 1. And they are displayed in Tables 1 and 2 of the Annex. Therefore where denotes when , , .
From Figure 5, it can be obtained thatwhere denotes when , , .
4.1.6. Scenario 6: , ,
This scenario can be rewritten as and , , . Thus the behavior of inventory can be described as Figure 6. In contrast to Scenario 5, shortage occurs since . HenceOther costs are the same as Scenario 5. Thereforewhere denotes when , , . From Figure 6, it can be obtained thatwhere denotes when , , .
4.1.7. Scenario 7: , ,
This scenario can be rewritten as and , , . Thus the behavior of inventory can be described as Figure 7. The alert signal is not sent in the first samples. Hence the preventive maintenance (PM) is implemented at the end of the (+1)th sampling interval (without sampling). Then the assignable cause is detected during the PM. Thus the PM is replaced by PRDM. Because , shortage does not occur.
The conditional expectations of all kinds of costs given that can be calculated using the same analogy as Scenario 1. And they are displayed in Tables 1 and 2 of the Annex. Thereforewhere denotes when , , .
From Figure 7, it can be obtained thatwhere denotes when , , .
4.1.8. Scenario 8: , ,
This scenario can be rewritten as and , , . Thus the behavior of inventory can be described as Figure 8. In contrast to Scenario 7, shortage occurs since . HenceOther costs are the same as Scenario 7. Thereforewhere denotes when , , . From Figure 8, it can be obtained thatwhere denotes when , , .
4.1.9. Scenario 9: , ,
This scenario can be rewritten as and . Thus the behavior of inventory can be described as Figure 9. In this scenario, the process shifts to out-of-control state in the (+1)th sampling interval. After the shift, due to the type II error, the alert signal is not sent until the th sampling, while the alert signal may be sent in the first samplings due to the type I error. After the th sampling, the PRDM is conducted for . Because , shortage does not occur.
The conditional expectations of all kinds of costs given that can be calculated using the same analogy as Scenario 1. And they are displayed in Tables 1 and 2 of the Annex. Thereforewhere denotes when .
From Figure 9, it can be obtained thatwhere denotes when .
4.1.10. Scenario 10: , ,
This scenario can be rewritten as and . Thus the behavior of inventory can be described as Figure 10. In contrast to Scenario 9, shortage occurs since . HenceOther costs are the same as Scenario 9. Thereforewhere denotes when . From Figure 10, it can be obtained thatwhere denotes when .
4.1.11. Scenario 11: , ,
This scenario can be rewritten as and . Thus the behavior of inventory can be described as Figure 11. No PRDM is conducted after samplings. Hence the preventive maintenance (PM) is implemented at the end of the (+1)th sampling interval (without sampling). Then the assignable cause is detected during the PM. Thus the PM is replaced by the PRDM. Because , shortage does not occur.
The conditional expectations of all kinds of costs given that can be calculated using the same analogy as Scenario 1. And they are displayed in Tables 1 and 2 of the Annex. Therefore where denotes when .
From Figure 11, it can be obtained thatwhere denotes when .
4.1.12. Scenario 12: , ,
This scenario can be rewritten as and . Thus the behavior of inventory can be described as Figure 12. In contrast to Scenario 11, shortage occurs since . HenceOther costs are the same as Scenario 11. Thereforewhere denotes when . From Figure 12, it can be obtained thatwhere denotes when .
4.1.13. Scenario 13:
This scenario can be rewritten as and . Thus the behavior of inventory can be described as Figure 13. In this scenario, the process shifts to out-of-control state in the (+1)th sampling interval. No PRDM is conducted after samplings. Hence the preventive maintenance (PM) is implemented at the end of the (+1)th sampling interval (without sampling). Then the assignable cause is detected during the PM. Thus the PM is replaced by PRDM. Because , shortage does not occur.
The conditional expectations of all kinds of costs given that can be calculated using the same analogy as Scenario 1. And they are displayed in Tables 1 and 2 of the Annex. Therefore