An agent-based approach for resources’ joint planning in a multi-echelon inventory system considering lateral transshipment

Computers & Industrial Engineering 138 (2019) 106098

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An agent-based approach for resources’ joint planning in a multi-echelon inventory system considering lateral transshipment

T

Zili Wanga, Bowen Cuia, Qiang Fenga, , Baiqiao Huangb, Yi Rena, Bo Suna, Dezhen Yanga, Zheng Zuoa ⁎

a b

School of Reliability and Systems Engineering, Beihang University, Beijing, PR China System Engineering Research Institute, China State Shipbuilding Corporation, Beijing, PR China

ARTICLE INFO

ABSTRACT

Keywords: Joint planning Multi-agent system Lateral transshipment Multi-echelon inventory system Spare parts Maintenance workers

A joint planning of spare parts and maintenance workers in cross-echelon and lateral transshipment can reduce total system cost; however, only a limited part of the literature considers this combination. In this paper, we propose an agent-based generic evaluation model for the joint planning of resources by examining the multiechelon inventory system with the operation management of multi-component equipment. First, a dual-clock failure modelling mechanism is provided to accurately generate the maintenance resource requirements. Afterwards, a resources scheduling process based on the contract net protocol is described. In addition, the dynamic change in the spare parts inventory is also analysed by considering whether the replaced failure components can be repaired or not. Moreover, a system cost assessment method is developed to compare different scheduling strategies and to select the one that can minimize the expected total cost. Finally, we examine an example of a two-echelon inventory system to verify the effectiveness of the proposed approach by using four representative scheduling strategies: fixed-pair, nearest distance, maximum inventory and minimum ratio. The results indicate that the approach can well support the joint planning of maintenance resources.

1. Introduction Maintenance logistics is an important discipline that has received considerable attention in both practice and the literature in recent decades. Effective maintenance depends on many tactical and operational aspects, such as the number of maintenance resources (e.g., maintenance workers) and the amount of spare parts inventory (Froger, Gendreau, Mendoza, Pinson, & Rousseau, 2016). However, many studies on the inventory management of maintenance resources focus only on spare parts (Çelebi, 2015; Feng, Li, & Sun, 2015; Gan, Zhang, & Zhou, 2015; Meissner & Senicheva, 2018; Sherbrooke, 1968; Zhang & Zeng, 2017; Zhang, 2015), and very few studies reported in the literature address the joint planning of maintenance resources. The preliminary results of such models can be found in Bijvank, Koole, and Vis (2010). In that work, the authors handled the repair kit problem, where several maintenance resources have to be dispatched simultaneously when a failure occurs. Furthermore, the model was extended in RahimiGhahroodi, Al, and Zijm (2017), Sleptchenko, Al, and Zijm (2018), Basten and van Houtum (2014) and Smidt-Destombes, Karin, and Heijden (2009), where the authors handle different scenarios with

partial or full backlogging of the necessary resources. The effort to guarantee a quick and efficient maintenance action led to the complex integrated multi-resource planning problem studied in this paper. However, in the maintenance logistics of capital-intensive equipment, a full backordering policy is not a common practice because the waiting time for a repair request can be long, and downtime is very expensive. For this reason, we shall assume that when a requirement arrives, if any resource is not readily available, the requirement will be satisfied via an external channel with a short replenishment time but at a high cost. Traditionally, the requirement is satisfied by shipments from cross-echelon warehouses to reduce shutdown time (Chu, You, Wassick, & Agarwal, 2015; Wang, 2016). Since lateral transshipment transports resources from same-echelon warehouses (Firouz, Keskin, & Melouk, 2016; Nakandala, Lau, & Zhang, 2017; Olsson, 2015) and takes advantage of the remaining inventory at each inventory warehouse, lateral transshipment has attracted increasingly more attention in recent decades (Paterson, Kiesmüller, Teunter, & Glazebrook, 2011; Axsäter, 2003). In this paper, we use lateral transshipment mainly to schedule maintenance resources in case of a shortage; otherwise, we use cross-echelon transshipment unless there are no available maintenance

Corresponding author. E-mail addresses: [email protected] (Z. Wang), [email protected] (B. Cui), [email protected] (Q. Feng), [email protected] (Y. Ren), [email protected] (B. Sun), [email protected] (D. Yang), [email protected] (Z. Zuo). ⁎

https://doi.org/10.1016/j.cie.2019.106098 Received 25 July 2018; Received in revised form 6 September 2019; Accepted 27 September 2019 Available online 28 September 2019 0360-8352/ © 2019 Elsevier Ltd. All rights reserved.

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resources in any of the same-echelon warehouses. The modelling approach for the joint planning of maintenance resources includes mainly mathematical programming, an approximation evaluation method and an intelligent search method. Such methods are usually based on certain simplified assumptions and require that the scale of the problem not be too large.

dynamic structures. These studies show that agent-based modelling and simulation have become an active research topic; however, the current available literature in the maintenance resources field does not present a generic model. Therefore, the agent-based model proposed in this paper for the joint planning of resources is extremely necessary. The main contribution of this paper can be summarized as follows: An agent-based approach for the joint planning of spare parts and maintenance workers in a multi-echelon inventory system with equipment having multiple components is analysed; an agent-based approach makes the framework more granular and flexible and can describe the complex logical relationships among various entities. Compared to previous works related to maintenance resource planning, this paper recognizes the importance of the uncertainty of maintenance requirements; therefore, we deal with the requirement generation at the component level and present a dual-clock failure modelling mechanism first. Afterwards, we describe the maintenance resource scheduling process based on the contract net protocol (El-Menshawy, Bentahar, El Kholy, & Dssouli, 2013). The selected inventory warehouses of a given resource requirement are alterable: because all warehouses are open and shared through lateral or cross-echelon transshipment, users can select the most suitable inventory warehouse according to the demand, the Time First – Inventory Second method, the Inventory First – Time Second method and so on. In addition, the dynamic change in the spare parts inventory is also analysed by considering whether the replaced failed components can be repaired or not. Furthermore, a system cost assessment method is developed to compare different scheduling strategies and to select the one that can minimize the total system cost. The remainder of this paper is organized as follows: We analyse the maintenance resource scheduling process and basic assumptions in Section 2. Section 3 is the analytical heart of this paper. This section describes the multi-agent-based resource scheduling process and a system cost assessment method. To verify the performance of the proposed approach, a case study of a general system is presented in Section 4. We end this paper with some brief concluding remarks and directions for further research in Section 5.

(1). Mathematical programming. This approach deals with the schedule planning of maintenance resources as an integer programming problem (Çelebi, 2015; Meissner & Senicheva, 2018). The resource scheduling process of the object is the decision variable, and the variable is usually valued at 0 or 1. However, this method is usually based on certain simplified assumptions: the optimal solution can be obtained within the bounded time, but the computational time is very long when the scale of the problem is large. (2). Approximation evaluation method. In combination with other methods, this approach can facilitate relatively large-scale planning. Due to its low computational complexity, high efficiency and good real-time performance, this method has been widely used in optimization research (Feng, Zhao, & Fan, n.d.; Liu, Liu, Zhao, & Xie, 2018; Wong, van Houtum, & Cattrysse, 2006) proposed a greedy method with a local search for multi-item, multi-location spare parts systems with lateral transshipments and waiting-time constraints. van Houtum and Kranenburg (2015) described a series of multi-item inventory models and presented exact and heuristic optimization methods. (Wong, Van Houtum, Cattrysse, and Van Oudheusden (2005) presented an efficient heuristic to deal with systems with more than two locations. This heuristic is based on a greedy initialization method combined with a local search improvement method. The results showed that the heuristic performs satisfactorily. Rahimi-Ghahroodi (2017) used an exact method and a greedy heuristic procedure for the evaluation of an optimize problem with more than five types of spare parts; the results demonstrate that approximate evaluations perform faster than exact evaluations as the system becomes complex. However, some special heuristic rules may be studied for a given problem. (3). Intelligent search method. In addition to solving the optimization problem with global optimization, this method has two advantages: it has a fast calculation speed, and it can be easily combined with other algorithms. Recently, Robinson (2007) investigated emergency transportation in real-life scenarios and formulated the problem of maintenance resource planning as an integer linear programming model; the authors combined a novel hybrid ant colony optimization-based algorithm to present a smart design of the ants' routing planning, thereby helping to simplify the solution.

2. Analysis of the maintenance resource scheduling process 2.1. Process description Consider a multi-echelon system with m equipment and n integrated inventory warehouses. Each inventory warehouse stores z types of spare parts, l spare parts of each type, q general maintenance workers and p skilled maintenance workers. Each piece of equipment contains r (r ≥ z) types of components. Note that there are some unreplaceable components; hence, the number of the types of total components is more than the number of the types of spare parts. In this study, the failure of a component generates the requirement for s (s ≤ l) spare parts, h (h ≤ q) general maintenance workers and f (f ≤ p) skilled maintenance workers. The process for selecting maintenance resources is based on the contract net protocol: users can select the most suitable inventory warehouse according to the variable demand. The objective is to minimize the total system cost. We take the components as the main flow to analyse the maintenance resource scheduling process. An illustration of the resource scheduling process during a maintenance activity is shown in Fig. 1. It is assumed that as long as there is a failed component, a resource requirement is generated. Furthermore, all inventory warehouses check their own remaining available inventory (NRij, i = 1,2,…,z, j = 1,2, …,n) and send their available inventory information to the command centre. According to NRij, the maintenance resource scheduling strategies (Ss = Ss1, Ss2,…, Ssk) and the contract net protocol, the command centre selects a suitable inventory warehouse, and the selected inventory warehouse provides the maintenance resources to replace the failed component on site (the basic level). Then, the replaced failed components are repaired at the basic or

In this paper, the system we consider has multiple inventory warehouses and multiple pieces of equipment with multiple repairable components, all of which can be regarded as different entities collectively responsible for transshipping spare parts or maintenance workers from inventory warehouses to the location of the machine failure (Cai, Shao, & Liu, 2019; Fasli & Kovalchuk, 2011; Ren, Fan, & Feng, 2019). Therefore, this system is essentially failure-flow-driven and dynamic in the operation of the resources flow. All of these characteristics add complexity to the system, so obtaining a feasible solution by using a single algorithm is difficult. To overcome the shortcomings of the traditional analytical methods, an agent-based simulation (Govindu & Chinnam, 2007; Othman, Zgaya, & Dotoli, 2017) is used. Agent-based simulations are one of the most effective tools for modelling the joint planning of maintenance resources. Amini, Wakolbinger, Racer, and Nejad (2012) applied agentbased modelling and simulation methodology to analyse the impact of alternative production–sales policies on the profit from a new product. Li and Chan (2013) proposed a common agent-based model for the simulation of make-to-stock and make-to-order supply chains with 2

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Resources scheduling process in maintenance

Component in equipment

Failed component

Spare part

Failed component

Component in equipment

Waiting for repairing

Standby

Working

No

Sufficient skilled maintenance workers at depotlevel?

Failure

Yes

Waiting for replacing

General maintenance workers occupied

Repaired

No

Sufficient general mai ntenance workers at basiclevel?

No

Yes

Spare parts inventory reduced

No

General maintenance workers idled

Successful?

Scrapped

Sufficient spare parts?

Yes Spare parts inventory increased

Skilled maintenance workers idled

Yes The used spare parts become components

Skilled maintenance workers occupied

Replaced Finishing replaced Fig. 1. Scheduling process of maintenance resources.

depot level. The maintenance workers at the basic level are less skilled. Although the maintenance cost is lower, the maintenance time is longer and the repair rate is lower, thereby possibly leading to a higher shutdown cost. While the maintenance workers at the depot level of maintenance are more skilled, they have a higher maintenance cost. Nevertheless, the repaired failed components can be taken to an inventory depot to supplement the inventory; hence, the order cost for new spare parts is reduced. In this paper, we consider the repairability of failed components and return the repaired components to the inventory warehouse, thereby enriching the original spare parts inventory management method; the costs (the holding cost, the ordering cost, the transshipment cost, the maintenance cost and the shortage cost) changed in this process are recorded. For the multi-echelon inventory system considered in this paper, by assessing the trade-off for the changed total cost entirely, we assume that all failed components are repaired at the depot level. Finally, according to the variable demands, the Time First – Inventory Second method, the Inventory First – Time Second method,

etc., we define some specific scheduling strategies. In addition, we develop a system cost assessment method is to compare different scheduling strategies and to select the one that can minimize the total system cost. 2.2. Basic assumption To model the maintenance resource scheduling framework, several basic assumptions are proposed below: (1). The failed components are replaced by ready-for-use spare parts on site (the basic-level) by idle general maintenance workers, then repaired by the skilled maintenance workers at the depot level. (2). All repairable components are as good as new after repair and are returned to the inventory warehouses to supplement their inventories. (3). The failed components of different equipment can be repaired simultaneously, but the maintenance times are independent. 3

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(4). Each maintenance requirement is handled by maintenance workers according to the first-come, first-served principle.

The equipment agent provides the location coordinates (Xie,Yie), i = 1,2,…,m), the type of components and the state of the equipment. Here, we consider two kinds of states: good and failure. When equipment enters the failure state, a maintenance resource requirement is generated. Because failure is a discrete event, the system clock (ts) used to describe the system’s functional state cannot directly express all the components’ failure. Hence, a new category of clock, the life clock, is used to accurately indicate the failure process. Based on the above analysis, a dual-clock-based failure modelling mechanism is proposed. Assuming Uij (i = 1,2,…m, j = 1,2,…,r) is component j in equipment i, the time to failure of Uij is TTFij; the life clock of Uij is denoted by Rij.

3. A multi-agent-based approach for the joint planning of maintenance resources 3.1. Multi-agent system framework In this paper, the system considered has multiple inventory warehouses, multiple pieces of equipment and multiple maintenance resources, which can all be regarded as different entities collectively responsible for transshipping spare parts or maintenance workers from inventory warehouses to the location of the equipment failure. In this section, we describe a multi-agent-based approach for the joint planning of maintenance resources. By using a multi-agent technique, we abstract the four classes of entities into the communication and cooperation of four classes of agents, namely, the equipment agent, the spare part agent, the maintenance worker agent and the management agent. Since the scheduling process is performed under a unified command, the management agent has priority over the equipment agent, the spare part agent and the maintenance worker agent. In addition, the internal reasoning mechanism plays an important role during the process: the mechanism determines the output results and affects the selection of the optimal scheduling strategy. The overall multi-agentbased framework, which is based on the above analysis, is shown in Fig. 2. The equipment agent, the spare part agent and the maintenance worker agent communicate with each other to obtain the maintenance resource requirement and determine the remaining available inventory. The management agent communicates with the other three agents to select suitable inventory warehouses from their own warehouse, sameechelon warehouses or cross-echelon warehouses based on contract net protocols and the scheduling strategy to obtain the minimal total system cost. To show the function of each agent used in the maintenance resource scheduling process, the major agents, the corresponding variables and the detailed modelling steps under the proposed agent-based framework are defined as follows:

Rij = 1

(1 TTFij )·(tije

vij · tij = 1

tijs )

(1)

where vij is the life clock consumption rate of Uij; this rate equals the reciprocal of TTFij. tij is the length of work time of Uij. is the system time when Uij starts working. tije is the current system time. For equipment i, each Rij is between 0 and 1. Set Rij = 1 when Uij begins working. For the component k, which has the minimal remaining life clock, when tike = tiks + TTFik , Rik = 0 and Uik enters failure status. At the same time, vij (j ≠ k) of the other Uij (j ≠ k) are set to 0. Equipment i stops working and sends a maintenance resource request to the command centre. After Uik is replaced by a new one, equipment i receives a repaired message and resumes working. Rik of the replaced Uik is reset to 1, Rij of the other Uij keeps the same values as before, vij of all the components equals the reciprocal of TTFij, and the life clock of all the components continues consuming. The time to failure (TTFij) can be sampled by the direct sampling technique (Hesterberg, 2002); the kth sampling result is expressed as follows:

TTFij (k ) = F

(2)

1 (Z )

where Z is the random number between 0 and 1, and F 1 (Z ) is the inverse function of the random distribution. Generally, TTFij obeys an exponential distribution, and the kth specific sampling result is given by the following function (Weimin & Yixing, 1990):

TTFij (k ) =

ln( )

(3)

ij

where is a random number between 0 and 1, and ij is the failure rate of Uij. The maintenance time of each repairable failed component is determined by the mean time to restoration (MTTRij), which is also sampled by the direct sampling technique, and the sampling process of

(1). Equipment Agent

Maintenance Worker Agent

Management Agent Spare part Agent

Equipment Agent

Fig. 2. Maintenance resource scheduling process based on a multi-agent. 4

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MTTRij is similar to that of TTFij. Commonly, MTTRij obeys a standard normal distribution, and the kth specific sampling result is given by the function:

MTTRij (k ) = where

1

2 ln

and

1 ·cos(2

2)

Then, using the following function, we can calculate the transshipment time (Tt):

dt = v0 Ts +

(4)

(2). Spare Part Agent The spare part agent provides information regarding the location coordinates (Xk,Yk), k = 1,2,…,l; the remaining inventory (NRij) of type i (i = 1,2,…,z) at warehouse j (j = 1,2,…,n); the initial inventory (NIij) of type i at warehouse j; and the repaired components (NBij) of type i returned to inventory warehouse j. The relationship between these variables satisfies the following equation: b

NRij = NIij

sf + NBij

(5)

f =1

where sf refers to the number of needed spare parts in the fth maintenance action. In addition, the spare part agent provides the time (teij) when spare part i (i = 1,2,…,l) is transported to inventory j (j = 1,2,…,n) and the time (toij) when i is scheduled to be transported from inventory warehouse j. The holding time (Thij) of spare part i is calculated by

Thij = toij

(6)

teij

3.2. Resource scheduling process In this section, we introduce the maintenance resource scheduling process based on a multi-agent approach. The resources can be scheduled to be shipped from their own warehouse or from same-echelon warehouses by lateral transshipment or from cross-echelon warehouses by direct transshipment in the case of a shortage. In addition, we analyse the reverse circulation process of resources by considering whether the failed components can be repaired or not and returning the repaired failed components to the inventory warehouse. The scheduling processes are related to the selection of the optimal warehouse, and in this paper, we use the contract net protocol principle to select the most suitable warehouse. We take the spare parts as an example to introduce the selection process, which is shown in Fig. 3. When a maintenance resource requirement is generated, the management agent calls the spare part agent for bids [demand | constraints], where demand represents s (the required number of spare parts) and

(3). Maintenance Worker Agent The maintenance worker agent includes general maintenance workers who are responsible for replacing failed components and skilled maintenance workers who are responsible for repairing failed components; both types of maintenance workers have busy and idle states. In the maintenance worker agent, NGj represents the available number of general maintenance workers at warehouse j, j = 1,2,…,n, and NSj represents the available number of skilled maintenance workers at warehouse j. Spare parts are taken by maintenance workers to replace the failed components; therefore, maintenance workers come from the same inventory warehouse of the spare parts. The location of the maintenance worker is (Xk,Yk), and the motion speed v of the maintenance worker and the spare parts are the same. To make the proposed model more robust and superior, we assume that the speed motion of the maintenance resource is variable. This assumption is reasonable in actual situation because the equipment used in many important economic sectors becomes increasingly more capital intensive. If an unplanned downtime due to failure occurs, it is of utmost importance to replace the failed components with ready-foruse ones. To do so, the schedule for shipping maintenance resources needs to be accelerated. The maximum speed (vmax) depends on the actual situation; once the maximum speed is reached, the acceleration becomes zero. The motion speed v is shown as follows:

v = v0 + a , v

(9)

Moreover, the management agent calculates the total system cost (CT) generated during the maintenance resource scheduling process by multiplying the time/distance and the coefficient; CT is used to select the most suitable scheduling strategy. After defining the specific agents, we list the detailed modelling steps as follows: Step 1: Abstracting the different specific agents according to the actual situation and designing the overall framework of the multiechelon inventory system. Step 2: Defining the complex interaction relationships among different agents and modelling the visual simulation framework. Step3: Based on the modelled visualization framework, injecting the simulation drive mechanism by using a dual-clock failure modelling approach. The maintenance resource requirement is generated at the component level; consequently, the failure modelling mechanism corresponds strongly to the actual situation. The maintenance resource requirement then assigns specific values to individual parameters. Step 4: Collecting and counting the real-time operating data and evaluating the system-related indicators (cost, reliability, etc.)

are random numbers between 0 and 1.

2

1 2 aTs 2

Equipment agent

Spare parts agent

failure re-bidding

Bidding for spare parts 1 [demand | constraints] m evaluate

(7)

vmax

where v0 is the initial movement speed of the maintenance resources, a is the acceleration and vmax is the maximum motion speed.

[No bid]

adjustment constraints

The management agent provides resource scheduling strategies (Ss = Ss1, Ss2,…, Ssk) for selecting the inventory warehouse. In addition, this agent is responsible for calculating the resource shortage time (Ts) and the transshipment distance (dt) between the failed equipment and the location of the selected maintenance resources using the following function:

(Xie

Xk )2 + (Yie

[demand shortage]

evaluate

(4). Management Agent

dt =

[optimal sorting]

[else]

Yk ) 2

n [No bid]

bid [cost , waiting time, etc.]

[n 1]

[veto bid]

1 n-1

1 [confirm | warrant] 1

1

[contract | scheduled]

Fig. 3. Inventory warehouse decision process under the contract net protocol.

(8) 5

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constraints include the scheduling strategy, the response time, etc. Then, the spare part agent examines NRij at warehouse j, j = 1,2,…,n. If, for all the warehouses, NRij < s, a resource shortage occurs with a high shutdown cost until the inventories of the warehouses are supplemented. If NRij s, the spare part agent plans bid information (response time, total system cost, etc.) and counters the bids of warehouse j (which were sent to the management agent) according to the total system cost. Afterwards, the management agent sorts the n (n 1) satisfactory counter bids according to the constraints, confirms the most suitable maintenance resource inventory warehouse to schedule maintenance resources and veto other bids. If none of the warehouses satisfies the constraints, the management agent should adjust the constraints and start the re-bidding process. The selected maintenance resource inventory warehouses respond to the management agent’s request to replace the failed component, and the states of the selected general maintenance workers are set to busy. In this process, NRij and NGj in warehouse j are reduced. Furthermore, the management agent records the transshipment distance dt and then calculates the transshipment time Tt according to Eq. (9). After the failed component has been replaced with a new one, the failed component is sent to the depot level by the general maintenance workers. Then, the workers return to their inventory warehouse to wait for the next maintenance mission and set their state to idle, and NGj is increased. Then, the failed component is repaired by the idle skilled maintenance workers at the depot level, and their state is set to busy. At the end of the maintenance, their state returns to idle. According to each repairable failed component’s MTTRi, the management agent records the maintenance time. According to the results of the maintenance, the repaired failed component sent to the inventory warehouse to supplement the inventory generates NBij, and correspondingly, the NRij at inventory warehouse j is increased.

b

Ct =

b

Cm =

r

i=1

j=1

(Thij· NRij ·coehij

where p is the profits generated per minute under normal working conditions. Then, we use the sequential Monte Carlo method to deal with the long-term expected cost, and the state duration time sampling method is used to determine the running and repaired time of each component. The specific steps of the simulation are as follows: Step 1: Data initialization. Set the initial state of all components to normal. Step 2: Based on the known probability distribution function of the state duration time of each component, sample the current state duration time of all components randomly. In Section 3.1, we sampled TTFij(k) and MTTRij(k); here, we selected the shortest TTFij(k) as the first failed component. Step 3: Repeat Step 2 during the simulation time, and record the random sample results of the state duration time of different components. For the sequential Monte Carlo method, the key problem is determining the warm-up period of one simulation experiment to ensure a more accurate estimate of the steady-state parameters of the model. The intuitive idea for dealing with this warm-up problem is to defer the observation time series of the output until it reaches a steady-state, delete the first y points from the jth simulation experiment, continue to run to z observation points, and use Cz(j) as a basic unit for analysis. The key challenge in this case is to determine the values for y and z (Pan, Nigrelli, & Ballot, 2014).

(10)

(11)

z

Cz (j) = [1 (z

(sif ·ui )

y )]

Ci (j ) i =y+1

(16)

It is recommended that the run length is at least twice as long as the estimated length of the warm-up period. Kelton and Law (1983) explained this assumption in detail. In this paper, failure generates the demand for maintenance resources; that is to say, the failure point is the data point that we recorded. In each simulation experiment, an initial estimate of the length of the warm-up period was obtained by simply inspecting a time-series of the output data to determine where the model appears to be in a steady-state. On completion of this stage, a time series of the key output data for each simulation experiment should have been collected; then, the average system cost can be calculated as follows:

b f =1

(15)

Cs = Ts· p

where Thij represents the holding time of spare part i at inventory warehouse j and coehij represents the holding coefficient of spare part i at inventory warehouse j. Co is due to maintenance requirements.

Co =

(14)

where h is the number of hired general maintenance workers, ugw is the unit price of the general maintenance workers, g is the number of hired skilled maintenance workers, usw is the unit price of the skilled maintenance workers, coesw represents the maintenance coefficient (unit price for repair per 1 min) of the skilled maintenance workers and MTTRi is the maintenance time of each failed component. As our analysis shows in Section 1, the replacement time is usually very short; hence, we ignore the cost generated by the replacement. Cs is generated by the shortage of maintenance resources; Cs is determined by multiplying the profits generated under normal operation per minute and the shutdown time.

Then, the calculation method for each type of system cost is given separately. Ch is generated by the remaining available spare parts, and the remaining inventory is determined by the spare part agent dynamically. n

(h ·ugw + g·usw + MTTRi · coesw f =1

In this paper, we propose a system cost assessment method to compare the different scheduling strategies and use the method to select the optimal scheduling planning. The expected total system cost (CT in the Chinese Yuan) includes the holding cost for spare parts (Ch), the ordering cost for spare parts (Co), the transshipment cost for spare parts and maintenance workers (Ct), the maintenance cost (Cm) and the shortage cost (Cs). The system cost function takes the following form:

Ch =

(13)

where Nt is the number of trucks, ut is the unit price of a truck, coet represents the transshipment coefficient (unit price for transport per 1 km) of the spare parts and the maintenance workers. Cm includes the cost for the hired general and skilled maintenance workers and the maintenance cost generated by the skilled maintenance workers.

3.3. System cost assessment method

CT = Ch + Co + Ct + Cm + Cs

(Nt ·ut + dt · coet ) f =1

(12)

where b are the failure times, sif is the number of spare parts i used in the fth failure and ui is the unit price of spare part i. Ct consists of the rental fees for trucks and the transportation cost; the latter includes the spare parts transportation cost and the maintenance worker transportation cost. 6

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Table 1 Variable information. Variable n m r s h f Nt

Ci (j ) x j=1 i =y+1

Description Number Number Number Number Number activity Number activity Number

z

C¯T =

of of of of of

Value inventory warehouses pieces of equipment key repairable components in all equipment spare parts required for one maintenance activity general workers required for one maintenance

x

z

j=1

i =y+1

((Chi (j ) + Coi (j) + Cti (j) + Cmi (j) + Csi (j)) x

=

5 3 5 1 1

4. Case study

of skilled workers required for one maintenance

1

4.1. Input data

of trucks required for one maintenance activity

1

A two-echelon system is considered in this paper, and the basic parameters related to our system are listed in Table 1. In the generic evaluation model for the joint planning of resources, every request can be satisfied by different warehouses according to the real-time status of the available inventory and the criteria for selection. The two most important factors taken into account are the available inventory at the warehouse and the distance between the failed equipment and the inventory warehouse. Accordingly, 4 categories of criteria are defined to show the practicality of the proposed model: the fixed-pair, the nearest distance, the maximum inventory and the minimum ratio (Ng, Li, & Chakhlevitch, 2001). Actually, there are many specific scheduling strategies for selecting the inventory warehouse; however, regardless of which scheduling strategy is selected, the scheduling strategy is nothing more than the combination of the normal employment, lateral transshipment and cross-echelon transshipment, and the best strategy may be determined specifically by real cases with different constraints and demands. The fixed-pair category is considered as a reference for the other criteria. As shown in Fig. 4, warehouse–equipment pairs are fixed, e.g., equipment 1 is satisfied by local warehouse 1, which is supplied by central warehouse 1, and so on. The fixed-pair strategy represents exactly the structure of today’s resource supply scenario despite the simplification. In addition to the fixed-pair strategy, the warehouse nearest the failed equipment will intuitively be selected to reduce the transportation cost. However, in this case, replenishment will not be dynamic if distance is the only criterion. In other words, the criterion is Distance First – Inventory Second. Correspondingly, the other criterion, Inventory First – Distance Second, is the maximum inventory scheduling strategy. Furthermore, we can consider distance and inventory to be equally important. The system is driven simultaneously by both the minimum distance and the available inventory level. The aim is to achieve a more uniform distribution of the available inventory, thereby decreasing the overall time of transshipment and the total holding cost; we define this schedule strategy as the minimum ratio. Here, we employ the ratio to formulate the interaction: the ratio is the distance to the available inventory. Until now, we have demonstrated the four categories of inventory warehouse selection strategies. They allow lateral transshipment: resources are scheduled to be shipped from same-echelon warehouses or, in the case of a shortage, from cross-echelon warehouses. Since resources scheduled to be shipped from cross-echelon warehouses usually have a high transshipment cost, we use a same-echelon warehouse

Central warehouses

Equipments

Local warehouses

Fixed-pair transshipment Lateral transshipment

Cross-echelon transshipment Normal replenishment

Fig. 4. Maintenance resources transshipment method.

Table 2 Failure rates and time parameters of repairable components. Parameter i

TTFij (min) MTTRij (min)

A

B

C

D

E

0.3 3300 100

0.9 1100 20

0.3 3300 110

0.7 1400 50

0.8 1250 40

Table 3 Related variables of spare parts. Maintenance resources

Variables

Center warehouse 1

Spare parts

type number ui (yuan/unit) coehij v0 (m/s) a (m/s2)

A 10 9 0.3 5 2

B 25 5 0.08 5 2

C 8 10 0.5 5 2

D 15 8 0.2 5 2

(17)

E 20 7 0.1 5 2

Table 4 Related variables of maintenance workers. Maintenance resources

Variables

Center warehouse 1

Maintenance workers

type number ug / us (yuan/unit) v0 (m/s) a (m/s2) coesw

General maintenance workers 8 12 5 2 /

7

Skilled maintenance workers 10 15 5 2 18

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Table 5 Transshipment prices of spare parts and maintenance workers. Maintenance resources

Schedule from same-echelon warehouses (yuan/unit)

Schedule from cross-echelon warehouses (yuan/unit)

Spare parts Maintenance workers

5 10

8 12

Fig. 5. Modelling of simulation process.

unless all the local inventory warehouses are out of stock. The variables, such as i , TTFij and MTTRij of the repaired component i (i = 1,2,…,r), are presented in Table 2. i and TTFij are correlated with the maintenance demand, and MTTRij affects the total system cost; MTTRij ranges from a few minutes to several hours. To compare the total system cost of different scheduling strategies, we use the same input data. Since the type and number of maintenance resources in different inventory warehouses are the same, as limited by this paper, we take Center warehouse 1 as an example to represent the spare parts information in Table 3 and list the maintenance workers information in Table 4. The transshipment price (transported per 1 km) of spare parts and maintenance workers is shown in Table 5, and the unit price of truck is ut = 200 (yuan/unit). Since this article does not consider the external inventory warehouse even if all inventory warehouses are out of stock, the consequences of downtime caused by the shortage of spare parts, general maintenance workers and skilled maintenance workers are the same,

the shortage cost are 1000 (yuan/minute). 4.2. Modelling of simulation results As analysed in Section 2.1, when there is a failed component, a resource requirement is generated, at the same time, Spare Part Agent and Maintenance Worker Agent check their own remaining available inventory (NRij, i = 1,2,…,z, j = 1,2,…,n) and feedback their available inventory information to the Management Agent. According to NRij, maintenance resources scheduling strategies (Ss = Ss1, Ss2,…, Ssk) and contract net protocol, the Management Agent selects appropriable inventory warehouse, the selected inventory warehouse provides the maintenance resources to repair the failed equipment. Due to the state transition is complex, here, we use the Anylogic software to model the process, partial design is shown in Fig. 5.

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Fig. 6. Cost composition under four scheduling strategies.

While the other composed cost is relatively smaller than those in other scheduling strategies, someone who does not care about shortage cost should select the Fixed-pair scheduling strategy. Since the Nearest distance scheduling strategy allows lateral transshipment to supplement maintenance resources from the nearest inventory warehouse, it greatly reduces transshipment cost and further reduce the total system cost. In order to balance the remaining inventories of different inventory warehouses, the Maximum inventory scheduling strategy schedules maintenance resources from inventory warehouses with the maximum remaining available inventory. Often, it will not schedule resources from the nearest inventory warehouses, and the selected inventory warehouse has strong dynamics, which will generate higher transshipment cost. Someone who pays more attention to balance the remaining inventory and wants to save holding cost and shortage cost should select the Maximum inventory scheduling strategy. The Minimum ratio scheduling strategy considers the remaining inventory and the transshipment distance simultaneously, which is relatively reasonable. From the simulation results, we can see that this scheduling strategy generates the lowest total system cost among the four scheduling strategies. We conclude some applicable scenarios for the different scheduling strategies in Table 6. Notably, for a specific maintenance requirement, the scheduling strategy is not unique, a cross-strategy or another scheduling strategy can be used according to the actual scenario. For example, the Nearest distance scheduling strategy can be selected when scheduling maintenance workers, the Maximum inventory scheduling strategy can be selected when scheduling spare parts, etc.

Table 6 Applicable scenarios of different scheduling strategies. Scheduling strategy

Applicable Scenarios

Fixed-Pair

Focus on the transshipment cost and do not care about shortage cost Focus on the transshipment cost and shortage cost Focus on the balance of the inventory in each warehouse Focus on the transshipment distance and remaining inventory

Nearest Distance Maximum Inventory Minimum Ratio

4.3. Result and analysis As analysed above, in order to obtain a more accurate estimate of the steady-state parameters of the simulation model, an initial estimate of the length of the warm-up period needed to be determined. By deferring observation time-series of the output of several simulation experiments, we find that the data from the first 5 points have greater fluctuations and the results from the later 15 observation points already have enough reference value. Hence, we delete the first 5 points, on completion of this stage, a time-series of the later 15 output data for each simulation experiment should have been collected, which are used as a basic unit for analysis. In order to obtain a more accurate result of the simulation model, we take 1000 simulation experiments of the above case, and use the average results as the final output. The five composed cost in the system cost assessment method under four scheduling strategies are shown in Fig. 6. From Fig. 6, we can conclude that different scheduling strategies have different impacts on the total system cost, especially the shortage cost. Allowing lateral transshipment (the latter three scheduling strategies) can greatly reduce shortage cost and further reduce the total system cost. The reduction of the average total system cost by using the Minimum ratio scheduling strategy is much better. Since the Fixed-pair scheduling strategy does not allow for scheduling resources from other inventory warehouses in case of a shortage, the failed equipment stops running until there is sufficient available inventory in the fixed warehouse, and the shortage cost are very large.

4.4. Discussion In order to further verify the effectiveness of the proposed approach, we perform sensitivity analyses on several key input parameters, such as the failure rate and the number of maintenance resources. Since these input parameters are estimated here, for the agent-based system model, the parameters are often estimated subjectively (See. Tables 7–9).

Table 7 Sensitivity analysis of the failure rate. Variable i

NI NG

Description

Original Value

Contrast Value

Failure rate of component A in all equipment Number of A spare parts at all warehouses Number of general maintenance workers at all warehouses

0.3 10 8

0.5 10 8

9

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Table 8 Sensitivity analysis of A spare parts number. Variable i

NI NG

Description

Original Value

Contrast Value

Failure rate of component A in all equipment Number of A spare parts at all warehouses Number of general maintenance workers at all warehouses

0.3 10 8

0.3 14 8

Description

Original Value

Contrast Value

Failure rate of component A in all equipment Number of A spare parts at all warehouses Number of general maintenance workers at all warehouses

0.3 10 8

0.3 10 12

Table 9 Sensitivity analysis of the number of general maintenance workers. Variable i

NI NG

Fig. 7. Comparison of total costs under the different failure rate.

Fig. 8. Comparison of total cost under the different number of spare parts.

Fig. 9. Comparison of total cost under the different number of general maintenance workers.

Fig. 7 shows that when the failure rate increases, the total maintenance cost increases significantly. The results are obvious since as the failure rate increases, both the number of failures and the total cost increase. From Figs. 8 and 9, we conclude that as the inventory of maintenance resources increase, the shortage cost reduces and the holding

cost increases, but whether the total system cost increases or decreases depends on which of the four scheduling strategies is used. When using the fixed-pair scheduling strategy, the total system cost decreases because the shortage cost is higher than the holding cost. Meanwhile, under the other three scheduling strategies, the total system cost increases because the use of lateral transshipment reduces the shortage 10

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cost. Furthermore, as the inventory increases, the holding cost significantly increases, while the shortage cost decreases to a relatively low level. The results obtained from the sensitivity analysis are similar to those obtained in an actual situation, thus ensuring that the proposed approach is robust.

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5. Conclusions In this paper, we analysed a multi-echelon inventory system with the operation management of multi-component equipment in which supply flexibility through lateral transshipments between local warehouses and emergency direct deliveries from the central warehouse and the plant may occur in response to stock-outs. We propose an agent-based generic evaluation model for the joint planning of resources by abstracting all entities into a team of agents and mapping their negotiations into agent interactions. The approach has strong applicability to different resource scheduling requests. More specifically, the main innovations and advantages of the scheduling approach are subsequently revisited here as follows: (1). The proposed agent-based generic evaluation approach makes the model more granular and flexible; therefore, the model can describe the complex logical relationships among various entities and support the large-scale, dynamic and complex system of scheduling the lateral or cross-echelon transshipment of spare parts and maintenance workers. (2). A dual-clock failure modelling mechanism is proposed. The maintenance resource requirement is generated at the component level, thereby making the failure modelling mechanism correspond strongly to the actual situation. (3). Considering the procedures for transforming failed components into available spare parts and returning them to supplement the inventory enriches the original inventory management method. (4). A system cost assessment method is developed to compare different scheduling strategies and is used to select the optimal strategy for minimizing the total system cost. Then, we use four specific strategies for scheduling the shipment of maintenance resources. The results of our simulation experiment show that different scheduling strategies have different impacts on the total system cost, especially the shortage cost. Lateral transshipment greatly reduces the shortage cost and, furthermore, reduces the total system cost. In addition, lateral transshipment performs quite satisfactorily in solving the problem of having identical target waiting times at all local warehouses. This research can be extended in several directions. One possible extension is to expand the proposed cost assessment model, which contains the basic cost (which is the sum of the holding inventory cost, the ordering cost, the transportation cost, the maintenance cost and the shortage cost), and other factors that can be considered in future studies, such as the safe stock level. Another extension is to analyse the location of the central warehouse, as the number and locations of central warehouses are important decisions affecting the whole system’s performance. In addition, the decision of whether to group several local warehouses into a pooling group served by a central warehouse needs to be made. The analysis becomes more complex as these additional decisions must be made jointly with the inventory decision. Instead of a controllable framework, as proposed in this paper, an optimization model may be applied in the future. References Froger, A., Gendreau, M., Mendoza, J. E., Pinson, É., & Rousseau, L. M. (2016). Maintenance scheduling in the electricity industry: A literature review. European Journal of Operational Research, 251(3), 695–706.

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