Author's Accepted Manuscript
A decision-making framework to integratemaintenance contract conditions with criticalspares management
David R. Godoy, Rodrigo Pascual, Peter Knights
PII: S0951-8320(14)00152-5DOI: http://dx.doi.org/10.1016/j.ress.2014.06.022Reference: RESS5082
To appear in: Reliability Engineering and System Safety
Received date: 1 October 2013Revised date: 30 May 2014Accepted date: 27 June 2014
Cite this article as: David R. Godoy, Rodrigo Pascual, Peter Knights, A decision-making framework to integrate maintenance contract conditions with criticalspares management, Reliability Engineering and System Safety, http://dx.doi.org/10.1016/j.ress.2014.06.022
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A decision-making framework to integrate maintenance
contract conditions with critical spares management
David R. Godoy∗,a, Rodrigo Pascuala, Peter Knightsb
aPhysical Asset Management Lab, Department of Mining Engineering, PontificiaUniversidad Catolica de Chile, Av. Vicuna Mackenna 4860, Santiago, Chile
bSchool of Mechanical and Mining Engineering, The University of Queensland, QLD4072, St. Lucia, Brisbane, Australia
Abstract
Maintenance outsourcing is a strategic driver for asset intensive industriespursuing to enhance supply chain performance. Spare parts managementplays a relevant role in this premise since its significant impact on equipmentavailability, and hence on business success. Designing critical spares poli-cies might therefore seriously affect maintenance contracts profitability, yetservice receivers and external providers traditionally attempt to benefit sep-arately. To coordinate both chain parties, we investigated whether the sparecomponents pool should be managed in-house or contracted out. This pa-per provides a decision-making framework to efficiently integrate contractualconditions with critical spares stockholding. Using an imperfect maintenancestrategy over a finite horizon, such scheme maximizes chain returns whileevaluates the impact of an additional part to stock. As result, an originaljoint value -preventive interval and stock level- sets the optimal agreementto profitably allocate the components pool within the service contract. Sub-sidization bonuses on preventive interventions and pooling costs are also es-timated to induce the service provider to adjust its policy when needed. Theproposed contractual conditions motivate stakeholders to continuously im-prove maintenance performance and supply practices, thus obtaining higherjoint benefits.
Key words: Maintenance outsourcing contract, Spare parts stockholding,
∗Corresponding authorEmail addresses: [email protected] (David R. Godoy), [email protected]
(Rodrigo Pascual), [email protected] (Peter Knights)
Preprint submitted to Reliability Engineering and System Safety July 2, 2014
Inventory pooling allocation, Imperfect maintenance, Supply chaincoordination.
1. Introduction
Maintenance outsourcing is a strategic means to improve business perfor-mance. Outsourcing creates value through the use of external resources byand for companies to acquire and sustain competitiveness [1]. The mainte-nance function is a main driver of outsourcing since it has excellent potentialto achieve cost benefits and enhance performance among partners [2]. Thisbusiness purpose is meaningful for asset intensive industries -such as min-ing, aeronautic, or defence- which face substantial investment in maintainingcomplex equipment and high demand on system availability. For these firms,main reasons to contract out maintenance tasks rather than perform themin-house are focusing on core business, accessing highly specialized services atcompetitive costs, and sharing risks [2, 3, 4, 5]. When dealing with outsourc-ing, effective supply chain coordination allows achieving a rewarding situationfor all stakeholders [3]. Accordingly, a model capable of coordinately opti-mizing performance can lead to successful maintenance contracting strategiesin capital intensive environments.
Spare parts management has a critical role toward operational efficiencyof asset intensive industries. Equipment criticality is defined by the mostrelevant assets that efficiently and safely sustain production [6]. The opera-tion of such equipment is consequently supported by critical spare parts [7].Major spare components are related to considerable investment, high reliabil-ity requirements, extended lead times, and plant shutdowns with importanteffects on operational continuity [8]. A method to prevent production lossevents is having inventories at hand, especially when either target service lev-els or backorder penalties are large [9]. This is the case of capital intensivefirms, wherein critical spares storage is directly linked to business successdue to the impact of stock-outs on assets utilization [7]. As an example, theaviation supply chain holds remarkable US$ 50 billion in spares inventories toprovide availability service [10]. Efficient critical spares stockholding is there-fore essential for companies in which success strongly depends on equipmentperformance.
Maintenance contracts profitability might be significantly affected by crit-ical spares policies. Particularly, the stock of critical repairable spares can
2
be interpreted as a pool of components from where replacements are satis-fied [7]. Consistently with the serious impact on operational and financialperformance, managing the pool of critical spare components becomes keyto improve profits within the service contract. However, as it depends on thedecision-maker’s position, both supply chain parties -service receiver (client)and external provider (agent)- traditionally intend to maximize benefits sep-arately. If the client controls the spare parts pool, there are scarce incentivesfor the provider to avoid an indiscriminate use of components aside from reg-ular restraints. Conversely, if the agent administers the pool, rational use ofcomponents turns reasonable. Critical spares stockholding is a supply chainlever to keep maintenance outsourcing viable for the parties involved.
In pursuit of coordinating the contracting parties, we investigated whetheror not the client should outsource the management of the pool of spare com-ponents to the agent. This paper provides a decision-making framework toprofitably integrate the contractual maintenance strategy with critical sparesstockholding. Such scheme is based on a joint value -preventive interval andstock level- that maximizes the supply chain returns while evaluates the im-pact of an additional part to stock. Using an imperfect maintenance strategyover a finite horizon, the model leads to an optimal decision to allocate thecritical spare components pool within the outsourcing contract. An inter-esting link is thus created between maintenance performance indicators andsupply chain practices.
Having introduced the importance of allocating critical spare parts man-agement within maintenance service contracts for asset intensive industries,the rest of the paper is organized as follows. Section 2 states the differencesbetween the enriched concept of the present paper and relevant existent re-searches. Section 3 describes the model formulation to integrate maintenanceand spares supply indicators. Section 4 presents a case study in the miningindustry, which holds substantial spares inventories to ensure system per-formance. Finally, Section 5 provides the main implications of applying thejoint model to coordinate the outsourcing strategy under an asset manage-ment perspective.
2. Literature review
The following literature review is structured as the importance of themanagement of the pool of critical spares within maintenance outsourcingcontracts.
3
As an interesting strategy to achieve cost-benefits, consolidating inven-tory locations by cooperative pooling has been addressed by [10, 11, 12, 13],among others. In the context of repairable spares pooling, the cost alloca-tion problem is analyzed using game theoretic models by [14]. As recentimplementations, virtual pooled inventory by managing information systemsis included in [15], and a calculation model of spare parts demand, storageand purchase planning in the coal mining industry is reported by [16]. Whendealing with cooperation in contractual alliances, the study of [17] statesthe relevance of interfirm trust to deter opportunistic behaviour in a sharedownership structure. Such trust is an important issue related to poolingstrategies. A widely applied scheme for repairable items stockholding fo-cused on system availability and spares investment is provided by [18]. Sincethe accuracy to determine the optimal inventory levels for both single-siteand multi-echelon techniques, such model is used to adapt the concept ofspare service level in the present paper.
Maintenance outsourcing under supply chain coordination is discussedby [3], study that deals with incentive contracts terms to coordinate agentsand clients by a maintenance policy seeking to optimize the total profit.The work of [4] extends this approach by incorporating realistic conditions,such as imperfect maintenance and finite time-span contract . That modeladapts the failure rate by using the system improvement model of [19]. Suchconcepts of profitable coordination and imperfect maintenance are also usedin the present paper to improve the practical applicability for asset intensiveoperations.
There are studies that specifically deal with allocation spare parts in ser-vice contracts. A paper intending to incorporate repair contract selection andspares provisioning under a multicriteria approach is presented in [20]. In[21], a profit-centric model is presented for spares provisioning under a logis-tics contract for multi-item and multi-echelon scenario. In [22], an inventorymodel is developed for a repairable parts system by varying failure and re-pair rates. A dynamic stocking policy to replenish the inventory to meetthe time-varying spare parts demand is proposed by [23]. A reliability-basedmaintenance strategy required for the spares inventory is described in [24],although the scope does not cover contract conditions. Since the relevanteffect of warranties as service contracting, a three-partite stochastic modelincluding manufacturer, agent, and customer is presented in [25]. However,none of these works has faced the pool management problem by using therealistic assumptions of imperfect maintenance, finite contract duration, or
4
profitable channel coordination.Regardless of the extensive literature, the present paper introduces new
contributions in terms of formulation and analytical properties. To the best ofour knowledge, a model capable of delivering profitable decisions to allocatethe pool of critical spare parts within maintenance outsourcing contracts -viathe inclusion of imperfect maintenance and the optimal conditions for supplychain coordination- has not been addressed in the literature.
3. Model Formulation
Consider a system belongs to a fleet of equipment whose operation issupported by a pool of repairable components. The proposed model op-timizes the management decisions of critical spare components within theoutsourcing service contract. The formulation is presented in three sectionsas follows: (i) preventive maintenance (PM) policy under the contractualconditions scheme, (ii) service level associated with the stock of critical spareparts, and (iii) decision-making model to integrate PM interval with optimalspares inventory to maximize global profits. The terms ”client” and ”agent”will henceforth be adopted to indicate service receiver and external provider,respectively.
3.1. Contractual preventive maintenance policy
Let the maintenance of the fleet system be contracted out by the clientto the agent. For sake of self-containment, relevant maintenance contractconditions -such as imperfect maintenance and finite contract horizon- de-veloped in [3, 4, 19] are described in detail. The scheme is set by the followingconditions.
• The interval between preventive interventions (PM interval) is T .
• The agent is free to select the age T at which PM will be performed.
• Direct costs and length of PM are, respectively, Cp and Tp.
• Direct costs and length of corrective interventions are, respectively, Crand Tr.
• The basic service fee to the agent is p.
• The net revenue of the client after production costs is r.
5
• The agent set a minimum expected profit π to participate in the game.
• The finite horizon is as the contract lasts from the beginning of a systemlife cycle to the end of the n-th overhaul.
The system has a Weibull distribution with shape parameter
β > 1. (1)
The inclusion of imperfect maintenance into the failure rate is based onthe system improvement model [19]. Each PM intervention restores the sys-tem condition to almost as good as new according to
hk(t) = αhk−1(t− T ) + (1− α)hk−1(t) (2)
where t denotes lifetime, k corresponds to the index of the k-th preventiveaction, and α ∈ [0, 1] is the maintenance improvement factor.
Before the first preventive intervention, the failure rate is:
h(t) = h0βtβ−1, t < T. (3)
The expected number of failures H after n overhauls is
H(nT ) =n∑i=0
(ni
)αn−i(1− α)i−1H0(iT ) (4)
where H0 =∫ nT
0h(t)dt.
For β integer, the expected number of failures is
H(nT ) = κh0Tβ (5)
where values of κ, some of them summarized in Table 1, depend on both αand n for different integer values of β.
As the duration of the contract is n (T + Tp), the expected maintenancedirect cost is
CM(nT ) =Cp +H(nT )Crn (T + Tp)
. (6)
In addition, the expected availability during the contract as a function ofmaintenance interventions is
AM(nT ) =nT −H(nT )Trn (T + Tp)
. (7)
6
Table 1: Values of κ as inclusion of imperfect maintenance and finite horizon.
β κ1 n2 n2(1− α) + nα3 n(n− 1)(n− 2)(1− α)2 + 3n(n− 1)(1− α) + n
From a perspective biased by single interests, it is clear that the clientfocuses on maximizing availability, whereas the agent focuses on minimizingmaintenance costs. To achieve the cooperation of both parties, the nextsections describe an optimal PM interval (T ) aiming to the entire chainbenefit while adding the influence of the critical spares inventory.
3.2. Spare components service level
The concept of spare components service level allows incorporating thepreventive maintenance policy described in the above mentioned section. Es-timation of system availability as a function of critical spare parts stock isadapted from the inventory model for repairable items developed in [18]. Forsake of conciseness, an one component case is treated but the extension tomulti-components is straightforward. The approach is as follows.
• The system belonging to the fleet of equipment requires I types ofrepairable spare components.
• The fleet size is N and the multiplicity of each type of spare componentsin the equipment is zi.
• Stock level of critical spare parts is S.
• Turn-around time, as the workshop repair cycle from removal of a com-ponent until readiness to use, is Tat.
We propose the following approach to incorporate the impact of PM in-terval on the critical spare parts demand to workshop. The demand λ(T ) isupdated as a function of each interval T from the maintenance policy by
λ(T ) =Nzi
MTBI(T ) + TpR(T ) + Tr (1−R(T ))(8)
7
where R(T ) is the reliability function at T and MTBI(T ) =∫ T
0R(t)dt is the
mean time between interventions.Expected backorders with spares stock level S, the unfilled number of
demands for not having sufficient inventory, is
EBO(s) =∞∑
j=S+1
(j − S)(λ(T )Tat)
je−(λ(T )Tat)
j!. (9)
Expected service level of equipment given by spares stock is then
AS(S) =I∏i=1
(1− EBOi(Si)
Nzi
)zi
(10)
where the aim is to maximize equipment availability, or analogously to mini-mize expected backorders, as a function of the optimal investment in criticalspare part inventories.
This service level usually corresponds to the fraction of time that equip-ment can operate because of critical spare parts are at hand. Nevertheless, inthis indicator it has been included the maintenance policy from the criticalequipment system under contracting. In the next section, both maintenancecontracts conditions and spare components service level are linked as an in-tegrated approach.
3.3. Optimal integrate maintenance policy with spares service level
The following model provides a decision-making framework to optimallydecide whether the spare components pool should be managed by the clientor the agent. Taking this premise into account, the system availability ofinterest is that which integrates the maintenance preventive policy with thespares service level, so that
A(nT, S) = 1−∏
(1− AM(nT )) (1− AS(S)) (11)
where AM(nT ) is given by Eq. 7 and AS(S) by Eq. 10.Expected global cost of spares inventory CG(S) during the contract is
CG(S) = Cv(S) + Ch(S) + Cd(S) (12)
where
8
• Cv(S) = ncu
(S0 +
∑j Sj
)CRF is the discounted acquisition cost of
investment in spare parts, where cu is the new spare acquisition cost,
i is the discount factor, and CRF =(
i(1+i)n(T+Tp)
i(1+i)n(T+Tp)−1
)is the capital
recovery factor across the contract horizon n(T + Tp).
• Ch(S) = cu
(S0 +
∑j Sj
)ch0CRF is the holding cost for keeping in-
ventories at hand, where ch0 is the holding cost rate.
• Cd(S) = cd0(1 − A(nT, S))∑
j Nj is the downtime cost given by theproduction loss period, where cd0 is the downtime cost rate.
This model is capable of efficiently integrating critical spare parts stock-holding with outsourcing contracts design. The main options to handle thespare components pool within the maintenance service contract are presentednext.
3.3.1. Option 1: Client manages the pool of spare parts
Option 1 sets the contractual framework in which the client agrees tomanage the pool of spare components. In this scenario, although agreementrestraints, there are no major incentives for the agent to avoid an indiscrim-inate use of components. Following the lead of [3] and [4], profits for thesupply chain can be adapted as follows.
Let Πc(nT, S) be the expected profit for the client. As the client man-ages the pool, its profit is affected by the entire spares global cost; that is,acquisition cost, holding cost, and downtime cost. Hence, this profit is
Πc(nT, S) = rA(nT, S)− p− CG(S). (13)
Moreover, let Πa(nT, S) be the expected profit for the agent. Under thisscenario, the profit for the agent is only affected by the service fee and thepreventive maintenance cost. That is
Πa(nT, S) = p− CM(nT ). (14)
3.3.2. Option 2: Agent manages the pool of spare parts
Option 2 sets the contractual framework in which the agent agrees tohandle the pool of spare components. If so, a policy based on rational useof components turns suitable for the agent. Profits for the supply chain arethe following.
9
Although the client does not cover the entire spares global cost, its benefitis still impacted by the related downtime cost. The expected profit for theclient is therefore
Πc(nT, S) = rA(nT, S)− p− Cd(S). (15)
As the agent manages the pool, its benefit is affected by both acquisitioncost and holding cost. The expected profit for the agent is hereby
Πa(nT, S) = p− CM(nT )− (Cv(S) + Ch(S)). (16)
Ultimately, the total expected profit for the service chain Π(nT, S) validfor both Option 1 and Option 2 is
Π(nT, S) = rA(nT, S)− CM(nT )− CG(S). (17)
Using this framework, the chain coordination can be achieved by selectingthe optimal joint value [T, S] that maximizes Π(nT, S). This policy profitablyallocates the spare components pool, while both contracting parties obtaininghigher benefits than pursuing single objectives separately.
3.4. Coordination mechanisms for optimal joint values
Coordination mechanisms can be used to ensure a cooperative settingunder the above mentioned Option 1 and Option 2. Following the lead of [3]and [4], subsidization bonuses on both PM intervals and spares acquisitionand holding costs can be adapted to set parties joint values [T, S] with theone of the supply chain.
3.4.1. Cost subsidization under Option 1
When the PM interval of the agent is higher than optimal T of the sup-ply chain, the client agrees to subsidize the direct cost of PM to create anincentive for the agent. If let ∆Cp be the PM subsidization bonus, the newpreventive cost is
C ′p = Cp −∆Cp. (18)
The expected profit for the client adding the bonus effect is
Πc(nT, S) = rA(nT, S)− p− CG(S)− n∆Cpn(T + Tp)
= rA(nT, S)− p− CG(S)− ∆CpT + Tp
. (19)
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The expected profit for the agent adding the bonus effect is
Πa(nT, S) = p− CM(nT ) +n∆Cp
n(T + Tp)
= p− CM(nT ) +∆CpT + Tp
. (20)
With the optimal selection of ∆Cp, the agent is encouraged to adjust itsPM interval as needed for chain coordination.
3.4.2. Cost subsidization under Option 2
Since under Option 2 the agent manages the pool, another mechanism isneeded to cope with its extra acquisition and holding costs. Although similarto the above mentioned PM bonus, this model is rather based on subsidizingthe spares pooling cost. The scheme creates an incentive for setting the stocklevel with the one of the chain, while it keeps the benefits of adjusting the PMinterval. Let ∆cu be the inventory subsidization bonus, the new acquisitioncost is thus
c′u = cu −∆cu. (21)
The expected profit for the client adding the bonus effect is
Πc(nT, S) = rA(nT, S)− p− Cd(S)− ∆CpT + Tp
−∆cu
(S0 +
∑j
Sj
). (22)
The expected profit for the agent adding the bonus effect is
Πa(nT, S) = p− CM(nT )− (Cv(S) + Ch(S)) +∆CpT + Tp
+ ∆cu
(S0 +
∑j
Sj
).(23)
The cost subsidization models for Option 1 and Option 2 induce the agentto optimally perform both maintenance and stockholding services. Such pol-icy ensures maximum supply chain performance.
4. Case study
In the following case study, the critical components of interest are prin-cipal alternators of a fleet of haul trucks operating in a copper mining com-pany. This client contracts out the fleet maintenance service to a specialized
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Table 2: Parameters for the joint maintenance-stockholding model.
Parameter Value UnitPreventive maintenance h0 0.001 (1/Kh)strategy β 3
Tp 1 (Kh)Tr .3 (Kh)Cp 8 (KUS$)Cr .4 (KUS$)r 1500 (KUS$)p 350 (KUS$)α 0.95n 5 (overhauls)
Spare components N 20 (trucks)stockholding zi 1 (alternator/truck)
Tat 933 (h)cu 80 (KUS$)cd0 5.3 (KUS$/h/truck)i 0.1
agent attempting to ensure high equipment performance. The parametersfor the preventive maintenance strategy and spare components stockholdingare shown in Table 2.
Figure 1 shows the system availability resulting of merging both the avail-ability related to maintenance strategy and the spares stockholding servicelevel. Higher service level can be provided as the spares stock level S in-creases, but higher investment is required. Moreover, the optimal PM inter-val T changes over the associated spares stock range. Under the proposedframework, the system availability A(nT, S) is clearly the performance indi-cator of interest and thereby it is used to coordinate the chain profits duringthe contract.
Figures 2 and 3 reveal the differences in profits depending on the allo-cating position of the critical spare components pool. The results of abovementioned Option 1 and Option 2 are obtained by solving Equations 13to 17 as follows. When the client manages the pool, the joint values [T, S]are [18000, 0] for the agent and [11000, 3] for the client. The correspondingsingle profits are Πa(nT, S) =US$ 287, 888 and Πc(nT, S) =US$ 935, 142.Conversely, when the agent manages the pool, the joint values are [18000, 0]
12
010
2030
0
5
10
0.8
0.9
1
PM interval (h x103)Stock (spares)
Inte
grat
ed a
vaila
bilit
y
0.85
0.9
0.95
Figure 1: System availability by integrating T and S.
for the agent and [10000, 10] for the client. The respective single profitsare Πa(nT, S) =US$ 1, 149, 772 and Πc(nT, S) =US$ 287, 888. It is consid-ered that p is set to fulfill the profit constraint π. Before subsidization, thecorresponding profits for the supply chain by using optimal parties T ∗ inter-vals are Π(nT ∗a , S) =US$ 1, 169, 230 and Π(nT ∗c , S) =US$ 1, 211, 243 for Op-tion 1, and Π(nT ∗a , S) =US$ 1, 169, 230 and Π(nT ∗c , S) =US$ 1, 206, 436 forOption 2. However, the optimal supply chain joint value [T ∗, S∗] is [15000, 3],which leads to a higher profit Π(nT, S) =US$ 1, 219, 018. Therefore, the op-timal duration of the contract is n(T ∗ + Tp) = 5(15000 + 1000) = 80000(h)
From the previous results, it is clear that taking into account the entiresupply chain is the best possible scenario. As expected, the joint valuesrelated to the client are closer to the optimal channel profit. The agentmust be motivated to adjust its PM interval and stock as needed for chaincoordination. To achieve this result, the cooperative mechanisms describedin Section 3.4 are used. Under Option 1, the interval of the agent is certainlyhigher than desired, thus the client subsidizes the PM cost. In this case,∆Cp = 2.853 sets the agent PM interval with the one of the chain, namelyfrom T = 18000 to T = 15000. Under Option 2, it is clear that the agent
13
decides to keep the stock level as low as possible since the extra acquisitionand holding costs. Hence, the client decides to subsidize those significantinventory costs. In this case, ∆cu = 55.030 sets the the stock level with theone of the chain.
After subsidization, profits for the whole supply chain by using optimalsingle intervals align with the maximum value Π(nT, S) =US$ 1, 219, 018.Nonetheless, as expected, the single profits change across options. For exam-ple, the client profit decreases from US$935, 142 to US$915, 065 due to to thesubsidization mechanism, and the agent profit increases from US$287, 882 toUS$303, 953. For further details on changes for both subsidization options,Figures 4 and 5 denote a sensitivity analysis for those optimal joint valuesthat maximize the profit for the entire channel. As above demonstrated, thesupply chain benefit is higher than those singly obtained by the contractingparties. Furthermore, the proposed framework motivates both chain partiesto improve their maintenance and supply services continuously.
5. Conclusions
This article has introduced a model for defining the optimal managerof the pool of components within outsourcing services. A decision-makingframework was defined to integrate preventive maintenance with criticalspares stockholding for contract profitability. This interesting policy hasbeen evaluated to induce the parties involved to perform maintenance andsupply activities cooperatively, rather than a separated non-optimal way.This aim is achieved by setting a joint value consisting of preventive mainte-nance strategy and spare parts stock level that maximizes the total expectedprofit for both client and agent.
It has been found the joint values that reach the supply chain coordi-nation for the two options under study, when the client handles the sparecomponents and when the pool manager is the agent. However, there arescenarios where the expected profit is not sufficient to drive changes in thepolicy. To provide an incentive to set parties’ joint values with the one of thesupply chain, subsidizing bonuses for additional PM performed and sensitiz-ing spares acquisition and holding costs are suitable methods. The procedureto estimate such values has been developed.
Finally, we have demonstrated that the model is capable of coordinatelyoptimizing business performance for the entire supply chain. Both clientand agent are encouraged to continually improve their maintenance services,
14
56
78 9
10
10 10 10
1111 11
11
12
12
12 12[T*,S*]=[15,3]
5 10 15 20 25 300
5
10Supply chain − Profit
566.
57
7.5 7.5 7.5
8
8 8 8
8.5
8.5 8.5 8.5
8.59
9 9 9
9
9[T*,S*]=[11,3]
Sto
ck (
spar
es)
5 10 15 20 25 300
5
10Client − Profit
0.5 1 1.5 2
2.5
[T*,S*]=[18,0]
PM interval (h x103)
5 10 15 20 25 300
5
10Agent − Profit
Figure 2: Study of optimal T and S when the client manages the pool of spare components.
15
56
78 9
10
10 10 10
1111 11
11
12
12
12 12[T*,S*]=[15,3]
5 10 15 20 25 300
5
10Supply chain − Profit
8.59 999.5 9.5 9.510 10 1010.5 10.5 10.5
11 11 11
[T*,S*]=[10,10]
Sto
ck (
spar
es)
5 10 15 20 25 300
5
10Client − Profit
0.50.5 0.5
11 1
1.51.5 1.5
22 2
2.5 2.5 2.5[T*,S*]=[18,0]
PM interval (h x103)
5 10 15 20 25 300
5
10Agent − Profit
Figure 3: Study of optimal T and S when the agent manages the pool of spare components.
16
56
78 9
10
10 10 10
1111 11
11
12
12
12 12[T*,S*]=[15,3]
5 10 15 20 25 300
5
10Supply chain − Profit
45
6
7
7 7 7
8
8 8 8
9
9
99 9
[T*,S*]=[15,3]
Sto
ck (
spar
es)
5 10 15 20 25 300
5
10Client − Profit
1 1.21.4
1.61.8 2 2.
2 2.42.6 2.
8 2.83 3
[T*,S*]=[15,0]
PM interval (h x103)
5 10 15 20 25 300
5
10Agent − Profit
Figure 4: Study of optimal T and S when the client subsidizes the PM cost of the agent.
17
56
78 9
10
10 10 10
1111 11
11
12
12
12 12[T*,S*]=[15,3]
5 10 15 20 25 300
5
10Supply chain − Profit
66.5
7
77
7 7.5 7.5 7.5
8
8 8 8
8.5
8.5 8.5
8.5
99
99 9
[T*,S*]=[15,3]
Sto
ck (
spar
es)
5 10 15 20 25 300
5
10Client − Profit
0.5
11.
5 2
2.5 3
3
[T*,S*]=[15,3]
PM interval (h x103)
5 10 15 20 25 300
5
10Agent − Profit
Figure 5: Study of optimal T and S when the client subsidizes both acquisition and holdingcosts of the agent.
18
thus obtaining higher joint benefits compared to those when no coordinationoccurs.
Acknowledgements
The authors wish to acknowledge the partial financial support of thisstudy by the FOndo Nacional de DEsarrollo Cientıfico Y Tecnologico (FONDE-CYT) of the Chilean government (project 1130234).
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