Review on Energy Resilience

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International Journal of Production Economics

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Optimal delivery due date for a supplier with an unreliable machine under outsourced maintenance☆

Moosa Sharafalia, Hakan Tarakcib,∗, Shailesh Kulkarnib,1, Raja Abdul Razack Shahul Hameedc,2 a 348 Clementi Ave 5, #11-48, 120348 Singapore b Department of ITDS, College of Business University of North Texas 1155 Union Circle #311160, Denton, TX 76203-5017, USA c The Naveen Jindal School of Management, The University of Texas at Dallas 800 West Campbell Road Richardson, TX 75080-3021, USA

A R T I C L E I N F O

Keywords: Maintenance Reliability Availability Outsourcing Channel coordination Game theory Nash and stackelberg equilibria

A B S T R A C T

In this paper, we analyse the due-date-to-promise problem for a supplier who supplies parts to his customer. The parts are produced on an unreliable machine with constant production rate. The machine is leased from a contractor who is also entrusted with its maintenance. The supplier's decision is the optimal due date to promise to his customer, taking into account the holding and tardiness costs on the customer side, as well as the transfer payment to the contractor for the lease and maintenance of the equipment. The contractor in turn has to decide on the frequency of pre-ventive maintenance. Both maximize their own profits. For this finite horizon optimi- zation problem, we employ a Non-Renewal-Theory based approach from the literature to derive the performance characteristics in the transient regime - first for a general setting involving general distributions and then, for a special case. We then use Game Theory to analyse the underlying two-member game for both the non-co- operative and cooperative cases. A numerical example drawn from the literature is used to illustrate the special case. We show that for the special case, the game has a unique Nash equilibrium. Another significant result is that the Nash solution dominates the Stackelberg solution when the supplier is the leader. The supplier is worse off when the contractor is the leader. The example further shows that the supplier and the contractor stand to gain under cooperation which is well documented in the literature. We point to relevant literature for strategies to enforce this cooperation.

1. Introduction

In this paper, we consider a supplier of parts to a JIT manufacturer (referred to as customer in the paper).

The supplier has already signed a contract with the customer to deliver a fixed quantity (Q0) of a product at an exact point in time in the future to be determined. To manufacture this product, the supplier needs to lease a special equipment from an outside contractor. Our supplier has to make the key decision on how much time to allow for production, usually called the flow time allowance (Grout and Christy (1993); Grout (1997); Grout (1998)) or promised lead time (Liu et al. (2007)). We assume that the duration of lease of the machine would be equal to the flow time allowance. We also call it the promised due date. This decision is exacerbated by the fact that the leased machine is un- reliable and hence requires maintenance. The contractor herself is

assigned the responsibility to perform the maintenance activities which include preventive maintenance (PM) and minimal repair (MR). Pre- ventive maintenance is an overhaul of the whole system, making it as good as it was at the start of the lease. It is performed at pre-determined intervals to minimize the frequency of failures, and the cost and the downtime associated with them. If the process breaks down while op- erating, minimal repair of the machine starts immediately which takes a random amount of time. On completion of this minimal repair the system is restored to the working state. The contractor selects the schedule of the PM operations and performs the appropriate MR actions when necessary. Thus, this paper considers a contractor-supplier service chain where each of the players are concerned with maximizing their own profits. One of our most important contributions is that our work is, to our best knowledge, the first paper to consider optimal lead time decisions and optimal maintenance decisions, made by two different

https://doi.org/10.1016/j.ijpe.2018.11.019 Received 4 April 2018; Received in revised form 25 October 2018; Accepted 21 November 2018

☆ *work carried out by the author when the author was an Associate Professor of Operations Management (Edn.), Singapore Management University (SMU). The author is now retired and thanks SMU for the excellent support received.

∗ Corresponding author. E-mail address: [email protected] (H. Tarakci).

1 We dedicate this work to our great friend and colleague, Professor Shailesh S. Kulkarni, who passed away on July 6, 2018. 2 work carried out by the author while he was an intern at the LKC School of Business, Singapore Management University during the summer of 2016.

International Journal of Production Economics 208 (2019) 53–68

Available online 24 November 2018 0925-5273/ © 2018 Elsevier B.V. All rights reserved.

T

entities. However, we propose system-optimal solutions where the two parties would coordinate, as well.

To place our work in context, the earliest works related to our model are Grout and Christy (1993), Grout (1997), and Grout (1998). In Grout and Christy (1993), the supplier needs to find the optimal time to allow for production, taking into account the penalty fee if actual production time turns out to be longer than the allowed time and the inventory holding cost if the production time is shorter than the allowed time. It is the supplier who incurs the holding cost since the JIT-customer would not hold any inventory. Therefore, the supplier needs to find a balance between producing too early and too late. Grout (1997) then gen- eralizes the findings of the earlier paper and points out that 100% on- time delivery is non-optimal in a vast majority of cases. Grout (1998) extends the model in the previous two papers so that delivery can be made within a range – called “delivery window” – rather than at a single point in time. There have been other papers which analyse de- livery times and random production durations, such as So and Song (1998), Palaka et al. (1998), Hill et al. (2000), So (2000), Boyaci and Ray (2003), Ray and Jewkes (2004), Ho and Zheng (2004), Urban (2009), Li et al. (2014), and Chen and Moinzadeh (2018); however, as far as we are aware, only the three papers described above specifically study JIT purchasing.

Our paper mostly resembles Grout and Christy (1993) and Grout (1997) in its assumptions, with the biggest difference is that we assume the production rate to be constant when the machine is up and running while their works assume random production rate. The randomness in our paper comes from downtimes due to unpredictable failures and scheduled preventive maintenance activities. Another major difference is that we study a contractor-supplier service chain where significant maintenance decisions are made while theirs is a supplier-customer supply chain.

Our focus on leasing and maintenance outsourcing is motivated by developments in practice. These activities have become important strategies for managing supply chains, especially for very expensive industrial equipment (Pongpech and Murthy (2006)). Most research in this area focus primarily on the lessor; however, our paper analyzes decisions made both the lessor and the lessee. One of the studies most similar to our work in this area is Tarakci et al. (2009) but with two major differences: i) the length of the production horizon is a decision variable in our paper, whereas it is fixed in their work, and ii) their work does not consider inventory-related costs.

Our mathematical model of the system is same as in Tadashi et al. (2001). However, there are three important distinctions: i) their work only considers a single decision maker whereas we have two players, ii) in their work, the up and down states of the equipment are governed by the busy and idle periods of an M/M/1 queue, whereas in our paper, these two periods are generally distributed, and iii) their analysis is for the steady-state regime whereas due to the short-term nature of the leasing contract, our analysis is for the transient regime.

The last difference mentioned above, which is about our focus on transient analysis of the performance characteristics of the 2-member service chain, is significant. There have been studies considering tran- sient analysis in supply chain literature as well as in maintenance and reliability literature. However, due to the complicated nature of the basic functions derived for the model, we have not been able to apply the commonly used Renewal Theory approach. We also note that even studies which employ transient analysis mostly resorted to using simple assumptions in order to derive tractable results. However, we have employed a different approach already available in the literature (Kletter (1996)), which helped derive the results even in general set- tings in our paper. This ability to find and utilize transient performance characteristics in a maintenance outsourcing setting with a production deadline is another substantial contribution of our paper to the opera- tions management literature.

Finally, we note that the transfer price the supplier pays to the contractor is dependent on the steady state availability of the

equipment leased. Further, the costs of minimal repair (MR) and pre- ventive maintenance (PM) are also assumed to depend on the mean rates of completion of MR and PM. From the foregoing, it is clear that our model and results are more general and more realistic than what has been considered in the literature.

The organization of the paper is as follows. In the next section, a detailed literature review is provided. In Section 3, we describe the model together with the assumptions and notation. Important perfor- mance characteristics like cycle time distributions, the renewal function and the availability of the machine are derived under a general setting. Although there exists an underlying regenerative stochastic process, due to the complicated nature of the functions involved, we follow the approach proposed in Kletter (1996). We then consider a special case by replacing the general distributions with known distributions in Section 4. Again as the functions are very complex, a numerical example to showcase the characteristics of the performance measures of the system is provided. This analysis brings out the importance of the use of transient results for optimizing the system performance. The game- theoretic modeling of this contractor-supplier system is taken up in Section 5. We first derive the respective profit and best response functions of the contractor and the supplier. We then introduce first the non-cooperative game models and derive results for the cooperative game model. Due to the complicated nature of the functions involved, we use a numerical example drawn from the literature to illustrate these models in Section 6. For this numerical example, we show that the game has a unique Nash equilibrium. We first identify this unique Nash equilibrium in the simultaneous move game. Another significant result is that the Nash solution dominates the Stackelberg solution when the supplier tries to be the leader. The supplier is worse off when the contractor is the leader. Then, results for the cooperative game model are derived. The example further shows that the supplier and the con- tractor stand to gain under cooperation which is well known in the literature. Hence, we point to relevant literature for strategies to en- force/adopt this cooperation. In Section 7, we provide some concluding remarks together with possible extensions.

2. Literature review

Our work in this paper is broadly related to two main areas: (i) Equipment leasing and maintenance outsourcing, and (ii) Production- maintenance decisions in finite time. Accordingly, we review below the important works related to these two areas.

2.1. Equipment leasing and maintenance outsourcing

Equipment leasing and maintenance outsourcing is very common in many industries, including office equipment, medical (hospital) equipment, airplane engines and brakes, computer hardware and net- works, and manufacturing processes (Jaturonnatee et al. (2006)). The main reasons for leasing equipment are their high costs and the ob- solescence due to rapid technological changes. For example, Mabrouk et al. (2016) discuss an example of a company in Tunisia and North and West of Africa which leases equipment such as separators, gas scrub- bers, process heating and cooling systems that are needed for produc- tion operations. These leases usually vary from 6 to 48 months. Another example is the leasing of CAD machines by small manufacturers (see United States Congress Office of Technology Assessment (1990)). As these machines will be used only intermittently during production, leasing is the best option for these companies instead of owning them.

Maintenance itself is a well-established research area. An integrated maintenance approach helps companies improve reliability, availability and performance. Consequently, it results in significant savings in time, money and other resources. Studies like Hora (1987) and Eti et al. (2006) have estimated that maintenance costs can make up 20%–40% of overall production costs. Ahuja and Khamba (2008) argue that in- adequate maintenance practices in the past have led to reduced quality,

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reduced throughput, ‘lowered availability of machinery, increased in- ventory and overall poor delivery performance’. Thus, for companies, excellence in maintenance is a competitive advantage to becoming world-class (Brah and Chong (2004)).

According to Tieman (2002), annual medical equipment main- tenance outsourcing is worth about $26 billion, and in a survey by Jensen (2006), 84% of the surveyed companies reported hiring main- tenance contractors and 64% believed contractors to be of high value and that contract maintenance has become a growing industry. In a recent article, Reed (2018) states that the largest US-based airlines, such as Southwest, American, Delta and United, outsource a third to more than a half of their maintenance activities.

For more information, we refer the reader to a number of surveys and reviews written over the years like Sherif and Smith (1981), Wang (2002), Nakagawa (2005), Ahuja and Khamba (2008), Alaswad and Xiang (2017), and Keizer et al. (2017), with the last two reviews fo- cusing on the increasingly more prevalent condition-based maintenance policies.

We note that a vast majority of papers on maintenance assume that maintenance is performed by the user itself; however, as maintenance outsourcing has become more common in practice, there has been an increase in the number of academic papers written on this subject. The first paper we find is by Murthy and Asgharizadeh (1999), in which the manufacturer outsources maintenance activities to a contractor. Using the principles of the Stackelberg game, the authors focus on maximizing the contractor's profit. In an extension of this work, Ashgarizadeh and Murthy (2000) analyse a system with one contractor and multiple homogeneous manufacturers.

Plambeck and Zenios (2000) apply the classic Principal-Agent model to maintenance outsourcing. Taking into account the reservation profit and incentives of the contractor (the Agent), the authors max- imize the discounted profit of the manufacturer (the Principal) over a multi-period contracting horizon.

Tarakci et al. (2006b) study various contracts that would coordinate the “service chain”, consisting of a manufacturer and a maintenance contractor. Assuming an infinite contracting horizon, they show that incentive contracts work better than forced contracts to ensure long- term collaboration. They then extend this research to the case where the manufacturer hires multiple contractors to perform maintenance activities for a system with multiple production stages. The purpose of this paper is to design contracts that would maximize the total system profit (Tarakci et al. (2006a)).

Tarakci et al. (2009) analyzed a finite horizon single manufacturer – single contractor model where learning takes place. The contractor, through repetition, is able to perform the preventive maintenance ac- tivities both faster and cheaper. The authors design a contract that maximizes the manufacturer's profit, provided the contractor's re- servation profit is met.

Hamidi et al. (2016) proposed game-theoretic models for usage- based lease contracts for maintenance. Their analysis is comprehensive and promotes a cooperative contract between the manufacturer and the maintenance contractor. But, the time horizon for their model is the life cycle of the equipment. Further, they assume the transfer price to de- pend on the total inventory at the manufacturer.

Alexander et al. (2017) study a two-party system (manufacturer and maintenance contractor) with a linearly increasing failure rate. They show that a robust contract can be written which will coordinate the system for any expected cost of minimal corrective repair by developing a complete characterization of the intervals for this expected cost.

2.2. Production-maintenance in finite horizon

Our paper in effect is a production-maintenance problem in a finite horizon. Research on maintenance over a finite horizon began to appear only in the 1970s. For example, Christer (1978) and Ansell et al. (1984) proposed an asymptotic cost approach to such finite horizon

maintenance problems. More recently, Huang and Guo (2011) used semi-Markov processes to analyse such problems. Wu et al. (2010) discussed maintenance policies for a degrading finite-life-cycle system with Poisson failures. Optimal maintenance outsourcing contract over a finite horizon with imperfect maintenance is studied by Pascual et al. (2012). Another significant and important work is Pascual et al. (2016). The authors study the positive and negative effects of fixed-term maintenance contracts. As the interests of the parties involved in such relationships are not aligned, the authors propose incentive schemes for sustainability of such contracts. Consequently, the fixed-term taken up are somewhat longer than the model considered in this paper. Joint optimal decisions of maintenance and production policies over a finite horizon have also been considered in the literature. Using mixed integer programming, Najid et al. (2011) obtain the optimal maintenance and production policies when demand shortages are expensive. Zied et al. (2011) resort to first finding the optimal production policy which then is used to obtain the optimal maintenance policy. Random yields is considered by Xiang et al. (2013) in their production and maintenance model. Bajestani et al. (2014) analyse a Markovian-deteriorating multi- machine production system over multiple periods. Using an integer programming model, they find the joint optimal decision of main- tenance and production scheduling in each period. In a recent article, Ekin (2018) optimized the joint maintenance and production decisions in the presence of uncertain yield, using the augmented probability simulation method.

In this paper, similar to Tarakci et al. (2009), we also maximize the profit of the supplier, who out-sources maintenance activities to a contractor, in a finite horizon setting. However, the major difference in our paper is that the length of the production horizon is a decision variable, whereas it is fixed in Tarakci et al. (2009). In addition, we consider holding cost (in case production is finished earlier) and tar- diness cost (when the promised delivery lead time is not met) for the supplier similar to a newsvendor model. We also differ from Hamidi et al. (2016) in the time horizon considered and also in the assumptions with regard to the transfer price. We assume the transfer price to be based on the steady state availability of the leased equipment. Hamidi et al. assume this cost (or transfer price) to be proportional to the total inventory at the manufacturer. It is our opinion that assuming the transfer price to be a function of the total inventory even in the non- cooperative model points to the existence of a tacit co-operation.

In summary, both of these research topics are well-established and have strong analytical foundations, as well as potential practical ap- plications. The major difference between them, from a practical/man- agerial standpoint, is that most of the papers in the equipment leasing/ maintenance outsourcing area focus on the interaction between a manufacturer and a maintenance/equipment contractor, and main- tenance decisions are emphasized over production decisions. On the other hand, the studies involved with production-maintenance deci- sions in finite time, mostly eschews the two-party analysis, instead fo- cusing on only the manufacturer; however, the maintenance and pro- duction decisions are assigned equal importance. We believe our paper provides a significant contribution to the literature by finding a good balance between the focus of these two areas by considering the actions of both the manufacturer and the maintenance contractor, as well as placing sufficient significance on both the production and maintenance decisions.

3. The model and its operating characteristics

At the center of our model is a supplier who manufactures and supplies parts to a customer. The supplier needs a special equipment to produce these parts in order to meet a specific demand (say Q0 units) from the customer. The special machine has to be leased from an out- side contractor who is in the business of leasing these machines. But, the machine is unreliable and thereby needs maintenance. The con- tractor is also awarded the contract to perform the maintenance. Both

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our supplier and the contractor are aware that the machine's lifetime is of increasing failure rate (IFR) type. So, the maintenance involves minor repairs and preventive maintenance, durations of both of which are random and are governed by given probability distributions. Now, for the customer, the timely response of our supplier is crucial as for her system to function, on-time delivery from the supplier is the key. As the customer insists on receiving all the Q0 units demanded in one lot, she is willing to wait a little longer than the promised due date in the event the production is delayed due to the unreliable machine. But, the cus- tomer will apply a penalty for this tardiness.

We now list the assumptions of our paper.

3.1. Assumptions

1. We assume a game theoretic model consisting of a supplier-cum- manufacturer and a contractor.

2. The supplier requires a machine to support production. Due to reasons of cost and/or intermittent usage of the machine, the supplier leases the machine from the contractor.

3. By appropriate normalization, we assume that the production rate at the supplier is unity.

4. The machine that is taken on lease is assumed to be in good working condition but failure prone.

5. The contractor is also entrusted with the maintenance of the ma- chine.

6. The machine requires minimal repairs (MR) and preventive main- tenance (PM) the durations of both of which are random, governed by known probability distributions.

7. It has been agreed that preventive maintenance will be performed once the cumulative up-time of the machine reaches V from the time of conclusion of the last preventive maintenance.

8. A MR restores the state of the machine to what it was at the be- ginning of that repair. A PM restores the state of the machine to what it was at the start of the lease.

9. The costs of PM and MR are assumed to depend on the mean rates of completion of PM and MR.

10. The contractor's decision is the optimal V that maximizes her profit. 11. The supplier's decision is the delivery due date (ℓ) to his customer

with the objective of maximizing his profit function. 12. As per the contract, the supplier agrees to pay the contractor a

transfer price of PA (V) per unit time of the contract duration. In fact, PA (V) is assumed to be dependent on the steady state avail- ability of the machine which in turn makes it a function of V.

13. It is further assumed that the supplier cannot breach the promised order quantity. That is, the supplier stops production only when the Q0 units demanded are produced. In this case the promised due date may not be fulfilled for which the supplier may incur a penalty cost or tardiness cost (βt (V) = r (ps − cs − PA (V)) per unit per unit time) with his customer for not meeting the promised delivery date.

14. In the event of an extension of contract with the contractor, the contractor charges a penalty of δPA (V) resulting in a changed transfer price (βe (V) = (1 + δ)PA (V) per unit per unit time) during the extension.

15. There will be no penalty for early termination of the contract in the event the promised quantity Q0 is finished production well before the due date. But, the supplier will incur a holding cost (βh (V) = i (cs + PA (V)) per unit per unit time) as his JIT customer will not hold inventory.

16. In view of the short duration of the time horizon, time is measured in days. The notation used in this paper is provided in Table 1.

3.2. Density and distribution functions of a typical cycle

A complete production-repair-preventive maintenance cycle is as shown in Fig. 1. As per our assumption, PM restores the state of the machine to what it was at the start of the lease. So, the duration of a

cycle is the time between two consecutive ‘up-from-PM’ states. Note that this random variable is the sum of the following:

1. The cumulative uptime V = ∑ += +X X̄i N V

i N V1 ( )

( ) 1, which is a constant and where N (V) is the random number of failures during the cycle.

2. The cumulative minimal repair times ∑ = Yi N V

i1 ( ) ,

The duration of PM which is Z. If T11 denotes the duration between two consecutive E1 events, then

clearly

∑= + + =

T V Y Z i

N V

i11 1

( )

(1)

Now, we have the following propositions: Proposition 1. Given that the duration of up-time is U, then N (U ), the number of minimal repairs required over this duration is distributed as

Poisson with rate Λ(U ) =∫ λ u du( ) U

0 . That is,

= = = −N U k Poi U k e U k

Pr { ( ) } (Λ ( ), ) Λ ( )

! U

k Λ ( )

(2)

Proof. Please refer to Barlow and Hunter (1960), p.96.

Proposition 2. The distribution function of cycle time T11 is given by

∫∑= ≤ = − − =

∞ −

F x Pr T x Poi U k F x V u dF u( ) { } (Λ ( ), ) ( ) ( ) k

x V

PM MR k

11 11 0 0

( )

(3)

with the mean given by

= + +− −E T V μ V μ[ ] Λ ( )MR PM11 1 1

(4)

Proof. By conditioning on N (V ) in (1), we get (3) and (4).

It is easily seen that the Laplace transform of the density function of T11 is

=∗ − ∗ − − ∗

f s e f s e( ) ( )sV PM V f s

11 Λ ( ) (1 ( ) )MR (5)

We observe that the above distribution function is a Poisson mixture of convolutions of the distribution functions of minimal repair and preventive maintenance. It is well known that if

∫ − − −

F x V u dF u( ) ( ) x V

PM MR k

0

( ) are all decreasing failure rate (DFR) func-

tions then F11(x) is also DFR. But, unfortunately, we are not sure whether this will be the case always. From the system point of view, we would like to have as many cycles as possible in order to satisfy the demand as early as possible. Hence, we would want to have F11(x) to be IFR. So, we need to choose the parameters of the distributions carefully in order for F11(x) to be IFR.

If now, T n11with T11 1 = T11 denotes the time between the (n – 1)-th

and the n-th E1 events, then, clearly, = …T n{ , 0, 1, }n11 is a renewal process. On closer examination, we observer that it is an alternating renewal process comprising of the two states: “under PM” and “not under PM”. In fact, it is a terminating alternating renewal process be- cause the process will be stopped as soon as Q0, the number of units demanded is produced.

It is now possible to apply renewal theory to derive the performance characteristics of the system but given the complicated nature of the distribution function of a cycle, the analysis becomes very tedious. But, it is easy to see that the arguments used above to derive the distribution function of the cycle length can be used to derive the distribution of the time to produce a given number of units (say Q0) and also the dis- tribution of the number of units produced in a fixed interval of time (say (0, T)) which suffice for the analysis of the system. We refer the reader to Kletter (1996) for an excellent presentation of such a method. Our work is markedly different from Kletter (1996) in the consideration of outsourced minimal repairs and preventive maintenance and in the

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game theoretic analysis of the decision making efforts of the supplier and the contractor. We take up this study in the following sections.

3.3. Distribution of L(Q0), the time to produce Q0 units, given V

Given V, in order to produce Q0 units, we should have N0(V) = Q V

0

full cycles and one partial cycle in which the remaining Q1(V) = Q0 − N0(V)V units will be produced. Therefore,

∑ ∑ ∑ ∑= + + + = = = =

L Q Q MR PM MR( ) j

N V

i

N V

i j

N V

j i

N Q V

i0 0 1

( )

1

( )

1

( )

1

( ( ) )j0 0 1

(6)

For ease of notation, in what follows, we will simply write N0(V) and Q1(V) as N0 and Q1. It is easy to write the following conditional dis- tribution of L Q( )0 :

≤ = …

= ∗ − ⎜ ⎛ ⎝

∑ ⎞

⎠ ⎟+

=

Pr L Q N Q N V j N

F F Q

{ ( ) ℓ| ( ), ( ) , 1, 2, , }

(ℓ )

j

MR

N Q N V

PM N

0 1 0

( ) ( )

0 j

N j1

1

0

0 (7)

Therefore, after unconditioning and then taking the Laplace trans- form, we get

=∗ − ∗ − + − ∗

f s Q V e f s e( | , ) ( ( ) )L sQ

PM N Q N V f s

0 [Λ ( ) Λ ( ) ] [1 ( )MR0 0 1 0 (8)

We note that in the above scheme, an uptime of Q0 units is guar- anteed due to our assumption that the production rate is unity.

3.4. Distribution of Q(T), the number of units produced given ℓ = T and V

Let GQ (q|T, V) be the distribution of Q (T), the random number of units produced during a fixed time of T. Then, for q < T,

= ≤ = ≥

⎧ ⎨⎩

− < ≥

G q T V q T q V q T

F T q V for q T for q T

( , ) Pr { ( ) } Pr {Time to produce units }

1 ( | , ) 1

Q

L

(9)

Assuming that the corresponding pdf exists for this distribution function, we can find it as

∫ =

⎧

⎨ ⎪

⎩ ⎪

− < <

≥

∂ ∂g q T V

f q V dL for q T

for q T ( | , )

(ℓ| , ) 0

0 Q q

T

q L

(10)

We highlight that Gq (q|T, V) is also the distribution of up time in (0, T) given V.

Table 1 Table of notation (in alphabetical order).

A (V) Steady state availability of the machine. A (t, V) Availability of the machine over the interval (0, t). α The shape parameter of the Weibull distribution function. βe (V) (1 + δ)PA (V), Transfer price charged by the contractor during extension of contract ($ per unit time). βh (V) i (cs + PA (V)), Holding cost incurred by the supplier ($ per unit per unit time). βt (V) r (ps − cs − PA (V)), Tardiness penalty charged by the customer ($ per unit per unit time). cs Supplier's production cost per unit ($ per unit) Cmr (μmr) Cost for minimal repair ($ per repair), a function of μmr. Cpm(μpm) Cost for preventive maintenance ($ per preventive maintenance), a function of μpm.

E1 The event that the machine goes into ‘up’ state after a preventive maintenance. f ∗(s) Laplace transform of f (t). fMR(x), FMR(x) pdf and cdf of Xi, fPM(x), FPM(x) pdf and cdf of Yi, λ The scale parameter of the Weibull distribution function. λ(x) The failure rate function of the machine and also the scale parameter. Λ(x)

∫ λ u du( ) x

0 L (Q0) Time to complete producing Q0 units, a random variable. ℓ: Due date, to deliver Q0 units, a decision variable for the supplier. MR Duration of a minimal repair. μMR Inverse of the mean time for minimal repair. μPM Inverse of the mean time for preventive maintenance.

N0(V) ,Q V

0 Number of cycles required to produce Q0 units, a function of V.

N (x) Number of minimal repairs over an up-time duration of x. ps Supplier's revenue per unit ($ per unit)

PA (V) Transfer price charged by the contractor, a function of steady state availability of the machine and thus, V. P oi (a, k) −e a a

k

k ! Q0 Quantity demanded by the JIT-customer. Q1(V), Q0 − N0(V)V The uptime during the last partial cycle, a function of V. Q (T) Number of units produced during a fixed time T, a random variable. T11 Duration of cycle time, the time between two consecutive E1 events. V Cumulative up-time since last PM that triggers the next PM, a decision variable for the contractor. Xi Duration of the i−th up time in a cycle, a random variable. X¯ Duration of the last up-time just prior to preventive maintenance. Yi Duration of the i−th minimal repair in a cycle, a random variable. Z Duration of a preventive maintenance, a random variable.

Fig. 1. Events and durations in one cycle with total uptime V = ∑ += +X X̄i

N V i N V1

( ) ( ) 1, Yi are the minimal

repair times, and Z is the duration of the preventive maintenance.

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3.5. Transient and steady state availability of the machine

From our supplier's point of view, the availability of the machine is an important characteristic. Availability of a system is defined as the probability that the system is available for its intended purpose whenever it is needed. The availability AV (t), over the interval (0, t) of the machine with minimal repairs and preventive maintenance as de- fined in this paper is given as

∫ =A t

I u du t

( ) ( )

V o

t

(11)

where

= ⎧ ⎨⎩

I t if the machine is up

otherwise ( )

1, 0,

We note that AV (t) is a function of V and t. Hence, the expected availability in (0, t) is

∫ =E A t

Pr the machine is up at u du t

[ ( ) ] { }

V

t 0

(12)

It is now well known that the steady state availability of the system is,

= = + +→∞ − −

A V A t V

V μ V μ ( ) lim ( )

Λ ( )t V

PM MR 1 1

(13)

Having derived above the necessary functions for the model, we illustrate the model with a numerical example in the next section. Let us call it the base case, which will bring out the behaviour of the perfor- mance characteristics of the model.

4. A special case

We now consider a special case in order to understand the beha- viour of the performance measures of our supplier-contractor system. Let us assume that the lifetime of the machine is distributed as a Weibull distribution with both repair time and duration of preventive maintenance following exponential distributions as given below:

= − −G x eLifetime: ( ) 1 λx( ) a

(14)

= − −x eRepair time: F ( ) 1 μ xMR MR (15)

= − −x ePM: F ( ) 1 μ xPM PM (16)

We highlight that the Weibull distribution is IFR when α > 1. Its hazard rate, h(x) = αλ(λx)(α−1) so ∧ =V λV( ) ( ) .α (17)

We also use the parameter values for this example as given in Table 2.

4.1. Density, distribution and related functions of cycle time

In this special case, the distribution of the cycle time T11 is given by the following proposition.

Proposition 3. For the special case, the distribution of the cycle time is, for x ≥ V,

∫= − +

−

− − − −

− − − −

[ ] [

]

F x e e

e e u λV

u I μ λV u

( ) 1 1

( ) (2 ( ) )

λV μ x V x V

μ x V u u u MR α

MR α

11 ( ) ( )

0

( ) 1

α PM

PM MR (18)

where I1(x) is the modified Bessel function of order 1 (see Abramowitz and Stegun (1965)).

Proof. It is clear that the total duration of minimal repair times is ac- tually a randomized Gamma distribution (see Feller (1971), p.58). Hence, by some algebraic manipulation as in (Feller (1971)), we get the above result. Please refer to Appendix A for a concise proof. It is interesting to note that the distribution function of the total duration of minimal repairs is a Bessel distribution.

From Proposition 3, we can now derive the density function for cycle time as

∫

⎜

⎟

=

⎧

⎨

⎪ ⎪ ⎪

⎩

⎪ ⎪ ⎪

<

⎛ ⎝

+ ⎞ ⎠

≥

− − −

− − − − −

f x

for x V

μ e e

e I μ λV u du for x

V

( )

0

(2 ( ) )

PM λV μ x V

x V μ x V u u u u λV

u MR α

11

( ) ( )

0

( ) ( ) 1

α PM

PM MR MR α

(19)

We now use the data in Table 2 to plot the functions derived above. For this purpose and also for the numerical analysis of our model in Section 6, we have used the open source programming language R (R Core Team (2017)), which has a good number of useful packages, such as Pracma (Borchers (2018)), Bolstad (Curran and Bolstad (2018)), and pspline (Ripley and Ramsey (2017)), for our purpose. Using these packages, it was convenient with R to numerically invert our complicated Laplace transforms of density, distribution, and renewal functions. They also proved useful for integration of functions and for spline interpolation.

The plots of the density and distribution functions of T11 together with the hazard rate and renewal functions are shown in Fig. 2, for various values of λ. We observe that as λ increases the support for the density also increases. The density functions all appear to be symmetric. The higher the λ the smaller is the mode.

Generally, the hazard rate function shows an increasing trend indicating increasing failure rate. But, for smaller values of λ and for larger t, the hazard rate function behaves oddly revealing a wavy pattern. The renewal function also has a wavy pattern. This may be due to the fact that the density and distribution functions are defined only for t ≥ V. Further, we observe that it takes longer for the renewal

4.2. Availability

Graphs for the transient and the steady state availability of the machine are presented in Fig. 3. The key observations are that avail- ability increases as λ decreases. Importantly, we observe that it takes quite some time for availability to reach the asymptotic availability. Once again, this indicates that transient results are important for the analysis of our model. In the next section, where we introduce costs, we will be considering the transfer price to be paid by the supplier to the contractor for leasing the machine. Naturally, the transfer price should be higher for a machine with higher availability. Hence, in the next section, we would assume the transfer price to be a function of the steady state availability of the machine.

4.3. Time to produce Q0 units and the number of units produced in a given time

For this base case example, the distribution and density functions

Table 2 Parameter values for the example.

Contractor Supplier

Λ Α μMR μ PM V Q0 T

Tmax

1 4.5 3 3 1 10 100 units 100 units (time) 200 units (time)

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58

for the time to produce Q0 units are given by the following proposition.

Proposition 4. The density function of the time to produce Q0 units is

∫=

⎧

⎨

⎪ ⎪ ⎪ ⎪

⎩

⎪ ⎪ ⎪ ⎪

≤

⎛

⎝

⎜ ⎜ ⎜ ⎜ ⎜

+ ⎞

⎠

⎟ ⎟ ⎟ ⎟ ⎟

>

−

− − − −

− − − − −

− − −

−

−

f t Q V

for t Q

μ e

e

e I μ Au du

for t Q

( | , )

0

(2 )

L

PM A

μ t Q μ t Q N

t Q μ t Q u u u u A

u MR

μ t Q u N

0

0

( ) ( ( ) ) ( 1) !

0

( ) 1

( ( ) ) ( 1) !

0

PM PM N

PM MR MR

PM N

0 0 0 1

0

0 0

0 0 1

0

(20)

where

= ∧ + ∧A Q N V( ) ( ).1 0 (21)

The distribution function is:

Fig. 2. Density, Distribution, Hazard and Renewal functions of T11 for various values of λ with V = 10, α = 3, μMR = 3, and μPM = 1.

M. Sharafali et al. International Journal of Production Economics 208 (2019) 53–68

59

∫

⎜

=

⎧

⎨

⎪ ⎪ ⎪ ⎪ ⎪

⎩

⎪ ⎪ ⎪ ⎪ ⎪

≤

⎛

⎝

⎜ ⎜ ⎜ ⎜ ⎜ ⎜

⎛

⎝ ⎜ − ∑

⎛ ⎝

⎞

⎠ ⎟+

⎡ ⎣

− ∑ ⎤ ⎦

⎞

⎠

⎟ ⎟ ⎟ ⎟ ⎟ ⎟

>

−

= − − −

−

− −

= − − − − −

−

−

−

F t Q V for t Q

μ e

e

e I μ Au

e

du for t

Q

( | , ) 0

1

(2 ) 1

L

PM A

i N μ t Q μ t Q

i

t Q u u u A

u MR

i N μ t Q u μ t Q u

i

0

0

1 ( ) ( ( ) )

( 1) !

0 1

1 ( ) ( ( ) )

( 1) !

0

PM PM i

MR MR

PM PM i

0 0 0 1

0

0 0 0 1

(22)

Proof. The proof follows along the same lines as for Proposition 3. For the chosen parameters, using numerical procedures, we have

obtained these functions. Their graphs are shown in Fig. 4. We note that the density function is almost symmetric and also clearly reveals that we need more than 100 units of time to produce 100 units of the pro- duct.

Now, using (9), the distribution function of the number of units produced in a given time T is given by

Proposition 5. For q < T,

∫

⎜

= ≤

= −

⎛

⎝

⎜ ⎜ ⎜ ⎜ ⎜ ⎜

⎛

⎝ ⎜ − ∑

⎛ ⎝

⎞

⎠ ⎟+

⎡ ⎣

− ∑ ⎤ ⎦

⎞

⎠

⎟ ⎟ ⎟ ⎟ ⎟ ⎟

−

= − − −

−

− −

= − − − − −

−

−

−

G q T V Q T q V

μ e

e

e I μ Au

e

du

( , ) Pr { ( ) }

1

1

(2 ) 1

Q

PM A

i N μ T q μ T q

i

T q u u u A

u MR

i N μ T q u μ T q u

i

1 ( ) ( ( ) )

( 1) !

0 1

1 ( ) ( ( ) )

( 1) !

PM PM i

MR MR

PM PM i

0 1

0 1

(23)

We note that N0 is a floor function involving q. Due to the compli- cated nature of functions involved in GQ (q|T, V), it is almost impossible to derive the density function gQ (q|T, V) by direct differentiation.

But, it is possible to find the density function numerically. For V = 10 and for T = 100, the distribution of the number of units pro- duced by T is plotted in Fig. 5. It is interesting to note that the number of units produced in 100 units of time is very much lower than 100. The dispersion also appears to be low.

5. Game theoretic models for optimal decisions of the contractor and the supplier

We now take up the issue of the decisions of the supplier and the contractor. We propose to use game-theoretic framework to model and analyse this two-member supply chain. We discuss both the non-

Fig. 3. Transient and steady state availability for the base case example functions to attain steady state. Hence, it becomes necessary to use transient results rather than steady state results in our analysis.

M. Sharafali et al. International Journal of Production Economics 208 (2019) 53–68

60

cooperative and co-operative game models. In our case, as highlighted earlier, the functions involved are very complicated due to the presence of the floor function in N0. Hence, we will illustrate these game models numerically in the next section using an example drawn from the lit- erature. Some sensitivity analysis will also be provided.

We recall that the contractor's decision variable is V. Her strategy xC = V belongs to the strategy set

= < ≤ ≤X V V V V{ ; 1 }.C min max (24)

Similarly, the supplier's decision variable is ℓ, the promised delivery due date to quote to his customer for delivering the Q0 units demanded. Note that ℓ is also the period of lease to be contracted with the main- tenance contractor. The supplier's strategy xS = ℓ belongs to the strategy set

= ≤ ≤X Q L{ℓ; ℓ }.S 0 max (25)

We first derive the payoff functions of both the contractor and the supplier. Then, to apply game theory, we derive the best response functions of the contractor and the supplier. We highlight again that the supplier is bound to deliver the promised Q0 units to the customer, the JIT manufacturer. This may result in not meeting the due date and hence an extension of the lease contract with the contractor. Consequently, the supplier incurs a tardiness penalty with his customer βt (V) = r (ps − cs − PA (V)) per unit per unit time. In addition, the supplier will face a different transfer price from the contractor, given by βe (V) = (1 + δ)PA (V) per unit of extended time, where δ may be positive (implying a penalty) or negative (implying an incentive). As highlighted earlier, if the production of the Q0 units demanded is

Fig. 4. Density and distribution functions of L (Q0) with (Q0 = 100; V = 10; α = 3; λ = 1/4.5; μMR = 3; μPM = 1).

Fig. 5. Distribution and density functions of Q (T) given V = 10 and T = 100.

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61

finished earlier than the due date, the supplier will hold these units in his inventory until the due date incurring a holding cost of βh (V) = i (cs + PA (V)) per unit per unit time. Note that all βt (V), βe (V) and βh (V) are functions of V.

5.1. Contractor's payoff function

We first take up the contractor's profit. The contractor's objective is to maximize her profit out of this contract. She will maximize her profit over ℓ, given ℓ and Q0.

Lemma 1. Given ℓ and Q0, the contractor's payoff function over (0, ℓ) is

∫= + − −

− +

∞

π V P V V t dF t Q V C μ N

C μ N V Q

( |ℓ) ( ) ℓ β ( ) ( ℓ) ( | , ) ( )

( ) [ Λ ( ) Λ ( ) ]

c A e L PM PM

MR MR

ℓ 0 0

0 1 (26)

Proof. The first term above is the revenue from the transfer pricing. The next term is the expected revenue she will earn in case of an ex- tension of the period of lease. Now, the contractor knows fully well that irrespective of the period of lease ℓ, the supplier's production run will be until the Q0 units are produced. Hence, with probability 1 there will be N0 preventive maintenances and the expected number of minimal re- pairs will be N0Λ(V) + Λ(Q1). This leads to the last two terms, which represent the costs incurred by the contractor for preventive main- tenance and minimal repairs, respectively.

5.2. Supplier's payoff function

The supplier's objective is to maximize his own profit. We recall that the supplier will have to supply the Q0 units demanded exactly even if the production run lasts longer than the period of lease with the con- tractor. Considering these possibilities, his payoff function is given by the following lemma.

Lemma 2. Given V, the supplier's payoff function is

= ⎡ ⎣⎢

− − − ⎤ ⎦⎥

π V p c P V

Q C V Q Q(ℓ| )

( ) ℓ (ℓ, | ) ,M s s

A

0 0 0

where

∫

∫

= −

+ ⎡ ⎣⎢

+ ⎤ ⎦⎥

− ∞

C V Q β V t dF t Q V

β V β V

Q t dF t Q V

(ℓ, | ) ( ) (ℓ ) ( | , )

( ) ( )

( ℓ) ( | , )

h L

t e

L

0 0

ℓ

0

0 ℓ 0

(27)

represents the cost that the supplier incurs for finishing the production run either early or late.

Proof. Proof is omitted as it is obvious.

We highlight that the supplier's payoff function clearly resembles the Newsvendor payoff. We now take up the issues of non-cooperation and cooperation in the following sections.

5.3. Contractor and supplier are non-cooperative

We now assume that the contractor and the supplier are non-co- operative, i.e., the contractor and the supplier act in their own best interests. The non-cooperative games for Nash and Stackelberg equili- bria require the best response functions of the players in the game. In deriving these functions, as is common in the literature (see Hamidi et al. (2016)), we assume N0 to be a continuous function of V.

5.3.1. Best response functions The contractor's best response function is obtained as follows:

Suppose, the supplier chooses ℓ = ℓ̂ from his strategy set XS and

announces it to the contractor. Now, the contractor would find her best response to this announcement from the supplier. The best response of the contractor is the solution of her optimization problem

= ∈

V arg π V(ℓ̂) max ( |ℓ̂)R V X

c C (28)

Similarly, the best response of the supplier is the solution of his optimization problem

= ∈

V arg π Vℓ ( ˆ ) max (ℓ| ˆ )R X

s ℓ S (29)

The propositions that follow present the best response functions of the contractor and the supplier.

Proposition 6. For a given ℓ, the contractor’s best response V R as a function of ℓ is given by the solution for V of the equation

∫+ − ∂ ∂

− − + = ≥

∞

−

dP V dV

β V t f t Q V

V dt

C μ α Q λ V C μ Q V

Q

ℓ ( )

( ) ( ℓ) ( | , )

( ) ( 1) ( ) 0, ℓ

A e

L

MR MR α α

PM PM

ℓ

0

0 2 0

2 0 (30)

Proof. Equating the first derivative of Πc (V| ℓ) in 26 with respect to V, we prove the proposition.

The supplier's best response is given by the Proposition 7.

Proposition 7. For given V, the supplier’s best response ℓ R as a function of V is uniquely given by

= ⎛

⎝ ⎜ ⎜

+ −

+ +

⎞

⎠ ⎟ ⎟

−V F β V

β V β V ℓ ( )

( )

( ) ( ) R

L t

β V Q

P V Q

h t β V

Q

1

( ) ( )

( )

e A

e

0 0

0 (31)

where F −1 is the inverse of the distribution of L (Q0). Proof. As usual, differentiating 27 with respect to ℓ, equating it to zero

and solving we get 31.

5.3.2. Nash equilibrium Nash equilibrium occurs when the supplier and the contractor an-

nounce their decisions simultaneously. In this case, no player can uni- laterally deviate from this strategy without being worse off while the other player sticks to his/her Nash strategy. If the Nash equilibrium exists, it is obtained by solving the equations (33) and (31).

5.3.3. Stackelberg equilibrium In Stackelberg equilibrium, one of the players moves first (as a

leader) with their best decision and the other follows that up with their own best decision, given the leader's position. As a rational player, the leader expects that the follower responds with the follower's own best response. So, the leader substitutes the best response of the follower in their objective function to determine their optimal decision.

5.3.3.1. Supplier as leader. Here, supplier as the leader announces his decision ℓst−s first, and the contractor as the follower takes it and then optimizes her problem. Thus, the optimization problem for the supplier is:

Proposition 8. The optimization problem for the supplier as the leader is

⎡ ⎣⎢

− − − ⎤ ⎦⎥

p c P V

Q C V Q Qmax

( ) ℓ (ℓ, | )

V s s

A

ℓ, 0 0 0

(32)

subject to

∫+ − ∂ ∂

− − + = ≥

∞

−

dP V dV

β V t f t Q V

V dt

C μ α Q λ V C μ Q V

Q

ℓ ( )

( ) ( ℓ) ( | , )

( ) ( 1) ( ) 0, ℓ

A e

L

MR MR α α

PM PM

ℓ

0

0 2 0

2 0 (33)

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62

5.3.3.2. Contractor as leader. Here, contractor as the leader announces her decision V st−c first. Then, the supplier as the follower takes it and then optimizes his own problem. Thus, the optimization problem for the leader is:

Proposition 9. The optimization problem for the contractor as the leader is:

∫+ − −

− +

∞

P V V t dF t Q V C μ N

C μ N V Q

max ( ) ℓ β ( ) ( ℓ) ( | , ) ( )

( ) [ Λ ( ) Λ ( ) ]

V A e L PM PM

MR MR

, ℓ ℓ

0 0

0 1 (34)

subject to

= ⎛

⎝ ⎜ ⎜

+ −

+ +

⎞

⎠ ⎟ ⎟

−V F β V

β V β V ℓ ( )

( )

( ) ( ) L

t β V

Q P V

Q

h t β V

Q

1

( ) ( )

( )

e A

e

0 0

0

We now embark on deriving the problem for cooperative solution.

5.4. Contractor and supplier are cooperative

In this case, the contractor and the supplier cooperate to find the optimal solution jointly for the system. The strategy set for this problem is

= ≤ ≤ ≤ ≤V V V Q LX { (V,ℓ); , ℓ }.min max 0 max (35)

The objective function here would be the sum of their individual payoff functions. The optimization problem is

= − − −

− +

π V p c C V Q Q C μ N

C μ N V Q

max (ℓ, ) ( (ℓ, | ) ) ( )

( ) [ Λ ( ) Λ ( ) ] V

J s s J PM PM

MR MR

ℓ, 0 0 0

0 1 (36)

where

∫

∫

= −

+ ⎡ ⎣⎢

+ ⎤ ⎦⎥

− ∞

C V Q β V t dF t Q V

β V β V

Q t dF t Q V

(ℓ, | ) ( ) (ℓ ) ( | , )

( ) ( )

( ℓ) ( | , )

J h L

t e

L

0 0

ℓ

0

0 ℓ 0

(37)

Note that in this case, the transfer price will be jointly determined to the satisfaction of both the supplier and the contractor. We will illus- trate these game-theoretic models in the next section.

6. Numerical example

To illustrate the game-theoretic models proposed in and to under- stand the behaviour of the performance characteristics, we use a nu- merical example from the literature. In particular, we use the data used for the case study in Pascual et al. (2016). In fact, that example is a scaled down version of the example from Tarakci et al. (2006b). We have additional considerations such that our model differs from theirs in the assumption of random durations for both minor repairs and preventive maintenance. We use the corresponding deterministic parameters in Pascual et al. (2016) as the mean for the exponential distributions we use to represent these random durations. In addition, the costs for minimal repair and preventive maintenance are assumed to be functions of these parameters. Besides this, the transfer price in our model is a function of the steady state availability. We also impose cost penalties when the supplier is tardy in meeting the promised de- livery date. Table 3 lists the values of all the parameters used in the numerical example.

6.1. Nash equilibrium

For the parameter values chosen, Fig. 6 presents the two players' best response functions. For this example, very clearly there exists a unique Nash equilibrium which is at the point of intersection of the two best response functions. Solving the two equations, we obtain the Nash equilibrium to be (V N, ℓN) = (11.2, 123.6181) days with the Nash payoffs (π π,c

N s N ) = ($252.6311, $1137.561) thousands.

6.2. Stackelberg equilibrium

6.2.1. Supplier As leader For our numerical example and from Fig. 6, we note that the con-

tractor's best payoff function is linear. We note that for some variations of the parameter values, it is actually piecewise linear. This indicates that (by the first condition of Theorem 1 of Serin (2007)) the Nash equilibrium itself is the Stackelberg equilibrium. It is because, whatever the strategy the supplier (the leader) chooses, the supplier knows fully well that the contractor will respond with her strategy of V = 11.2. Thus, the Stackelberg solution for the supplier as leader coincides with the Nash solution.

6.2.2. Contractor As leader In this case, for the example considered, we obtain the Stackelberg

equilibrium as (V st−c, Lst−c) = (11.6, 124.1206) days with the payoffs as ($252.6742, $1137.068) thousands. We note that the contractor as leader is not worse-off but the supplier is worse off. The system-wide profit is also lower than that in the Nash equilibrium. Also, both V and ℓ are higher than that in the Nash equilibrium forcing the supplier to sign for a longer lease of the equipment. Hence, it will be difficult to im- plement this solution.

Table 3 Model parameters (monetary units are in $1000) We now present the numerical analysis of our model.

λ Α μMR μPM 1 10 3 3 1

i Q0 r Tmax

0.20 100 units 1.20 200 days 365

CPM CMR PA (V) cs $(3 + 5 μPM) $(0.10 + 0.9

μ MR) $(1.25 + 1.8A (V)) $10

βe βh βt ps $1.0PA (V) i (cs + PA (V)) r (ps − cs − PA (V)) $25

Fig. 6. Best response functions and unique Nash equilibrium for the data in Table 3.

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63

6.3. Cooperative game

For the numerical example, we have graphed the joint objective function in Fig. 7. Clearly, the function appears to be concave.

The optimal solution is given by (V J, ℓJ) = (11.2, 125.6281) with the joint profit πJ = $1392.408 thousands. The optimal V is same as in Nash equilibrium but the optimal ℓ is two days longer than that of Nash equilibrium. Consequently, the system-wide profit is also higher.

Table 4 compares all the solutions discussed so far. As pointed out above, Stackelberg solution is not attractive to the supplier. So, the selection reduces to choosing between the Nash and Joint solutions. Very clearly, if the two cooperate, then the system stands to gain. While this is so, we also note that for the joint optimal solution, the contract period has to be two days longer.

Hence, the implementation of the joint solution needs cooperation and agreement on the way the extra profit can be shared. The me- chanism required to enforce coordination and cooperation is a well- researched topic. One can use the cost-subsidization schemes employed in Tarakci et al. (2006b), and Pascual et al. (2016) according to which the supplier's payment to the contractor has two components: a fixed payment and a cost subsidization (CS) scheme for every PM and MR operation that the contractor performs. For other schemes, we refer the readers to the recent work of Hamidi et al. (2016) and the references therein.

6.4. Some sensitivity analysis

In this section, we study the impact of changes in parameter values on the optimal solutions, as well as the profit values. Since the machine plays a central role in our system, we have chosen to focus on the parameters related to the machine only. The other reason for choosing

these parameters only is that some of the remaining parameters, such as the transfer price, the minimal repair cost, and the preventive main- tenance cost are functions of these chosen parameters. In particular, the parameters chosen for these numerical studies are: i) λ, the scale parameter of the Weibull distribution used in our model, ii) α, the shape parameter of the Weibull distribution, iii) μPM, the inverse of the mean time of preventive maintenance (also can be referred to as the mean rate of preventive maintenance completions); and iv) μMR, the inverse of the mean time of minimal repair (also can be defined as the mean rate of minimal repairs). We vary the values of these parameters, one at a time, around the base values given in Table 3. This allows us to see how changing the value of one parameter affects the outcome. We sum- marize our findings below, where we present the results of the Nash equilibrium model only for the sake of brevity. This is also because, the other models (Supplier as leader, Contractor as leader, and Cooperative games) have very similar trends and outcomes.

6.5. The effect of the weibull scale parameter, λ

Here, we varied λ from 0.01 to 0.2 and the results are presented in Fig. 8. Some trends are obvious; for example, as λ increases, ℓ, the supplier's decision on the production lead time (which is also the period of lease of the machine) increases, as well. This makes intuitive sense since a larger λ implies a higher failure rate, which means more ex- pected failures and more expected downtime due to minimal repairs. Therefore, to make up for this loss of production time, the supplier needs to increase the lead time. On the other hand, V, the contractors decision of how long should the system be up before performing a preventive maintenance, decreases as λ increases. This also should be expected because with a higher λ, the contractor needs to perform more minimal repairs, on average. However, this can become quite costly. So, the contractor would prefer to perform preventive maintenance sooner to reduce the frequency and cost of minimal repairs; hence, the lower V. One final point we would like to make is that both profit values (con- tractor and supplier) are inversely related to the value of λ. Once again, this makes sense because higher λ values indicate a more failure-prone production system and so the parties cannot make as much profit as they could with a more reliable, or better, machine.

6.6. The effect of the weibull shape parameter, α

Here, α varies from 2 to 4. The results, which can be seen in Fig. 9 are quite similar to those of λ, albeit at a more muted level. As α in- creases, implying a higher failure rate in general, the supplier increases his production lead time, ℓ, to account for the expected increase in minimal repair times. The contractor, on the other hand, reduces the preventive maintenance trigger of V in order to execute preventive maintenance more often and thus, negate the effects of potentially more costly and time-consuming repairs due to the increasing failure rate. We also note that similar to the case of λ, both profit values are inversely related to α since a higher failure rate reduces the profit margins of both parties.

6.7. The effect of the mean rate of PM completions, μPM

The results are presented in Fig. 10. For this study, we varied μPM from 0.8 to 2. Please note that since μPM represents the rate of pre- ventive maintenance (PM) execution, an increase in μPM actually im- plies a lower PM duration. We can observe that as μPM increases, the supplier reduces the production lead time, ℓ, presumably because the downtime lost to PM is now less. This decision increases the supplier's profit slightly, probably because the supplier needs to pay the con- tractor for a shorter period. The contractor, on the other hand, now quotes a higher V as μPM increases. This makes sense because CPM also increases with μPM and the contractor, now facing more expensive preventive maintenance activities, would prefer to perform fewer of

Fig. 7. Joint profit function for the data in Table 3.

Table 4 Optimal solutions - Nash, Stackelberg and Joint.

ℓ days V days Expected profit (in thousands)

Contractor Supplier System/Joint

Nash 123.6181 11.2 $252.6311 $1137.561 $1390.1921 Supplier as leader 123.6181 11.2 $252.6311 $1137.561 $1390.1921 Contractor as

leader 124.126 11.6 $252.6742 $1137.068 $1389.7422

Cooperative/Joint 125.6281 11.2 $1392.408

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64

them; hence, the higher V. We also note that the contractor's profit goes down because the production lead time, ℓ, is now lower. So, the con- tractor gets paid for a shorter period. In addition, the PMs are more expensive, which brings the profit even further down.

6.8. The effect of the mean rate of minimal repairs, μMR

The results of changing the rate of minimal repairs are presented in Fig. 11, where the range for μMR goes from 2 to 4. Similar to the case of μPM, a higher μMR actually implies a lower minimal repair (MR) dura- tion. We can see that for the supplier, the outcome is similar to the preventive maintenance case: a higher μMR results in a shorter pro- duction lead time, ℓ (because there is less downtime lost to repairs) and a higher expected profit (because the supplier needs to pay the con- tractor for a shorter period). Recall that as μMR increases, so does CMR, implying that the minimal repairs become more expensive. The con- tractor, then, performs preventive maintenance sooner to contain the increased cost of repairs; thus, we see a lower optimal V in the output. The contractor's profit goes down due to the shorter contract period (a lower ℓ) as well as higher MR costs.

7. Conclusion

In this paper, we have discussed the joint problem of production and maintenance outsourcing by a supplier of parts to a customer. This is an important problem in supply chain management because if proper maintenance is not carried out, it is highly likely that the supply chain performance will suffer. We have used game theory to analyse our model as the supply chain involves two players, the maintenance con- tractor and the supplier each pursuing their own objectives.

The main goal of the paper is to highlight the requirement of transient analysis in these kinds of problems where the time horizon for decision making is not long. Although there are many maintenance related works in the literature with a finite time horizon, most of the models use assumptions that facilitate the application of renewal theory. In general, such models require renewal functions and their N- fold convolutions, which are difficult to derive. Besides, even if they can be derived in closed form, the analytical and numerical treatment of the resulting performance measures will be almost impossible unless the model itself has simple assumptions. Hence, much of the literature re- sort straight to using the steady state results. Moreover, many models in this area mainly focus on the optimal maintenance policy but do not really study in detail the system performance per se.

This paper has taken a different route in deriving the system per- formance characteristics like cycle time distributions, renewal function and availability. We have adapted another simple, easy and straight forward method already available in the literature to derive the per- formance characteristics of the system. Although, we have derived the renewal function for the underlying renewal process, our results do not require its use. First, we have derived the results for the general case and then for a special case. Due to the stochastic nature of the failures, repairs and preventive maintenances assumed, even for the special case the relevant functions are very complicated. Hence, we have illustrated our model for the special case with a numerical example. We highlight once again that the analysis of our model requires transient results.

Then, our game theoretic model in the non-cooperative case shows that there is a unique Nash equilibrium. We also note that our game model is a game with the contractor's best response function being piecewise linear. Hence, the Nash solution dominates the Stackelberg solution (see Serin (2007)). To our knowledge, no other model in the

Fig. 8. Sensitivity due to changes in λ

Fig. 9. Sensitivity due to changes in α

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literature has exhibited this feature. As is common in the literature, we also show that if the contractor and the supplier cooperate and jointly make decisions then this two-entity supply chain will reap more profits than if they acted independently.

A possible extension to consider would be to analyse the co- ordination mechanisms in detail. Using various profit-sharing/arbitra- tion schemes from the literature (for example, Leng and Parlar (2005)), we can allocate the additional profit gained through coordination among the parties involved. Including the customer of the supplier in the supply chain relationship would be another interesting extension. That would be in effect a three-entity supply chain. To this end, we can introduce a lead-time and price induced demand into the model so as to glean insights on the performance of the supply chain. Another sig- nificant extension would be to consider imperfect maintenance as considered in Pascual et al. (2016).

7.1. Discussion about managerial and practical implications

We have provided an analytical framework to a very complex pro- blem involving two parties, and production, maintenance and lead time decisions. However, we believe our research also has significant

potential practical applications, similar to Pascual et al. (2016) where the authors applied their analytical framework to a case study. Our focus on the optimal decisions made by both the manufacturer and the maintenance contractor makes our study relevant to managers on either side of the contract. Moreover, by making the production lead time a decision variable, we invite production managers to base their sche- duling decisions on a solid analytical foundation. We also provide re- ferences as to how to coordinate the two parties through various con- tract designs, which can come handy for negotiators on both sides.

Acknowledgement

The authors would like to thank the anonymous reviewers whose comments and suggestions greatly improved the quality and presenta- tion of the paper.

One of the authors (MS) would like to thank Prof. Parlar, M., McMaster University, for the many fruitful discussions he had with him on this paper. Another author (HT) would like to acknowledge the summer research grant (Junior Faculty Summer Research Fellowship) he received from the University of North Texas to work on this project.

Appendix

A. Proof of Proposition 3

Equation (3) becomes

= − +− − −[ ]F x e e( ) 1λV μ x V11 ( ) ( ) α

PM

Fig. 10. Sensitivity due to changes in μPM.

Fig. 11. Sensitivity due to changes in μMR.

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66

∫∑ − =

∞ −

− − − − − −

−

[ ]e λV k

e μ e ( )

! 1

k

λV αk x V

μ x V u MR

μ u μ u

k du

1

( )

0

( ) ( )

( 1) ! α

PM MR MR

k 1

− +− − −[ ]e e1λV μ x V( ) ( )α PM

∫∑ +

− =

∞ −

+ − − − − −[ ]e λV

k e μ e( )

( 1) ! 1

k

λV α k x V

μ x V u MR

μ u μ u

k du

0

( ) ( 1)

0

( ) ( )

( ) ! α

PM MR MR

k

(38)

Let us now consider

∑ +=

∞ +λV k

μ μ u

k ( )

( 1) ! ( )

( ) !k

α k

MR MR

k

0

( 1)

∑= +=

∞ +

u k μ λV u

k 1

( 1) ! ( ( ) )

( ) !k MR

α k

0

1

∑= +=

∞ +

u μ λV u

k k 1 ( ( ) )

( ) ! ( 1) !k

MR α k

0

2 2

∑= +=

∞ +μ λV u

μ λV u k k

( ) ( ( ) ) ( ) !Γ ( 2)

MR α

k

MR α k

0

2 1

∑= +

⎛

⎝ ⎜

⎞

⎠ ⎟

=

∞ +μ λV u k k

μ λV u( ) 1 ( ) !Γ ( 2)

2 ( ( ) ) 2

MR α

k

MR α k

0

2 1

= μ λV

u I μ λV u

( ) 2 ( ( ) )MR

α

MR α

1 (39)

Now, using (39) in (38) and simplifying, we prove Proposition 3.

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