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International Journal of Production Economics

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Coordinating failed goods collecting and repair capacity policies in the maintenance of commoditized capital goods Henny van Ooijena,∗, J. Will M. Bertranda, Nasuh C. Buyukkaramiklib a Eindhoven University of Technology, 5600 MB, Eindhoven, Netherlands b Erasmus University Rotterdam, Netherlands

A R T I C L E I N F O

Keywords: Servitization Periodic admission Repair capacity policy Collecting policy

A B S T R A C T

We study a maintenance organization serving industrial customers operating a commoditized capital good. The maintenance organization provides for availability of the capital good. Upon failure, a substitute good rented from a third party is installed. A logistics service provider picks up the failed good and delivers it to the repair shop. On arrival at the repair shop the failed good is repaired, returned, and installed for use. Rent for the substitute ends at installation of the repaired good. Idle repair shop capacity can be used for other revenue generating tasks, depending on the time the capacity is available. We consider three failed good collection modes. First, a failed good is immediately collected, transported, and immediately admitted to the repair shop. Second, the logistics service provider is allowed a certain slack time for the delivery of the failed good to the repair shop. Third, a failed good is only admitted periodically to the shop and collection and transportation ensures that the failed good is available in the repair shop at the start of the first period possible. We ask under what conditions a transportation slack time, either as a constant slack or as a periodic admission to the repair shop, can be beneficial. We present analytical solutions for immediate transportation and transportation with a constant time slack considering costs. We find that (some) transportation time slack can be beneficial for high failure rates and high transportation costs.

1. Introduction

During the past decades, services have become a major section of activities in the developed economies, and research into service op- erations has led to knowledge now available in teaching programs (Bitner and Brown (2006); Rai and Sambamurthy (2006)). This includes not only pure services, such as hotel, travel and personal care, but also enhancements of the deliverables of the manufacturing industry, as is exemplified by the service contracts that capital goods manufacturers offer for maintenance and repair of the goods after their delivery, in- stallation and use. More and more manufacturers of capital goods ex- tend their offerings by providing services with regard to their existing products in order to increase the customer experienced value (so-called servitization, see e.g. Vandermerwe and Rada (1988); Kastalli and Van Looy (2013); Smith et al., (2014)). As compared to goods, services are characterized by a more intimate customer relationship, often involving the customer in the delivery process, and a higher customer specificity of the deliverables. Setting up a system for delivering services therefore involves very different resources and very different coordination me- chanisms as compared to the setup for manufacturing systems.

Moreover, in services as well as in manufacturing, companies have increasingly outsourced and subcontracted activities they considered not be non-core, resulting in lower investments in resources at the costs of higher coordination effort. As discussed in Cavalieri et al., (2018), the servitization shift needs a shift in business models, organizational structure and mindset, operational processes as well as their relation- ships with the end-customers, the suppliers and their eco-systems.

This paper deals with the setup of system for maintenance and re- pair of capital goods involving one core activity, being the maintenance and repair activity, and two outsources non-core activities, being the transportation of the goods to be serviced or repaired from the custo- mers to the central repair facility and back, and the renting of a sub- stitute capital good to enable continued processing during transport and service of the good to be repaired. Demand for maintenance and repair activities is notoriously variable in timing and amount of resources required (Keizers et al., (2003); Guide et al., (2000); Reményi and Staudacher (2014)). Repair shops therefore have to cope with highly variable workloads, while simultaneously facing pressure for fast re- pair. Installing high capacity levels is one way to solve this problem, although an expensive one. More advanced is the strategy to adapt the

https://doi.org/10.1016/j.ijpe.2018.11.001 Received 6 July 2017; Received in revised form 17 October 2018; Accepted 3 November 2018

∗ Corresponding author. E-mail addresses: [email protected] (H. van Ooijen), [email protected] (J.W.M. Bertrand), [email protected] (N.C. Buyukkaramikli).

International Journal of Production Economics 208 (2019) 29–42

Available online 07 November 2018 0925-5273/ © 2018 Elsevier B.V. All rights reserved.

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capacity in response to the workload. However repair activities are highly specialized and require experienced operators for effective and efficient execution. Thus a repair shop must operate with a sufficient number of experienced operators to be able to cope with peak demand while still satisfying delivery time requirements. The high costs of peak demand capacity can be mitigated if the repair operators can perform other revenue generating work for a third party during periods with low demand for repair work. In this paper we assume that such ‘fill’ work exists and that revenues from doing fill work depend on predictability of capacity being available.

In setting up the maintenance and repair system, the company has to decide about the optimal configuration of resources to implements its business model. In particular it must decide about: 1) the repair capacity level, 2) the contract for third party fill work, 3) the contract for trans- porting failed and repaired goods to and from repair shop, 4) the contract for renting a substitute good during repair. Some of the trade-offs that exist in this decision problem are quite straightforward. A higher repair shop capacity level leads to a shorter repair time, which leads to a shorter renting period of substitute goods. Similarly, a faster pickup and delivery in transportation leads to a shorter renting period. And longer and more predictable period of capacity available for fill work lead to higher rev- enues per time unit. However some detailed elements of contract options can have multiple and counteracting effects. This in particular pertains to the transportation contract, where we consider two options. Under the first option, the transportation firm is allowed a specific time allowance on top of the pure transportation time to collect or return a good. Under the second option, the transportation company must deliver the good to be serviced at the repair shop at the start of regular intervals of a specific length. Transportation costs are smaller for larger time allowance or larger interval length. Using a delivery interval instead of a fixed time allowance brings a higher predictability of free capacity in the repair shop and therefore results in higher fill work revenues per time unit. However, it also brings a more irregular arrival process of repair work, resulting in longer repair shop throughput times, and leading to higher renting costs. In this paper, we focus on the comparison of these two types of transportation contracts, and want to know under which con- ditions the fixed time allowance contract is to be preferred over the periodic delivery contract.

This research was inspired by the existence of regionally centralized maintenance and repair shops for the aircraft industry, like StandardAero in Tilburg (see http://standardaero.com/) and logistics service providers that offer transportation services for this industry, like Jan de Rijk Logistics in Roosendaal (see https://www.janderijk.com/). Other examples might be equipment used in the building and con- struction industry (cranes (small) draglines etc.), or agro industry (heavy machines for cultivating the land and harvesting, etc).

We show that there are realistic situations (parameter settings for the transportation cost function and the use of idle time revenue function) for which the combination of two control domains, trans- portation control and capacity control, leads to total costs benefits compared to independent control of these two domains. However we also show that there are also many situations for which this leads to a small increase of the total costs. Based on this we conclude that it can be worthwhile for a MRSP to consider to use periodic admission (as a re- sult of the integration of the two control domains), especially if there are also other positive (cost) effects than the ones shown in this study.

The remainder of this paper is organized as follows. In section 2 we will review the relevant literature. In section 3 we model the base si- tuation and provide the analytical solution to the problem of setting the repair capacity so as to minimize total relevant costs. We also present the model for the fixed time slack for transportation and provide the analytical solution. In section 4 we present the cost model for the periodic job admission mode and present the computational model for determining the job throughput time distribution as a function of ar- rival process, admission interval length, and repair shop capacity. In section 5 we discusses the computational optimization approach for the

periodic admission mode and, for a set of realistic model and cost parameter settings, we compare the minimal costs that can be obtained under the fixed time slack mode and the periodic admission mode for identical average slack values, to identify the optimal mode for each setting. Finally, in section 6 we will give the conclusions.

2. Literature review

In this section, we review the relevant literature on repair shops and goods collection. We exclude studies on leasing, and focus on short- term rental in our literature search. We also review relevant literature on periodic admission/release and capacity management.

System performance in the repair shop environment is measured by how long it takes to replace a failed system in the field. It is a function of (Scudder (1985)):

- the initial spares inventory levels - the capacity to repair parts - the priority scheduling in the repair shop

Research on repairable items and repair shops have a long tradition. The stream on repairable items mostly concerns (initial spares) in- ventory aspects of these items, e.g. see Sherbrooke (1968) that develops a model to determine optimal stock levels for recoverable items in a two-echelon setting, Muckstadt (1978) that gives some approximations for the problem studied by Gross and Ince (1978) that determines the optimal number of spares and repair channels for a population of sto- chastic failing units and Schneeweiss and Schröder (1992) where a hierarchical model is developed for determining the number of spares to guarantee a certain service level and the schedule for the repair shop in order to obtain that service level. A comprehensive review can be found in Guide and Srivastava (1997).

With regard to repair shop capacity (allocation) a lot of research is done on resources required for the repair and maintenance of aircraft related items. Hodgson and McDonald's case study (1981) is one of the first that takes into account the capacity in an explicit manner. Kurz (2016) studies capacity planning for a maintenance service provider, however, this concerns overhaul activities. Gatland et al., (1997) use simulation to get a better understanding of the production capability of an engine maintenance facility and furthermore a number of studies investigate resource planning for aircraft maintenance, e.g. Dijkstra et al., (1991), Yan et al., (2004).

With respect to the repair shop capacity Büyükkaramikli et al., (2013a) and Büyükkaramikli et al., (2013b), investigate repair shops where orders arrive continuously (so immediate transportation of failed goods) and the effects of (flexible) periodically repair capacity contracts are studied. Büyükkaramikli et al., (2015) studies situations where the MRSP has a pool of critical components to replace failed goods at the customer (so the good is not hired from a third party) and the repair shop has an amount of permanent capacity and can make use of a fixed amount of contingent capacity if necessary. The repair shop capacity flexibility thus consists of using or not using the contingent capacity. Most of the papers on the use of the capacity concern using revenue management with regard to order acceptance. However these are not applicable to our problem.

We found a number of papers on using overtime in repair shops see for instance Scudder (1985)). For more general non-repair shops the use of overtime has been studied by Holloway and Nelson (1974) and Goodwin et al., (1978). The objective is to minimize overtime usage subject to meeting all due dates. Papers investigating the use of idle capacity were hard to find. Literature that considers idle capacity only investigates the role of unused (service) capacity (Ng et al., (1999)), or opportunity costs of idle capacity (Balakrishan et al., (2001)), but not the revenues of alternative usage of capacity. We haven't found any paper where the production control of a repair shop is related to the arrival process (immediate or periodic admission) of failed goods and the alternative use of idle capacity.

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The collection of goods is comparable with the distribution of goods. Problems investigated with regard to distributing goods often concern the number of vehicles that are necessary and determining the optimal routes. Such problems are called Vehicle Routing Problems. It is well known that combining a number of transports leads to economies of scale effects. In this research we are, in first instance, interested in finding (lower bounds for) the cost savings that are obtained when failed goods are collected in one trip instead of having one trip for each failed good. There is a whole stream of literature on Periodic Vehicle Routing where the purpose is to minimize the total number of routes, i.e. the total transportation. An overview of this research is given in Campbell and Wilson (2014). In that research customers require visits on one or more days within a planning period and there are a set of feasible visit options. Customers must be assigned to a feasible visit option and a VRP is solved for each day in the planning period. The typical objective is to minimize the total distance travelled over the planning period. Examples can be found for instance in Coene et al., (2010), Yang and Chu (2000) and Alegre et al., (2007). However, in the periodic vehicle routing problem, the places and times to visit are known in advance, which is not the case in our research.

Minimizing the costs of collecting failed goods that have failed at dif- ferent locations in a region is equivalent to finding the route, starting at the repair shop and ending in the repair shop, that minimizes the distance travelled. In Beardwood et al., (1959) and Christofides (1967) the expected length of the optimal travelling salesman route through N+1 points (N customers and a depot) is derived and we will use this result in our research.

A stream of research that comes close to our way of transportation is the research on consolidation of transport. Papers investigating this topic are, amongst others, Taherian (2014), Cetinkaya and Bookbinder (2003), Ülkü and Bookbinder (2012a), Ülkü (2012), Ülkü and Bookbinder (2012b), Berling and Eng-Larsson 2016)). New in our study is that a substitute has to be rented during the consolidation, trans- portation and repair time and that we explicitly take into account the effect of bulky arrivals on the throughput time of the repair shop.

Since we investigate situations with periodic admission it is also relevant to review the literature on periodic admission or periodic re- lease. Büyükkaramikli (2012) studies situations where orders are peri- odically released to the repair shop. However, he studies a more or less isolated situation since only capacity (cost) effects in situations where capacity if it runs idle is sold back and repair renting costs are con- sidered. Periodic admission or release is further mainly studied in re- search on work load control but these studies are on controlling the work load in the shop by selective releasing orders to the shop. This is not the case in our situation where periodically all orders are released to the shop, independent of the work load in the shop.

So, although a number of the topics we consider in this paper is (somehow) studied thus far, these topics are not studied in an integrated way. In this study we focus on the situation where substitute goods can be rented with zero lead time and repair shop capacity is owned, but under certain conditions can be used for other purposes when it runs idle and thus can bring some extra revenue. We integrate two control domains: the combination of transportation and the use of the repair capacity and in- vestigate whether the interaction of these two can lead to total cost ben- efits. Given the focus on the integration we don't look inside the repair shop, but consider it as a black box. Contrary to the consolidation research we don't determine a fixed time period (fixed transportation frequencies) that only benefits the LSP, but a fixed time period that leads to lower total costs taking into account the costs of transportation, the renting of a substitute good and the repair of the failed good.

3. Base case and fixed time slack models

In this section we will discuss the details and the (relevant) costs of the base case: failed goods are immediately picked up, transported and ad- mitted to the repair shop, and repaired goods are immediately picked up and returned to the plant. We assume that idle capacity can be used for

other tasks and leads to additional revenue. We extend this model to in- clude the case with a positive fixed time slack for transportation. Transportation is subcontracted to a logistics service provider (LSP), which is assumed to be located in the neighborhood of the MRSP, under a con- tract that requires action after being informed about a failure at a plant or a repair at the shop. The contract offered by the LSP gives a rebate on the transportation costs as a function of the time slack allowed for the trans- portation. Under zero time slack, the time and costs of transportation are linear functions of the distance travelled. The repair shop operates with a capacity that allows for a processing rate μ. Capacity costs are proportional with processing rate, c.μ, and idle capacity can be used for doing ‘fill work’ that brings a revenue that is proportional with processing rate, r.μ, 0 < r < c. We assume exponentially distributed repair times.

In case of a failure, the MRSP rents a substitute good from a supplier who installs the good at the plant as soon as possible, and de-installs the substitute when the repaired good arrives from the repair shop. There are costs for the transportation, installation and de-installation of the rented good, but we consider these to be taken into account in the renting rate. Transportation, installation and de-installation are as- sumed to be a responsibility of the supplier of the rented good, and thus cannot be influenced by the MRSP. Therefore these costs are not ex- plicitly taken into account in our models.

Substitute renting costs are linear in the renting time. We assume that the capital good is in the mature phase of its life cycle and has a large installed base in a region around the repair shop. As a result, it is reasonable to assume that the failure rate per good is small, and that the aggregate failure rate for the whole region is stationary and in- dependent of the number of failed goods at any point in time.

A schematic overview (timeline) under zero time slack for trans- portation, of the timeline of transportation to the shop, repair, and transportation to the plant of a failed good, is given in Fig. 1, where the red line is an Example time line. In the next subsection we present the cost functions for the base problem.

3.1. The substitute renting costs

The renting starts as soon as a failed good at a plant is replaced by a substitute and ends when the repaired good is back at the plant. So substitute renting time consists of

- the time from failure at the plant to arrival of the failed good at the repair shop, TS,

- the time from arrival at the shop to the time of repair of the good, STPT,

- the time from repair to the arrival of the repaired good to the plant, TP.

The first part covers the renting costs during the transportation time of the failed good from the plant to the MRSP, the second part covers the time at the MRSP and the third part covers the transportation time from the MRSP to the plant.

We next model the average transportation time for an arbitrary failed good at a plant located at an arbitrary position in the service region of the MRSP.

3.1.1. Transportation process We assume that the repair shop is located in the middle of a circular

region with radius r and area size S km2, and that the plants that are serviced are located at arbitrary positions in the region. The expected distance from such a position to the center of the region, using the Euclidian norm, is then equal to:

=E d r S( ) 2 3

0.376i

For transportation of a failed good the LSP needs to drive to the plant and from the plant to the repair shop. Thus the expected distance

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to cover is:

= =E d S S( ) 2 0.376 0.752i

Let v denote the driving speed in km per unit time of the trans- portation vehicle. Then the expected time to pick up an arbitrary failed good and bring it to the repair shop is equal to: S

v 0.752 . If, under the

transportation contract, a slack time equal to A time units is permitted for delivering the failed good to the repair shop, the expected time is equal to + f A( )Sv

0.752 , where f(A) stands for the average time slack used by the LSP. The LSP will search for possibilities to combine the transportation of the, at time t, failed good with other transportation activities in the time interval (t, t + A). We assume that such an op- portunity will be found on average halfway the interval. Thus the average delivery time of the failed good to the repair shop is equal to

+ .S v

A0.752 2 For returning a repaired good to the plant, the average

distance to travel from the repair shop to the plant is equal to S0.376 , and if also a time slack of A is allowed, the average return delivery time is equal to equal to + A/2Sv

0.376 . Driving times are exogenous to the problem. They are determined

by the locations of the plants and the repair shop, and the driving speed,

all of which are unaffected by the decision variables in this paper. Thus, the relevant part of the substitute renting times are the parts resulting from the slack used. The average slack used per repair cycle is equal to A/2 + A/2 = A.

3.1.2. Repair process We model the repair shop as a single server queueing system with

arrival rate λ and service rate μ. We assume a Poisson arrival process and a Poisson repair process. Then the expected time to repair a failed good once it has arrived at the repair shop, STPT, is equal to: 1

µ Adding these three components up, we get the average time during

which a substitute must be rented. Let h denote the renting cost per substitute per unit time. Then the system wide average relevant renting cost per unit time, E(RC) is equal to:

+h µ

A( 1 ) (1)

3.2. The transportation costs for zero and positive slack

In the base case with zero slack, the driving times and driving costs

PLANT:

TRANSPORT TO REPAIR SHOP:

REPAIR SHOP:

Pdf of arrival time at plant

Arrival defect good at repair shop Defect good admitted

Pdf repair time

Transportation timeTransportation time

T

Substitute rental cost period

Good fails, replaced by a substitute Arrival repaired Good at plant

Failed good picked up

Delivered to repair shop

Pick up repaired good

Good repaired

Fig. 1. A schematic overview of the processes between the time of a specific failure of a good and the time the repaired good arrives at the plant; T is an Example repair shop throughput time. Transportation time is time needed to pick up the failed item and bring it to the repair shop + extra time used (≤A).

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are out of scope. However, if time slack is allowed for delivering failed goods to the repair shop and to the plants, we expect the transportation costs to decrease and the delivery times to increase, the latter leading to higher renting costs. We now model the transportation costs as a function of the slack.

Let k denote the cost of transporting one good per unit time under a contract with zero slack (A = 0).

Then, given the average distance travelled per failure occurrence, the system wide average cost of transportation (from plant to repair shop and form repair shop to plant) per unit time is:

=E TC k S v

( ) 2 0.752i

We assume that the slack in the contract is used by the LSP to combine transportation of failed or repaired goods with the transpor- tation of one or more other goods, thus achieving efficiency and that a part of this efficiency is transferred to the MRSP in the form of a dis- count on the cost per distance travelled. How many goods can be combined by the LSP within the time slack given, depends on the size of the business of the LSP and this is not explicitly modeled in this paper. This would require modeling the LSP in detail which would go beyond the purpose of this paper and make the findings LSP dependent. We model the efficiency gain for the LSP implicitly in the following way: we assume that for the LSP, the marginal value of slack is a decreasing function of the slack, and that the maximum value is achieved if the slack in infinite.

In line with economic theory we assume a discount function that is based on the law of diminishing returns. The parameters of the function are determined by the environment of the MRSP: for instance, whether there are many companies in the region that can make use of the LSP or not. Therefore, the discount, due to the efficiency gain of the LSP due to his bundling process, is assumed to be an exponential function of the slack, A, with a maximum of α and a sensitivity factor γ. The resulting average transportation costs per failure occurrence per unit distance, with a time slack A for both delivery to the repair shop and back to the plant, are:

=k k e(1 (1 ) )A A (2)

(we will call e(1 (1 ) )A the transportation costs multiplication function)

Then, the average (total) transportation cost per unit time is given by:

k S v

2 0.752A

3.3. Repair shop capacity costs

We assume that the capacity related cost per unit time is a linear function of the processing rate μ. Let c denote the cost of capacity per unit of processing rate per unit time. We exclude the baseline capacity needed for the repair process itself, since repair times are not affected by the decision variables we consider, so we only take into account (μ- λ). We assume that part of the equipment, workers and infrastructure can also be used for other products or services, although not with the same efficiency. Using this multipurpose capability of (parts of) the production resources can lead to a better utilization of the resources. This in turn may result in a lower cost per product or service produced. A repair system is generally highly specialized for the type of good that it has to repair. Specialization mostly resides in specific tools and in the knowledge of the workers, but many of its resources often can also be used for performing other tasks. In this paper we assume that the repair shop has the option to use some of its production resources for other products or services. We furthermore assume that the repair shop al- ways gives absolute priority to the ‘own’ repair jobs over the alternative jobs. We model the effect of using the option as a revenue per unit of

capacity available for performing the alternative jobs. The capacity available for alternative jobs is equal to the idle capacity if only own jobs were performed. The revenue per unit of capacity available we denote as r, with 0 < r < c, where c denotes the cost of capacity. The value of r reflects the value of the production system resources for performing other tasks. A low value indicates that only a few resources can be used or that the efficiency of the resources is quite low for the other processes. A value close to c indicates that the costs of capacity can be almost totally earned back by performing other tasks. However, we may assume that in most situations, a value close to c is very un- likely since the capacity is only available if there are no repair jobs, and will be immediately withdrawn again as soon as repair jobs arrive. For the relevant cost of capacity per unit time we then get: c r µ( ) ( )

Now we can formulate the optimization problem for the base case. For the base case, the slack (A) is zero, and transportation times are minimal, as are the components TP and TS of the substitute renting times. The only variable component left is STPT, which is the time the failed good spends in the repair shop. In the base case we have only one decision variable, μ, and for the total relevant costs per unit time we have:

= + +TRC c r h µ

k S v

(µ) ( ) (µ ) ( 1 ) 2 0.752B A

(3)

The optimal capacity level µ , follows from: = c r h0, so ( )RC

µ µ (µ) 1

( ) B

2 = 0 and thus:

= + h c r

µ (4)

We next formulate the problem for positive slack. For positive time slack, the renting cost have to be extended with a component h.A, and the reduction in transportation costs that result from a positive time slack for transportation has to be included.

With respect to the stay of the failed good in the repair shop under a positive time slack for the transportation, the arrival of failed goods at the repair shop is still a Poisson process with arrival rate λ, on the condition that the actual delays in the arrivals due to the use of the slack time are uncorrelated in time and uncorrelated with the travelling times. Proof is given in Appendix A.

Now we have two decision variables, A and μ, and the average system wide relevant costs per unit time under positive time slack are then given by:

= + +

+

TRC A c r h µ

A

k S v

e

(µ, ) ( ) (µ ) ( 1 )

2 0.752 ( 1 (1 ) )

I

A (5)

From this cost function we see that the slack time and shop capacity can be optimized independently, since the slack time only influences the slack related part of the renting costs and the transportation costs, and the shop capacity only influences the throughput time related part of the renting costs. Since under a Poisson arrival failure arrival process at plants, and uncorrelated arrival delays due to the use of the positive time slack, the aggregate repair job arrival process at the repair shop is still a Poisson process with the same parameters an under immediate transportation, the shop throughput time is the same as under zero slack time, and therefore the optimal value for the shop capacity is the same as under zero slack time (see (3)). The optimal value for the slack A as a function of h, α and γ time is given by:

=A ln h

E TC( )i

If α and γ are such that λh/( E TC( )i ) > 1 then there is no positive optimal value for A, and thus for these values of α and γ, a positive

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value of A leads to higher costs. This might be the case when the hiring costs h are that high, compared to the maximum transportation benefits with a change in the slack allowance A (αγE(TCi)), obtained for A → ∞, that it is not beneficial to give the LSP some slack for transporting the failed good to the MRSP.

Therefore, the optimal value of A is given by:

=A ln

max ( , 0) h

E TC( )i

(6)

We see that under rather weak assumptions regarding the renting, transportation and repair processes, the optimal values for transporta- tion slack time and repair shop capacity can be analytically derived from processes that can be realistically estimated for real life situations. We will use these optimal results to investigate the possible benefits that can be obtained with a different way to allow some transportation slack to the LSP, but that can also bring advantages for the use of ca- pacity not used for repair. This different way consists of organizing the repair shop such that repair jobs are only admitted to the shop at reg- ular times with interval time D. In the next section we discuss this way of organizing the shop.

4. Periodic admission to the repair shop

Periodic admission to the repair shop leads to the situation where the LSP experiences a (now variable) slack in the time available to deliver the failed good at the repair shop. The LSP must deliver the failed good at the earliest possible period start, given the pure travel time needed to bring the failed good from the plant to the repair shop. With periodic admission, separate optimization of the slack and the shop capacity is no longer possible, since periodic admission changes the nature of the repair job arrival process, and also introduces an additional benefit with respect to the revenues to be obtained from idle repair shop capacity, which both are determined by the value of the admission interval, D. Fig. 2 shows the various components of the renting, transportation and repair processes under periodic admission with admission interval D.

4.1. The substitute renting costs

We next model the average transportation time for an arbitrary failed good at a plant located at an arbitrary position in the service region of the MRSP under periodic admission to the repair shop.

4.1.1. Transportation process In Appendix B we show that the average time slack per failure oc-

currence is equal to half the admission interval: D/2. For symmetry reasons, we assume that the LSP is required to start transportation of a repaired good at latest at the end of the period in which the good was repaired. Thus also for return transportation, on average D/2 time slack is available. This (now variable) time slack can be used by the LSP to realize efficiency in the transportation process. Since orders are peri- odically admitted, the average delay in the transportation process is equal to the average slack available: D/2. The average delay in re- turning the repaired good to the plant, however, is equal to halve the average time slack available: D/4 (see section 3). Also under periodic admission the driving times are exogenous to the problem. They are determined by the locations of the plants and the repair shop, and the driving speed, all of which are unaffected by the decision variables in this paper. Thus, the relevant part of the substitute renting times are the parts resulting from the slack used for transportation. This leads to the following system wide substitute renting costs due to transportation, in case of periodic admission (TS + TP):

h D( 3 4

)

4.1.2. Repair process In case of periodic admission, the arrival process at the repair shop

is no longer Poisson, but periodic, with a Poisson distribution of the number of failed goods that arrive each period. This results in a bursty arrival process and therefore in a stochastically larger distribution of the repair time if the capacity is the same. An analytical expression for the repair shop throughput time as a function of μ and D is not available and therefore we must resort to a computational approach for finding near optimal solutions.

Let us denote the average repair shop throughput time under peri- odic admission, given a capacity μ, by S D¯ (µ, ), then the average system wide relevant renting costs per unit time in case of periodic admission is equal to:

+h S D D( ¯ (µ, ) 3 4

) (7)

4.2. The transportation costs for zero and positive slack

Due to the periodic admission with interval D the LSP experiences on average a time slack D/2 to deliver the failed goods to the repair shop since the failed good has to be delivered to the shop at the start of the first period that follows the time of failure plus the transportation time. As in the base case situation, described in Section 3, this allow- ance gives opportunities to combine transport and thus to reduce transportation costs. We assume that this reduction is reflected in the price paid by the MRSP: due to this reduction in transportation costs, the MRSP gets a discount of the price it has to pay per unit time that depends on the period length D.

Using the same basic equation for the relationship between trans- portation cost per unit of distance and slack as for the fixed time slack case, we can derive the following expression for the average transpor- tation costs as function of D:

=E TC k S v

( ) 0.752D D

(8)

where kD is equal to:

=k k e(1 (1 ) )D D/2

4.3. Repair shop capacity costs

What remains to determine is the revenue from the use of idle ca- pacity under periodic admission under periodic admission. With respect to the periodic admission policy we may expect that this, compared to immediate admission, constitutes an advantage with respect to the value of idle capacity. Under the periodic admission policy, it is certain that idle capacity, if it occurs, will be available until the end of the period. Knowing with certainty that capacity is available for a certain amount of time, makes it more attractive to use it for other purposes then in the situation where it is not known how long the capacity will be idle. In the latter situation we don't know how long the capacity can be used and it might be that work that has started must be disrupted, which makes the use of the capacity less attractive. We assume that the more attractive it is to use the capacity, the higher the revenues will be. We model this effect by assuming that the revenue of alternative usage (when not needed for the repair of the failed goods) per unit of capacity is an increasing function of the period length D, with a revenue of r for period length 0, and a value of M, r < M < c, for D goes to infinity. Again in line with economic theory we assume that the law of dimin- ishing returns holds. We therefore model the function as an ex- ponential, which in a continuous way reflects the diminishing value of an increase in period length. The parameters of the function are de- termined by the environment of the MRSP: for instance, are there many other, more or less comparable, high tech firms in the environment, such that it is easy to find other work for the operators or not, etc.

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34

Therefore we get for the revenue per unit idle capacity due to the coverage of the repair capacity costs by the filling-work:

= +s r M e c( (1 ) )D (9)

with: r the revenue in case idle capacity is used without knowing for how long it can be used (immediate admission situation)

M the maximum revenue we can get for doing substitute work (0 < M < 1) Ε sensitivity parameter, indicating the increase in revenue of sub- stitute work by an increase of the period length D

We will call (9) the coverage function. In the situation with periodic admission we have two decision

variables, D and μ, and the total relevant costs per unit time are:

= + +

+

TRC D c s µ h S D D

k S v

(µ, ) ( ) ( ) ( ¯ (µ, ) 3 4

)

2 0.752

A

D

(10)

Using sample path analysis it can be shown that for D > 0, F(S ( D xµ, ), ) is stochastically greater than F(S(µ, 0),x) (F(.,.) denotes the CDF). So if we replace S̄(µ, D) by the S̄ (µ, 0), we can find a lower bound

for the total relevant costs, and upper bounds for the optimal values of μ and D (these bounds still depend on D because D is in the transportation costs and in the renting costs during the transportation time). When doing so we find as lower bound for μ, given D:

= + +

h c X M e c

µ ( (1 ) )D

and for D, given μ, the solution of:

+ =c Me h k S v

e(µ ) 3 4

0.752 0D D/2 (11)

We were not able to find explicit optimal solutions for problem (10). Therefore we followed a two-step approach (which has been developed in Büyükkaramikli (2012)):

- based on a trade-off between capacity costs and repair time renting costs we first determined the optimal values of μ, and corresponding costs for certain values of D. Since the capacity costs and the repair time renting costs are convex functions of μ, we used a golden search method to find the optimal values for μ and the corresponding S̄. Given a certain value for D, we used (see appendix C for more de- tails):

PLANT:

TRANSPORT TO REPAIR SHOP:

REPAIR SHOP:

Failed good delivered to repair shop

(n+i)D

Transportation time

(n+2i+j)D (n-1)D (n)D

Substitute rental cost

Good fails, replaced by a substitute

(n+2i+j-1)D

Failed good picked up

(n+1)D (n+2i+j)D(n+i-1)D

(n+i+j)D T

Good repairedArrival at repair shop Admission to repair shop

(n+i-1)D (n+i)D

Pick up repaired good

Arrival repaired good at plant

(n+k)D

Transportation time

Fig. 2. A schematic overview of the processes between the time of failure of a good and the time the repaired good arrives at the plant; T is an Example throughput time. D-discretized means that all times between xD and (x+1)D are counted as (x+1)D.

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35

= = = = = =

P S x P N i D

P S x O n{ | µ } { } { | µ, } n g n i i

n

q

e D g

1 0

1 ( ) !

D g

where O is the position of an arbitrary arriving customer (in a batch) in the queue and

= = =

P S x O n e x j

( | µ, ) (µ )

!j

n x

j

0

1 µ

to construct a discretized CDF of the throughput time (steps of 0.0005; so F(S( =D xµ, ), ) +P S x P S xC C{ 0.0005| } { | }d d )). Herewith we determined S̄, given D and μ.

- next, for each of the values of D considered and the corresponding optimal μ determined in the first step, we added the transportation costs and transportation hiring costs to the corresponding costs de- termined in the first step. The value of D leading to the lowest total costs is the optimal value of D.

5. Computational study

To investigate for what kind of environments periodic admission and/or doing substitute work with the capacity can be beneficial, we performed a computational study. In this study we normalize the mean arrival rate of failed goods (not for one capital good but for the ag- gregate of all capital goods installed in the whole region) to = 1 (failures per unit time). We assume that there are 500 installed capital goods which each fails on average once every two years. If the capital good is in use for 250 days per year then on average we have one failure per day. For the surface area S we used 1.000.000 km2 (Western Europe, Central Europe), the transportation costs tc have been set equal to € 90 per hour and for the average speed we used 60 km per hour, so the average speed per unit time (a day of 8 h in our case), v, is 480 km. The parameter values in the base case scenario are found from the following situation: Suppose that the capital good has a value of

500.000. The capital good is used in the production process of other products and the economic lifetime of the capital good is assumed to be 5 years. If the capital good is in use for 250 days per year and 8 hours per day, then the capital good related costs are =

× × 50500.000

(250 8 5) per hour.

For the cost of the (specialized) workforce, machines and tools of the repair shop, we will use a cost of 80 per hour and we assume that a repair of a failed good takes about 40 hours. As a reference we use the data in Keizers (2000) who reports on an empirical study of a central repair facility for preventive and curative maintenance of advanced defense systems, and finds average repair processing times for different

shops ranging between 25 and 45 h; curative maintenance jobs, who have priority in the shops, account for 30–55% percent of the workload of the shops. Preventive repair job throughput times for such shops ranges between 10 and 20 weeks. Curative repair jobs, having priority over preventive jobs, may expected to have waiting times of a few days only, since most shops consists of up to five parallel work centers, and a curative job is started as soon as the first work center becomes available (the average job time is about 5 working days). Using an average repair time of 40 h, the repair of a failed good on average costs 3200. Next, we derive the values for rental costs. We assume that the rental supplier adds on a 100% premium on top of the capital good related costs. Therefore, the rental cost per hour is equal to: = 100x X X

2 500.000 250 8 5 per hour.

Based upon this more or less real-life setting, scaling the parameter for the cost of workforce per repair to €1, so dividing the repair costs by 3200, and expressing the values for the rental costs as a multiple of this normalized cost (by dividing the costs by 3200), we get the base case scenario, where =c 1, and h= 0.25 per day. In the numerical ex- periments we used 3 different h values: €0.2, €0.25 and €0.3.

The reduction function of the transportation costs due to the al- lowance and periodic collection uses two parameters: α and γ: α re- presents the maximal reduction we can get and therefore we used two values: α = 0.3 and α = 0.6 (resulting in 30% resp. 60% reduction in transportation costs when the transportation company gets a lot of time to combine) and γ represents the speed with which the maximal re- duction is obtained. We determined γ by assuming two extreme cases: 1/6 of the optimal reduction resp. 1/2 of the maximal reduction is obtained when the time to collect failed goods and to deliver them to the repair shop equals the average time needed for collecting and de- livering a failure: S

v 0.752 (1.57 in this study). This results in the fol-

lowing four functions for the transportation costs multiplicator in the situation with periodic admission (see also Fig. 3a):

e kTC1: (1 0.3 (1 ) ) D0.2328

2

e kTC2: (1 0.3 (1 ) ) D0.8849

2

e kTC3: (1 0.6 (1 ) ) D0.2328

2

e kTC4: (1 0.6 (1 ) ) D0.8849

2

With TC0 we denote the situation with periodic admission without a reduction of the transportation costs.

Although our parameter values seem to be arbitrarily chosen, we conclude from Fig. 3a that they cover a large part of what we assume

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

TC1

TC2

TC3

TC4

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

ACR1

ACR2

ACR3

ACR4

Fig. 3. a and b. The different transportation costs multiplicator functions (3a) and alternative capacity usage revenue functions used in our study as a function of the slack. 3a: y-axis is 1-reduction (kA/k in (2) or kD/k in (8); 3b: y-axis revenues due to alternative capacity usage.

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36

that in practice could be observed. The straight lines illustrate the si- tuation where the maximum efficiency benefits for the LSP are low and the sensitivity of the LSP for having some slack is low resp. high. The dashed lines illustrate the situation where the maximum efficiency benefits of the LSP are high and the sensitivity of having some slack is low resp. high.

For the function used for the alternative use of idle capacity, (9), we used the values 0.1c for r (always a coverage of 10% of the idle capa- city), 0.2 resp. 0.6 for M (in total a coverage of the idle capacity of 30% resp. 70% for period length D →∞) and 0.1054 resp. 0.6931 for ε (an extra coverage of 0.1M resp.

0.5M if the period length is equal to 1 day). This gives the following alternative capacity usage revenue functions (see also Fig. 3b):

ACR0: 0.1

+ e cACR1: 0.1 0.2 (1 )D c0.1054

+ e cACR2: 0.1 0.2 (1 )D c0.6931

+ e cACR3: 0.1 0.6 (1 )D c0.1054

+ e cACR4: 0.1 0.6 (1 )D c0.6931

Again, the parameter values seem to be arbitrarily chosen but we conclude from Fig. 3b that they cover a large part of what we assume that in practice could be observed.

Since the upper bounds for D, for all situations we consider are smaller than 2 (see equation (11)), we used the following values for D: 0.5 (half a day), 1, 1.5, 2, and 2.5.

5.1. Immediate admission

Table 1 gives the results for the base case (immediate admission) with revenues for alternative usage of the capacity, X, equal to 0.1. In the case of immediate transportation (with or without a transportation time allowance) we assumed that, since it is not known how long the idle capacity will stay idle, that a minimum revenue of 10% is obtained by alternative usage.

For all situations considered, A* turned out to be positive only for the situations with α = 0.6 and γ = 0.8849 (see (5)): for h = 0.2, A* = 0.71; for h = 0.25, A* = 0.46 and for h = 0.3 A* = 0.25. For all other situations considered A* is equal to 0, so for these situations a positive value of A does not decrease the total costs in case of im- mediate transportation. For most realistic parameter settings (as argued before we think that due to our parameter values a large part of what we assume that in practice could be observed, is covered) we therefore postulate that in case of immediate admission it is not beneficial to give the LSP some slack for transporting the failed good to the MRSP. Table 2 gives the effect of using an optimal delivery time allowance, which is only positive in case we use transportation costs function TC4.

5.2. Periodic admission

The effects of periodic admission for the different transportation cost function and alternative capacity usage revenue can be found in Table 3. For each combination of h, transportation costs function and

alternative usage of idle capacity revenue function, we give the value of D that gives the lowest system wide costs. Moreover, we show the percentage decrease of total system wide costs. This percentage is based on the costs of the similar base case: immediate transportation and (minimal) alternative capacity usage.

Inspection of the data in Table 3 reveals that, for most of the si- tuations, the optimal value of the period length D is equal to 0.5. This is in contrast to immediate release where only for α = 0.6 and γ = 0.8849 we get a positive value for the slack. As can be expected for TC0 and ACR0, where there are no period length related transportation benefits or alternative capacity usage benefits the use of periodic admission leads to a substantial increase of total costs (5.3%–7.03%). This is caused by the additional hiring costs (due to the periodic admission) and the bulky arrival of failed goods. The increase declines as the maximum reduction α increases (TC0→TC1→T3) or the sensitivity γ increases (TC1 →TC2, TC3→TC4). Also for increasing value of the maximum value of the alternative usage of capacity (ACR0→A- CR1→ACR3) or increasing value of the sensitivity parameter (AC- R1→ACR2, ACR3→ACR4), the negative effects of the additional renting costs and the bulky arrival are diminished.

For high values of the maximum transportation reduction and/or revenues of alternative usage of capacity, and high sensitivity para- meter values, most of the situations show costs reductions when using periodic admission instead of immediate admission; especially if all values (α, M, γ, and ε) are high the cost benefits of periodic admission are substantial (5%–10%).

Example. If we consider in Table 3 the setting (TC4, ACR4 and h = 0.3), we see that compared to the base case (immediate transportation to the repair facility, and minimal revenues for the alternative usage of capacity), the total costs are 5.22% lower. It turns out that in this situation the processing rate is equal to 1.7076 (1.58 in the base case) which gives a utilization rate of around 59% and an average waiting time of 0.486 days. The average throughput time then equals 40 + 0.486*8 h = 43.88 h. In our discussion of Keizers (2000) earlier in this section, we expected an average waiting time of a few days, based on the priority of curative jobs over preventive jobs. In our model, no preventive jobs are present and waiting times will be the result of competition between curative jobs only. For the shops in Keizers (2000) operating at 60% utilization with a processing rate of 1.7 in the absence of preventive jobs, an average waiting time of half a day is to be expected.

5.3. Bursty arrival effect

To investigate the effect of the bursty arrival and alternative capa- city usage, we used in the cases with immediate admission an allowance equal to the optimal average allowance in the corresponding situation with periodic admission: A = D/2 if the optimal period length is D. This gives the results as in Tables 4–6. In these situations the transportation costs under immediate admission are equal to the transportation costs for periodic admission, so these tables show the capacity effects (bulky arrival and alternative capacity usage) of periodic admission related to

Table 1 Optimal capacity levels and costs for different values of the renting costs for the base case (no delivery time allowance; alternative capacity usage). TC = transportation costs; E(RHC) = renting costs during the repair time; CRC = capacity related costs.

μ TRC TC E(RHC) CRC

h = 0.2 1.47 1.55 0.705 0.424 0.424 h = 0.25 1.53 1.65 0.705 0.474 0.474 h = 0.3 1.58 1.74 0.705 0.520 0.520

Table 2 Optimal capacity levels and costs for different values of the renting costs for the base case with transportation time allowance and alternative usage of capacity. TCi = transportation costs multiplicator function i.

Immediate admission: optimal delivery time allowance and alternative usage of capacity

μ TC0 TC1 TC2 TC3 TC4

h = 0.2 1.47 1.55 1.55 1.55 1.55 1.43 (7.7%) h = 0.25 1.53 1.65 1.65 1.65 1.65 1.56 (5.5%) h = 0.3 1.58 1.74 1.74 1.74 1.74 1.69 (2.9%)

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37

immediate admission. Table 4 gives the transportation cost functions. We see that for the different transportation costs multiplicator func- tions, for each of the values of the renting costs the difference between the system wide costs of immediate admission and periodic admission more or less stays the same. Since the costs of immediate admission are lower than the costs of periodic admission the relative costs difference increases with increasing value (TC0 →TC1→TC3) of the maximal cost reduction and increasing value of the sensitivity parameter (TC0 →TC2→TC4) as can be seen in Table 5. The difference being negative can be explained by the bursty arrival of failed goods to the shop and the lower renting cost during transportation in case of immediate ad- mission with transportation slack: if in case of periodic admission the optimal value of the period length is D then the total average (trans- portation) renting period in case of immediate admission is D/2 (see equation (1)) and in case of periodic admission 3D/4 (see equation (7)). Table 6 gives the different cost components for immediate admission and periodic admission in the case with no transportation cost reduc- tions (TC0) and a minimal alternative capacity usage revenue (ARC0). For the effect of the bursty arrival we have to inspect column 5. We see that there is a minor negative effect of the bursty arrival in case we have

only a minimal revenue of the alternative usage of capacity.

5.4. Alternative capacity usage effects

Now the interesting question is whether there are situations where this negative effect disappears, and possibly even becomes positive, for other values of the alternative capacity usage revenues. In Table 7 we see the relative differences in costs between immediate admission and periodic admission in case there are no transportation cost reductions, for different values of the alternative capacity usage revenues. We see that there is a significant impact of the alternative capacity usage function: due to the fact that with a D > 0 more time can be spent on alternative usage (without being interrupted due to the arrival of new failed goods) and especially with a high value for the sensitivity factor, the differences have almost disappeared or have even become in favour of periodic admission. So, even in case of no transportation cost bene- fits, there are situations were periodic admission leads to benefits (be- tween 3% and 4%) compared to immediate release.

Table 8 gives the combined effects of the different values of alter- native capacity usage revenues and different values of transportation

Table 3 (Immediate with allowance = 0 - Periodic)/(Immediate with allowance = 0) for the optimal value of D in case of periodic admission.

h TC0 TC1 TC2 TC3 TC4

D % D % D % D % D %

ACR0 0.2 0.5 −5.30 0.5 −4.53 0.5 −2.59 0.5 −3.76 0.5 0.12 0.25 0.5 −6.19 0.5 −5.46 0.5 −3.64 0.5 −4.73 0.5 −1.10 0.3 0.5 −7.03 0.5 −6.34 0.5 −4.62 0.5 −5.66 0.5 −2.21

ACR1 0.2 0.5 −4.99 0.5 −4.22 0.5 −2.28 0.5 −3.45 0.5 0.43 0.25 0.5 −5.86 0.5 −5.13 0.5 −3.32 0.5 −4.41 0.5 −0.77 0.3 0.5 −6.69 0.5 −6.00 0.5 −4.28 0.5 −5.32 0.5 −1.87

ACR2 0.2 0.5 −3.49 0.5 −2.72 0.5 −0.78 0.5 −1.95 1 2.04 0.25 0.5 −4.29 0.5 −3.56 0.5 −1.74 0.5 −2.84 0.5 0.81 0.3 0.5 −5.06 0.5 −4.37 0.5 −2.64 0.5 −3.68 0.5 −0.24

ACR3 0.2 0.5 −4.36 0.5 −3.59 0.5 −1.65 0.5 −2.81 0.5 1.06 0.25 0.5 −5.19 0.5 −4.47 0.5 −2.65 0.5 −3.75 0.5 −0.10 0.3 0.5 −6.00 0.5 −5.31 0.5 −3.59 0.5 −4.63 0.5 −1.18

ACR4 0.2 0.5 0.33 0.5 1.10 1 4.08 1 2.19 1.5 10.01 0.25 0.5 −0.27 0.5 0.46 1 2.44 0.5 1.18 1.5 7.03 0.3 0.5 −0.89 0.5 −0.20 0.5 1.53 0.5 0.49 1 5.22

Table 4 Optimal costs for immediate admission (I) and periodic admission (P) with a value of D equal to the optimal value of D in case of periodic admission and minimal alternative capacity usage (ACR0) for different transportation costs multiplicator functions.

h TC0 TC1 TC2 TC3 TC4

D costs D costs D costs D costs D costs

I 0.2 0.5 1.6035 0.5 1.5916 0.5 1.5616 0.5 1.5796 0.5 1.5196 0.25 0.5 1.7162 0.5 1.7042 0.5 1.6742 0.5 1.6923 0.5 1.6322 0.3 0.5 1.8192 0.5 1.8073 0.5 1.7773 0.5 1.7953 0.5 1.7353

P 0.2 0.5 1.6322 0.5 1.6202 0.5 1.5902 0.5 1.6083 0.5 1.5482 0.25 0.5 1.7521 0.5 1.7401 0.5 1.7101 0.5 1.7281 0.5 1.6681 0.3 0.5 1.8623 0.5 1.8503 0.5 1.8203 0.5 1.8383 0.5 1.7784

Table 5 (Immediate with allowance = D/2 - Periodic)/(Immediate with allowance = D/2) with optimal value of D in case of periodic admission and minimal alternative capacity usage revenues for different transportation costs multiplicator functions.

h TC0 TC1 TC2 TC3 TC4

D % D % D % D % D %

ACR0 0.2 0.5 −1.79 0.5 −1.80 0.5 −1.83 0.5 −1.82 0.5 −1.88 0.25 0.5 −2.09 0.5 −2.11 0.5 −2.14 0.5 −2.12 0.5 −2.20 0.3 0.5 −2.37 0.5 −2.38 0.5 −2.42 0.5 −2.40 0.5 −2.48

Table 6 Cost components of immediate admission and periodic admission in case there are no transportation cost reductions and there is only a minimal alternative capacity usage revenue; RCT = renting cost during transportation; E(RHC) +CRC = renting cost during repair + capacity costs; TC = transportation costs.

ACR0/TC0 h D RCT E(RHC)+CRC TC Total cost

Immediate 0.2 0.5 0.05 0.8485 0.705 1.6035 0.25 0.5 0.0625 0.9487 0.705 1.7162 0.3 0.5 0.075 1.039 0.705 1.8192

Periodic 0.2 0.5 0.075 0.8522 0.705 1.6322 0.25 0.5 0.09375 0.9533 0.705 1.7521 0.3 0.5 0.1125 1.0448 0.705 1.8623

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costs. We see that the negative effect of the bursty arrival in most of the situations is almost completely compensated by the revenues of the alternative capacity usage: the costs of periodic admission are only 1%–2% higher than in the cases with immediate admission. If the maximum revenues of alternative capacity usage and the sensitivity are rather high, the capacity effects of periodic admission lead to significant lower costs compared to immediate admission (between 3% and 7%, depending on the values of the transportation costs).

To summarize our findings: we are not searching for the whole range for which our approach is beneficial, but we want to show that there are circumstances, which to our belief are realistic, for which the approach is beneficial. Since nothing can be found in literature on (parameters of the) transportation cost reduction function and the (parameters of the) alternative usage of capacity revenue functions, we have chosen values that in our opinion are realistic, however this could not been verified. We have used a rather low value and a rather high value for the maximal reduction in transportation costs (if there is unlimited time to consolidate) and a rather low sensitivity value (the reduction gradually increases) and a rather high sensitivity value (a lot of the maximal reduction with a short period to consolidate and then slowly decreasing with increasing consolidation time); for the para- meters of the alternative usage of capacity revenue function the same holds.

Overlooking the results of our numerical study, we conclude that there indeed exist complex interactions between the transportation processes and transportation costs, on the one hand, and the repair processes and repair costs, on the other hand. This calls for careful consideration of the specific characteristics of individual situations where the use of periodic collection of failed goods is considered. One condition for application seems to be that the revenue of alternative use of repair capacity must be high enough to compensate for the (minor) costs caused by the bursty arrival of jobs. Whether or this suffices for justify application of periodic admission depends whether the costs

savings from combined transportation exceed the increase in renting costs. If this is the case, application is beneficial; if not, resulting loss must at least be compensated by the (possible) reduced costs of running the repair shop.

6. Conclusions

In this paper we studied a maintenance service and repair provider being responsible for the availability of a certain type of capital good at a number of industrial customers. The MRSP takes responsibility for renting substitute goods, the transportation of the failed goods to the repair shop, the repair of the failed goods and the transportation of the repaired goods to the customers. We considered a transportation mode where the failed good immediately has to be collected and transported to the repair shop, so there is no time slack for the logistics service provider, and a mode where the logistics service provider gets some time slack for collecting and transporting the failed good. This slack can be used to get efficiency benefits, which we assume are partly trans- ferred to the repair shop. Moreover, we concentrated on repair shops where idle repair shop capacity can be used for other tasks and then bring some revenue. These revenues of using the capacity for other work depends on the time the capacity is available.

It turned out that for many realistic parameter settings the optimal value for the slack in case of immediate admission turned out to be negative. In case of immediate admission we therefore postulate that for most realistic situations it is not beneficial to give the LSP some slack for transporting the failed good to the MRSP.

The main purpose of this research was to investigate whether there are situations where periodic admission can be beneficial. Periodic ad- mission leads to a bursty arrival effect, however, we have seen that there is only a small negative effect of bursty arrivals and that there is a significant impact of the alternative capacity usage function: due to the fact that with a D > 0 more time can be spent on alternative usage (without being interrupted due to the arrival of new failed goods) and especially with a high value for the sensitivity factor, the negative ef- fects have almost disappeared or have even become positive in favour of periodic admission. So, even in case of no transportation cost bene- fits, there are situations were periodic admission leads to benefits (be- tween 3% and 4%) compared to immediate release.

If there are also transportation costs benefits, we showed that there can be realistic situations where periodic admission of failed goods to the repair shop leads to significant total cost benefits compared to immediate admission (up to 7% lower costs). However, also many si- tuations exist where periodic admission leads to an increase of total costs, although the increase is often rather small (1%–2%). So, if be- sides the transportation benefits and alternative capacity usage there are other benefits when using periodic admission, it is worthwhile considering to use periodic admission. The optimal value of the length of the interval depends on the specific situation.

This study shows that it can be worthwhile for a MRSP (or equivalent company) to consider working with periodic admission, especially if there are also other positive (cost) effects of using periodic admission then the ones used in this study, this can be of much value to the company. Especially if the company is surrounded by many com- parable high tech industries (increases the possibility to find work in case the own capacity runs idle) and repairs relatively small items (easy to consolidate), periodic admission can be beneficial. However, the costs effects carefully have to be determined since there are situations where this can be rather beneficial, but there are also situations where the total system wide costs increase. We used parameter values for the different functions that in our opinion reflect what could be seen in practice, however, it could be interesting to see what the effects are in more extreme situations. It would also be worthwhile to investigate the transportation cost reduction function and the alternative capacity usage revenue function in practice; do they really look like shown in Fig. 3a and b or are they totally different? If the latter holds, what are

Table 7 (Immediate with allowance = D/2 - Periodic)/(Immediate with allowance = D/2) with optimal value of D in case of periodic admission and no transportation cost benefits for different values for the alternative capacity usage revenues.

h ARC0 ARC1 ARC2 ARC3 ARC4

D % D % D % D % D %

TC0 0.2 0.5 −1.79 0.5 −1.48 0.5 −0.04 0.5 −0.87 0.5 3.66 0.25 0.5 −2.09 0.5 −1.77 0.5 −0.27 0.5 −1.14 0.5 3.60 0.3 0.5 −2.37 0.5 −2.05 0.5 −0.48 0.5 −1.39 0.5 3.51

Table 8 (Immediate with allowance = D/2 - Periodic)/(Immediate with allowance = D/2) with optimal value of D in case of periodic admission (with different transportation costs multiplicator functions and different values of alternative capacity revenues).

h TC1 TC2 TC3 TC4

D % D % D % D %

ACR1 0.2 0.5 −1.50 0.5 −1.52 0.5 −1.51 0.5 −1.57 0.25 0.5 −1.79 0.5 −1.82 0.5 −1.80 0.5 −1.87 0.3 0.5 −2.05 0.5 −2.09 0.5 −2.07 0.5 −2.14

ACR2 0.2 0.5 −0.04 0.5 −0.03 0.5 −0.04 1.0 −1.07 0.25 0.5 −0.26 0.5 −0.27 0.5 −0.27 0.5 −0.28 0.3 0.5 −0.48 0.5 −0.49 0.5 −0.49 0.5 −0.51

ACR3 0.2 0.5 −0.88 0.5 −0.90 0.5 −0.89 0.5 −0.92 0.25 0.5 −1.14 0.5 −1.17 0.5 −1.15 0.5 −1.20 0.3 0.5 −1.39 0.5 −1.41 0.5 −1.40 0.5 −1.45

ACR4 0.2 0.5 3.69 1.0 5.77 1.0 5.67 1.5 6.91 0.25 0.5 3.62 1.0 5.48 0.5 3.65 1.5 6.23 0.3 0.5 3.53 0.5 3.60 0.5 3.55 1.0 5.39

H. van Ooijen et al. International Journal of Production Economics 208 (2019) 29–42

39

then the consequences for our findings? Moreover, it might be inter- esting to investigate whether there are other benefits related to periodic

admission then the ones taken into account in this research. If possible these should be expressed in costs and/or revenues.

Appendix A

It is well-known that if there is a large number of goods that can fail with the same failure rate, the aggregate failure process can be approximated by a Poisson process. Assuming that there is more or less immediately a truck available to transport a failed good, the transporting process can be seen as a M/G/∞ queueing process. Since the throughput time of such a process follows a negative exponential distribution (see Daley (1976)), failed goods arrive at the repair shop according to a Poisson process. If we allow a certain delay in the transporting process this influences the “service” time of the process, the assuming that these delays are independent are uncorrelated in time and uncorrelated with travelling times the former still holds.

Appendix B

Again we assume that the repair shop is located approximately in the middle of the region it serves, that this region has a, more or less, circular shape with radius a and a size of S km2 and that the transportation company is located (more or less) close to the MRSP. Furthermore we assume that occurrence of failure goods that have to be transported follow a Poisson process with a constant arrival rate λ and that the speed of a truck is on average v km per unit time. Due to the periodic admission with interval D the LSP experiences (besides the transportation time), some extra time A to deliver the failed goods to the MRSP, since the failed good has to be delivered to the MRSP at the start of the first period that follows the time of failure plus the transportation time. As in the base case situation, described in 3.2, this allowance gives opportunities to combine transport and thus to reduce transportation costs. Due to this reduction in transportation costs, the MRSP gets a discount of the price it has to pay per unit time that depends on the period length D.

Of course, the decrease in transportation costs, due to the possibility to combine the transportation of a number of goods, depends, amongst others on the size of the goods and the fleet of the transportation company. However we think that the decrease in costs due to combined trans- portation of goods in a general way, can be modeled in the following way:

We assume that the transportation company uses the average extra time Ā to determine the discount in transportation costs that is given to the MRSP. The extra time A of course depends on the period length D, the geographic place where the failure takes place and the moment the failure takes place. Suppose the radius of the environment of the MRSP is R, that the company uses a period length D, that the failure takes place in a point with a distance r from the MRSP and that it takes place at a time nD + x for some n with x < D. Given r and the transportation speed v it takes r

v 2

units of time to pick up the failed good and to deliver it to the MRSP. Then for Ā we have:

= + + + = = =

A r R D

entier x r v

D D x r v

dxd dr¯ 1 { [ ( 2 )/ ] ( 2 ) } r

R

x

D

0 0

2

2 0

Suppose for given x and r that k is such that +entier x D[ ( )/ ]r v

2 = kD. Then

+ + + =

entier x D D x dx{ [ ( )/ ] ( ) } x

D

D r

v r

v 0

1 2 2 = + + = =

+

+ kdx k dx( 1)

x

k D

k D

D

0

( 1)

( 1)

r v

r v

2

2 + + + + =k k D k D k D( ( 1) ) ( 1) ( ( ( 1) ) )r

v r

v r

v 2 2 2 (which

is independent of k!) and

+ = =

D D x r

v dx D r

v 1 ( ( 2 ) ) 1

2 2

x

D

0

This gives:

= = = = = = = =

A r R

r v

D r v

d dr D R

rd dr D R

R D¯ ( 2 ( 1 2

2 ) ) 2

1 2 2

r

R

r

R

0 0

2

2 2 0 0

2

2 2

Appendix C

Sojourn Time Distribution in a D M/ /1X Queue In this appendix we analyze the sojourn time distribution of the D M/ /1X queuing model. Note that “customers” refer to the number of failed

goods and the capacity level of the repair shop corresponds to the processing rate of the queue, μ. The length of the period between two admission points is equal to D.

At any time point t , the total number of customers, N t( )d , is the sum of the number of customers in the queue (including the one in the service), N t( )q , and the number of customers outside the queue, N t( )o , that are waiting to be admitted. At the start of each period, all customers outside the queue are admitted into the queue based on their arrival order, therefore we have =N nD N nD( ) ( )d q and =N nD( ) 0o for =n 1,2, ... =P N i{ }q and

=P N i{ }o are the limiting probabilities that there will be “i” customers after “t ” time units from the start of an arbitrary interval, in the queue and outside the queue waiting to be admitted, respectively, under policy C = D µ[ , ] for = …i 0,1,2, and t D (see Büyukkaramikli (2012)).

In the derivation of the sojourn time distribution of an arbitrary customer, we are primarily interested in the number customers when: t D , i.e. at the end of an interval, just before the admission time point. =P N i{ }q can be derived under any capacity policy = D µ[ , ] (see Büyukkaramikli (2012)) We also know that =P N i{ }o is independent from the capacity level µ and is Poisson distributed with a mean equal to D.

The position of an arbitrary arriving customer in the queue, determines that customer's sojourn time in the queue. The position of an arriving customer in the queue depends on the number of customers left over from the previous period as well as the size of the group of customers, which the arriving customer belongs to. Under a given capacity policy C = D µ[ , ], if the position of a given arriving customer in the queue is n, then the

H. van Ooijen et al. International Journal of Production Economics 208 (2019) 29–42

40

sojourn time of that arriving customer is the sum of n service times in the queue. Since the service time of each customer is exponentially distributed with mean ( µ1/ ), the sojourn time of an arriving customer whose position in the queue upon arrival is n, will be Erlang (n µ) distributed. Due to its critical importance in the derivation of the sojourn time distribution, next we derive the steady state probability that an arbitrary arriving customer's position in the queue is n.

Lemma. Suppose G denotes the size of the group of customers that an arbitrarily chosen customer belongs to. The probability that an arbitrary arriving customer belongs to a group of customers that has a size of g , =P G g{ } can be written as:

= = × =

= ×

P G g g P N g

E N

g

D { }

( ) ( )

o

o

e D g ( ) !

D g

(1)

Proof. Suppose the number of the customer group sizes that are admitted to the queue are recorded during a large number (M ) of intervals. By the law of large numbers (Ross, 1983), for large enough M , the fraction of the groups of size g would go to =P N g{ }o , the number of groups having a size of g would be equal to × =M P N g{ }o , and the total number of customers coming from a group having size g would be × × =g M P N g{ }o . Similarly, the sum of all of the customers from these M groups would be × × == g M P N g{ }g o0 . Without loss of generality, assuming that =( )01 0, if M goes to infinity, and a randomly customer is picked, the probability that the picked customer is coming from a group of size g can be written for

= …g 0,1,2, as below:

= =

× × =

× × = =

×

=

P G g

M g P N g

g M P N g

g

D { } lim

{ } 1

{ } 1

M

o

g o

e D g

0

( ) !

D g

From Lemma 1, we can proceed to the derivation of the probability of the order of an arbitrary arriving customer.

Theorem. Suppose O denotes the position of an arbitrary arriving customer in the queue. The probability that an arbitrary arriving customer’s position is equal to n for >n 0 can be written as follows:

= = = = = = = = = =

P O n P N i P G g g

P N i D

{ } { } { } 1 { } g n i i

n

q g n i i

n

q

e D g

0

1

0

1 ( ) !

D g

(2)

Proof. The position of an arriving customer in the queue consists of the number of customers who are left over from the previous period (Nq) and the newly arrived group of customers, G, of which the corresponding customer belongs to. In order to have an arriving customer having a position of n, we need to have <N nq and >G n N( )q and the order of the arbitrary customer in the arriving group should be equal to n N( )q . Note that if the size of a group is equal to g ( =G g ), the probability that a given customer’s order is n N( )q within that group is g1/ . By summing all possible Nq and G combinations, we obtain =P O n{ } in (2).

From this Theorem, we can derive P S x C{ | } for any given policy C = D µ[ , ] as follows:

= = = = = = = = = =

P S x P O n P S x O n P N i D

P S x O nC C C{ | } { } { | & } { } { | & } n n g n i i

n

q

e D g

1 1 0

1 ( ) !

D g

(3)

where = = =P S x O n eC( | & ) j n x x

j0 1 µ (µ )

!

j , which is the tail distribution of an Erlang (n µ) distributed random variable.

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