A MODEL FOR DESIGNING MULTI-ECHELONINVENTORY NETWORKS WITH FINITE CAPACITY

Ben D. Van Roo, James A. Rappold1

Doctoral Candidate, University of Wisconsin, Madison, WI 53706, bdvanroo@wisc.edu

Abstract: We propose an approach to model and solve the joint problem of facility location, inventoryallocation, and capacity investment when demand is stochastic. The objective of the decision problem isto minimize the total expected costs of (1) opening a number of repair facilities, (2) assigning each fieldservice location to an opened facility, (3) assigning capacity levels to each of the opened repair facilities,and (4) determining an inventory allocation among the opened repair facilities and field service locations.

Keyword: Multi-echelon inventory optimization; Facility location; Finite capacity; Supply chain design;Service parts logistics.

1 Introduction

When designing a multi-echelon supply chain network for reparable parts that balances costefficiency with effectiveness, several questions must be answered and decisions made. Some ofthese questions are strategic and establish the basic infrastructure of the supply chain. Suchstrategic questions include: How many repair facilities are needed to support the system andmeet customer requirements? How much capacity, in terms of equipment and labor, are neededat each repair facility? Where should these facilities be located? Which field stocking locations(or customer service centers) should be served by which repair facilities? While there are manyother additional strategic questions, the few that we have listed are themselves not easy decisionsto make. They are not easily changed and can establish the basic operational environment foryears. Moreover, they must be robust to contend with a constantly changing environment andset of customer requirements.

The alternatives to these strategic decisions will have tactical operating consequences interms of cost and responsiveness. Some tactical consequences are: What will the transportationcosts be of a supply chain design? How much inventory and working capital will be necessaryto support customer requirements? Does sufficient inventory even exist to support customerrequirements associated with a particular supply chain design? The answers to these questionsfundamentally depend on the strategic design of the supply chain.

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One Capacitated Repair Facility

Depot Stocking Location

Field StockingLocations

Ite

m R

eco

very

Random Item Demand Random Item Demand

Ite

m R

eco

very

Ite

m R

eco

very

TwoCapacitated

Repair Facilities

Supply Chain Design #1 Supply Chain Design #2

Figure 1: An example of two alternative supply chain designs.1Dissertation Advisor, University of Wisconsin, Madison, WI 53706, jrappold@bus.wisc.edu

1

The strategic and tactical considerations are therefore inextricably linked. For example,Figure 1 illustrates two alternative supply designs. Items that require repair fail at field stockinglocations and are sent back to an assigned repair facility through an item recovery process. If areplacement item is available at the depot stocking location, it is dispatched to the field stockinglocation. If not, the item is backordered and one is sent when one becomes available. Itemsarrive at the repair facility, enter a repair queue, and are repaired by a repair team (or crew).The repaired item is stocked at the depot until one is required at a field stocking location.

On the one hand, an organization could consolidate (or outsource) its repair capacity intoone facility as in Supply Chain Design 1, seeking repair economies-of-scale and perhaps a highcapacity utilization; however, if the customer base is geographically dispersed, transportationcosts may overwhelm operating costs, depending on the particular economic circumstances.Moreover, the pipeline and safety stock necessary to support long lead times (of item recovery,queueing, repair, and transport) may lead to high working capital requirements throughout thenetwork.

An alternative design, Supply Chain Design 2, is to split the repair capacity into two differentfacilities (one of which may be co-located with a field stocking location) and assign field stockinglocations based on transportation and inventory considerations. Due to the interdependentnature of strategic and tactical considerations, we believe that it is imperative to include thetactical consequences explicitly in the strategic decision-making process.

Unfortunately, in practice this seems more often an exception rather than the rule. Thereason is because these decisions are complicated by many quantitative and qualitative factorsthat are difficult for groups of decision and policy makers to consider and render an informedand rational decision. Further complicating matters, there are an enormous number of possi-ble supply chain design alternatives to evaluate and usually limited time and data to do so.Therefore, decision models can play a valuable role in evaluating competing alternatives.

Our objective in this research is to develop a set of integrated quantitative and computa-tionally efficient decisions models that can be used to assist in this complicated decision process.Our models and modeling framework must be flexible enough to handle a wide variety of con-siderations, yet computationally efficient enough to support a rapid, iterative decision process.

2 An Application

While we have observed this problem in industrial firms, the motivation for our particularproblem grew out of research performed by one of the authors in conjunction with RANDProject Air Force and the US Air Force (USAF). The issue is how to best configure the supplychain for the repair and deployment of aircraft engines and other complex aircraft subsystems.

The repair of aircraft engines is a notoriously difficult and complicated task that requiresteams of skilled and specialized technicians and equipment. The demand for repair is highlyuncertain and extremely difficult to forecast accurately. On the one hand, the repair of theseitems may be performed locally at each air force base (field stocking location), thereby requiringduplication of repair facility overhead across the supply chain. This design has the advantagethat the turnaround times for failed items is fast and the pipeline inventories necessary tosupport operations is small; however, it has the disadvantage that it is extremely expensive tooperate.

Studies at The RAND Corporation have shown that economies-of-scale do exist as the repaircapacity is increased at a facility. It should be noted that repair capacity is increased in discreteblocks, consisting of a team of technicians and their equipment. An alternative supply chaindesign is to centralize the repair of these items at one repair facility to serve the repair needs ofall field stocking locations. While the labor and equipment costs would be significantly lower,the transportation costs, inventory requirements, and response times would be higher.

2

A unique characteristic of the repair of aircraft engines for the US Air Force is that, depend-ing on the specific aircraft, there is likely a fixed number of engines in the system at any time.This quantity has been determined a priori, often years in advance, and is not in changeablein the short-term. Therefore, the reparables supply chain design must include an inventoryconstraint on the average number of aircraft engines in the system. It is not immediately clearhow to approach thinking about this problem, how to quantify the unique operating character-istics and requirements of each field stocking location and repair facility, nor how to arrive atan informed set of decisions that can be justified in front of a Congressional committee.

3 Literature Review

Although significant advances have been made in multi-echelon inventory theory, facility loca-tion problems, and queueing theory, research opportunities exist at the intersection of theseareas. In particular, most multi-echelon models are concerned with optimizing inventory per-formance in the presence of uncertainty given a network of facilities. On the other hand, facilitylocation models have been approached from a combinatorial optimization perspective and onlyrecently has uncertainty been incorporated explicitly. Though we know of no work that directlyencompasses the joint location, capacity, and inventory aspects of our problem, we believe thatintegrating many of the insights gained from facility location theory with the design and opera-tion of multi-echelon inventory systems presents opportunities for both research and application.

Beginning with the queueing aspects of our problem, we use the results from Prabhu (1965)for M/M/k queueing systems. The multi-echelon inventory aspects of our problem are linkedto the seminal paper Sherbrooke (1968). Sherbrooke applied Palm (1938) in his formulation ofa model for Multi-Echelon Technique for Recoverable Item Control (METRIC). His techniqueoptimizes inventory levels in the presence of demand uncertainty by taking into considerationthe time delays caused by material shortages at upstream locations.

We leverage an important extension made by Graves (1985), which incorporated a newmethod of computing both the mean and the variance of the expected number of outstandingorders at the base level. This greatly improved the accuracy of the overall approach. To ourknowledge, Sleptchenko (2002) was the first to extend Graves (1985) by explicitly consideringfinite repair capacity. In his paper, Sleptchenko reformulates Graves (1985) by specifying arepair shop model and selecting a multi-class M/M/k repair queue found in van Harten andSleptchenko (2003). He formulated the model by using the convolution of the arrival processwith the repair queueing time when formulating the mean and variance of the expected numberof outstanding orders in the system.

The facility location aspects of our problem are a special case of general facility locationproblem. We determine the best choices of facilities given a set of candidate locations, andassign demand and capacity to these best choices. Our problem is formulated as a mixed-integerprogram. Similar models have been studied and, even in the case of hundreds of locations, itis not computationally expensive to find the mixed-integer solution. A comprehensive surveyof facility location models can be found in Drezner (1995). Location models under uncertaintyare reviewed in Snyder (2004). More specific to our problem, Eppen and Schrage (1981) andDaskin et al. (2002) developed models with both location and inventory considerations, andthey study impacts on inventory as well as benefits of “risk pooling.” Our work differs fromtheir work in that we are interested in a multi-echelon system and while selecting an optimallocation, we also seek to determine its ideal level of capacity investment.

We stress that the focus of our model is strategic and tactical in nature. It has been shownthat real-time execution decision models can outperform tactical first-come, first-served policiessubstantially. See Pyke (1990) and Caggiano et al. (2002) for further discussion.

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4 The Problem

We formulate our combination facility-location, multi-echelon inventory optimization prob-lem as a mixed-integer program with non-linear partial expectation cost functions. The objectivefunction is to minimize the long-run expected costs of the system. Computational efficiency is arequirement of our model. We examine the problem structure and determine solution methodsthat balance solution speed with accuracy.

We consider a single-item, single-indentured parts network. This is a reasonable startingpoint for our situation, as specific aircraft platforms often require specialized and dedicated crewand equipment; however, it is worthwhile to consider possible extensions to address a multi-item, multi-indentured environment. The operating cost drivers in our case are: (1) fixed repairfacility costs, (2) repair crew and equipment (repair center) capacity costs, (3) transportationcosts, (4) inventory holding costs, and (5) backorder costs at field stocking locations (FSLs).

4.1 Assumptions and Notation

We assume that the problem is for a single-item, single-indentured parts network. The failuresat field stocking locations follow a Poisson distribution. Repair facility times are assumed tohave an exponential repair distribution and are independent and identically distributed. Usingthe exponential service distribution time is not only convenient, it also closely approximates ourempirical data. We assume constant transit times between locations, and a first-come-first-serverepair, one-for-one replenishment policy with no fixed cost of transportation or setup times inrepair. Assignments between field stocking locations and repair facilities are unique, and thereare a finite number of identical, parallel repair servers at each repair facility.

As mentioned earlier, the total amount of system inventory is determined a priori and is ahard constraint in the decision model. We assume that the total amount of inventory allocatedacross all locations must be exactly equal to the total amount of inventory available. In certaincases, this amount of system inventory may be less than desirable and the model will increaserepair capacity to reduce the need for inventory. In other cases, this amount of system inventorymay be too high and the model will have to determine the best location to hold excess inventory.

Let G be the full set of possible repair facility locations in the network, with index j ∈ G.Let F be the set of FSLs, with index i ∈ F . The notation F j will refer to the set of FSLsassigned to open repair facility j ∈ G. The failures at i ∈ F j , follow a Poisson process withrate λji, where λj0 is the total demand rate assigned to open repair facility j. Repair timesfor each repair center at j are assumed to be exponentially distributed µj , and are independentand identically distributed (i.i.d.) across the repair servers. Each server within repair facilityj is assumed to have identically distributed repair times. Failed units from various FSLs arerepaired and distributed in a first-come, first-served manner. We assume constant transit timesbetween locations, lji. Inventories are controlled via a one-for-one replenishment policy, and theholding costs and backorder costs associated with the inventory are independent of the numberof servers at a repair facility. We assume that there are no fixed costs of transportation or setuptimes in repair. Let [x]+ = max{x, 0} for x real. We define the following notation:

Sets and Indexes:

G = set of possible repair facility locations, with index j ∈ G;

F = set of field stocking locations (FSLs), with index i ∈ F ;

F j = set of field stocking locations assigned to repair facility j ∈ G; and

Z = set of non-negative integers, with index k ∈ Z.

Decision Variables:

sj0 = base stock level of the item at facility j;

4

sji = base stock level of the item at field stocking location i ∈ F j ;

xj ={

1 if facility j ∈ G is opened;0 otherwise;

yji ={

1 if field stocking location i ∈ F is assigned to be supplied from facility j ∈ G;0 otherwise;

zjk ={

1 if facility j ∈ G is assigned k ∈ Z or more repair machines;0 otherwise; and

zj =∑

n∈Z zjn, the total number of repair machines installed at facility j.

Model Parameters:

λji = the mean failure rate of the item at a field stock location i ∈ F j ;

λj0 =∑

i∈Fj λji = the mean demand rate for reparables assigned to facility j ∈ G;

µj = the mean repair rate per machine at facility j ∈ G;

µ̂j = zjµj , the total repair capacity of j ∈ G based on the number of assigned machines;

lji = the constant transit lead time from facility j to field stocking location i;

cji = the transportation cost per unit from facility j to field stocking location i;

fj = the fixed facility cost of opening facility at j ∈ G, amortized annually;

ajk = the annual cost of adding the kth repair capacity increment (to capture repaircost economies-of-scale) at facility j ∈ G;

hji = the holding cost rate for the item at field stocking location i ∈ F j , where hj0 isthe holding cost rate at facility j ∈ G, in dollars per unit per year;

πji = the backorder cost rate for the item at i ∈ F j in dollars per unit per year; and

S = the steady-state total amount of inventory in the network.

Key Relationships: For a given set of FSL-to-repair assignment decisions [yji], we define:

ρj(yji, zj) =(∑

i∈Fj yjiλji

)/zjµj = the repair capacity utilization of facility j ∈ G;

Dj0(λj0, zj) ∼ Fj0, the random variable (r.v.) representing the item demand at j duringthe repair lead time (the queue time plus the service time) in steady-state;

Ψj(yji, zj , sj0) = E[sj0 −Dj0(λj0, zj)]+ = the expected number of units on-hand at facilityj associated with sj0, in steady-state;

E[Dj0(λj0, zj)− sj0]+ = the expected number of outstanding replenishment backorders atj associated with sj0, in steady-state;

τj(yji, zj , sj0) = E[Dj0(λj0, zj) − sj0]+/λj0 = the expected replenishment delay from j toi ∈ F j , steady-state;

Dji(λji[τj + lji]) ∼ Fji(λji[τj + lji]), the r.v. representing the demand during the replen-ishment lead time to FSL i, from repair facility j, in steady-state;

Γi(yji, zj , sj0, sji) = E[sji−Dji(λji[τj + lji])]+ = the expected number of units on-hand atFSL i, served by facility j given sji, in steady-state; and

Φi(yji, zj , sj0, sji) = E[Dji(λji[τj + lji]) − sji]+ = the expected number of backorders atFSL i, from facility j given sji, in steady-state.

5

An important aspect of our problem is the opportunity to reduce labor costs througheconomies-of-scale in consolidating the repair facilities. At the strategic level, due to theirhigh cost, we first consider which repair facility locations to open. The amortized annual costof opening facility j is fj . To capture the repair cost economies-of-scale from consolidation, letzj be number of allocated parallel repair centers (servers), if facility j is opened. The cost ofadding the kth repair center increment is ajk. In this way, we can model the repair facility costas a concave cost function.

At a tactical level, our objective is to jointly optimize the repair capacity and the allocationof available inventory across the network, given the open repair facilities. We will do this bydecomposing the supply chain into a collection of sub-networks, each comprised of one repairfacility j and its associated FSLs, F j . Within each sub-network, the multi-echelon inventorydecisions are concerned with the base stock levels sj0 at each repair facility j, as well as thebase stock levels sji at each FSL i ∈ F j . To determine these values, we will leverage the work ofGraves (1985) along with results from the queueing literature. Our model deviates from Graves’work since we cannot properly invoke Palm’s Theorem (Palm (1938)) based on an assumptionof infinite repair capacity in Sherbrooke (1968).

Because of the parallel repair centers, each with a finite capacity, replenishment lead timesto the FSLs will not merely be variable, but will also be positively correlated – especially forthe single server case. We will model the queue and distribution of repair times as an M/M/kqueueing system and use results of Prabhu (1965). Let ωj(yji, zj) be the steady-state expectednumber of items in repair facility j, either awaiting repair or in repair, as a function of theassigned FSLs and number of repair servers. Let ω2

j (yji, zj) be its second moment. We employresults from Prabhu (1965) for the exact steady-state distribution of the number of units in anM/M/k system to compute, ωj , and ω2

j .This distribution is important for computing the expected number of backorders at the

repair facility associated with a base stock level of sj0 units, and its associated time delay to theFSLs. This dependence between inventories at the repair facility and inventories at the FSLsprevents us from separating the problem by location.

An overview of the solution strategy is as follows. We will determine the first two moments,ωj and ω2

j , for a set {zj} of possible repair capacities at j, and fit them to a negative bino-mial distribution. While we could employ the exact steady-state probability distribution of anM/M/k system for the number of units in repair, we choose to use a two-moment negativebinomial approximation to this distribution for computational speed. This approximation isextremely close for a wide variety of number of servers k and capacity utilizations. Most im-portantly, this approximation yields virtually identical estimates of the time delay to the FSLsfor a given sj0.

To optimize the FSL base stock levels sji, we extend the argument of Graves (1985). Wetake the convolution of the FSL demand over the fixed transportation lead time from the repairfacility plus any time delay associated with repair facility backorders. We assume that therepair facility fulfills FSL demand in a first-come, first-served manner. We estimate the firsttwo moments of the number of outstanding orders at each FSL. Again, we fit this to a negativebinomial distribution. This provides a method for optimizing the tactical inventory allocationwithin each sub-network between its repair facility j and among the FSLs, F j . We use theserelationships to estimate the expected cost associated with carrying an amount of inventorySj = sj0 +

∑i∈F j sji in sub-network j, assuming an optimal allocation of inventory within sub-

network j. Finally, linking the inventories across the sub-networks, we impose a tight allocationconstraint on the total amount of system inventory, S =

∑j Sj .

Written as a non-linear program, the objective function is the minimization of the sum of(1) fixed repair facility costs, (2) transportation costs, (3) holding costs at the repair facilities,(4) backorder costs at bases, and (5) holding costs at bases. The Master Problem (MP) is:

6

(MP) minyji,zjk,sji,sj0

∑

j∈Gxjfj +

∑

j∈G

∑

i∈Fλjiyjicji +

∑

j∈G

∑

k∈Zzjkajk

+∑

j∈G

∑

i∈F

(hj0Ψj(yji, zj , sj0) + hjiΓi(yji, zj , sji, sj0) + πjiΦi(yji, zj , sji, sj0)

)

s.t.∑

j∈Fyji = 1 ∀i ∈ F (4.1)

yji ≤ xj ∀i ∈ F , ∀j ∈ G (4.2)∑

i∈Fyjiλji ≤ µj

∑

k∈Zzjk ∀j ∈ G (4.3)

∑

j∈G

(sj0 +

∑

i∈Fsji

)= S (4.4)

zj(k+1) ≤ zjk ∀j ∈ G, ∀k ∈ Z (4.5)

zj =∑

n∈Zzjn ∀j ∈ G (4.6)

sji, sj0 ≥ 0 ∀i ∈ F , ∀j ∈ G (4.7)

Constraint (4.1) creates an exclusive assignment between field stocking location and repairfacilities, while (4.2) assigns FSLs to open repair facilities. Constraints (4.3) is used to setcapacity at a repair facility that corresponds with the total demand assigned to that repairfacility. Constraint (4.4) is the tight bound on the total network inventory. Constraints (4.5)and (4.6) insure incremental increases in capacity levels and capture the total number of repairservers in repair facility j, respectively. Constraints (4.7) insure non-negative stock levels.

5 Two Step Solution Approach

The full formulation (MP) is difficult to solve for large-scale systems with dozens of locationsand hundreds of items; however, for practical purposes, computational efficiency is essential andthe problem structure lends itself well to decomposition into subproblems.

Our problem carries a few dominant features. The empirical data from the USAF indicate:(1) fixed-facility costs, transportation costs, and costs of capacity dominate other costs, (2) costeconomies-of-scale exist through consolidating demand and labor, and (3) the system inventoryis finite. Given these observations, we have developed a two-step solution approach as follows:

1. solve the facility location problem that minimizes the cost of opening facilities, trans-portation costs, and the minimum cost of capacity at each facility; and

2. using solutions from Step 1 (locations, assignments, and minimum capacity), balanceincremental capacity with the cost associated with an optimal allocation of the networkinventory.

5.1 Step 1: Location, Assignment, and Capacity

The first step focuses on the primary cost drivers of the system. To limit capacity utilizationat the repair facility, we introduce an upper bound, ρ̂j , to guarantee system stability of Step 2in the approach. We may set ρ̂j arbitrarily close to, but strictly less than one; however, theremay be other considerations for setting ρ̂j . For example, USAF doctrine stipulates a ρ̂j of 77%.This is done for responsiveness and robustness in case the forecasts of the mean demand ratesshift suddenly and dramatically.

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In addition, we will include a constraint on the number of units dedicated to the trans-portation pipelines of the network. We use V , to represent the total number of units that canbe dedicated to the transit pipelines. The Step 1 linear formulation for the facility locationsubproblem is:

(Step 1) minxj ,yji,zjk

∑

j∈Gxjfj +

∑

j∈G

∑

i∈Fλjiyjicji +

∑

j∈G

∑

k∈Hzjkajk

s.t.∑

j∈Fyji = 1 ∀i ∈ F

yji ≤ xj ∀i ∈ F ,∀j ∈ G∑

i∈Fyjiλji ≤ ρ̂jµj

∑

k∈Zzjk ∀j ∈ G (5.8)

∑

i∈F

∑

j∈G(ljiλjiyji) ≤ V (5.9)

0 ≤ zjk ≤ zj(k+1) ∀j ∈ G, ∀k ∈ Z

Constraint (5.8) now reflects the upper bound on capacity utilization at the repair facility.Constraint (5.9) is included to limit the number of units assigned to the transportation pipelines.The upper bound of V = S, which would permit all of the system inventory to be tied up intransportation pipelines. The tactical costs of such a solution will be revealed in Step 2.

After solving Step 1, we have decomposed the supply chain into M independent sub-networks, where M is the number of opened repair facilities. The repair facilities, a minimumnumber of servers at each repair facility, and the assignments for the field stocking locations torepair facilities are chosen. They are denoted as x∗j , zj =

∑n∈Z z∗jn, and y∗ji, respectively. Let

the set of opened facilities be denoted j ∈ G∗.

5.2 Step 2: Incremental Capacity Gains and Inventory Allocation

The basic topology of the supply chain is determined in Step 1. In Step 2, we balance the costsof incremental repair capacity with the costs of total inventory within each sub-network. Weuse the network design (x∗j , y∗ji, zj) to decide how much additional capacity to assign to therepair facilities, and how to allocate the inventory among the sub-networks (and their locations)optimally. To accomplish this, we must characterize the expect cost for each sub-network j ∈ G∗as a function of its total amount of inventory and repair servers.

For sub-network j, let the total amount of inventory designated to that sub-network beSj = sj0 +

∑i∈Fj sji. For a capacity level zj ≥ zj , we define Cj(zj , Sj) as the minimum

expected cost of repair capacity and inventory holding and backorder costs for sub-network jassociated with zj repair servers and inventory Sj . We will refer to Cj(·) as the capacity-dependent inventory cost function for sub-network j. That is,

Cj(zj , Sj) = minsji,sj0

zj∑

n=0

ajn +∑

i∈Fj

hj0Ψj(y∗ji, zj , sj0)

+∑

i∈Fj

hjiΓi(y∗ji, zj , sji, sj0) +∑

i∈Fj

πjiΦi(y∗ji, zj , sji, sj0)

s.t. Sj = sj0 +∑

i∈Fj

sji (5.10)

sj0, sji ≥ 0 ∀i ∈ F j . (5.11)

8

Constraint (5.10) insures that amount of sub-network inventory Sj is allocated to all loca-tions within the sub-network. Constraints (5.11) insure non-negative base stock levels. Notethat Cj(·) is defined only for zj ≥ zj , corresponding to a sub-network repair capacity utilizationthat is strictly less than 1. Note also that Cj(·) is not convex in Sj for a given zj , although it isnear convex. This issue of non-convexity is well-known and is discussed in Sherbrooke (1968).

Starting with zj = zj , sub-network j has the highest possible utilization (but still lessthan 1) based on an integer number of repair teams. Note that for an M/M/k system, theaverage number of items in the system is monotonically decreasing in the capacity utilization.Consequently, this is the condition in which the expected repair lead time will be longest. Asthe number of repair teams zj increases, the repair cost will increase (concavely) and the repairlead time will decrease, asymptotically approaching the mean repair time µj . There is thereforea limit to the inventory benefit associated with adding repair capacity.

Alternatively, observe that as the amount of inventory Sj increases, the cost advantage of areduced average repair lead time arising from additional repair capacity decreases. This is be-cause additional repair capacity decreases the expected backorder costs for a given Sj ; however,as Sj increases, the expected backorder costs also decrease, and the inventory cost savings ofadding capacity become smaller than the incremental cost of adding capacity. Therefore, fora given sub-network, the capacity-dependent cost curves can be combined to form an optimalcost hull, which we will call SPj(Sj). Define the minimum expected repair and inventory costsfor sub-network j for a given Sj as,

SPj(Sj) = minzj ≥ zj

Cj(zj , Sj).

An illustration of the relationships between Cj(·) and SPj(·) is shown in Figure 2 for in-creasing repair servers (zj , zj + 1, zj + 2). From the convexity of the repair waiting time in zj

and from the near-convexity of the expected inventory holding and backorder costs in Sj , wehave the following important observation: associated with every level of sub-network inventorySj, there is a corresponding cost-minimizing number of repair servers z∗j (Sj). We define,

z∗j (Sj) = arg minzj≥zj

Cj(zj , Sj).

For each sub-network j, we have a means to analytically compute SPj(Sj) to provide anestimate of the long-run expected costs of repair capacity and inventory costs associated withcarrying Sj of inventory. Because the cost functions SPj(·) are separable by sub-network, theproblem lends itself well to a parallel computing environment. Using the SPj(·), we can statethe global optimization problem (GP) of allocating the S units of total system inventory amongthe sub-networks as:

(GP) min∑

j∈G∗SPj(Sj)

s.t.∑

j∈G∗Sj = S.

It is important to again emphasize that the goal of our framework is to integrate strategicand tactical aspects of the problem in a computationally efficient manner. Ideally, we would liketo solve (GP) for the optimal balance of capacity and inventory via a fast marginal allocationalgorithm or Lagrangian relaxation approach. Unfortunately, SPj(·) is not convex in Sj , thushindering such fast approaches.

To deal with the non-convexity of SPj(·), we propose constructing a fast, convex, andaccurate approximation to SPj(·), which we denote SP j(·). There are several attributes ofthe problem that allow us to achieve increased computational performance. Our approach forconstructing fast and accurate approximation to SPj(·) for sub-network j is described in thefollowing steps:

9

Sub-network

Cost vs. Inventory

Inventory

Exp

ecte

d C

ost

),( jjj SzC)( jj SSP

),1( jjj SzC),2( jjj SzC

00

jS

Figure 2: Sub-network cost curves and the cost hull.

1. Identify an upper bound on the repair servers, zj. Consider the sub-networkinventory to be Sj = 0. The expected inventory costs are the average backorder costrate multiplied by the average demand rate multiplied by the average time through thenetwork. This is where additional repair capacity provides the largest cost advantage inthe form of reduced backorder costs.

Begin by setting the number of repair servers zj = zj . If Cj(zj , 0) > Cj(zj + 1, 0),increment zj = zj + 1. Continue until the condition is false, thus determining the zj atwhich there is no longer an economic incentive to add another increment of capacity. LetBj = {zj , ..., zj}, be the set of possible values of zj . Note that SPj(0) = Cj(zj , 0) is theinitial point on the optimal hull.

2. Compute the first two moments of the number of items in repair. The number ofrepair servers affects the steady-state distribution of the number of items in repair. Usinga recursive method based on Prabhu (1965), we compute the first two moments of thenumber of units in repair in an M/M/k system, ωj(zj) and ω2

j (zj), for each zj ∈ Bj . Weapproximate the exact distribution by fitting these two moments to a negative binomialdistribution with very little loss of accuracy.

3. Compute the sub-network cost incrementally. Interestingly, in computing SPj(·),we do not need to calculate Cj(zj , Sj) for all zj ∈ Bj . There are two important attributesassociated with the capacity-dependent cost functions along the optimal hull: (1) onlysuccessive increments of capacity intersect each other in order of increasing Sj , and (2)two capacity-dependent cost curves intersect at most at one point. Therefore, for eachincreasing value of Sj , we only need to compare the current zj and zj−1. Set zj = zj , andset SPj(Sj) = min{Cj(zj , Sj), Cj(zj−1, Sj)}. If Cj(zj−1, Sj) ≤ Cj(zj , Sj), set zj = zj−1.Repeat until Sj = S, where S is the total amount of inventory in the system.

4. Construct a convex approximation, SP j(·). While computing SPj(·), we track in-formation that used to create the approximation curve. For each zj ∈ Bj such thatCj(zj , Sj) = SPj(Sj) for some Sj , let S∗j (zj) be the corresponding cost minimizer ofSPj(·). This is the local minimum of SPj(·) associated with zj . As Sj is incremented,we store the corresponding number of servers zj , the local minimizer S∗j (zj), and the lo-

10

cal minimum cost value SPj(S∗j (zj)). We also obtain the global minimum, SP (S∗∗j ) andglobal optimizer S∗∗j .

Set SP j(0) = SP j(0). For Sj = 1, ...., S, let

∆s(Sj) = |SP j(Sj)− SP j(Sj − 1)|,

or the cost differential between the convex approximation at the prior value of inventory,Sj − 1, and the true cost minimum SPj(Sj). Let

∆g(Sj) =|SP j(Sj)− SPj(S∗∗j )|

|Sj − S∗∗j |,

or the slope between Sj and the global minimum. Let

∆l(Sj) =|SP j(Sj)− SPj(S∗j (zj − 1))|

|Sj − S∗j (zj − 1)| ,

or the slope between Sj and the local minimum along the next level of capacity S∗j (zj−1).Lastly, we let ∆p(Sj) := ∆s(Sj−1), or the slope of the cost approximation at Sj−1. Thisis used to preserve convexity between iterations of Sj . We can then determine SP (Sj)conditionally upon its relative position with respect to the local and global minimums:

SP (Sj) =

min(max{∆s(Sj), ∆g(Sj),∆l(Sj)},∆p(Sj)

), if Sj ≤ S∗∗j and Sj ≤ S∗j (zj − 1);

min(max{∆s(Sj), ∆g(Sj)}, ∆p(Sj)

), if Sj ≤ S∗∗j and Sj ≥ S∗j (zj − 1);

max(min{∆s(Sj), ∆g(Sj)}, ∆p(Sj)

), if Sj ≥ S∗∗j and Sj ≤ S∗j (zj − 1);

max(min{∆s(Sj), ∆g(Sj),∆l(Sj)},∆p(Sj)

), if Sj ≥ S∗∗j and Sj ≥ S∗j (zj − 1).

We then set ∆p(Sj) = SP (Sj) and increment Sj until Sj = S.

We chose this approximation technique over creating the convex lower bound of the cost hullbecause this approximation procedure is both accurate and very fast. The approximation pro-cedure is O(N) versus O(N2) for the exact convex lower bound.

Utilizing the collection {SP j} of convex cost approximations (one for every sub-network),we are able to solve a reasonably accurate approximation to (GP) and simultaneously optimizerepair capacity and the allocation of the entire network inventory S to the sub-networks via asimple marginal allocation algorithm:

(Step 2) min∑

j∈G∗SP j(Sj)

s.t.∑

j∈G∗Sj = S.

We will examine the accuracy of this approximation under various system conditions througha comprehensive numerical study.

6 Numerical Study

To assess the quality and accuracy of our approach, we perform two numerical studies. Thepurpose of the first numerical study is to validate the accuracy of our deviation from Palm’sTheorem. The second numerical study focuses on the two step approach itself. We study itsability to reach the optimal solution as well as the beneficial aspects of the approach.

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6.1 Numerical Study of Model Accuracy

It is unclear how the cost accuracy of the model is affected by incorporating the estimated queuetime of an M/M/k queue and invoking Graves’ approximation. We emphasize that while ouralgorithm is a computationally efficient way to unify the strategic and tactical aspects of thedecision problem, it is not appropriate for day-to-day operational purposes.

In addition to a general understanding of when the approximation is reasonable and whenit is not, we are interested in the following questions: How well does the negative binomialapproximation represent the steady-state probability distribution of the number of items inrepair? How well would Graves’ conditioning argument approximate the first two moments ofthe units outstanding at the FSLs when repair capacity is finite?

6.1.1 Design of Numerical Study

For a series of scenarios, we compare our estimated costs with those observed through a simula-tion. The test network is comprised of one repair facility and five field stocking locations. TheFSL holding cost rate to $1 per unit per period. The factors of the study are listed in Table 1:

Factor Low Medium High

Inventory Shortage Optimal Excess

Capacity Utilization (%) 75 85 95

Number of Parallel Servers 1 3 5

Transit Lead Time (days) 1 3 5

Depot Holding Cost ($) 0.1 0.5 0.9

Backorder Costs ($) 9 - 99

Table 1: Factors examined to validate estimation technique.

The two-step decomposition optimization algorithm is developed using GAMS and MicrosoftExcel. Step 1 is solved by GAMS in fractions of a second. The data tables are built in MicrosoftExcel. Step 2 then determines the optimal allocation of inventory using marginal analysis.Step 2 solution times range from a few seconds to a half minute, depending on the numberof units in the network, and are dominated by the creation of the tables SPj(·) and SP j(·).We develop a continuous-time simulator to simulate actual costs for comparison purposes. Wesample 500,000 repair events from sixteen random input streams and their antithetic stream asa variance reduction technique.

6.1.2 Accuracy Study Results

Across all test scenarios, the total expected costs under-estimated the actual simulated costs by3.2% on average. We believe that given the strategic and tactical nature of our approach, thisis a reasonable level of accuracy under a wide variety of test cases.

Separating the test cases by the backorder costs, with a backorder to holding cost ratio of 9:1,the expected costs are within 0.52% of the simulated costs. Under these nominal conditions, theapproximation is quite accurate. We are therefore interested in determining the circumstancesin which the approximation performs poorly.

For the backorder to holding cost ratio of 99:1, Table 2 details the expected costs, actualcosts, and relative difference for varying depot holding costs, FSL backorder costs, FSL holdingcosts, and total costs. In this case, all backorder estimation errors are significantly magnifiedby the high backorder costs, and our approach under-estimates the total cost by 5.06%. Webelieve that this case is a reasonable worst-case performance.

To further explore conditions that negatively impact our cost estimate, we found that cou-pling the high backorder cost ratio with increases in repair server utilization leads to the greatest

12

Cap.

Util.

Num.

of

Serv.

Depot

Holding

Costs Expected

Actual

Simulated

Relative

Difference Expected

Actual

Simulated

Relative

Difference Expected

Actual

Simulated

Relative

Difference Expected

Actual

Simulated

Relative

Difference

.1 3.21$ 3.21$ 0.02% 19.45$ 19.47$ 0.08% 24.06$ 24.37$ 1.26% 46.73$ 47.05$ 0.68%

.5 13.00$ 13.00$ 0.02% 17.99$ 18.00$ 0.08% 42.29$ 42.53$ 0.56% 73.28$ 73.53$ 0.35%

.9 19.77$ 19.77$ 0.02% 19.87$ 19.88$ 0.08% 48.98$ 49.21$ 0.48% 88.61$ 88.87$ 0.29%

.1 3.37$ 3.37$ -0.06% 19.46$ 19.48$ 0.08% 23.99$ 24.64$ 2.70% 46.83$ 47.49$ 1.41%

.5 13.96$ 13.94$ -0.10% 18.65$ 18.67$ 0.12% 39.49$ 40.19$ 1.76% 72.09$ 72.80$ 0.98%

.9 21.28$ 21.24$ -0.16% 19.97$ 20.01$ 0.16% 45.56$ 46.18$ 1.37% 86.81$ 87.43$ 0.72%

.1 3.47$ 3.47$ -0.05% 19.43$ 19.45$ 0.14% 24.81$ 25.46$ 2.64% 47.70$ 48.38$ 1.43%

.5 15.08$ 15.06$ -0.11% 19.38$ 19.42$ 0.21% 34.09$ 34.87$ 2.28% 68.55$ 69.35$ 1.17%

.9 23.38$ 23.33$ -0.20% 20.45$ 20.51$ 0.29% 44.42$ 44.94$ 1.16% 88.25$ 88.78$ 0.60%

.1 4.38$ 4.38$ 0.00% 20.66$ 20.68$ 0.11% 23.29$ 25.24$ 8.39% 48.33$ 50.30$ 4.09%

.5 18.14$ 18.14$ 0.00% 20.09$ 20.11$ 0.12% 36.39$ 38.43$ 5.62% 74.62$ 76.69$ 2.77%

.9 28.16$ 28.16$ 0.00% 22.48$ 22.50$ 0.12% 42.56$ 44.82$ 5.31% 93.20$ 95.48$ 2.45%

.1 4.55$ 4.54$ -0.05% 20.72$ 20.75$ 0.15% 21.62$ 24.21$ 12.01% 46.89$ 49.51$ 5.60%

.5 19.05$ 19.04$ -0.07% 19.70$ 19.73$ 0.17% 37.10$ 39.54$ 6.58% 75.84$ 78.30$ 3.24%

.9 29.89$ 29.86$ -0.11% 22.18$ 22.22$ 0.20% 40.57$ 43.30$ 6.73% 92.64$ 95.38$ 2.96%

.1 4.67$ 4.67$ -0.09% 20.66$ 20.69$ 0.19% 22.71$ 25.53$ 12.43% 48.04$ 50.90$ 5.95%

.5 20.02$ 20.00$ -0.10% 19.79$ 19.82$ 0.17% 33.81$ 36.74$ 8.68% 73.62$ 76.57$ 4.01%

.9 31.46$ 31.40$ -0.17% 22.17$ 22.23$ 0.26% 41.37$ 44.36$ 7.22% 95.00$ 97.99$ 3.15%

.1 12.12$ 12.14$ 0.23% 25.62$ 25.81$ 0.73% 22.22$ 28.07$ 26.37% 59.95$ 66.03$ 10.13%

.5 49.45$ 49.57$ 0.24% 27.63$ 27.86$ 0.85% 40.83$ 47.52$ 16.39% 117.90$ 124.95$ 5.98%

.9 79.51$ 79.70$ 0.24% 31.48$ 31.75$ 0.84% 49.26$ 56.52$ 14.73% 160.26$ 167.97$ 4.81%

.1 12.52$ 12.55$ 0.25% 25.15$ 25.33$ 0.71% 20.05$ 28.82$ 43.69% 57.73$ 66.70$ 15.54%

.5 50.25$ 50.39$ 0.26% 27.68$ 27.93$ 0.89% 37.76$ 48.08$ 27.32% 115.69$ 126.39$ 9.24%

.9 80.81$ 81.02$ 0.26% 31.06$ 31.33$ 0.90% 47.64$ 58.14$ 22.04% 159.50$ 170.49$ 6.89%

.1 12.91$ 12.93$ 0.16% 24.65$ 24.79$ 0.54% 18.13$ 30.81$ 69.96% 55.69$ 68.52$ 23.05%

.5 44.40$ 44.48$ 0.17% 25.31$ 25.45$ 0.55% 98.31$ 105.82$ 7.63% 168.03$ 175.75$ 4.59%

.9 81.60$ 81.73$ 0.15% 31.04$ 31.28$ 0.79% 46.26$ 63.38$ 37.01% 158.90$ 176.39$ 11.01%

25.94$ 25.97$ 0.10% 22.69$ 22.78$ 0.40% 37.32$ 41.54$ 11.33% 85.95$ 90.30$ 5.06%

3

Base Holding Costs

5

Overall

75

85

95

1

3

5

Base Backorder Costs Total Costs

3

1

Depot Holding Costs

1

5

Table 2: Actual versus expected costs when base backorder to cost ratio is 99:1.

under-estimates of the total cost. For example, when increasing the utilization from 75% to95% in the 99:1 case, the gap in our estimate increases from 0.78% to 8.5% on average.

As can be expected, the absolute variance of the number in queue and in repair at a repairfacility greatly impacts the estimate of the variance of the number of outstanding orders at aFSL. More specifically, when the variance of the repair process (in queue and in repair) is high,as in the 95% utilization case, our use of the Graves (1985) approximation under-estimates thevariance of the outstanding orders at a FSL, and subsequently, we under-estimate the numberof backorders at the FSLs. This in turn causes the under stocking of inventory at the FSL.The root cause is due to the finite repair capacity that we model as a M/M/K queue versusGraves’ M/G/∞ assumption. Further research is needed to develop a better approximation ofthe second moment of the number of units outstanding to a FSL.

Overall, we feel that our approximation for the number of units in repair over the repairtime delay is reasonably accurate and provides the necessary foundation to link the strategicsupply chain design decision with the tactical repair capacity and inventory allocation decision.We have explored the circumstances in which it performs well, and have assessed worst-casescenarios.

6.2 Numerical Study of the Two Step Optimization Approach

Using data that are representative of actual operating USAF data, we construct a numericalstudy with five candidate locations (under current operating structure, repair facilities are co-located at each operating location). We then create a table of factors that we use to analyzethe approach under a myriad of conditions. Finally, we enumerate each candidate solution inStep 1, solve Step 2, and compare are two step solution with each possible enumerated solution.

The goal of this study is three-fold: (1) to assess the economic value of consolidating repaircapacity and locations; (2) to determine the performance of the two step approach in reaching

13

the optimal solution; and (3) to examine the cost improvements gained by including the tacticalconsiderations in the strategic model formulation (Step 2).

6.2.1 Design of the Numerical Study

The current supply chain locations as well as the table of factors under study are listed Figure3. There are 864 total combinations of system inventory, demand, transit lead times, fixedfacility costs, economies-of-scale (EOS), transportation costs, repair facility holding costs, andbase backorder costs.

The daily failure rates at the locations are: (1) 0.571; (2) 0.326; (3) 0.202; (4) 0.127;and (5) 0.106. The average one-way transportation cost is $1,500, and the average one-waytransportation time between locations is four days. The cost of a unit is $3,500,000, and weassume the annual holding cost at a base is 20% of the unit cost. The annual amortized fixedfacility costs are assumed to be $1,000,000, and incremental capacity costs are $900,000. Werepresent economies-of-scale with a function that is concave in the number of repair teams at arepair facility.

Factor Low Medium High

Inventory Shortage Optimal Excess

Demand 1 - 2

Lead Time 1/3 1 3

Fixed 1 - 2

Productivity Normal - EOS

Transit Cost 1 2 5

Depot Hold .5 - .9

BO Cost 9 - 99

5

1

2

3

4

Figure 3: Enumeration factors and the supply chain candidate locations.

6.2.2 Two Step Approach Study Results

The first exploration in this experiment is to justify the consolidation of repair facilities andcapacity. Across the 864 combinations, the average annual savings of consolidation is $7,500,000.This represents a total annual cost reduction of 24%. This is a significant annual savings giventhe scale of this five location experiment.

The two step approach performs well in its ability to choose the optimal solution. SinceMaster Problem is a non-linear integer program, it is very difficult to solve by itself directly.We addressed this problem in each scenario by enumerating over all possibilities in Step 1, andthen solving Step 2. Of the 864 scenarios, there are 84 scenarios in which the two step algorithmdid not achieve the minimum solution. Of those 84 cases, the relative difference between theminimum value and the two step solution is 1.65%, with a maximum gap of 7.35%. We believethis is an acceptable cost gap given the strategic nature of the model.

The justification for incorporating tactical considerations is shown in Figure 3. Across allscenarios, 107 scenarios add additional repair capacity in Step 2 of our approach. For thissubset, we compare the two step approach to an approach that solves Step 1, and determinesthe inventory allocation as a function of zj . We do so to quantify the value of incorporating Step2. As a result of solving Step 2, average of 1.94 increments of capacity are added, decreasingthe average FSL backorders. The average cost savings associated with Step 2 is approximately$900,000, which represents an average of 3% of the total cost, with a maximum of 12%. As aresult of this study, we assessed the ability of the two step approach to find the optimal solution.The relative value will depend on the specific economic costs associated with a particular aircraft

14

Economies

Fixed

Facility ($)

Depot

Holding

Costs ($)

Savings of

Approach ($)

Savings

Relative to

Costs (%)

Ave. Number of

Servers Added

EOS x1 0.5 1,170,546.75$ 4.24% 1.92

EOS x1 0.9 1,088,145.76$ 4.12% 2.29

EOS x2 0.5 1,105,016.31$ 3.70% 1.85

EOS x2 0.9 1,123,947.80$ 3.88% 2.29

Norm x1 0.5 591,241.61$ 1.85% 1.73

Norm x1 0.9 665,808.34$ 2.19% 2.00

Norm x2 0.5 730,494.99$ 1.93% 1.55

Norm x2 0.9 689,030.68$ 1.94% 1.92

Average 895,529.03$ 2.98% 1.94

Table 3: Economic value of two step approach.

subsystem. Our expectation is that when this method is applied to a wide variety of productfamilies, the annual cost savings will be significant.

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