Integrated inventory and transportation mode selection

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Transportation Research Part E 44 (2008) 665–683

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Integrated inventory and transportation mode selection:A service parts logistics system

Erhan Kutanoglu *, Divi Lohiya

Graduate Program in Operations Research and Industrial Engineering, Department of Mechanical Engineering,

The University of Texas at Austin, United States

Received 9 August 2006; received in revised form 4 January 2007; accepted 21 February 2007

Abstract

We present an optimization-based model to gain insights into the integrated inventory and transportation problem for asingle-echelon, multi-facility service parts logistics system with time-based service level constraints. As an optimizationgoal we minimize the relevant inventory and transportation costs while ensuring that service constraints are met. Themodel builds on stochastic base-stock inventory model and integrates it with transportation options and service respon-siveness that can be achieved using alternate modes (namely slow, medium and fast). The results obtained through differentnetworks show that significant benefits can be obtained from transportation mode and inventory integration.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Inventory management; Transportation mode selection; Service parts logistics; Integer programming

1. Introduction

The availability of technically advanced systems like heavily automated production systems, military sys-tems, medical equipments and computer systems increasingly affects daily operations as they play an impor-tant role in society. Downtime of critical equipments due to part failures may have serious consequences, e.g.in terms of loss of production, ineffective military missions, or quality reduction in health care. Various mea-sures are therefore taken to reduce the amount of system downtime, such as providing system redundancy,providing appropriate preventative maintenance and effective corrective maintenance. Especially with respectto the latter, it is essential to have fast supply of service parts, where we define a service part as the part neededto replace the failed part.

In today’s marketplace, when selecting suppliers of technically advanced systems, high quality after-salesservice through availability of spare parts and quick response time has become important criteria. A surveyby Cohen et al. (1997) on 14 companies with service logistics operations indicated that after-sales service rev-enues are equal to 30% of the product sales and the service parts inventories equal to 8.75% of the value of the

1366-5545/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.tre.2007.02.001

* Corresponding author. Tel.: +1 512 232 7194; fax: +1 512 232 1489.E-mail address: [email protected] (E. Kutanoglu).

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666 E. Kutanoglu, D. Lohiya / Transportation Research Part E 44 (2008) 665–683

product sales. The results of this survey indicated following characteristics and trends common in the servicedelivery organizations:

• There are large number of service parts to be stocked, varying between 2500 and 300,000.• The cost of these service parts is increasing due to increasing complexity and modularity of the products.• The service parts have high obsolescence rate caused by short product life cycles.• The fraction of slow moving service parts is large and increasing. This is caused by increased system cus-

tomization and design improvements. Average inventory turnover for such parts equals .87 parts per year,hence there are many parts with demand rates less than one per year.

In the realm of service part logistics, a service provider establishes relations with its customers through ser-vice agreements that extend over a period, usually in years. These agreements typically specify the level of ser-vice as well as the time window within which the service will be provided. Customers’ agreements may involvemultiple customer locations, hence an overall service level to be achieved across multiple locations. A serviceprovider therefore faces a problem of determining optimal stocking levels of service parts while providing thedemanded parts to the customers within the specified time windows. The overall service parts logistics problemtherefore brings forth several tradeoffs – as insufficient stock can lead to extended customer system downtimeand hence the lost customer goodwill and sometimes heavy penalties, while maintaining excessive service partscan lead to large inventory carrying costs. The time dimension which is usually called response time addsanother unique feature to the problem, which is the interaction between inventory and transportation. Sincethe response time is a function of distance to the customer, which in turn depends on which facility in the net-work the demand is satisfied from and which transportation mode is used to satisfy the demand, the transpor-tation decisions and associated costs are affected by inventory decisions.

Service parts are often supplied via a multi-echelon distribution network. One reason to have a multi-ech-elon network is the need to achieve quick response time and the need for stock centralization to reduce holdingcosts. There is a trend however to reduce the number of echelons and the number of locations per echelon inorder to reduce fixed location costs and service parts obsolescence costs. This striving for an efficient networkis facilitated by stocking essential parts close to the customers and by using the fast transportation modes.However, when multiple modes with different costs and speeds (say fast, medium, slow) or competing alter-natives for transportation are available, the integrated problem becomes rather unique in the sense that thereare many opportunities to investigate for total cost savings and the overall efficiency.

In this paper, our objective is to develop a tactical optimization-based model to gain insights into the inte-grated inventory/transportation problem for a single-echelon multi-facility service parts logistic system withtime-based service level constraints. We view the use of this model at the tactical level, in which service levelsand demands of individual customer locations are aggregated according to overall service agreements to con-trol the size and the granularity of the model. In this model, the service levels are defined for a group of cus-tomers (which in turn could be a collection of individual customer sites). As an optimization goal, weminimize the relevant inventory, and normal and emergency transportation costs. The time-based service levelis defined as the portion of total customer demand that is satisfied within a specified period of time (service timewindow), and the model seeks the minimum-cost solution that satisfies the target service levels for each serviceregion. (Note that in this tactical model, aggregation may lead to solutions in which some customers are notsatisfied within the time window due to the transportation mode assignment and/or stocking level. The modeltries to keep the demand portion unsatisfied within the time window under control.) Finally, note that themodel is capable of handling multiple modes in the overall system (typically more than 2). However, due tothe cost and service structure inherent in the problem, the choices for individual customers effectively boil downto selecting between the least costly mode that delivers parts within the time window and the least costly modethat does not deliver within the time window, although these two can be different across different customers.

The paper is organized as follows: Section 2 discusses the literature on inventory management for serviceparts with specific focus on base-stock models for lost sales and integrated inventory and transportation mod-els with service level considerations. In Section 3, we formulate the problem and present a detailed descriptionof the model. The experimental design used to test the model and the computational results are discussed inSections 4 and 5, respectively. Section 6 lists some extensions and improvements possible on the current model.

E. Kutanoglu, D. Lohiya / Transportation Research Part E 44 (2008) 665–683 667

2. Literature review

We focus our attention to the literature on inventory policies used for spare/service parts management, andmodels that seek to integrate transportation mode decisions into inventory policies.

Optimization of inventory systems to find stock levels of service parts has been a subject of research formany years. Of special interest and assumed in this report are high cost, low demand, critical parts. Due totheir nature, a base-stock policy (also called one-for-one replenishment), usually denoted by the (S � 1,S)model where S represents the base-stock level, is an appropriate replenishment policy. Here, the base-stocklevel S is defined as the desired maximum number of parts or the inventory stocking level for the part. It isalso common to assume demand to have Poisson distribution, which is quite accurate for real service partssystems. Early papers such as Feeney and Sherbrooke (1966) considered (S � 1,S) continuous reviewinventory policies and derived steady state probabilities for various state variables of the system. Thefamous METRIC model developed by Sherbrooke (1968) derived the stock levels for a multi-echelonsystem with multiple parts, with complete backlogging for stock-outs. Sherbrooke and others laterimproved and generalized the method to consider more complex issues (see, e.g. Muckstadt, 1973; Mucks-tadt and Thomas, 1980; Sherbrooke, 1986). Muckstadt (2005) is a reference for inventory research inservice parts logistics. The literature on service parts logistics inventory management is extensive, but otherselected service parts-related studies include Cohen et al. (1990), Caglar et al. (2004) and Wong et al.(2006).

Smith (1977) demonstrated how to evaluate and find optimal (S � 1,S) policies for an inventory systemwith zero replenishment costs and generally distributed stochastic lead times with lost sales. He assumed, ifa nominal stock S is exhausted before the necessary replacement part arrives, a penalty cost L is incurredfor each unsatisfied demand and will have to be filled on an emergency basis (corresponding to penaltyincurred due to lost sales). The emergency handling procedure corresponds to priority shipping using premiummode of transport like special air shipment. He used known formulas for the steady state probabilities of thequeuing system associated with the inventory system to formulate a cost function for the inventory system inthe steady state. Silver and Smith (1977) used the result from this model to develop a methodology to con-struct a set of indifference curves which permitted the exact determination of optimal inventory level in a sys-tem. Andersson and Melchiors (2001) presented a heuristic method for evaluation and optimization of(S � 1,S) policies for a two-echelon inventory model with lost sales. Their heuristic uses the METRIC approx-imation as framework for finding cost effective base-stock policies.

Although limited, a more relevant line of research focuses on incorporating multiple transportation anddelivery modes into inventory models. A recent example is Jaruphongsa et al. (2005) that considers the avail-ability of multiple modes within a dynamic lot sizing model. Another study by Klincewicz and Rosenwein(1997) considered multiple transportation modes in the context of warehouse outbound logistics planning withthe goal of making timely and consolidated deliveries to customers and keeping the warehouse workload bal-anced. This is not exactly an inventory and transportation mode integration, but the underlying mode-basedplanning decisions affect the inventory levels. There are similar studies that incorporated mode-selection issuesinto a higher level supply chain design and strategy models, some of which include long term inventory policiesand costs, but their focus and the underlying goals are different from ours as they focus on strategic designchoices and high-level planning decisions. In this category, Monahan and Berger (1977) analyzed the tradeoffbetween the transportation cost and the maximum time until a shipment reaches a consolidation point or acentral warehouse within a high-level distribution system. Van Roy (1989) considered transportation options(truck fleet or outside carriers) within a facility location planning model with an approximate inventory costfunction, as applied to petrochemical industry. Kiesmuller et al. (2005) investigated the value of delayingtransportation mode decisions and the value of using a slow mode along with a fast mode, especially withan explicit consideration of manufacturing lead times and inventory policies. Eskigun et al. (2005) explicitlyconsidered transportation mode choices along with lead times and inventory costs within a supply chain net-work design model, as applied to an outbound vehicle distribution system. Rao et al. (2000) presented a noveldecomposition-based optimization application redesigning the supply chain of a production line with routingand inventory decisions, with choices that can be viewed ‘‘regular’’ and ‘‘expedited’’ transportation modes.Also, several inventory-focused papers considered expediting individual deliveries to increase service levels


as part of an overall problem. Most relevant ones are by Aggarwal and Moinzadeh (1994) and Moinzadeh andAggarwal (1997).

3. Model formulation

3.1. Problem description

The inventory/transportation system under consideration is a single-echelon system with multiple facilitiesfor stocking service parts. These facilities also serve the purpose of dispatching parts to their customers whichare dispersed within a geographic area. If any of the facilities are out of stock, the demand for that part isfulfilled through an emergency shipment from the central warehouse (which is assumed to be outside the timewindow). The facilities’ inventories are controlled by a one-for-one replenishment policy (also called (S � 1,S)policy). We assume that the central warehouse has infinite stock, i.e., can satisfy any replenishment or directemergency shipment request without any extra delay due to stock-outs (hence shipments from the centralwarehouse take constant time). The replenishment lead time is assumed to be the same for all facilities.The emergency shipment transportation time between the central warehouse and any customer in need isassumed to be outside the service time window hence does not count toward the time-based service level.As these shipments typically handled as the next day delivery and are handled by outsourced service providerswith a flat rate, we assume that these shipments cost the same across different customers. The local deliveries(facilities to customers) introduce cost and time differentials that are critical to the overall problem, and hencehandled more explicitly in the model.

The demand for the parts from the customers follows independent Poisson distributions, which is a com-mon assumption in almost all service parts models as reviewed in the literature review. From the manage-ment’s point of view, the service provider may have further divided the geographic area into serviceregions. Each service region must respect the time-based service levels that are derived from the service agree-ments signed with the customers in that region. The time-based service level is defined as the percentage oftotal service region demand satisfied within a time window. We require the actual percentage satisfied withinthe time window (after considering the fill rates of the facilities within the region) to be at least a certain targetservice level. If the overall geographic area represents a single region, then a system-wide service level shouldbe satisfied. To meet these constraints, the company uses three different transportation modes for delivery(called slow, medium and fast) to customers which are distinguished based on their speed and cost. A tradeofftherefore exists between the cost savings using the less costly modes and the responsiveness achieved using thefaster modes. Furthermore, in order to achieve the required time-based service level, it is essential that thefacility has the requested part when a customer needs it. Hence, inventory stocking decisions should be inte-grated into the mode choice decisions. The problem would therefore involve two types of decisions: The firsttype is the stocking of the service parts that the facilities should maintain. The second type is the assignment oftransportation modes to the customers, so as to satisfy the time-based service constraints. As a first step toanalyze the general problem with multiple parts, we consider one service part version in this paper althoughthe multi-part extension is relatively straightforward. We develop a model to study the behavior of the system,i.e., how the optimal stocking policies and mode choice decisions change with the changes in various param-eters of the system, including cost structure and service levels.

Note that in our model fill rates are decision variables which could be varied to achieve the overall time-based service level. Hence, the defined (time-based) service level is a function of fill rates which were traditional‘‘service’’ measures in the literature. In this sense, the time-based service levels are similar in sprit and form tothe demand-weighted composite fill rates in Muckstadt (2005) and Hopp et al. (1997). As all demands areeventually satisfied within the system (either directly from stock on hand at a facility or with an emergencyshipment from the central warehouse), the level of demand satisfaction is 100%.

3.2. Model

The mathematical model’s objective is to minimize the total cost of the system. The objective functiontherefore comprises of the following components:


• Holding cost (HC) – Cost of stocking the service part at all stocking facilities• Transportation cost (TC) – Cost of transporting the part from facilities to the customers.• Emergency shipment cost (EC) – Cost of fulfilling the demand from the central warehouse through direct

emergency shipments, needed when the main facility responsible from the customer’s demand is out ofstock at the time of the demand.

This is subject to the constraints that ensure:

• Demand of each customer is met and is fulfilled by one mode, and• Target time-based service levels are met.

First, we introduce the notation used in the model (with a little abuse of notation, we use one symbol todenote both the set and its size, e.g., J is the set of customers and the number of customers).

Sets and indices

I set of modes, indexed by i = 1,2,3

J set of customers, indexed by j = 1,2, . . . ,J

K set of regions, indexed by k = 1,2, . . .,K

M set of stocking facilities, indexed by m = 1,2, . . .,M

Jk set of all customers located in service region k

Jm set of all customers assigned to facility m

Demand and service parameters

kj mean demand rate (per unit time) of customer j

km mean demand rate at location m, km ¼P

j2Jmkj

k total mean demand rate from all customers in the overall area, k ¼P

j2Jkj

s replenishment lead time from the central warehouse to a facility, same for all facilities

ak time-based service level for region k

h time within which customers must be served to be considered part of the time-based service level

dij 1 if customer j can be reached within h time units from its assigned location using mode i, 0otherwise

Cost parameters

cij unit transportation cost for delivering the part to customer j using mode i from its assigned facility

hm holding cost per unit time at facility m

e unit emergency shipment cost for direct deliveries from the central warehouse to customers (assumedto be the same for all customers)

Decision variables

Xij 1 if mode i is used to deliver the part to customer j from its assigned location, 0 otherwise

Sm base-stock level at facility m

bm long run fill rate of facility m (percentage of demand satisfied directly from stock on hand at locationm)

The following is the formulation for the minimization of the steady state total cost per unit time subject totime-based service level constraints:


MinimizeX

m

hmðSm � bmkmsÞ þX

m

Xi

Xj2Jm

cijkjbmX ij þX

m

ekmð1� bmÞ ð1Þ

subject to bm ¼ 1� gðSmÞ=GðSmÞ 8m ð2ÞXm

Xi

Xj2Jk\Jm

dijkjbmX ij P ak

Xj2Jk

kj 8k ð3ÞX

i

X ij ¼ 1 8j ð4Þ

X ij ¼ 0 or 1 8i; j ð5ÞSm P 0 and integer 8m ð6Þ

Objective function (1) is the total cost, which is composed of the three components mentioned earlier, namely,the total holding cost as the sum of average holding cost at all the facilities, the total transportation cost fordelivering the part to customer locations using the assigned mode, and the cost for satisfying demands passedfrom the facilities and satisfied by direct shipment from the central warehouse. Note that the second term inthe objective is non-linear due to the multiplication of fill rates and mode-selection variables.

Constraints (2) calculate the fill rates for all facilities, using the lost sales case formula. Here, gm(Sm) andGm(Sm) are respectively the probability mass function and the cumulative distribution function of the lead timedemand at facility m. The lead time demand at each facility has also Poisson distribution (with mean kms),because individual customer demands are independent Poisson random variables (see Zipkin (2000) for morediscussion and derivation of the lost sales fill rate formula).

Constraints (3) satisfy the time-based service level constraints, each written for a service region. Here, theleft hand side of the constraint calculates the total demand satisfied directly from stock from a facility withinthe time window (h) using the corresponding transportation mode. This amount divided by the total systemdemand must be at least ak. Note that the left hand side is non-linear because of multiplication between the fillrates bm’s (which are themselves functions of stock levels that are in turn decision variables) and mode-selec-tion variables, Xij.

Constraints (4) ensure that all customers are served and only one mode is assigned to each customer.Finally, constraints (5) are the binary restrictions on mode-selection variables, and constraints (6) are thenon-negativity and integrality restrictions on the stock levels.

3.3. Special case: one facility, one mode setting

We can obtain more direct insights on the model and its solution when we analyze the special case with onefacility (hence one service region) and one transportation mode. Note that in this case, all the demand isassigned to a single facility using one mode of transportation. Hence, the only real decision variable is thefacility’s base-stock level that minimizes the total costs such that the service level is satisfied. This special casecan be represented with the following model (omitting the indices for location (m), service region (k), andmode (i)):

Minimize hðS � bksÞ þ bXj2J

cjkj þ ekð1� bÞ ð7Þ

subject to b ¼ 1� gðSÞ=GðSÞ ð8ÞbXj2J

djkj P ak ð9Þ

S P 0 and integer ð10Þ

First, for this problem to be feasible, we must have a 6P

j2Jdjkj=k, because the demand rate within the timewindow of the facility divided by the total demand rate provides an upper bound on the maximum fractionthat can be satisfied within the time window after considering the facility’s fill rate. We now show that the totalcost function is convex under most conditions, hence it is not sufficient to find the smallest stock level thatsatisfies the service level (i.e., find S such that b P ak=

Pj2Jdjkj):


Proposition 1. The total cost (7) is (discretely) convex function of stock level S if ek PP

j2J cjkj � hks.

Proof. Note that the lost sales fill rate is a (strictly) concave function of S because the Erlang loss proba-bility (g(S)/G(S)) is strictly convex in S (Karush, 1957). The objective function can be rewritten ashS þ bð

Pj2J cjkj � ek� hksÞ � ek. Hence, showing that the objective is convex reduces to showing that coef-

ficient of b in the cost function is non-positive which leads to the condition in the proposition. h

Note that under our assumptions (and most realistic conditions) this condition is already satisfied becausethe unit emergency shipment cost (e) from the central warehouse is greater than any of the unit transportationcosts from a stocking facility (cj). Then, to find the optimal stock level (denoted by S*) subject to the servicelevel constraint, we first find the stock level that minimizes the cost function (7) (denote this stock level by S 0

and its fill rate by b 0(S 0)), and compare its service level with the target: if b0ðS0ÞP

j2Jdjkj P ak, then S* = S 0

(from convexity of the cost function, S 0 is the smallest non-negative integer such that b0ðS0 þ 1Þ�b0ðS0Þ 6 h=ðhksþ ek�

Pj2J cjkjÞÞ. Otherwise, S* is the smallest stock level that satisfies the service level. Fig. 1

shows an illustrative example where k = 4,P

j2Jdjkj ¼3.6, h = 100, e = 150, s = 1, andP

j2J cjkj ¼ 20. Here, ifthe target service level a is set to a level larger than 90%, the problem is infeasible; for any a 6 72%, S* = 5 isoptimal. For a = 85%, S* = 8.

3.4. Special case: multiple facilities, one mode

A multi-facility extension of this model (still with one region and one mode) can be solved with a Lagrang-ian-relaxation method. The model in this case reads as follows:

MinimizeX

m


m

bm

Xj2Jm

cjkj þX

m

ekmð1� bmÞ ð11Þ

subject to bm ¼ 1� gðSmÞ=GðSmÞ 8m ð12ÞXm

bm

Xj2Jm

djkj P aXj2J

kj ð13Þ

Sm P 0 and integer 8m ð14Þ

Fig. 1. Costs and service levels for an illustrative example with one facility and one mode.


Relaxing constraint (13) with multiplier u, we obtain the relaxed problem:

MinimizeX

m


m

bm

Xj2Jm

cjkj þX

m

ekmð1� bmÞ þ u ak�X

m

bm

Xj2Jm

djkj

!ð15Þ

subject to bm ¼ 1� gðSmÞ=GðSmÞ 8m ð16ÞSm P 0 and integer 8m ð17Þ

This problem decomposes into subproblems, one for each facility, very similar to the one introduced earlier(Eqs. (7)–(10)). The main difference here is that the transportation costs are now modified with respect tou: Instead of cij, we use cij � udj. Suppose the stock level that solves the subproblem for location m for givenu is S0mðuÞ. Starting with u = 0, if S0mð0Þ values lead to fill rates that satisfy the service level constraint (13), thenthese stock levels are optimal. Otherwise, we seek to find the smallest value for u such that the service levelconstraint is satisfied. We can use bisection search or subgradient optimization to find the optimal u.

3.5. Special case: One facility with a given stock level

For the problem with one facility with multiple transportation modes available, we first make the stocklevel a parameter. Hence, given a stock level S = s, and its fill rate b, we formulate the problem as follows(Note that in this case the inventory and emergency shipment costs are constants):

MinimizeX

i

Xj2J

cijkjX ij ð18Þ

subject toX

i

Xj2J

dijkjX ij P ak=b ð19ÞX

i

X ij ¼ 1 8j ð20Þ

X ij ¼ 0 or 1 8i; j ð21Þ

For the setting with two transportation modes (mode 2 being faster and more expensive), we can furthersimplify the problem with substitution X1j = 1 � X2j:

MinimizeXj2J

c1jkj þXj2J

ðc2j � c1jÞkjX 2j ð22Þ

subject toXj2J

d1jkj þXj2J

ðd2j � d1jÞkjX 2j P ak=b ð23Þ

X 2j ¼ 0 or 1 8j ð24Þ

Here, one can view the reformulation as all the customers are (temporarily) assigned the slow (and less costly)mode (note that d2j P d1j). Assuming that this is not sufficient for the target service level, the remaining prob-lem is to decide which of the customers (that can take advantage of the fast mode, i.e., d2j = 1, and d1j = 0)should be switched to the fast mode such that the ‘‘switching’’ cost is minimized and the target service levelis satisfied. This reduced problem can be as hard as the Knapsack problem in general (when there is no clearpattern in transportation costs (or distances) and customer demand rates), which is known to be NP-hard.One can extend this analysis to include more transportation modes, but it is clear that even the problem withone facility, two transportation modes, and given stock level is not trivial.

3.6. General case

Note that even though the single-mode problem (with single or multiple facilities) is directly handled, theproblem with multiple modes (even for a single facility with a given stock level) does not lend itself to suchmethods, mainly due to discrete mode choices available to customers. To remove non-linearity in the originalmodel in such general cases and make it amenable to be solved by direct optimization solvers such as CPLEX(ILOG, 2005) or Xpress-MP (Dash Optimization, 2006), we introduce the following additional notation:


Parameter

bms fill rate level at location m with stock level s

Decision variableYms 1 if location m uses stock level s, 0 otherwise

Here, bms represents the tabulated values of fill rates, calculated at potential stock levels s = 1, . . . ,Smax, foreach location m, where Smax is the stock level that practically gives approximately 100% fill rate. This valueis usually a small number, between 3 and 10, for most of the practical cases, due to extremely low demandlevels experienced in SPL systems. Hence, bms values are preprocessed fill rates for potential stock levels, eval-uated with known lead time demand for each facility, i.e., bms = 1 � gm(s)/Gm(s), for all s and m, each evalu-ated using the location’s corresponding mean lead time demand kms. With potential fill rates in parameters, wecan calculate the actual fill rate variables at facilities (depending on the stock levels chosen) as

bm ¼X

s

bmsY ms8l

We can also represent the stock level variable with the expression Sl ¼P

ssY ls8l. By substituting these twointo the model, we obtain the following version:

MinimizeX

l

hl

Xs

sY ls �X

s

blsY lskls

!þX

l

Xs

Xi

Xj2Jl

cijkjblsX ijY ls þX

l

ekl 1�X

s

blsY ls

!

ð25Þsubject to

Xl

Xs

Xi

Xj2Jk&j2Jl

dijkjblsX ijY ls P ak

Xj2Jk

kj 8k ð26Þ

Xi

X ij ¼ 1 8j ð27ÞX

s

Y ls ¼ 1 8l ð28Þ

X ij ¼ 0 or 1 8i; j ð29ÞY ls ¼ 0 or 1 8l; s ð30Þ

Here, constraints (28) make sure that only one of the available stock levels is chosen for each facility. This newversion of the model internalizes the potential fill rates as parameters, but it still has non-linear terms in theobjective and the constraints, but these terms are effectively handled by another substitution easily: Define Zijls

as the decision variable that is 1 if customer j is assigned to mode i and the customer’s location l uses stocklevel s, 0 otherwise. This way, Zijls = XijYls. To obtain the correct representation, we add the following con-straints into the model (25)–(30):

Zijls 6 X ij 8i; j; l; s ð31ÞZijls 6 Y ls 8i; j; l; s ð32ÞZijls P X ij þ Y ls � 1 8i; j; l; s ð33ÞZijls ¼ 0 or 1 8i; j; l; s ð34Þ

Now with the model fully linearized, one can use any of the standard integer programming solvers.

4. Experimental settings

To investigate if a significant interaction exists between mode decisions and inventory and to obtain insightinto how the behavior of the system changes with the changes in the system parameters, we designed four net-works by randomly locating 100 customers and 3 facilities on a grid of 100 · 100 units, representing an area tobe served by the facilities. For each of these customers and facilities, two random numbers were generated


between 0 and 100 and these were taken as the x and y coordinates of customer/facility locations. Using thecoordinates, the Euclidean distances of the customers from facilities were calculated. These networks were alsosolved for the single facility case. The single facility was located at the centroid of the three facilities. In reallife, this would represent the case, when the decision has to be made between opening one large facility tocover the demand of the entire area or opening three smaller facilities to meet the demand of the segregatedcustomers in the area. We focused on the single service region instances.

In generating the rest of the data, we base our choices on the characteristics of a real life industrial data set.For each of these networks, five demand scenarios were randomly ‘‘simulated’’. The demand range was chosenso as to incorporate slow moving characteristic of service parts. Each customer is assumed to be assigned to itsnearest facility. The behavior of the system was analyzed by observing how the costs of the system, modeusage, and stocking decisions change with the changes in the following factors:

• mean customer demand rates,• time-based service levels,• emergency shipment cost.

In the experimental study, we used the data shown in Table 1.The system was analyzed for low and high value of penalty level defined as a scalar multiple of fixed holding

cost. For our experiments we chose low and high levels to be 5 and 10 times the holding cost, respectively.Increasing the value of penalty level indicates higher level of cost resulting from stock-outs at the facilitiesand consequently paying more for direct emergency shipments from the central warehouse to the customers.For networks 1–3, we generated low demands, while for network 4 was characterized by relatively highdemand. Networks 1–3 are generated to represent randomness in facility locations and unequal demand pat-tern at the facilities. Network 4 was generated in a more controlled way, such that the facilities are centrallylocated and each of the facilities shared similar demand pattern.

The holding cost was kept constant at $20/unit/week, while the mode speed factors were fixed at 4, 10 and20 units of distance per unit time (hour) for the slow, medium and fast modes, respectively. We assigned ship-ment cost from the facilities to the customer location as $1, $1.5 and $2.5 per unit distance for slow, mediumand fast modes, respectively. Average replenishment lead time from the central warehouse to the facilities wasfixed at 1 week and the service delivery time window from the facilities to the customer locations was chosen as4 h. From the locations of customers and facilities and their distances, and speeds of the modes, we computedij’s.

Note that there is a certain bound on how much of demand one can cover within a time window (which isour defined service level, with one or more transportation modes) even if we assume 100% fill rate (part avail-ability) at every facility. This bound, given by

Pj2Jdijkj=k, is a function of how the customers are geograph-

ically dispersed, what their demand rates are, and where the facilities are located (with respect to customerlocations). For example, requiring 95% service level may be infeasible given that maximum one can ‘‘cover’’within the time window is 85%, because only 85% of the total demand is achievable within the time windowfrom all the facilities. Using multiple modes gives some flexibility as one may be able to cover more of thedemand by using faster modes. In any case, in the random problems generated, the high time-based servicelevels (80%) are demanding given the maximum ‘‘coverage’’ bounds. The low levels (60%) are consideredfor comparison.

One expectation is that for slow moving service parts, one can save substantially by reducing inventory atthe facilities by using appropriate combination of modes while maintaining the same service level with the

Table 1Data used for experimental study

Factor Unit Low High

Demand/customer Units/week 0.00–0.02 0.00–0.06Time-based service level % Demand satisfied in the region 60% 80%Penalty level factor – 5 10


customers. We also investigate the cases in which it may not be beneficial to stock an item at the facilities. Theperformance is measured in terms of the total cost of the system, the base-stock levels at the facilities, and themode usage for the given service level. The mode usage is defined as the percentage of demand satisfied by eachmode.

5. Computational results

The model was solved for single facility and three-facility cases in GAMS modeling language (Brooke et al.,2000; Rosenthal, 2006) using CPLEX integer programming solver (ILOG, 2005) for the four networks gener-ated. Effectively this resulted in 200 problem instances, with each of the four networks being tested for twolevels of the emergency transportation cost, two levels of the time-based service level, and five demand scenar-ios, solved for a single facility model and a three-facility model. These problem instances were run on PentiumM 1.3 GHz Centrino processor, with Windows based operating system and the solver was able to converge tothe solution for the specified instances of the problem within 1–10 min per instance. For the comparison pur-pose and to see the effectiveness of integrating mode choice decisions with the inventory decisions on the over-all cost savings, we also used the model’s version with only one mode choice available.

The results based on the average of five demand scenarios for the single facility and three-facility models,both with multiple modes are presented in groups of four charts. Figs. 2–10 illustrate how stocking decisions,mode usage and total cost structure change with changes in service level and penalty level for the four net-works. More detailed results in tabular form are available from authors.

5.1. Network 1

From Fig. 2, it is clear that at fixed service level of 60%, in the one-facility network, an increase in penaltylevel is handled by increasing the base-stock level at the facility, which leads to increase in the holding cost.Higher base-stock levels lead to decrease in emergency shipment costs because we have more parts available tosatisfy demands directly from the facility. At the low service level, the model suggests using slow mode to sat-isfy 60–68% of the demand. The fast mode is not used at all. However, when the service level is set at 80%, the

Fig. 2. Single facility network #1.

Fig. 3. Three-facility network #1.


model suggests an increase in both the base-stock level as well as in the use of medium mode and the fast modeto satisfy customers. Intuitively this makes sense, as we need to reach more customers in time to achieve 80%service level, which calls for faster modes and stocking more parts at the facility.

In Fig. 3, the results for the three facilities are presented. The customer assignment based on nearest dis-tance leads to satisfying approximately 82% of the demand from facility 2. The model suggests that for thebase case (low service level/low penalty level), we are better off by not stocking the part at facilities 1 and3, but making more direct shipments from the central warehouse. Also, facility 2 (with location coordinates(45,30) being closer to the majority of customers) is able to satisfy 80% of the demand by using the slow mode.For the base case (low service/low penalty level), the overall cost performance is closer to the one-facility loca-tion case. The difference in the transportation cost is due to the location of facility 2.

An increase in penalty level deteriorates the performance as we incur higher emergency shipment cost. Alsoto remain feasible, we are forced to stock at facility 3. The total cost goes up with both the higher penalty leveland the higher target service level, when compared to the single facility case. Higher penalty levels force themodel to start looking at the options of stocking the part at all the facilities while increasing service level forcesthe model to satisfy more demand by using faster modes. Thus, for such a network we are better off by stock-ing one facility rather than three facilities.

5.2. Network 2

Fig. 4 presents a network when we have facilities located far from the customers. Since the distancesbetween the facilities and the customers are large, the model suggests using the fast mode to satisfy thetime-based service level constraint. Hence, transportation costs become significant portion of the total cost.The base-stock level follows a similar pattern; an increase in service or penalty level makes the facility stockmore. However, when we switch the service level to 80%, the model suggests turning both levers (maintainlarger stock and use faster modes) to satisfy greater demand within the time window for optimal performance.It is interesting to note that the emergency shipment cost forms the small portion of the overall cost for such anetwork.

Fig. 4. Single facility network #2.


Fig. 5 summarizes the results obtained for network 2 with three facilities. The customer assignments basedon nearest distance lead to a high demand at facilities 1 and 2, jointly covering 96% of the demand. Very lowdemand at facility 3 indicates an average part requirement of less than one. In the three-facility case, althoughwe stock more parts than the single facility case, the total costs are about the same. For low penalty level, thethree-facility network gives slightly lower cost while with high penalty level the single facility network haslower total cost. This is because we are closer to customers with three facilities; hence we can use cheaper med-ium mode instead of fast, and still satisfy the time-based service constraint. Since we do not stock at facility 3,an emergency shipment cost is incurred for all demand assigned to this facility, leading to an increase in totalcost of the system. For such a network, the variable cost is the same and the decision to operate three facilitiesor one facility should be driven by fixed cost of opening a facility, customer demands, and cost of transpor-tation modes.

5.3. Network 3

Fig. 6 summarizes results for the single facility version of network 3, which shows similar results to the sin-gle facility version of networks 1 and 2 in general behavior. At high penalties and service levels, we increaseour average base-stock level from 2.4 to 3.2 and use faster modes to satisfy 54.72% of demand as compared to39.79%.

Fig. 7 presents the results for the three-facility case for network 3. Again, there is an imbalance in the dis-tribution of demand: only 6% of the total demand goes to facility 1, while the remaining demand is assigned tofacilities 2 and 3. For the 60% service level, the model suggests not to stock at facility 1 but supply its custom-ers through emergency shipments from the central warehouse. The costs for the three facilities are almost sameas those of the one-facility model for this network. The holding cost increases while the transportation costgoes down due to proximity to the customers. At higher service levels, we stock even at facility 1. This isbecause the amount saved on transportation while satisfying demand from this facility more than compensates

Fig. 5. Three-facility network # 2.

Fig. 6. Single facility network # 3.


Stock Distribution-N3-3L



the higher emergency shipment costs from the central warehouse or the cost of transportation to meet othercustomers’ demands from the other two facilities with faster modes.

5.4. Network 4

Network 4 was designed such that facilities were located centrally around the customers and each facilityfaces about the same level of demand from its customers. The demand from a customer ranges from 0 to 3parts per year. Due to higher demand rates, the model suggests maintaining higher base-stock levels. Sincethe inventory moves faster, the average inventory costs do not increase in proportion to the increase in thebase-stock level as shown in Fig. 8.

Even though the facility is centrally located, we see that the transportation cost is very high portion of thetotal cost. This is because when the demand rate is high, we incur higher frequency of shipments to the cus-tomers. The mode usage pattern indicates that at the 60% service level we tend to use the fast mode to satisfyonly 13.76% of the demand. At the 80% service level, this goes up to 30.42%. The medium mode usage remainsunchanged with changes in service and penalty levels. This is because at the 60% service level some customerswhich are far off and are not served within the time window are now served using faster modes to reach the80% service level.

In the three-facility scenario (Fig. 9), the total cost of the system is less than that of the single facility case.With more facilities close to customers, the transportation cost reduces by approximately 56–60% as facilitiescan use slower modes and still satisfy the time-based service constraint. The reduction in transportation costmore than compensates increase in emergency shipment cost as well as holding cost due to stocking inventoryat all three locations. In all above cases, the optimal solutions do not suggest use of the fastest mode. Hence,understanding the mode options available and using the right one in conjunction with the inventory stockingdecisions would result in otherwise hidden cost savings for a company.

5.5. Comparative analysis

For comparative analysis, we rerun the three-facility models with integrated inventory and transportationdecisions and service constraints but with making only single mode available for delivery. To remain feasible

X Cordinate

Penalty Cost, SLPenalty Cost, SL

Mo

de

Usa

ge

Fig. 8. Single facility network # 4.



and obtain a gross estimation of potential improvement, the fastest mode was assumed to be the only availablemode. Fig. 10 presents the costs for all the networks and scenarios for the single and multi-modal cases. From

Fig. 10. Cost comparison – single mode and multiple modes.


the relative costs, it seems that the maximum benefit from the multiple modes occurs when the network hashigh service levels and low penalty levels. Also in all the cases, we find that integrating mode choice decisionswith the inventory decisions helps us operate our network for lower cost, i.e. we are able to meet our networkconstraints at lower cost. Note that the single-mode solution can be improved by selecting the modes for indi-vidual customers more carefully (slower and less costly modes that deliver the parts within the time window).Hence, the improvement due to the usage of multiple modes here represents a gross estimation.

5.6. Discussion

In our analysis, we designed four networks to study how inventory decisions and mode choices are affectedby changes in required time-based service levels and in penalty level in service parts logistics. Integrating multi-ple mode options in the inventory and transportation decisions for all the networks resulted in lower cost,while ensuring service level constraints. Our model determines optimal inventory and customer-mode assign-ment decisions. While all four networks are different with different solutions obtained, they share some com-mon results and observations:

• An increase in penalty level leads to stocking more in all four networks. However it is not always optimal tostock all parts at all facilities. Our model provides insights whether it is better to stock at a particular facilityor ship directly from the central warehouse. This would help in determining major and non-major facilitiesfor different parts. Stocking only major facilities while letting faster modes satisfy demands such that servicelevels are achieved will reduce both inventory costs and warehouse capacity pressures.

• Increase in time-based service levels requires adjusting both inventory and mode assignment decisions.• Operating three smaller facilities or one large facility with fast mode to satisfy service constraints depends

on customer demand rates and the given network. While for networks 1–3, one could chose between open-ing one or three facilities, network 4 clearly suggests that three-facility design performed better.

6. Conclusions

Service parts are characterized as parts with extremely sporadic and low demand, and high costs. In thispaper, we developed a tactical optimization model that builds on the base-stock inventory model and inte-grates it with transportation options and service responsiveness that can be achieved using alternate modes.


The model captures the cost tradeoffs at different service levels and penalty levels. We also analyzed severalspecial cases (such as single facility, single-mode problems) that can be solved efficiently by taking advantageof the special structure of these problems. We showed that a slight generalization toward multiple modes (evenfor a single facility with a known stock level) leads to a hard Knapsack type problem. Hence, we proposed andshowed the feasibility of a linearization scheme for the general case, which makes it possible to obtain optimalintegrated solutions using a standard integer programming solver. We tested our approach on four differentnetworks for a single region with up to three facilities, 100 customers, and three modes.

There are several extensions possible for this model:

• Extending the model beyond single region. This would require determining partitioning of the overall servicearea based on customer demands and potentially different service levels in each region. Once the data forpartitioning the area is known, the model can be easily extended to handle this extension.

• Extending the model for multiple-product case. In such a model, each product can have its own desired ser-vice level and penalty cost depending on criticality. This modeling extension too is relatively easy.

• Integrating facility location decisions into the model. In our model, we assumed the locations of facilitieswere fixed. This extension can be used to analyze the cost changes when we optimally decide the locationof facilities. (See Candas and Kutanoglu (2007) for such an integrated location and inventory model withone transportation mode.)

• Integrating customer assignment decisions into the model. Here, the transportation modes would play evenmore crucial role in determining the inventory at each facility.

• Extending the model to a multi-echelon structure. Our model is valid for single echelon and assumes infinitecapacity at the central warehouse. From a practical perspective, most service providers use at least two ech-elons in their distribution network where both levels are capacitated in terms stocking levels.

These extensions would make the model richer (and harder to solve due to the increased complexity andsize) and would show how effective it would be in solving real world problems. The maximum time to solvethe model for an instance of the problem was around 10 min of run time. It would be interesting to see how thecomputational time increases as we try to solve larger and more complex networks.

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Business

Integrated inventory and transportation mode selection