Integrated inventory and transportation mode selection

Available online at www.sciencedirect.com

Transportation Research Part E 44 (2008) 665–683
www.elsevier.com/locate/tre

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 a
single-echelon, multi-facility service parts logistics system with time-based service level constraints. As an optimization
goal we minimize the relevant inventory and transportation costs while ensuring that service constraints are met. The
model 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 different
networks show that significant benefits can be obtained from transportation mode and inventory integration.
Ó 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 respect
to the latter, it is essential to have fast supply of service parts, where we define a service part as the part needed
to replace the failed part.
In today’s marketplace, when selecting suppliers of technically advanced systems, high quality after-sales
service through availability of spare parts and quick response time has become important criteria. A survey
by 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

*
Corresponding author. Tel.: +1 512 232 7194; fax: +1 512 232 1489.
E-mail address: erhank@mail.utexas.edu (E. Kutanoglu).

1366-5545/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved.
doi:10.1016/j.tre.2007.02.001

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 service
delivery 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 involve
multiple customer locations, hence an overall service level to be achieved across multiple locations. A service
provider therefore faces a problem of determining optimal stocking levels of service parts while providing the
demanded parts to the customers within the specified time windows. The overall service parts logistics problem
therefore brings forth several tradeoffs – as insufficient stock can lead to extended customer system downtime
and hence the lost customer goodwill and sometimes heavy penalties, while maintaining excessive service parts
can lead to large inventory carrying costs. The time dimension which is usually called response time adds
another unique feature to the problem, which is the interaction between inventory and transportation. Since
the 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 holding
costs. There is a trend however to reduce the number of echelons and the number of locations per echelon in
order to reduce fixed location costs and service parts obsolescence costs. This striving for an efficient network
is 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 there
are 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 with
time-based service level constraints. We view the use of this model at the tactical level, in which service levels
and 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, we
minimize the relevant inventory, and normal and emergency transportation costs. The time-based service level
is defined as the portion of total customer demand that is satisfied within a specified period of time (service time
window), and the model seeks the minimum-cost solution that satisfies the target service levels for each service
region. (Note that in this tactical model, aggregation may lead to solutions in which some customers are not
satisfied within the time window due to the transportation mode assignment and/or stocking level. The model
tries to keep the demand portion unsatisfied within the time window under control.) Finally, note that the
model is capable of handling multiple modes in the overall system (typically more than 2). However, due to
the cost and service structure inherent in the problem, the choices for individual customers effectively boil down
to selecting between the least costly mode that delivers parts within the time window and the least costly mode
that 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 service
parts 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 description
of the model. The experimental design used to test the model and the computational results are discussed in
Sections 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, and
models 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 for
many years. Of special interest and assumed in this report are high cost, low demand, critical parts. Due to
their nature, a base-stock policy (also called one-for-one replenishment), usually denoted by the (S À 1, S)
model where S represents the base-stock level, is an appropriate replenishment policy. Here, the base-stock
level S is defined as the desired maximum number of parts or the inventory stocking level for the part. It is
also common to assume demand to have Poisson distribution, which is quite accurate for real service parts
systems. Early papers such as Feeney and Sherbrooke (1966) considered (S À 1, S) continuous review
inventory policies and derived steady state probabilities for various state variables of the system. The
famous METRIC model developed by Sherbrooke (1968) derived the stock levels for a multi-echelon
system with multiple parts, with complete backlogging for stock-outs. Sherbrooke and others later
improved 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 in
service parts logistics. The literature on service parts logistics inventory management is extensive, but other
selected 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 À 1, S) policies for an inventory system
with zero replenishment costs and generally distributed stochastic lead times with lost sales. He assumed, if
a nominal stock S is exhausted before the necessary replacement part arrives, a penalty cost L is incurred
for each unsatisfied demand and will have to be filled on an emergency basis (corresponding to penalty
incurred due to lost sales). The emergency handling procedure corresponds to priority shipping using premium
mode of transport like special air shipment. He used known formulas for the steady state probabilities of the
queuing system associated with the inventory system to formulate a cost function for the inventory system in
the 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 À 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 and
delivery 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 with
the 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-based
planning decisions affect the inventory levels. There are similar studies that incorporated mode-selection issues
into a higher level supply chain design and strategy models, some of which include long term inventory policies
and costs, but their focus and the underlying goals are different from ours as they focus on strategic design
choices and high-level planning decisions. In this category, Monahan and Berger (1977) analyzed the tradeoff
between the transportation cost and the maximum time until a shipment reaches a consolidation point or a
central 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 cost
function, as applied to petrochemical industry. Kiesmuller et al. (2005) investigated the value of delaying
¨
transportation mode decisions and the value of using a slow mode along with a fast mode, especially with
an explicit consideration of manufacturing lead times and inventory policies. Eskigun et al. (2005) explicitly
considered 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 novel
decomposition-based optimization application redesigning the supply chain of a production line with routing
and 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 and
Aggarwal (1997).

3. Model formulation

3.1. Problem description

The inventory/transportation system under consideration is a single-echelon system with multiple facilities
for stocking service parts. These facilities also serve the purpose of dispatching parts to their customers which
are dispersed within a geographic area. If any of the facilities are out of stock, the demand for that part is
fulfilled through an emergency shipment from the central warehouse (which is assumed to be outside the time
window). The facilities’ inventories are controlled by a one-for-one replenishment policy (also called (S À 1, S)
policy). We assume that the central warehouse has infinite stock, i.e., can satisfy any replenishment or direct
emergency shipment request without any extra delay due to stock-outs (hence shipments from the central
warehouse 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 is
assumed 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 providers
with 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 hence
handled 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 service
regions. 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 of
total service region demand satisfied within a time window. We require the actual percentage satisfied within
the time window (after considering the fill rates of the facilities within the region) to be at least a certain target
service level. If the overall geographic area represents a single region, then a system-wide service level should
be 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 tradeoff
therefore exists between the cost savings using the less costly modes and the responsiveness achieved using the
faster modes. Furthermore, in order to achieve the required time-based service level, it is essential that the
facility 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 first
type is the stocking of the service parts that the facilities should maintain. The second type is the assignment of
transportation modes to the customers, so as to satisfy the time-based service constraints. As a first step to
analyze the general problem with multiple parts, we consider one service part version in this paper although
the 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 to
the demand-weighted composite fill rates in Muckstadt (2005) and Hopp et al. (1997). As all demands are
eventually satisfied within the system (either directly from stock on hand at a facility or with an emergency
shipment 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 function
therefore 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 of
stock 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 to
denote 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
P
km mean demand rate at location m, km ¼ j2J m kj
P
k total mean demand rate from all customers in the overall area, k ¼ j2J kj
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, 0
otherwise

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 (assumed
to 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 location
m)

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

X XXX X
Minimize hm ðS m À bm km sÞ þ cij kj bm X ij þ ekm ð1 À bm Þ ð1Þ
m m i j2J m m

subject to bm ¼ 1 À gðS m Þ=GðS m Þ 8m ð2Þ
XX X X
dij kj bm X ij P ak kj 8k ð3Þ
m i j2J k J m j2J k
X
X ij ¼ 1 8j ð4Þ
i

X ij ¼ 0 or 1 8i; j ð5Þ
Sm 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 for
delivering the part to customer locations using the assigned mode, and the cost for satisfying demands passed
from the facilities and satisfied by direct shipment from the central warehouse. Note that the second term in
the 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) and
Gm(Sm) are respectively the probability mass function and the cumulative distribution function of the lead time
demand 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 more
discussion 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, the
left hand side of the constraint calculates the total demand satisfied directly from stock from a facility within
the time window (h) using the corresponding transportation mode. This amount divided by the total system
demand must be at least ak. Note that the left hand side is non-linear because of multiplication between the fill
rates 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 the
non-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 one
facility (hence one service region) and one transportation mode. Note that in this case, all the demand is
assigned to a single facility using one mode of transportation. Hence, the only real decision variable is the
facility’s base-stock level that minimizes the total costs such that the service level is satisfied. This special case
can be represented with the following model (omitting the indices for location (m), service region (k), and
mode (i)):
X
Minimize hðS À bksÞ þ b cj kj þ ekð1 À bÞ ð7Þ
j2J

subject to b ¼ 1 À gðSÞ=GðSÞ ð8Þ
X
b dj kj P ak ð9Þ
j2J

SP0 and integer ð10Þ
P
First, for this problem to be feasible, we must have a 6 j2J dj kj =k, because the demand rate within the time
window of the facility divided by the total demand rate provides an upper bound on the maximum fraction
that can be satisfied within the time window after considering the facility’s fill rate. We now show that the total
cost function is convex under most conditions, hence it is not sufficient to find the smallest stock level that
P
satisfies the service level (i.e., find S such that b P ak= j2J dj kj ):

P
Proposition 1. The total cost (7) is (discretely) convex function of stock level S if ek P j2J cj kj À 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 as
P
hS þ bð j2J cj kj À ek À hksÞ À 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 because
the unit emergency shipment cost (e) from the central warehouse is greater than any of the unit transportation
costs from a stocking facility (cj). Then, to find the optimal stock level (denoted by S*) subject to the service
level constraint, we first find the stock level that minimizes the cost function (7) P (denote this stock level by S 0
0 0
and its fill rate by b (S )), and compare its service level with the target: if b ðS Þ j2J dj kj P ak, then S* = S 0
0 0

(from convexity of the cost function, S 0 is the smallest non-negative integer such that b0 ðS 0 þ 1ÞÀ
P
b0 ðS 0 Þ 6 h=ðhks þ ek À j2J cj kj ÞÞ. Otherwise, S* is the smallest stock level that satisfiesP service level. Fig. 1
P the
shows an illustrative example where k = 4, j2J dj kj ¼3.6, h = 100, e = 150, s = 1, and j2J cj kj ¼ 20. Here, if
the target service level a is set to a level larger than 90%, the problem is infeasible; for any a 6 72%, S* = 5 is
optimal. 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:
X X X X
Minimize hm ðS m À bm km sÞ þ bm c j kj þ ekm ð1 À bm Þ ð11Þ
m m j2J m m

X X X
bm dj kj P a kj ð13Þ
m j2J m j2J

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:
!
X X X X X X
Minimize hm ðS m À bm km sÞ þ bm c j kj þ ekm ð1 À bm Þ þ u ak À bm dj kj ð15Þ
m m j2J m m m j2J m

S m 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 to
u: Instead of cij, we use cij À udj. Suppose the stock level that solves the subproblem for location m for given
u is S 0m ðuÞ. Starting with u = 0, if S 0m ð0Þ values lead to fill rates that satisfy the service level constraint (13), then
these stock levels are optimal. Otherwise, we seek to find the smallest value for u such that the service level
constraint 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 stock
level 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):
XX
Minimize cij kj X ij ð18Þ
i j2J
XX
subject to dij kj X ij P ak=b ð19Þ
i j2J
X
X ij ¼ 1 8j ð20Þ
i

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 further
simplify the problem with substitution X1j = 1 À X2j:
X X
Minimize c1j kj þ ðc2j À c1j Þkj X 2j ð22Þ
j2J j2J
X X
subject to d1j kj þ ðd2j À d1j Þkj X 2j P ak=b ð23Þ
j2J j2J

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 level
is satisfied. This reduced problem can be as hard as the Knapsack problem in general (when there is no clear
pattern 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 with
one 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, the
problem with multiple modes (even for a single facility with a given stock level) does not lend itself to such
methods, mainly due to discrete mode choices available to customers. To remove non-linearity in the original
model 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 variable
Yms 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, for
each location m, where Smax is the stock level that practically gives approximately 100% fill rate. This value
is usually a small number, between 3 and 10, for most of the practical cases, due to extremely low demand
levels 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 À 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, we
can calculate the actual fill rate variables at facilities (depending on the stock levels chosen) as
X
bm ¼ bms Y ms 8l
s
P
We can also represent the stock level variable with the expression S l ¼ s sY ls 8l. By substituting these two
into the model, we obtain the following version:
! !
X X X XXXX X X
Minimize hl sY ls À bls Y ls kl s þ cij kj bls X ij Y ls þ ekl 1 À bls Y ls
l s s l s i j2J l l s

ð25Þ
XXX X X
subject to dij kj bls X ij Y ls P ak kj 8k ð26Þ
l s i j2J k & j2J k
j2J l
X
X ij ¼ 1 8j ð27Þ
i
X
Y ls ¼ 1 8l ð28Þ
s

X ij ¼ 0 or 1 8i; j ð29Þ
Y 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 new
version of the model internalizes the potential fill rates as parameters, but it still has non-linear terms in the
objective 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 stock
level s, 0 otherwise. This way, Zijls = XijYls. To obtain the correct representation, we add the following con-
straints into the model (25)–(30):
Z ijls 6 X ij 8i; j; l; s ð31Þ
Z ijls 6 Y ls 8i; j; l; s ð32Þ
Z ijls P X ij þ Y ls À 1 8i; j; l; s ð33Þ
Z ijls 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 insight
into 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 to
be 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 the
coordinates, the Euclidean distances of the customers from facilities were calculated. These networks were also
solved for the single facility case. The single facility was located at the centroid of the three facilities. In real
life, this would represent the case, when the decision has to be made between opening one large facility to
cover the demand of the entire area or opening three smaller facilities to meet the demand of the segregated
customers 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 chosen
so as to incorporate slow moving characteristic of service parts. Each customer is assumed to be assigned to its
nearest facility. The behavior of the system was analyzed by observing how the costs of the system, mode
usage, 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 facilities
and 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 high
demand. 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 centrally
located 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 and
20 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, medium
and fast modes, respectively. Average replenishment lead time from the central warehouse to the facilities was
fixed at 1 week and the service delivery time window from the facilities to the customer locations was chosen as
4 h. From the locations of customers and facilities and their distances, and speeds of the modes, we compute
dij’s.
Note that there is a certain bound on how much of demand one can cover within a time window (which is
our defined service level, with one or more transportation modes) even if we assume 100% fill rate (part avail-
P
ability) at every facility. This bound, given by j2J dij kj =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 customer
locations). 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 window
from all the facilities. Using multiple modes gives some flexibility as one may be able to cover more of the
demand by using faster modes. In any case, in the random problems generated, the high time-based service
levels (80%) are demanding given the maximum ‘‘coverage’’ bounds. The low levels (60%) are considered
for comparison.
One expectation is that for slow moving service parts, one can save substantially by reducing inventory at
the facilities by using appropriate combination of modes while maintaining the same service level with the

Table 1
Data used for experimental study
Factor Unit Low High
Demand/customer Units/week 0.00–0.02 0.00–0.06
Time-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. The
performance is measured in terms of the total cost of the system, the base-stock levels at the facilities, and the
mode usage for the given service level. The mode usage is defined as the percentage of demand satisfied by each
mode.

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 two
levels 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 Pentium
M 1.3 GHz Centrino processor, with Windows based operating system and the solver was able to converge to
the 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 penalty
level 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 to
satisfy 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 mode
to 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 the
base case (low service level/low penalty level), we are better off by not stocking the part at facilities 1 and
3, 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. Also
to remain feasible, we are forced to stock at facility 3. The total cost goes up with both the higher penalty level
and the higher target service level, when compared to the single facility case. Higher penalty levels force the
model to start looking at the options of stocking the part at all the facilities while increasing service level forces
the 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 distances
between the facilities and the customers are large, the model suggests using the fast mode to satisfy the
time-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 stock
more. However, when we switch the service level to 80%, the model suggests turning both levers (maintain
larger 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 a
network.


Fig. 4. Single facility network #2.

Fig. 5 summarizes the results obtained for network 2 with three facilities. The customer assignments based
on nearest distance lead to a high demand at facilities 1 and 2, jointly covering 96% of the demand. Very low
demand at facility 3 indicates an average part requirement of less than one. In the three-facility case, although
we stock more parts than the single facility case, the total costs are about the same. For low penalty level, the
three-facility network gives slightly lower cost while with high penalty level the single facility network has
lower 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 total
cost of the system. For such a network, the variable cost is the same and the decision to operate three facilities
or 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 increase
our average base-stock level from 2.4 to 3.2 and use faster modes to satisfy 54.72% of demand as compared to
39.79%.
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 to
facilities 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 same
as those of the one-facility model for this network. The holding cost increases while the transportation cost
goes down due to proximity to the customers. At higher service levels, we stock even at facility 1. This is
because 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 other
customers’ 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 facility
faces about the same level of demand from its customers. The demand from a customer ranges from 0 to 3
parts per year. Due to higher demand rates, the model suggests maintaining higher base-stock levels. Since
the inventory moves faster, the average inventory costs do not increase in proportion to the increase in the
base-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 the
total 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 satisfy
only 13.76% of the demand. At the 80% service level, this goes up to 30.42%. The medium mode usage remains
unchanged with changes in service and penalty levels. This is because at the 60% service level some customers
which are far oﬀ and are not served within the time window are now served using faster modes to reach the
80% 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 facilities
can use slower modes and still satisfy the time-based service constraint. The reduction in transportation cost
more than compensates increase in emergency shipment cost as well as holding cost due to stocking inventory
at 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 stocking
decisions 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 transportation
decisions and service constraints but with making only single mode available for delivery. To remain feasible


X Cordinate
Mode Usage

Penalty Cost, SL Penalty Cost, SL

Fig. 8. Single facility network # 4.


and obtain a gross estimation of potential improvement, the fastest mode was assumed to be the only available
mode. 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 has
high service levels and low penalty levels. Also in all the cases, we find that integrating mode choice decisions
with the inventory decisions helps us operate our network for lower cost, i.e. we are able to meet our network
constraints 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 affected
by 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 to
stock all parts at all facilities. Our model provides insights whether it is better to stock at a particular facility
or ship directly from the central warehouse. This would help in determining major and non-major facilities
for different parts. Stocking only major facilities while letting faster modes satisfy demands such that service
levels 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 this
paper, 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 several
special cases (such as single facility, single-mode problems) that can be solved efficiently by taking advantage
of the special structure of these problems. We showed that a slight generalization toward multiple modes (even
for a single facility with a known stock level) leads to a hard Knapsack type problem. Hence, we proposed and
showed the feasibility of a linearization scheme for the general case, which makes it possible to obtain optimal
integrated solutions using a standard integer programming solver. We tested our approach on four different
networks 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 service
area based on customer demands and potentially different service levels in each region. Once the data for
partitioning 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 facilities
were fixed. This extension can be used to analyze the cost changes when we optimally decide the location
of facilities. (See Candas and Kutanoglu (2007) for such an integrated location and inventory model with
one transportation mode.)
• Integrating customer assignment decisions into the model. Here, the transportation modes would play even
more 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 infinite
capacity 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 and
size) and would show how effective it would be in solving real world problems. The maximum time to solve
the model for an instance of the problem was around 10 min of run time. It would be interesting to see how the
computational time increases as we try to solve larger and more complex networks.

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