IE572
Production Planning Systems Design
Research Proposal
Multi-Item Location Inventory Models:
Consideration of Multi-Item Concept on
Locating DCs
by
Evren Sarınalbant
Bilkent University
Department of Industrial Engineering
May 16, 2003
Problem Definition
Logistics network design is one of the most important problems in supply chain management
studies. Firms which aim to be profitable, successful and permanent should have long term
strategic goals. One of these strategic goals is the design of logistics network design which
has long term influences such as cost reductions due to the decrease in fixed order and setup
costs, reduction on safety stock levels and increase in fill rate. An appropriate definition of
logistics design may be determination of optimal locations of suppliers, warehouses,
distribution centers and retailers, optimal number and size of distribution centers, and finally
the allocation of retailers (or customers) to distribution centers.
Most general problem in logistics network design concept is the determination of optimal
distribution center location and number. Most of the studies on this concept assumed that
location, size and number of suppliers and retailers are known. These studies focused on
finding location of DCs while minimizing transportation, installation cost of DC subject to
allocation constraints.
One of the most important issues in the design of logistics network is the customer demand.
While designing supply chains, customer demand should be carefully examined to have a
wide view of problem, since it is the driver of the firms. In order to be more effective and
more responsive to the customer demand, companies should consider their effects on the
design. Demand for different items may vary in different retailer locations and for covering a
significant share of this demand while minimizing the costs is a problem in the design of the
supply chain. This problem can be solved by location-allocation models which was a very
common area of study.
Location-inventory models constructed in OR literature has similar components. The model
proposed by Daskin et al. (2002) is shown below:
fX
Min
jJ
i
ij


    d ij  i Yij  
 iI jJ



  2hF j   g j      i Yij    a j    i Yij   hz 


iI
jJ
iI
jJ
 jJ

Subject to
Y
jJ
ij
1
Yij  X j
X j and Yij are binary var iables
i I
 i  I, j J
 L Y
iI
i ij
The first term in the objective function is the fixed cost of locating facilities and the second
term represents the local transportation cost for each candidate location j. The third term
represents the total working inventory cost and finally fourth term is the formulation of safety
stock cost. Moreover, first set of constraints states that each demand point should be assigned
to a DC and second constraint ensures that demand assignments are done to the locations
where there is an installed DC.
Another research subject which may be another issue in design of supply chain is multi-item
concept. As discussed previously, demand of different items can affect the location of
distribution centers and this problem should be examined in a detailed manner. Unfortunately,
the location inventory models are a very fresh subject in the operations research area and there
are no studies which analyzes the impacts of multiple items on locating DCs. In OR literature,
location-inventory models constructed up to now use single item cost formulations and
demand patterns or simply aggregate the independent demands of multiple items and ignore
the effects of different demand patterns. The specific problem which will be discussed in this
research proposal is investigating the impacts of multiple items in design of logistics network
design. In this study, a location inventory model which incorporates multi-item inventory
costs and demand patterns will be constructed by assuming fixed supplier and retailer
locations.
Literature Review
In searching for the relevant literature for the problem, the nature of the problem should be
considered. Since the problem is a multi-item location inventory model, the review can be
divided into two streams, which are existing studies on location-inventory models and multiitem inventory models. Main structure of the model is based on location-inventory concept;
therefore it is better to start on these models.
Location-Inventory Models:
One of the earliest studies on location-inventory models is the study of Erlebacher and Meller
(2000) in which they analyzed the interactions between location and inventory. They
proposed an analytical model and solution methods for determining number of DCs, their
location and the allocation of customers to them while considering service level issues, fixed
costs of installing and operating DCs, inventory holding costs at the DCs, transportation costs
between suppliers-DCs and DCs-retailers. In order to understand the nature of the study,
assumptions should be analyzed. They assumed that customers are located on a unit-square
grid structure and demand is uniform across any grid. Distance calculations are done
rectilinearly and the location of each plant is known. The objective function of the model can
be divided into two main parts. The first part symbolizes the costs, which are fixed costs of
operating a DC, DC inventory costs, transportation costs from plant to DC and from DC to
customers. The constraints can also be divided into three categories. The first two constraints
make sure that every customer is assigned to an open DC. In the second category of
constraint, the demand for a particular DC is fully satisfied by the plants. The last constraint
set considers the capacity of each plan and checks whether it is adequate or not. Since the
problem is NP-hard, they proposed two heuristics to obtain a good solution and they tested
these heuristics by conduction a computational study which compares these heuristics and the
optimal solution for small size of problems.
Another study done on location-inventory models is Nozick and Turnquist (1998). In their
study, they integrated inventory costs into a fixed-charge DC location model. They
formulated inventory costs as cycle stock and safety stock, then calculated the safety stock by
resembling the inventory system to a queuing system where stock-out rate is represented by
the waiting time of a customer where the server is not available. They conducted a
probabilistic analysis to determine the stock-out rate. Moreover, they have showed a linear
function between safety stock and number of DCs to be located and found an expression
showing the sum of all safety stock in a number of DCs. In addition, they represented a fixed
charge facility-location model where inventory costs added implicitly.
After three years, Nozick and Turnquist (2001) developed their model to a different one
which formulated the location model as a maximization of demand coverage and cost
minimization while considering inventory costs and a number of facilities on a large network.
This methodology is applied on an automotive manufacturer which serves the continental US
through discrete demand areas.
Third study conducted on the location-inventory models is the paper of Daskin et al (2002). In
this paper, a DC location model which incorporates working inventory and safety stock costs
is built. The system is assumed as three-tiered which means one or more suppliers,
distribution centers and retailers. The location problem is modeled differently compared to the
conventional models in the sense that there are I number of retailers and J number of
candidate sites determined for constructing DC’s. The model tries to find the optimal
locations of DC’s from this set. The retailer locations are constant and the model behaves
these retailers as demand points and tries to allocate these demand points to the DC’s.
The objective function minimizes the fixed costs of locating DC’s, costs of local
transportation, working inventory costs and cost of holding safety stock. There are two sets of
constraints, one of which ensures each demand point is allocated to a DC and the other
constraint set forces these allocations to be appropriate in the sense that the demand points
must be allocated to an open DC. After that, model was represented as a Lagrangian dual
problem and solved by branch-and-bound algorithm.
A further study conducted on this subject is the paper of Shen et al (2001) in which a joint
location inventory model is constructed. Their study was motivated by a case study done for a
local blood bank where the product is perishable and the transportation time should be small
enough to cover urgent demands. They have done similar analysis as Daskin et al did (since
the authors are the same); however, they formulated their decision problem as a set-covering
model and solved using column generation.
Multi-Item Inventory Models:
After a review on existing location-inventory studies, second stream of literature review is
done on inventory systems with multiple items. The subjects that I have searched for are the
formulation of multi-item inventory costs and the demand structure constructed for multiple
items, especially demand correlation issue is considered.
First paper I found on multi-item inventory systems is Hausman et al. (1998) which conducted
a probabilistic analysis on joint demand fulfillment in a multi-item inventory system with
correlated demand structure. They assumed identical periodic review order-up to policy for
each item but ordering decisions for each item are made on the basis of the inventory position
of that particular item. They further assumed that item demands are correlated across different
items in a given period but autocorrelation is not considered meaning demand of an item for a
particular period does not affect the demand of other periods. Then, they calculated the
probability of no stockouts for joint distribution of demands by use of a multivariate normal
distribution and calculated the mean and covariance of each item demands.
Another study on multi-item inventory systems is conducted by Mitchell (1988) who
investigated multi-item inventory systems considering a system wide service objective. In his
model, he formulated expected average inventory cost and the probability of no-stockouts for
infinite horizon with a service level α.
Specific Considerations of the Model
In the design of supply chains, researchers generally ignore the impact of inventory costs,
because most of them thought that the effect of inventory costs on the location of a
distribution center is insignificant to consider. However, after the development of locationinventory models, it is seen that inventories should be considered while locating DCs.
Nevertheless, they did not consider the effects of multiple item systems which are most
frequently encountered in real world. Therefore, the issues which may show the effects of
multiple items should be analyzed. These issues are stated below:
Correlated Product Demands:
In multi-item systems, if demand of one product affects the demand of other, one can think of
a correlation in demands of these items. Demand correlation can be divided into two
categories namely negative and positive correlation. For positive correlation, an increase or in
the demand of one product causes an increase in the demand of other demand considered, vice
versa. This situation generally occurs in case of complementary products such as assembly
products. An example to these situations can be the relation between hard disk drivers and the
mainboards. It is obvious that if the demand of hard disk drivers increases significantly, the
demand of main boards will increase, but the demand of floppy disks may not necessarily be
affected by this change. In a similar manner, negative correlation means that an increase in
the demand of a product makes the demand of other product reduce. These situations are
generally faced in substitute products such as hard disk drivers and CD rewriters. In these
kinds of situations, customers may buy additional hard disk divers or buy a CD rewriter to
store information. Therefore, if we see an increase in the demand of CD rewriters, we expect
that the demand of hard disk divers would decrease. This issue may affect the location of
distribution centers, because inventory location models try to minimize the costs, for this
situation inventory holding costs and holding costs of different items can be different.
Therefore, the changes in the demands of correlated products affect the total inventory
holding cost, thus the location of the DC have a probability to change due to this reason.
Autocorrelation:
Autocorrelation is the term used when explaining the effect of the change in the demand of
one product for a particular period on the demand of other periods. An example of this
concept can be given from durable goods sector. In this sector, if manufacturer sells more than
he expected, in the other period he should expect that the demand of that item would be lower.
Specifically, if many customers buy refrigerator, they would not buy for a long period and by
this way the demand of item would be affected negatively.
Prioritization:
In supply chain management, suppliers generally give different importance to different items.
There may be many reasons for this situation. One of the reasons for prioritization can be the
profit margin concern. Because of the marketing issues, pricing of items may change.
Supplier can’t simply set the same profit margin for all of the items. Therefore, managers give
more importance to these items by setting higher safety stocks or service level for these items.
Another concern on the prioritization issue is the goodwill cost. If the customers of a
particular item are sensitive to backordering, a higher priority should be given to that item to
provide higher service levels.
Costs and Penalties:
In multi-item inventory models generally assume different holding and shortage costs for
different items. If this situation is analyzed and integrated with prioritization issue, one can set
higher holding cost for an undesired item or higher shortage cost for an item with high profit.
Supply contracts can be considered as a reason of this situation, where high penalties are
incurred for backlogs.
The topics stated above which are correlated demands, autocorrelation, prioritization, costs
and penalties are the main considerations that may be used while constructing multi-item
location-inventory models. However, we do not know if there will be a significant difference
or cost savings if the supply chain is designed using multi-item concepts. Therefore, the
specific hypothesis to be tested for this research can be “whether the cost saving achieved by
integrating correlated demand, autocorrelation, prioritization issues on the location-inventory
model is significant or not.”
Solution Methodology
In the formation of the model, most important factor is the integration of the multi-item
demand pattern to the location-inventory model. Therefore, in order to integrate this pattern
the study of Hausman et al. (1998) should be carefully analyzed. In their paper, for the
formulation of correlated demands, they used a multivariate normal distribution for the
demands of different products and formulated the mean and covariance of the products as:
 i  li  1EDit  and Vij  min li , l j   1 CovDit , D jt  , where li is the replenishment lead


time for item I, Dit is the demand for item i for period t. This formulation can be appropriate
for introducing multi-item demand into the location-inventory model. The working inventory
cost and the local delivery costs represented in the location-inventory model proposed by
Daskin et al. (2002) can be altered to conform to the correlated multi-item demand pattern
Working Inventory Costs
Safety Stock costs
where L is the lead time from suppliers to DCs, L i2 is the lead time demand variance, Yij is
the decision variable which determines whether the demand at retailer i is assigned to DC
location j, F is the fixed cost of placing an order,  is the coefficient used to turn daily
demand to annual demand,  and  are weights, h is the inventory holding cost,  is the
stockout probability and finally D is the expected annual demand.
Conclusion
Logistics network design is one of the most important issues in logistic companies, in order to
have a better starting point to the business. A crucial question in the design of efficient
logistics is the identification of locations for distribution centers. However, the optimization
of these location decisions requires careful attention to the natural trade-offs among facility
costs, inventory costs, transportation costs and customer responsiveness. The research in OR
literature have considered these trade-offs and come up with some solution techniques.
However, the impacts of multi-item logistics have been usually ignored. By using the points
in this intuitive research proposal, a multi-item location-inventory model can be developed.
References
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Daskin, M., Coullard, C. R., Shen, Z. M., 2002. An Inventory Location Model:
Formulation, Solution Algorithm and Computational Results. Annals of Operations
Research 110,83-106.
Erlebacher, S., Russel, M.,2000. The Interaction of Location and Inventory in
Designing Distribution Systems. IIE Transactions 32, 155-166.
Hausman, W. H., Lee, H. L., Zhang, A. X., 1998. Joint Demand Fulfillment
Probability in a Multi-Item Inventory System with Independent Order-up-to Policies.
European Journal of Operational Research 109, 646-659.
Mitchell, J. C., 1998. Multi-item Inventory Systems with a Service Objective.
Operations Research 36, 747-755.
Nozick, L. K., Turnquist, M. A., 1998. Integrating Inventory Impacts into a FixedCharge Model for Locating Distribution Centers. Logistics and Transportation Review
3, 173-186.
Nozick, L. K., Turnquist M. A., 2001. Inventory, Transportation, Service Quality and
the Location of Distribution Centers. European Journal of Operational Research 129,
362-371.
Shen, Z. M., Coullard C., Daskin, M. S., 2001. A Joint Location-Inventory Model.