A Dynamic Inventory Model with Supplier Selection
in a Serial Supply Chain Structure
José A. Ventura(1) and Victor A. Valdebenito(2)
Harold and Inge Marcus Department of Industrial & Manufacturing Engineering
The Pennsylvania State University
University Park, PA 16802, USA
Telephone: (814) 865-3841
Fax: (814) 863-4745
Emails: (1) jav1@psu.edu and (2) vav112@psu.edu
and
Boaz Golany
William Davidson Faculty of Industrial Engineering and Management
Technion - Israel Institute of Technology
Technion City, Haifa 32000, Israel
Telephone: (972-4) 829-4512
Fax: (972-4) 829-5688
Email: golany@ie.technion.ac.il
Extended Abstract
Purchasing is one of the most strategic activities involved in supply chain logistics, because it
provides opportunities to reduce costs across the entire supply chain, and improves the value and
performance of finished products by acquiring high quality raw materials and component parts.
Given that the cost of materials represents the largest percentage of the total product cost in
many industries, an essential task within the purchasing process is supplier selection. For
instance, in high technology firms, purchased materials and services account for up to 80% of the
total cost of a product.
In addition to purchasing, inventory is recognized as one of the major drivers of a supply
chain. High inventory levels increase the responsiveness of the supply chain but decrease its cost
efficiency because of the cost of holding inventory. Hence, a relevant problem in supply chain
logistics is to determine the appropriate levels of inventory at the various stages involved in a
supply chain. Given the prevalence of both supplier selection and inventory management
decisions in a supply chain, this study addresses both problems simultaneously by considering
the production and distribution of a single product or an aggregated unit representing a family of
products with similar process plans in a serial supply chain structure. An example of this
situation is a manufacturer that purchases raw parts from various preferred suppliers. These raw
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parts are stored at the manufacturing facility or processed into finished products. These products
are either stored at the manufacturer level or transported to a warehouse. At the warehouse stage,
products are either stored there or transported to a distribution center (DC). In general, the DC
may serve products to an entire market area or to a set of retailers. Considering the possible
impact of transportation costs on both supplier selection and inventory replenishment at each
stage of the supply chain in today’s enterprises, the proposed mixed integer nonlinear
programming (MINLP) model considers purchasing, production, inventory, and transportation
costs over a planning horizon with time varying demand taking into account quality constraints
for the suppliers, capacity constraints for suppliers and the manufacturer, as well as inventory
capacity constraints. All these constraints need to be satisfied for each planning period. The
proposed model can be represented by a time-expanded transshipment network, which is defined
by the sets of nodes and arcs that can be reached by feasible material flows. The model is
developed to determine an optimal inventory policy that coordinates the transfer of products
between consecutive stages of the supply chain from period to period while properly placing
purchasing orders to selected suppliers and satisfying customer demand on time.
Taking into account the transportation lead-times between consecutive stages of the supply
chain, potential initial pending orders of raw material and inventory levels, and possible final
inventory requirements at the various stages of the supply chain, theoretical results are developed
to identify a reduced time-expanded network, and determine necessary feasibility conditions
regarding customer demand at the last stage. An efficient approach to reduce the problem size
consists in eliminating from the analysis all arcs and nodes that are not feasible to the problem.
In this context, an arc or node is considered infeasible, if it cannot be used or accessed by any
feasible raw material or finished product flow due to the lead-times and finite planning horizon.
Notice that, since only infeasible arcs and nodes are eliminated, the original problem can be
formulated in a reduced network, which will have the same optimal solution.
Nowadays most manufacturing companies rely on third-party providers for the transportation
of products through their supply chain. Assuming trucks as the common means of
transportation, freight can be transported using the full-truck-load (TL) or less-than-truckload
(LTL) options. Even though, TL is frequently the less expensive transportation option, when
other supply chain costs are simultaneously considered in the analysis, the LTL option can
provide more flexibility in the definition of the optimal order size to allow reduction of the
overall supply chain cost. This situation is especially true when the shipment sizes are relatively
small. TL rates are usually expressed on a per-mile basis, while LTL rates are commonly
expressed per hundredweight, defined for a given origin and destination. Justifiably, LTL
transportation cost functions are similar to ordering cost functions for inventory replenishment
with quantity discounts, where the unit transportation cost progressively decreases as the number
of units increases. In the proposed model, transportation costs can be either represented exactly
by piecewise-linear functions, or approximated by quadratic or power functions.
Even thought the general MINLP model does not necessarily assume that cost functions are
linear, to guarantee computational tractability, production and ordering cost functions between
stages can be approximated by a fixed (setup) cost component and a linear cost component, and
inventory holding and in-transit costs can be represented by linear functions. In addition, if the
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transportation costs are characterized by piecewise-linear functions, the proposed formulation
becomes a mixed integer linear programming (MILP) model, which is much easier to solve.
In this study, an illustrative example with three raw material suppliers, three stages, and a
planning horizon with six periods has been analyzed. Stages 1 y 2 represent the raw material and
finished product warehouses in the manufacturing facility, and stage 3 represents a DC that ships
products to customers. Computational results show that each instance of the MINLP model with
transportation costs approximated with quadratic or power functions and the MILP model with
(exact) piecewise linear transportation cost functions can be solved to optimality in GAMS 21.7
in less than 5 seconds on a Pentium 4 with 3.40 GHz and 1 GB of RAM. Fourteen different cost
scenarios have been evaluated in order to capture the effect of production and distribution setup
costs, and inventory holding costs over raw material supply decisions, and determine the relative
errors in total transportation cost when quadratic or power approximation functions are used.
According to the experimental results, inventory holding and production/distribution setup
costs have a significant impact on supplier selection and order lot sizing decisions. Even though
these parameters may not appear directly related with raw material supplier decisions, the results
show that they play an important role in the selection of the optimal sourcing strategy. The
results show that, by reducing inventory holding costs, the total number of raw material orders is
also reduced, with the consequent increase in the average raw material lot size. Moreover, by
changing production and distribution set up costs, decisions related with order allocation by raw
material supplier may also change, producing important changes in the raw material sourcing
strategy. These results demonstrate the need to develop integrated models to correctly minimize
the total cost of the entire supply chain network.
The experimental results also show that the use of approximation functions for the
transportation cost may lead to suboptimal solutions, even though the optimal solution of the
MINLP model is apparently reached. The power and quadratic functions produce average
transportation cost errors of 3.1% and 2.2 %, respectively, in relation to the actual transportation
costs. In addition, when their optimal solutions found with the MINLP model are compared with
the optimal solutions produced by the MILP model (piece-wise linear approximation), average
solution gaps of 1.2% and 0.5% are obtained. Consequently, when approximation functions are
used to incorporate transportation costs under an all-unit discount structure, analysts need to be
aware of potential solution gaps that cannot be easily detected without an alternative exact model
formulation.
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