IE 572, PRODUCTION PLANNING SYSTEMS
DESIGN
Term Paper
A Research on the Inventory Routing Problem
by
I. Esra Buyuktahtakin
May 2003, Bilkent
Abstract
The Inventory Routing Problem (IRP) is one of the core problems that arises under a vendor
managed resupply policy where the vendor decides when to visit each customer and how much
to deliver. The Inventory Routing Problem adresses the integration of inventory management and
transportation. IRP is hard especially if a large number of customers is involved.
In this paper we aim to discuss and analyze the inventory routing problem. We are going to
review different variants of the inventory routing problem studied by the previous researchers
and investigate the basics of inventory routing problem.
1. Introduction
Recently many companies has adopted Vendor Managed Inventory Replenishment which is an
emerging trend in logistics. The implementation of the VMI requires the solution of one of the
core problems, inventory routing problem (IRP). VMI refers to the situation in which vendor
monitors the inventory levels of its customers and decides when and how much inventory to
replenish at each customer. VMI differs from the conventional inventory management in such a
way that in conventional inventory management, the customers monitor their own inventory
levels and the reorder time is determined by the customer. Also under conventional inventory
management policy, the supplier receives the order, prepares the product for delivery and makes
deliveries by using the fleet of vehicles.
VMI has many advantages over conventional inventory management. Under VMI policy,
vendors can usually obtain a more uniform utilization of production resources which leads to
reduced production and inventory holding costs. Correspondingly, vendor can achieve more
uniformly utilized transportation resources, which in turn leads to reduced transportation costs
(Kleywegt, Nori and Savelsbergh, 2002).Furthermore using more efficient routes which combine
close customers into a vehicle trip and utilizing cost-effective full truck loads also reduce the
transportation costs.
VMI has also advantages for the customers. Customers devotes less resources for monitoring the
inventory and making orders as long as the vendor provides the assurance that the product would
be available when required. Also customers may reach an increased service level, since the
availability of the product increases as a result of better and smoother demand forecast of the
supplier (Kleywegt, Nori and Savelsbergh, 2002).
To successfully implement the VMI, the supplier should be able to access to a lot of relevant
information, such as the current inventory levels at all the customers, the customer’s demand
behaviour, the customer’s locations relative to the supplier and relative to each other and the
resulting transportation costs, and the capacity and availability of vehicles and drivers for
delivery (Kleywegt, Nori and Savelsbergh, 1999). The increased amount of information makes it
difficult to make good decisions.
The implementation of VMI is not an easy task as the involved decision problems are very hard
and interrelated. One of the main problems of VMI, is the coordination of inventory
replenishment and transportation decisions. The inventory routing problem (IRP) is directly
related with this problem which adresses the integration of inventory management and
transportation decisions. The objective here is to maximize the total profit (sales revenues minus
production costs, transportation costs, inventory holding costs, and the shortage penalties) or
minimize the overall costs over the planning horizon.
The aim of this term paper is to present and discuss the inventory routing problem (IRP). This
term paper is organized as follows. We discuss the inventory routing problem in section 2. In
section 3, the variants of the inventory routing problem studied by different researchers are
analyzed and some of the representative solution approaches are presented. We investigate the
answers of some basic questions related to the inventory routing problem and provide some
possible answers to these questions in section 4. Section 5 includes the discussion of a threecustomer problem and a proposed delivery approach different from direct and multiple delivery
strategies. Finally, in section 6 conclusions are provided.
2. The Inventory Routing Problem
The inventory routing problem captures the basic characteristics of situations where vendor
managed resupply may be used, and methodologies developed for its solution could become
building blocks for logistics planning systems (Campbell et al.,1998).
The Inventory Routing Problem (IRP) is concerned with the repetitive distribution of a set of
products from a single facility to a set of customers over a given planning horizon. The facility
produces these products at given rates and have ample storage capabilities for the products. The
customers consume products at a given rate and have limited storage capabilities. A fleet of
vehicles is available at the facility as well as a set of drivers. The objective is to minimize the
overall costs including transportation, shortage and inventory holding costs or maximize the total
profit (revenue minus costs) over a planning period during the planning period. The IRP has
numerious industrial applications such as air products, automotive parts and foods distribution as
well as blood collection.
In the Vehicle Routing Problem (VRP), a fleet of vehicles coordinated by distribution center
visits many delivery points while satisfying their capacity and time constraints. The feature
differentiating the IRP from the VRP is the inventory constraints. The distribution company
would have to take into account the inventory level to decide when to visit the customers and
how much to deliver. The decision maker should be able to reach the precise information at any
time on the inventory levels and location in order to maximize the benefits of the system.
IRP is based on customer’s usage rather than customer’s orders and is a multi-period dynamic
problem. For every day of the period three decisions have to be made to three sub-problems:
 Which customers to visit
 How much to deliver
 Which delivery routes to use
The global optimization of these sub-problems constructs better results than fixing the solution
of one problem before resolving the second problem (Cousineau-Ouimet, 2002). Since the
solutions to the first two problems are the inputs for the vehicle routing problem which has
already been NP-hard, the combined modelling of these problems and finding solutions to these
interrelated sub-problems is a very complex task.
Golden, Assad & Dahl (1984) discussed the difficult problem resulting from the addition of the
inventory constraint to the Vehicle Routing Problem : “ The multi-period nature of the problem
is largely responsible for its complexity. Since decisions taken on different days interact, myopic
decision-making can cause serious problems […]”
3. Literature Review
Since the inventory routing problem is a long term dynamic and stochastic control problem in
practice, it is extremely difficult to solve the real life IRP. As a result, the studies in the literature
have simplified the problem by making different assumptions. In table 1, the variants of the
inventory routing problem studied by different researchers and the contributions that have made
are catagorized. The column headings in the table represent some key problem characteristics,
which will be briefly described below. Customer demands, have been modeled as stochastic and
deterministic. In real life customer demands are not known before the usage takes place. The
number of available vehicles, i.e. fleet size, is assumed to be unlimited by some researchers,
although it is, in practice, limited. Another key issue is the planning period. While some
researchers approach to analyze the IRP as a short term problem, the others study IRP as a long
term problem or consider a reduced horizon approach in which the effects of the short term
period is reflected to the long term. In practice, the objective is to maximize the profit overall
long period of time. Another issue is the number of customers visited on a vehicle trip. In the
table while direct refers to a single customer, multiple refers to a set of customers. Finally, a
distinguishing feature of research contributions is whether policies are presented that specify
when to deliver to each customer, how much to deliver to each customer, and how to deliver to
customers, or whether bounds on the profits (or costs) are presented.
Table 1: Problem characteristics considered by previous researchers
Reference
Fisher et al. (1983)
Federgruen and Zipkin (1984)
Golden, Assad and Dahl (1984)
Burns et al. (1985)
Dror, Ball and Golden (1985)
Dror and Levy (1986)
Dror and Ball (1987)
Chien, Balakrishnan and Wong (1989)
Anily and Federgruen (1990)
Gallego and Simchi-Levi (1990)
Trideau and Dror (1992)
Anily and Federgruen (1993)
Bramel and Simchi-Levi (1995)
Viswanathan and Mathur(1997)
Campbell et al. (1998)
Kleywegt, Savelsbergh and Nori (1999)
Demands
Deterministic
Stochastic
Deterministic
Deterministic
Deterministic
Deterministic
Stochastic
Deterministic
Deterministic
Deterministic
Stochastic
Stochastic
Deterministic
Deterministic
Deterministic
Stochastic
Vehichles
Limited
Limited
Limited
Limited
Limited
Limited
Limited
Limited
Unlimited
Unlimited
Unlimited
Unlimited
Unlimited
Unlimited
Limited
Limited
Horizon
Short
Short
Short
Short
Reduced
Reduced
Reduced
Reduced
Long
Long
Reduced
Long
Long
Long
Long
Long
Delivery
Multiple
Multiple
Multiple
Direct, Multiple
Multiple
Multiple
Multiple
Multiple
Multiple
Direct
Multiple
Multiple
Multiple
Multiple
Multiple
Direct
Contribution
Policy
Policy
Policy
Bound
Policy
Policy
Policy
Policy
Bound and Policy
Bound
Policy
Bound and Policy
Bound and Policy
Bound and Policy
Policy
Policy
Here a comperensive review of the literature is not provided, but rather, papers that are
representative of the solution approaches have been proposed and investigated.
Federgruen and Zipkin (1984) analyze the inventory routing problem as a single period(day)
problem and utilize many of the ideas from vehicle routing. Their problem consists of a plant
with a limited amount of available resource and the demand for this resource at a customer per
day is assumed to be a random variable. In a period(day),the problem is to allocate the inventory
among the customers so as to minimize transportation plus inventory and shortage costs at the
end of the day while satisfying capacity constraints and meeting customer demands. The problem
here is modeled as a nonlinear integer program. Since the inventory and shortage costs are
charged and a limited amount of inventory is available, some of the customers will not be visited
every day and for the customers not receiving a delivery, a dummy route is used in the model.
The solution approach of the problem is decomposing the non-linear integer program into an
inventory allocation problem which determines the inventory and shortage costs and a TSP for
each vehicle which yields the transportation costs. Therefore while using separate calculations
for the allocation and routing decisions, they are coordinated appropriately. The idea beyond the
interchange heuristic used in the model is to construct an initial feasible solution and iteratively
improve the solution by switching customers between routes. Evaluation of such a switch in this
problem clearly requires more computational time than that in the standart vehicle routing
algorithms. Each switch identifies a new customer to route assignment, which in turn identify a
new inventory allocation problem and new TSPs.
Golden, Assad, and Dahl (1984) develop a heuristic that tries to minimize costs on a single day
while maintaining an adequate level of inventory for all customers. The heuristic starts with
determining the potential customer for delivery by computing urgency which is based on the
ratio of remaining tank inventory level to tank capacity for each customer. All customers with
the ratio smaller than a certain point are considered as potential customers. The urgency so the
profit attached to each potential customer increases with the decrease of this ratio. The
appropriate profit matrix and the time matrix which consists of travel times between potential
customers are set up for the time constraint travelling salesman problem(TCTSP). To solve this
large TSP, initially, a time limit for the total travel time of the tour, say TMAX, is set to the
number of vehicles multiplied by the length of a day. Customers are added until this limit is
reached or there are no more customers left. The final tour is partitioned into a set of feasible
routes by enforcing that each customer must be filled up when it receives a delivery. If this turns
out to be impossible, the heuristic can be re-run with a smaller value for TMAX.
Same with the Golden, Assad, and Dahl (1984), Chien, Bakrishnan, and Wong (1989) also
develop a single day approach. However different from the study of Golden et. al which treats
each day completely separate, the system studied here links each single period to the following
one, by passing information from one day to the next, and solves a series of single period
inventory allocation and vehicle routing problems in order to find appropximate solution for the
multiperiod vehicle routing problem. Based on the maximum usage per day for each customer,
the daily profit is defined a revenue per unit delivered minus a penalty per unit of unsatisfied
demand. Unsatisfied demand today is projected to tomorrow as an increased revenue. The
objective of their heuristic is to maximize the total profit on a single day. Once a solution for one
day is found, the results are used to modify the revenues for the next day. A mixed integer that
considers the allocation of the limited inventory available at the plant to the customers, the
customer to vehicle assignments, and the routing, is formulated. To solve this mixed integer
program a Lagrangean dual ascent method is used.
Fisher et al.(1983) study the inventory routing problem at Air Products, an industrial gases
producer. The objective of the model considered here is maximizing the profit from product
distribution over the next two-to-five day horizon. In this research demand is not considered to
be a random variable or completely deterministic but demand is given by upper and lower
bounds on the amount to be delivered to each customer for every period in the planning horizon.
The route selection model is formulated as an integer program that considers delivery volumes,
assignment of customers to routes, assignments of vehicles to routes, and assignment of start
times for routes. This integer program is analyzed and solved using a Lagrangean relaxation to
dualize a set of constraints whose removal greatly simplifies and reduces the problem to a
knapsack problem which can be solved by dynamic programming.
Dror and Ball (1986) and (1987),approach the inventory routing problem in such a way they take
into account what happens after the short-term planning period. In one of these two companion
papers, Dror and Ball (1986), consider the problem of reducing a long-term problem to a single
time period problem. Demand is assumed to be deterministic. Two solution strategies are
proposed and compared in detail, one which first allocates deliveries to days and then solves a
vehicle routing problem for each day and a second which treats the multiday problem as a
modified vehicle routing problem. Then to obtain an improved solution, inter-route and inter-day
customer exchanges are executed.
Dror and Ball (1987) propose single customer deterministic and stochastic models and also a
multiple customer model. The optimal replenishment day t’ minimizing the expected total cost is
determined for each customer by using the probability that a customer will run out on a specific
day in the planning period, the average cost to deliver to the customer, and the anticipated cost of
a stockout. If t’ falls within the short-term planning period, the customer will definitely be
visited, and a value ct is computed for each of the days in the planning period that reflects the
expected increase in future cost if the delivery is made on day t instead of on t’. If t’ falls outside
the short-term planning period a future benefit gt can be computed for making a delivery to the
customer on day t of the short-term planning period. These computed values reflect the long term
effects of short term decisions. To assign the customers to a vehicle and a day, or just day, an
integer program which minimizes the sum of these costs plus the transportation costs, is solved.
Then either TSP or VRP problems are solved in the second stage.
Trudeau et al. extended and improved some of the ideas of Dror and Ball (1987). Dror and Levy
(1985) use a similar analysis to make a weekly schedule, but then apply node and arc exchanges
to reduce costs in the planning period.
The first integrated infinite horizon model appears to be studied by Burns et al. (1985). Variable
transportation costs are assumed to be proportional with Euclidean distances and demands at the
retailers occur at a constant deterministic rate. The focus of this research is on an analytic
approach to solving distribution problems. In this study two distribution strategies, direct
shipping and peddling are analyzed and compared. It is shown that the cost trade-off depends on
the shipment size. The results indicate that, for direct shipping, the optimal shipping size is given
by the economic order quantity (EOQ) model, while for peddling, the optimal shipment size is a
full truckload. This trade-off is evaluated analytically and graphically.
Anily and Federgruen (1990) studies on minimizing long run average system-wide transportation
and inventory cost by the determination of long term integrated replenishment strategies
(i.e.inventory rules and routing patterns). Inventory is allocated at the retailers not at the depot.
To determine the routing patterns, a modified circular regional partitioning scheme is used. After
partitioning the customers, customers facing identical demand rates are all located at the same
region within a partition. If a customer is seemed to be in more than one region, a certain
percent of his demand is allocated to each region. The replenishment policy is restricted to a
class of strategies in which whenever a customer in a region gets a delivery, all other customers
in that region are also visited. Using this restricted class of strategies, they reduce the original
problem to a Euclidean Vehicle Routing problem with a cost function depends on the length of
the route and the number of points visited. In this paper, also a lower bound for the average cost
is determined to evaluate the performance of their routing patterns. Some variants consisting of
uncapacitated models, nonidentical vehicles, sales volume constraints and backlogging options
are applied to the same model with some modifications.
Viswanathan and Mathur (1997) study a multiproduct generalization of the distribution model
considered by Anily and Federgruen (1990). The heuristic developed here provides a nested joint
replenishment policy in which replenishment intervals of different items are multiples of each
other with equal replenishment intervals for each of the items. Unlike the methods discussed
before , the proposed heuristic is capable of solving problems involving distribution systems with
multiple products.
Gallego and Simchi-Levi (1990) also used similar ideas to those of Anily and Federgruen [1990]
to evaluate the long run effectiveness of direct shipping (separate loads to each customer) . The
results of their study shows that direct shipping is at least 94% effective over all inventory
routing strategies whenever minimal economic lot size is at least 71% of truck capacity. This
indicates that direct shipping is not a good policy when many customers require significantly
less than a truck load, making more complicated routing policies the appropriate choice.
Bramel and Simchi-Levi (1995) study another adaptation of these ideas. Different from the
earlier studies, they consider that customers can hold an unlimited amount of inventory. Firstly
they transform the problem to a capacitated concentrator location problem (CCLP), solve the
CCLP, and then transform the solution back into a solution to the IRP. The solution to the CCLP
will partition the customers into disjoint sets, which in the IRP, will become the fixed partitions.
Delivery to these partitions are made with a similar approach considered for the regions of Anily
and Federgruen.
In the study of Anily and Federgruen (1993), the analysis of Anily and Federgruen (1990) is
extended to the case where central inventories may be kept in the warehouse. The same
replenishment strategy is improved to integrate the problems of determination of an appropriate
replenishment strategy for the warehouse, which is optimally coordinated with that of each
retailer and parallel with the transportation schedules.
The research of Campbell, Clarke, Kleywegt and Savelsbergh (1998) focuses on developing a
flexible system capable of handling large instances that properly balances short-term and long
term goals and that considers all of the important factors, i.e., geography, inventory, capacity,
and usage rate. Three issues are addressed here: How to create a finite horizon approximation of
an infinite horizon problem, and how to make effective use of aggregation in integer
programming. Their solution methodology is in two phases and uses both integer programming
and new heuristics. Their study is applied to an industrial problem at Praxair, an international
industrial gases company.
Kleywegt, Savelsbergh and Nori (1999) consider the stochastic and direct delivery version of the
problem studied by Campbell, Clarke, Kleywegt and Savelsbergh (1998).They formulate the
inventory routing problem as a Markov decision process, and they propose approximation
solutions to the problem.
4. Basic Questions related to the inventory routing problem
In this term paper, we investigate the answers of the following basic questions related to the
inventory routing problem and propose some answers to these questions.
How would you decide which customers should receive a delivery on a day to make sure none of
them would run out of product?
The current inventory levels of each customer and the distance of each customer from the plant
and from each other are the two factors that can be considered while making a delivery schedule.
Since the location of each customer is known and constant, the lead time required to ship a
delivery to a customer is also known and placed to the schedule as a constant. Therefore we
focus on the current inventory level which is determined by the initial inventory minus the
consumed amount. We can determine the candidates for a possible delivery by setting an
urgency level for each of the customers. This can be realized by not considering the customers as
potential customers unless they have a ratio of remaining tank level to the tank capacity below
some predetermined threshold as in the study of Golden et al (1984).
In real life the consumption amount is stochastic which leads us to maintain an adequate level of
inventory at all points in time. We can use this safety stock level as a signal to start planning a
delivery to a given customer. Then a question comes to mind :
How can we set this level?
The level of safety stock depends on the service level you select, on the replenishment lead time
and on the stochastic information about the customer usage rate, such as customer specific
deviation. Based on the expected cost (inventory holding and shortage costs), the service level
can be determined, then the safety stock level can be determined to satisfy this target service
level. We should consider the trade-off between the inventory holding cost and the shortage cost
when we make the delivery plans. If we deliver to a customer too late, then we will run a high
risk of paying the stockout penalty associated with the customer. On the other hand, if we deliver
too early, the inventory holding cost will be incurred at the customer site. Therefore we should
consider this trade-off while we calculate the expected cost and set the safety stock level.
Which customers do you think would be good choices to be on a route together? What factors
would you use to make such a decision?
We make the choices in order to minimize the total cost of a route. The cost of serving a route
does not only depend on the geographic locations of the customers in that route, but also on the
compatibility of the inventory capacities and usage rates of the customers in a route.
In the study of Savelbergh et al.(1999), a cluster is defined as to be a set of the customers on the
same route and the following approach is used to identify a good set of disjoint clusters covering
all customers :
1. Generate a large set of possible clusters.
2. Estimate the cost of serving each cluster.
3. Solve a set partitioning problem to select clusters.
If you were a planner trying to make a schedule for these customers, is there any other
information do you think to be helpful?
Other than the inventory, demand and location information of the customers, the information
including the relative importance of the customers may be useful. The capacity of each of the
customers is also important for the plant. Another issue is the selling price of the same product at
different customers (retailers). Think about two customers one of which receives a smaller batch
size and sells with a higher selling price than that of the other. The priority may be given to the
customer with the smaller batch size when we consider the total system profit maximization or a
negotiation between this customer and the vendor which requires the customer to guarantee to
buy a fixed batch size each time.
If you were making a schedule for delivering to these customers, which actions would you take
to make sure you wouldn’t let anyone run out of product?
Since the scheduling problem here is hard, it is very difficult to make an optimal schedule. So,
firstly, we try to make a feasible schedule in order not to let any customers run out of product.
The lead times of the deliveries are important features for a feasible schedule. If we can not
satisfy the lead time constraints so as to prevent the stockouts, we try to make a feasible
schedule with making some shipments earlier than their due dates or increase the number of the
vehicles. Here since we assume that the facility has an ample storage capacity, the bottleneck
seems to be the number and the capacity of the vehicles. However we should consider the tradeoff between the inventory holding cost of the customers and the transportation cost of the
supplier and take an action that will minimize the total cost. If we increase the number of our
vehicles, then we will increase transportation costs. On the other hand, if we deliver too early,
the inventory holding cost will be incurred at the customer site.
For each of the costs -stockout cost, driver cost, and vehicle cost- if it represented the only cost
involved, how would this change your delivery policy?
Intuitively, if the stockout cost is higher than the driver or vehicle costs, we increase the number
of the vehicles and the number of the drivers to prevent any stockout. If the case is the vehicle
costs, we decrease the number of vehicles, try to aggregate the customers on a route as much as
possible and we permit the stockouts to be occur. If the case is the driver costs, we do not change
the number of vehicles but increase the capacity of vehicles and try to aggregate the customers
on a route as much as possible.
How do small delivery time windows for customers complicate the problem?
A delivery time window restricts when a customer can receive a delivery. Small delivery time
windows, of course, complicate the problem, since extra constraints are added to the problem.
Another reason for why time windows make the problem more complex may be the congestion
of the demands on the delivery days. For instance, a customer receives delivery on monday and
tuesday, and if there is a necessity to ship another customer on these days and there are not
enough vehicles to make the delivery, the plant may have to choose one of customers to ship and
let the other to stockout or increase the vehicle number. Here we look at the trade-off between
the extra vehicle procurement cost and shortage costs.
5. Direct Delivery or Peddling or Sometimes Direct Delivery- Sometimes Peddling
A key issue while making delivery is the number of customers visited on a single trip. In many
situations vehicles can visit multiple customers on each route. Several researchers have also
studied variants in which a single customer is visited on each route, which is called the direct
delivery case. On the other hand some researchers have considered a set of customers on a single
trip. In the study of Burns et al.(1984), two distribution strategies, direct shipping(separate loads
to each customer) and peddling(multiple customers are shipped in a single trip) are analyzed and
compared. It is shown that the cost trade-off depends on the shipment size. The results of
Gallego and Simchi-Levi (1990)’s study shows that direct shipping is at least 94% effective over
all inventory routing strategies whenever minimal economic lot size is at least 71% of truck
capacity. As mentioned in the literature review, this indicates that direct shipping is not a good
policy when many customers require significantly less than a truck load.
Other than direct shipping and peddling, another approach comes to mind: sometimes visit the
customer together and sometimes visit them by themselves. The efficiency of this approach has
not been studied and compared with the performance of the direct shipping and peddling before.
Therefore this approach has the potential to be a research topic of a future work.
The three-customer problem
In this term paper, the tree-customer problem is considered and the tour efficiency of the
proposed solutions to the problem is discussed. This problem and the proposed solutions are
illustrated on the sketchs in two parts. In the first part, some of the results of the previous studies
considering direct shipping and/or peddling are applied to the problem. In the second part, a
different strategy, sometimes visit the customer together and sometimes visit them by
themselves, is proposed and the application of this approach is presented on the sketchs.
The three customer problem is concerned with the distribution of a set of products from a plant
to three customers over a single day in the first part and over two-days period in the second part.
The plant has an ample storage capability for the products. The customers consume products at a
given rate and have limited storage capabilities.The distances of customers from the plant and
from each other are shown in the figures. Any stockout is not permitted. Beginning inventory is
assumed to be zero. Two vehicles are available at the plant with the capacity of 500 units in part
one, and 400 in part two. The vehicle may be shipped with less than its capacity. The objective
here is to find the cost-efficient tours. In this problem not only do we have to decide which
customers to visit next, but also how to combine them into vehicle tours, and how much to
deliver at each customer. Even if there are only three customers, these decisions may not be easy.
For simplicity, the problem is considered on the specific examples.
Part 1
In part one, we consider a single period problem and do not consider what happens after the
single period. We assume that the customers require as much as their capacity and also we have
enough vehicles on hand to prevent any stockout. Here only transportation costs are considered.
So this single period problem turns into a vehicle routing problem.
In figure 1, the capacity and the delivery batch size is the same for each customer. Obviously,
direct shipment is the best strategy here since demand of each customer is equal to the maximum
truckload. In figure 2, since the maximum truckload is equal to the sum of the demands of A and
B, we combine A and B on a single tour and save from the transportation cost. Here, we make a
direct delivery to C.
Capacity: 400
Delivery: 400
Capacity: 500
Delivery: 500
Vehicle
Capacity:500
Vehicle
Capacity:500
A
A
80 miles
100
miles
Capacity:
500
Delivery:
500
B
25
miles
C
Capacity:
100
Delivery:
100
100
mile
s
B
25
miles
Capacity: 500
Delivery: 500
C
Plan
Plant
10 miles
Figure 1
Capacity:
500
Delivery:
500
10 miles
Figure 2
While, in the third figure, combining B and C on the same route would be a cost-effective
strategy, in figure 4 we consider to combine all of the customers on the same route since the
total requirement is equal to our vehicle capacity.
In figure 5, all customers require less than a truckload, so the strategy must be to combine two of
them. Here the transportation cost is the factor which will determine our decision. The
transportation cost of combining B and C and making a direct delivery to A is greater than the
transportation cost of combining A and B and making a direct delivery to C. Therefore our
strategy should be the latter one as shown in figure 5.
Capacity: 100
Delivery: 100
Capacity: 500
Delivery: 500
Vehicle
Capacity:500
Vehicle
Capacity:
500
A
A
80 miles
Capacity:
400
Delivery:
400
100
mile
s
Capacity:
200
Delivery:
200
100
miles
B
25
miles
B
20 miles
20 miles
C
Plan
Capacity:
100
Delivery:
100
C
Capacity: 200
Delivery: 200
Plant
10 miles
10 miles
Figure 3
Figure 4
Capacity: 300
Delivery: 300
Vehicle
Capacity:500
A
80
miles
Capacity:
200
Delivery:
200
100
mile
s
B
25
miles
C
Plan
10 miles
Figure 5
Capacity:
200
Delivery:
200
Part 2
In part two we extended the single day problem to a two-day problem to apply our approach. Our
vehicle capacity is 400 units in this version. Here we assume that the usage rate is 100 units per
day for each of the three customers. Here also we do not permit any stockout. In the example
below, we apply another strategy. We can call this strategy sometimes direct, sometimes multiple
delivery. This strategy refers to a situation in which sometimes we visit the customers together,
and sometimes visit them by themselves. In this part, we have two costs: the inventory holding
cost and transportation costs. It is assumed that the inventory holding cost is much less than the
transportation cost. Our objective here to minimize the overall cost. We have an example which
is illustrated in figure 6 and 7.
In Figure 6 we consider the first day of the delivery, and the second day of the delivery is
illustrated in figure 7.
As shown in figure 6 and 7, we combine A, B and C on the first day of the two-day period. On
the remaining day, we only make a shipment to A and B on a single route.
Capacity: 100
Delivery: 100
Capacity: 100
Delivery: 100
Vehicle
Capacity:400
Vehicle
Capacity:400
A
A
80
miles
80
miles
Capacity:
100
Delivery:
100
100
mile
s
B
Capacity:
100
Delivery:
100
100
mile
s
B
20 miles
C
Plan
Capacity:
200
Delivery:
200
25
miles
C
Plan
Capacity:
200
Delivery:
200
10 miles
Figure 6
Figure 7
As a result, we can say that the proposed delivery strategy may be effective in the cases in which
the inventory holding cost is less than the transportation costs as we want to minimize the overall
costs. Intuitively, when one customer has much higher usage rate or a much smaller tank size
than the others, we visit that customer by itself several times and occasionally visit that customer
with the others.
Conclusions
The inventory routing problem is of special interest because it integrates two components of
supply chain management: inventory control and vehicle routing. This type of integration is
essential to improve overall system performance.
In this term paper, we presented and discussed the inventory routing problem (IRP). Some of the
solution methods proposed by the previous researchers to solve the IRP are also analyzed. We,
then investigate answers to some of the basic questions related to IRP. Then an alternative
approach to direct and multiple delivery is proposed and the results of the applications showed
that this approach may be useful for the cases in which transportation costs are higher than
inventory holding costs. Also another result is that it is reasonable to ship a customer by itself
several times and sometimes visit her with the others, when she has a much higher usage rate or
a much smaller tank size. Sometimes direct - sometimes multiple approach may be presented as a
future research since the effectiveness of this approach has not been studied and compared with
the other shipping strategies before.
References
A. Federgruen and P.Zipkin(1984). A combined vehicle routing and inventory allocation
problem. Operations Research, 32(5):101 –1036
Ann Campbell, L.Clarke, A. Kleywegt and M.Savelsbergh (1998)The Inventory Routing
Problem. In Fleet Management and Logistics. T.G. Crainic and G. Laporte (editors). Kluwer
Academic Publishers, 95-113
B. Golden, A. Assad, and R. Dahl(1984). Analysis of a large scale vehicle routing problem
with an inventory component. Large Scale Systems, 7(2-3):181–1 0, 1 84.
G. Gallego and D. Simchi-Levi.(1990) On the effectiveness of direct shipping strategy for
the one-warehouse multi-retailer-systems. Management Science, 36(2):240–243,
Lawerence D. Burns,Randolph W. Hall, Dennis E.Blumenfeld and Carlos F. Daganzo (1985).
Distribution Strategies that Minimize Transportation and Inventory Costs. Operations Research,
33(3):469-490
J. Bramel and D. Simchi-Levi(1995). A location based heuristic for general routing problems.
Operations Research, 43(4):649 –660
Kleywegt, Savelsbergh and Nori (1999), The Stochastic Inventory Routing Problem with Direct
Deliveries, Transportation Science, 36(1)
Kleywegt, Nori and Savelsbergh (2002).Dynamic Programmming Approximations to A
Inventory Routing Problem, Transportation Science
M. Dror and L. Levy.(1985) Vehicle routing improvement algorithms: Comparison of a ’greedy’
and a matching implementation for inventory routing. Computers and Operations Research,
13(1):33–45.
M. Dror, M. Ball, and B.Golden(1985) Computational comparison of algorithms for the
inventory routing problem. Annals of Operations Research, 4(1-4):3–23
M. Dror and M. Ball(1987) . Inventory/routing: Reduction from an annual to a short
period problem. Naval Research Logistics Quarterly, 34(6):8 1– 05
S. Anily and A. Federgruen.(1993)Two-Echelon Distribution Systems with Vehicle Routing
Costs and Central Inventories. Operations Research, 41(1):37-47
S. Anily and A. Federgruen(1990) One warehouse multiple retailer systems with vehicle routing
costs. Management Science, 36(1): 2–114
S. Viswanathan and K. Mathur(1997). Integrating Routing and Inventory Decisions in OneWarehouse Multiretailer Multiproduct Distribution Systems. Management Science, 43(3): 294
312
P. Trudeau and M. Dror(1992). Stochastic inventory routing: Route design with stockouts
and route failures. Transportation Science, 26(3):171–184
T. Chien, A. Balakrishnan, and R. Wong(1989) An integrated inventory allocation and
vehicle routing problem. Transportation Science, 23(2):67–76
W. Bell , L. Dalberto, M. Fisher, A. Greenfield, R. Jaikumar, P. Kedia, R. Mack, and P.
Prutzman(1983) Improving the distribution of industrial gases with an on-line computerized
routing and scheduling optimizer. Interfaces, 13(6):4–23