Location, Routing, and Inventory
Site: | Saylor Academy |
Course: | BUS606: Operations and Supply Chain Management |
Book: | Location, Routing, and Inventory |
Printed by: | Guest user |
Date: | Saturday, September 7, 2024, 7:55 PM |
Description
Read this article. In it, a model is presented to help determine the number of distribution centers, their locations, and capacity among other factors. Among the 15 assumptions presented, which do you feel are most important and least important?
Abstract
This paper considers a single-sourcing network design problem for a three-level supply chain. For the first time, a
novel mathematical model is presented considering risk-pooling, the inventory existence at distribution centers
(DCs) under demand uncertainty, the existence of several alternatives to transport the product between facilities,
and routing of vehicles from distribution centers to customer in a stochastic supply chain system, simultaneously.
This problem is formulated as a bi-objective stochastic mixed-integer nonlinear programming model. The aim of
this model is to determine the number of located distribution centers, their locations, and capacity levels, and
allocating customers to distribution centers and distribution centers to suppliers. It also determines the inventory
control decisions on the amount of ordered products and the amount of safety stocks at each opened DC,
selecting a type of vehicle for transportation. Moreover, it determines routing decisions, such as determination of
vehicles' routes starting from an opened distribution center to serve its allocated customers and returning to that
distribution center. All are done in a way that the total system cost and the total transportation time are minimized.
The Lingo software is used to solve the presented model. The computational results are illustrated in this paper.
Keywords: Stochastic supply chain; Inventory control; Risk-pooling; Uncertainty; Capacity levels
Source: Reza Tavakkoli-Moghaddam, Fateme Forouzanfar, and Sadoullah Ebrahimnejad, https://www.econstor.eu/handle/10419/147160
This work is licensed under a Creative Commons Attribution 4.0 License.
Background
Nowadays, rapid economic changes and competitive pressure in the global market make companies pay more attention on supply chain topics. The company whose supply chain network structure is more appropriate has higher competitive advantage. This structure helps to overcome environmental disturbances. Analyzing location issues and decision making about facility location is considered as one of the important issues of decision making in companies. Certainly, appropriate facility location has high effects on economic benefits, appropriate service, and customer's satisfaction. Propounding the supply chain because of its effect on factors of operational efficiency, such as inventory, response, and lead time, specific attention is focused on how to create a distribution network. Facility location and how to relate them with customers are an important factor in designing a distribution network.
As nowadays living conditions have changed due to increasing world changes, mutually, situations have changed where supply chains are confronted with and influenced by them. The manager is confronted with more unknown conditions and new risks. Customers' demands have been more uncertain and various, and the lead time on their services is very effective. The demand variety can be recognized as one of the important sources of uncertainty in a supply chain. Hence, inventory and product holding in a distribution center are an important issue in the supply chain. The inventory existence in these centers can lead to a great success in reaching the risk-pooling advantage in order to overcome the variability of customer demands. The proposed risk-pooling strategy and centralizing the inventory at distribution centers are considered as one of the effective ways to manage such a demand uncertainty to achieve appropriate service levels to customers. The lead time is one of the effective factors in safety stock levels due to customer demand uncertainty. For sure, whether the amount of the level is low for the product, it is considered an additional value that one can gain a long-term or short-term competitive advantage in the market.
The lead time is dependent on different factors, such as transportation mode. Different modes of transportation include a reverse relation between cost and time. It contains different routes for any type of vehicles. The implicit assumption is that a faster transportation mode is also the more expensive one, creating a trade-off between cost and time affecting the distribution network configuration. In the recent decades, the topic of multi-depot heterogeneous vehicle routing problem is presented in order to increase the productivity and efficiency of transportation systems, in which this model leads to the least cost function by minimizing the number of vehicles.
Literature review
One of the important factors of the total
productivity and profitability of a supply chain is to consider its
distribution network, which can be used to achieve variety of the supply
chain objectives. Designing a distribution network consists of three
subproblems, namely, location allocation, vehicle routing, and inventory
control. In the literature, there are some research studies
amalgamating two of the above subproblems, such as location-routing
problems, inventory-routing problems, and location-inventory problems.
These three subproblems of a distribution network design are considered
in few papers simultaneously. Location-routing problems are surveyed and
classified by Min et al. and Nagy and Salhi.
Inventory-routing problems are studied in several studies. In
addition, a number of studies have considered location-inventory
problems. Finally, Ahmadi Javid and Azad presented a new model for a location-routing-inventory problem.
They considered one objective for their model and did not consider
transportation time and risk-pooling. However, in this paper, we present
a multi-objective model to concurrently optimize location, allocation,
capacity, inventory, selection of vehicles, and routing decisions with
risk-pooling in a stochastic supply chain system for the first time.
These decisions are made in a way that the total system cost and the
total transportation time are minimized.
Problem Formulation
Problem description
The trade-off
between cost and time creates a bi-objective problem. One criterion
tries to minimize the fixed cost of locating the opened distribution
centers, the safety stock costs of distribution center by considering
uncertainty in customer's demand, inventory ordering and holding costs,
the transportation costs from a plant to its allocated distribution
centers, and also vehicle routing costs beginning from a distribution
center (DC) with the aim of replying to and covering the devoted
customer's demands to the DC by considering risk-pooling. The other
criterion looks for the reduction of the time to transport the product
along the supply chain. It is desired to minimize the transportation
time from a plant to customers. The important assumptions in this paper
are as follows:
- One kind of product is involved.
- Each distribution center j is assumed to follow the (
,
) inventory policy.
- The inventory control is to be conducted only at DCs in this paper.
- A single-sourcing strategy is considered in the whole supply chain.
- It is considered that the customers' demands after reaching the retailer are independent and follows a normal distribution.
- Each plant has a limited capacity.
- We consider different capacity levels for each distribution center, and finally, one capacity for each of them is selected.
- Each DC with the limited capacity carries on-hand inventory to satisfy demands from customer demand zones as well as safety stock to deal with the mutability of the customer demands at customer demand zones to attain risk-pooling profits.
- All customers should be served.
- The number of available vehicles for each type and the number of allowed routes for each DC are limited.
- There are several modes of transportation between two consecutive levels.
- Between two nodes on an echelon, only one type of vehicle is used.
- A faster transportation mode is the more expensive one.
- The amount of products is transported from each plant to each distribution center that is associated with it, and an equal amount of products has been ordered from the desired distribution center to that plant.
- To determine all feasible routes, the following factors are taken into account:
- Each customer should be visited by only one vehicle.
- Each route begins at a DC and ends at the same DC.
- The sum of the demands of the customers served in each route must not exceed the capacity of the associated vehicle.
- Each of the distribution center and the vehicle have various limited, and determined capacity.
Model formulation
Following are the notations introduced for the mathematical description of the proposed model.
1. Indices
-
, set of plants indexed by
, set of candidate DC locations indexed by
, set of customer demand zones indexed by
, set of capacity levels available to
, set of all feasible routes using a vehicle of type
from
, set of vehicles
between nodes
and
, set of vehicles
between nodes
and
2. Parameters
, yearly fixed cost for opening and operating distribution center
with capacity level
, cost of transporting one unit of product from plant
to distribution center
using vehicle
, cost of sending one unit of product in route
using vehicle
(These costs include the fixed cost of vehicle plus the transportation cost of each demand unit in route r. The mentioned transportation cost for each demand unit is not related to customer demand zone, and it is considered fixed for all locations in each route
).
, time for transporting any quantity of a product from plant
to
using vehicle
, time for transporting any quantity of a product from
on route
using vehicle
, safety stock factor of
, unit inventory holding cast at
, (annually)
, mean demand at customer demand zone
, variance of demand at customer demand zone
, fixed inventory ordering cost at
, capacity with level
for
, capacity of plant
, number of available vehicles of each type
, number of routes associated with each distribution center
3. Binary coefficients
4. Decision variables
if distribution center
is opened with capacity level
, and 0 otherwise
- \(A_{ijl1}, binary variable equal to 1 if vehicle
connecting plant
and
is used, and equal to 0 otherwise
, binary variable equal to 1 if vehicle
connecting
and customer
is used, and equal to 0 otherwise
, 1 if and only if route
is selected, and 0 otherwise
, quantity transported from plant
to
using vehicle
5. Mathematical model
(a) The problem formulation is as follows:
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
(21)
In
this model, the first objective function minimizes the total expected
costs consisting of the fixed cost for opening distribution centers with
a certain capacity level, transportation costs from plants to
distribution centers, annual routing costs between distribution centers
and customer demand zones, and expected annual inventory costs. The
second objective function looks for the minimum time to transport the
product along any path from the plants to the customers.
Constraint
(1) ensures that each distribution center can be assigned to only one
capacity level. Constraints (2) and (3) are the capacity constraints
associated with the distribution centers, and also, constraint (4) is
the capacity constraints associated with the plants. Constraint (5)
states that if distribution center with
capacity is opened, it is
serviced by a plant. Constraint (6) represents the single-sourcing
constraints for each customer demand zone. Constraints (7) and (8)
ensure that if two nodes on an echelon are related to each other, one
type of vehicle transports products between them. Constraint (9) makes
sure that if the distribution center j gives the service to the customer
k, that center must get services at least from a plant. Constraint (10)
ensures that if the distribution center j is allocated to customer k by
vehicle
, that center should certainly be established with a
determined capacity level. Constraint (11) is the standard set covering
constraints, modeling assumption 9. Constraints (12) and (13) impose
limits on the maximum number of available vehicles of each type and the
maximum number of allowed routes for each DC, modeling assumption 10.
Constraint (14) makes sure that if plant i gives the service to the DC j
, the amount of transported products from that plant to the desired
distribution center would be more than one. Constraint (15) implies that
customers' demands of zone
are more than 1. Constraint (16) is the
capacity constraint associated with plant i. Constraints (17) to (20)
enforce the integrality restrictions on the binary variables. Finally,
constraint (21) enforces the non-negativity restrictions on the other
decision variables.
Solution method
Optimization is a mathematical procedure to
determine devoting the optimal allocation to scarce resources, and it
helps to get the best result from the model solution. In this paper, we
consider five examples, and then they are solved by the Lingo 9.0
software to show that this model works well. This software is a
comprehensive tool designed to make building and solving linear,
nonlinear, and integer optimization models faster, easier, and more
efficient. It provides a completely integrated package that includes a
powerful language for expressing optimization models, a full featured
environment for building and editing problems, and a set of fast
built-in solvers. Objective functions have been normalized
between zero and one. In other words, they have been without any
dimension (i.e., scaleless). By using the following formula, these
objectives are converted to a single objective function, where
and
are the normalized forms of
and
objective functions.
To
minimize deviations from the ideal, this function is reduced. As the
first objective function () is more important than the second
objective function (
) in the given problem, the coefficients of the
above formula are considered in the form of
and
.
This
problem is implemented by this software, and a global optimal solution
is obtained. The computational results are shown in Tables 1, 2, 3, 4,
and 5.
Table 1 is 1 if distribution center
is opened with capacity level
. and 0 otherwise
DC1 | DC2 | |||
---|---|---|---|---|
Capacity 1 | Capacity 2 | Capacity 1 | Capacity 2 | |
Example 1 | 1 | 0 | 0 | 1 |
Example 2 | 0 | 1 | 1 | 0 |
Example 3 | 0 | 1 | 1 | 0 |
Example 4 | 1 | 0 | 0 | 1 |
Example 5 | 0 | 1 | 1 | 0 |
Table 2
Route 1 | Route 2 | Route 3 | Route 4 | Route 5 | Route 6 | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Truck | Airplane | Truck | Airplane | Truck | Airplane | Truck | Airplane | Truck | Airplane | Truck | Airplane | |
Example 1 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | - | - | - | - |
Example 2 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | - | - | - | - |
Example 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 |
Example 4 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
Example 5 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | - | - | - | - |
Table 3
Plant 1 | Plant 2 | |||||||
---|---|---|---|---|---|---|---|---|
DC1 | DC2 | DC1 | DC2 | |||||
Train | Airplane | Train | Airplane | Train | Airplane | Train | Airplane | |
Example 1 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
Example 2 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 |
Example 3 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 |
Example 4 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
Example 5 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 |
Table 4
DC1 | DC2 | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Demand 1 | Demand 2 | Demand 3 | Demand 1 | Demand 2 | Demand 3 | |||||||
Truck | Airplane | Truck | Airplane | Truck | Airplane | Truck | Airplane | Truck | Airplane | Truck | Airplane | |
Example 1 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
Example 2 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
Example 3 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 |
Example 4 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
Example 5 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
Table 5
Plant 1 | Plant 2 | |||||||
---|---|---|---|---|---|---|---|---|
DC1 | DC2 | DC1 | DC2 | |||||
Train | Airplane | Train | Airplane | Train | Airplane | Train | Airplane | |
Example 1 | 4 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
Example 2 | 0 | 0 | 2 | 0 | 8 | 0 | 0 | 0 |
Example 3 | 0 | 6 | 0 | 0 | 0 | 0 | 0 | 2 |
Example 4 | 5 | 0 | 0 | 6 | 0 | 0 | 0 | 0 |
Example 5 | 0 | 0 | 0 | 0 | 6 | 0 | 0 | 1 |
Conclusions
In this paper, a new mathematical model to design a
three-level supply chain has been presented by considering the inventory
under uncertain demands, risk-pooling, vehicle routing, transportation
time, and cost. The decision related to the transportation options has
an impact on the transportation time from plants to customers. The
trade-off between cost and time is considered in the formulation of a
mathematical model that minimizes both criteria. Therefore, this model
holding two objectives has been formulated for the first time as a
location-inventory-routing problem with a risk-pooling strategy in a
three-level supply chain. The Lingo software has been used to solve the
given problem. Some future studies are as follows: considering each
parameter as a fuzzy, multi-period planning and solving the presented
model by using heuristic or meta-heuristic algorithms.