Coordinated Location, Distribution, and Inventory Decisions in Supply Chain Network Design
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Course: | BUS606: Operations and Supply Chain Management |
Book: | Coordinated Location, Distribution, and Inventory Decisions in Supply Chain Network Design |
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Date: | Saturday, September 7, 2024, 7:31 PM |
Description
Read this article. The goal is to understand optimum product allocation and distribution locations so products are delivered at the lowest possible cost. As you read Part 2, what are some other problems associated with supply chain allocation and distribution?
Abstract
This research presents an integrated multi-objective distribution model for use in simultaneous strategic and operational food supply chain (SC) planning. The proposed method is adopted to allow use of a performance measurement system that includes conflicting objectives such as distribution costs, customer service level (safety stock holding), resource utilisation, and the total delivery time, with reference to multiple warehouse capacities and uncertain forecast demands. To deal with these objectives and enable the decision makers (DMs) to evaluate a greater number of alternative solutions, three different approaches are implemented in the proposed solution procedure. A detailed case study derived from food industrial data is used to illustrate the preference of the proposed approach. The proposed method yields an efficient solution and an overall degree of DMs' satisfaction with the determined objective values.
Source: G. Reza NasiriI and Hamid Davoudpour, http://www.scielo.org.za/scielo.php?script=sci_arttext&pid=S2224-78902012000200015
This work is licensed under a Creative Commons Attribution 4.0 License.
Introduction
The distribution planning decision (DPD) is one
of the most comprehensive strategic decision issues that need to be
optimised for the long-term efficient operation of a whole supply chain
(SC). The DPD involves optimising the transportation plan for allocating
goods and/or services from a set of sources to various destinations in a
supply chain. An important strategic issue related to the
design and operation of a physical distribution network in a supply
chain system is the determination of the best sites for intermediate
stocking points, or warehouses. The use of warehouses provides a company
with flexibility to respond to changes in the marketplace, and can
result in significant cost savings due to economies of scale in
transportation or shipping costs.
The major task of DPD is the
determination of distribution costs, customer service level (safety
stock holding), resource (warehouse space) utilisation, and the total
delivery time, with reference to multiple warehouse capacities and
uncertain forecast demands.
In
previous studies, both deterministic and stochastic customers' demands
have been considered, but more attention has been paid to the
deterministic cases.
Generally, the applied
constraints in modelling DPDs are the capacity limitation and single
source constraints. In some cases, in
addition to the capacity constraints, some other restrictions on the
number of covered demands and the service levels of the warehouses are
also defined.
Recently some authors have incorporated inventory
control decisions into DPD models. For example, Miranda & Garrido, Daskin et al., and Shen et al. present similar versions
of the DPD model incorporating the inventory control decisions.
Erlebacher
& Meller present a location-inventory model for designing a
two-level distribution system serving continuously represented customer
locations. They develop a stylised analytical model to provide some
intuition and basic results for the problem. The stylised model also
motivates bounds for the problem, which they use to develop a heuristic.
They show that the heuristic performs very well on test problems that
considered variation in customer demand and spatial dispersion.
In
these works, the ordering decisions are based on the classic economic
order quantity (EOQ) model, and a normal distribution is assumed for the
demand pattern.
Additionally, researchers have developed various
methods to solve multi-objective DPD problems. Liang develops an
interactive fuzzy multi-objective linear programming method for solving
the fuzzy multi-objective DPD problem with piecewise linear membership
functions. Selim & Ozkarahan suggest an interactive fuzzy goal
programming (FGP) for the supply chain distribution network design. The
goal of their model is to select the optimum numbers, locations, and
capacity levels of the plants and warehouses to deliver the products to
the retailers at the least cost while satisfying the preferred customer
service level.
In most of the past research studies like Gourdin
et al., Jayaraman, Pirkul & Jayaraman, and
Tragantalerngsak et al., one major drawback is that they limit the
number of capacity levels available to each facility to just one.
However, in real case studies, there are usually several capacity levels
to choose for each facility. This flexibility in capacity levels makes
the problem more realistic and, at the same time, more complex to solve.
Another major drawback in some previous studies is that they limit the
number of opening facilities to a pre-specified value. Moreover, these
studies fail to describe how this value can be determined in advance.
Amiri represents a significant improvement over previous research
by presenting a unified model of the problem that includes the numbers,-
locations, and capacities of both warehouses and plants as variables to
be determined in the model. In addition, he develops an efficient
heuristic solution procedure based on Lagrangean Relaxation (LR) of the
problem, and reports extensive computational tests with up to 500
customers, 30 potential warehouses, and 20 potential plants.
In
this paper we develop a new non-linear multi-objective DPD model,
consisting of one manufacturer and multiple distribution centres
(warehouses), that integrates the location/allocation and distribution
plans. In this model, to improve over previous research, we also
incorporate tactical/operational decisions - such as inventory control
decisions -into the DPD problem. Then we propose an efficient goal
programming approach to solve the developed model. We consider three
important objective functions:
- investment in opening distribution centres/warehouses (location costs);
- total cost of logistics, such as the costs of transporting products from the plant to the opened warehouses, and from the opened warehouses to the retailers, and holding costs (inventories and safety stocks); and
- total delivery time.
The
problem is particularly motivated by consulting work that was done for a
large food industry company owning one production site and multiple
distribution centres. Since the transportation cost constitutes the main
part of the unit cost, and since delivery time and limited storage
capacity are also very important, implementing the distribution planning
system for the DCs' location and inventory control decisions is of
particular interest.
The paper has two important applied and
theoretical contributions. First, it presents a new comprehensive and
practical, but tractable, optimisation model for distribution network
designing. Second, it introduces a novel solution procedure for finding
more non-dominated and efficient compromise solutions to a stochastic
multi-objective mixed-integer programme. In our literature survey we
have found a lack of studies in this field, which is understandable,
given that large mixed integer programming is known to be complex even
when all data is certain and precise.
The remainder of this paper
is organised as follows. In Section 2 we consider a summary of key
challenges in agro-food supply chains. In Section 3 we define our
notation, state our assumptions, and propose a new multi-objective
stochastic non-linear program (MOSNLP) for the proposed DPD problem.
After applying appropriate strategies for converting the stochastic
model into a multi-objective nonlinear model (MONLP), in Section 4 we
propose a novel interactive payoff approach to solve this MONLP and find
an efficient compromise solution. The proposed model and the solution
method are validated through numerical tests in Section 5. The data for
these numerical computations have been inspired by a real life food
industrial case study, as well as randomly generated data. Concluding
remarks on computational results and further research directions are the
subject of Section 6.
Important Problems in Supply Chain Management in the Food Sector
Almost without exception, all well-known industries are redesigning
their distribution networks in a bid to meet global trends and continue
to meet customer demands. Considering the competitive market, key food
industries must work on the optimal supply chain structure. However
there is no doubt that supply chain networks are confronting new
challenges. Hunt et al. summarise these challenges:
- More instances of multi-site manufacturing
- Increasingly cut-throat marketing channels
- The maturation of the world economy
- Heightened demand for local products
- Competitive pressures to provide exceptional customer service
- Quick, reliable delivery
- Commonality of turbulence and volatility in markets
- Time-to-market for new products
Based
on these challenges, in this paper we focus on supply chain costs,
customer service level, and delivery time in food distribution planning.
Problem Scope and Formulation
We consider a firm that owns a manufacturing plant that is capable of producing multiple products. The products are then delivered to different distribution centres (warehouses/wholesalers) in order to satisfy their associated dynamic demands. The network is illustrated in Figure 1.
![](https://learn.saylor.org/pluginfile.php/5619137/mod_book/chapter/50026/15f01.jpg)
Assume that in the company the
logistics centre seeks to determine the right transportation plan to
allocate multiple commodities from the source (factory) to J warehouses
(DCs). Each destination has a forecast demand of each commodity to be
received from the plant. The forecast demand of each warehouse depends
on uncertain demands of allocated customers to this warehouse. This work
focuses on developing a MOSNLP method to optimise distribution
decisions such as location/allocation and inventory policy in a food
industry company.
This problem in fact integrates three decision
sub-problems: (1) selecting the optimum numbers, locations, and capacity
levels of the warehouses to deliver products to retailer/customer at
the least cost while satisfying desired service level to retailers; (2)
allocation of these retailers to the open warehouses; and (3) inventory
decisions for the supply chain.
Decision-making in such a complex
supply chain network requires the consideration of conflicting
objectives and of different constraints imposed by the manufacturer and
distributors. Moreover, in practical situations, due to the variability
and/or uncertainty of required data over the strategic and mid-term
horizon, most of the parameters embedded in a DPD problem are frequently
stochastic in nature, and can be obtained through probabilities or
subjectively in a fuzzy environment. For example, in a real decision
problem, market demands, cost/time coefficients, and the amounts of
available resources are usually imprecise over the planning horizon, and
therefore assigning a set of crisp values for such ambiguous parameters
is not appropriate. We rely on probability theory to model this
uncertainty. This theory uses statistical distributions to handle this
inherently ambiguous phenomenon in the problem parameters.
Problem description, assumptions, and notation
The stochastic multi-objective DPD problem examined here can be described as follows:
- In the analysed case study, the plant location is known and fixed. The network considered encompasses a set of retailers with known locations, and a discrete set of possible location zones/sites where the plant and warehouses are located.
- The final products have stochastic retailer demand over the given finite planning horizon with mean dil and variance vil (note that our customers are mostly retailers, not end consumers).
- There are multiple values for storage capacity at each warehouse (five level storage capacities).
- The distribution costs and delivery time on the given route are directly proportional to the shipped units.
- Products are independent of each other, related to marketing and sales price.
- The number of potential DCs and their maximum capacities are known.
- Retailers receive each product only from a single DC.
- No inventory is held in the plant.
- Decisions are made within a single period.
The indices, parameters, and variables used to formulate the mathematical problem are described as follows:
Indices:
index set of customers/customer zones (i=1,....., I)
index set of potential warehouse sites (j=1,....,J)
index set of products (l=1,...,L)
index set of capacity levels available to the potential warehouses (h=1,_,H)
index for objectives for all g=1,2,3
investment cost performance index [0,1]
delivery time performance index [0,1]
customer service performance index [0,1]
normal distribution value for system service level
Parameters:
unit cost of supplying product l to customer zone i from warehouse on site j
unit cost of supplying product l to warehouse on site j from the plant
delivery time per unit delivered from warehouse j to customer zone i for each product l
delivery time per unit delivered from the plant to warehouse j for each product l
the elapsed time between two consecutive orders of product l for site j
fixed cost for opening and operating warehouse with capacity level h on
site j per time unit
mean demand of product l from customer zone i per time unit
variance demand of product l from customer zone i per time unit
holding cost of product l in warehouse on site j per time unit
ordering cost of product l from warehouse on site j to the plant
capacity of warehouse on site j with capacity level h
space requirement of product l at any warehouse
planning horizon
Decisions variables:
it takes value 1, if a warehouse with capacity level h is installed on potential site j, and 0 otherwise;
it takes value 1, if the warehouse on site j serves product l of customer i, and 0 otherwise;
mean demand of product l to be assigned to warehouse on site j per time unit
variance demand of product l to be assigned to warehouse on site j per time unit
Stochastic multi-objective non-linear programming model
Objective functions
We
have selected the multi-objective functions for solving the DPD problem
by reviewing the literature and considering practical situations. In
particular, these objective functions are normally stochastic or fuzzy
in nature owing to incomplete and/or uncertain information over the
planning horizon. Accordingly, three objective functions are
simultaneously considered in formulating the original stochastic DPD
problem, as follows:
This objective function contains (see Appendix):
- Transportation cost of products from the plant to the warehouses and from the warehouses to the retailers
- Holding cost for mean inventory and safety stocks
- Minimise total delivery time (TDELT)
Constraints
- Constraints that ensure that each retailer is served exactly for each product by one warehouse (single source):
- Constraints of the warehouse capacity:
- Constraints that compute the served average demand by each warehouse:
- Constraints that indicate the total variance of the served demand by each warehouse:
Implicitly,
we assume that the demands are independently distributed across the
retailers, and thus that all the covariance terms are zero.
The Proposed Sgp-based Solution Approach
Defining the goals of the objective functions
As
we know, stochastic goal programming (SGP) needs an aspiration level
for each objective. These aspiration levels are determined by DMs. In
addition to the aspiration levels of the goals, we need max-min limits
( ,
) for each goal. While the DMs decide the max-min limits, the
linear programming results are starting points, and the intervals are
covered by these results. Note that in non-linear programming (with a
minimisation objective) the minimum limit of any non-linear objective
may be calculated by the results of the other objectives. This situation
may occur because the optimum value may be its local optimum.
Generally
the DMs find estimates of the upper () and lower (
) values for each
goal using payoff table (Table 1). Thus the feasibility of each
stochastic goal is guaranteed.
Here,
denotes the gth objective function, and
is the optimal solution of
the gth single objective problem. Solving the problem with
for each objective, a payoff matrix with entries
can be formulated as presented in Table 1. Here,
and
.
Using the interactive paradigm can improve the flexibility and robustness of multi-objective decision making by:
- Providing a learning process about the system, whereby the DMs can learn to recognise good solutions,
- The DMs can control the search direction during the solution procedure and, as a result, the efficient solution achieves their preferences,
- Various scenarios could be generated, based on a systematic procedure.
Solution methodology
To
deal with multi-objectives and enable the DMs to evaluate a greater
number of alternative solutions, three different approaches are
implemented in this section.
Solution Approach 1. The weights of objective and
are specified with
and
as follows:
Note
that, based on the three presented objective functions and preferred
DMs' service level (K), in this approach we generate several scenarios
and the TDELT objective is not considered. (A more detailed explanation
about the service level of the system is presented in Appendix) So
problem 1 can be summarised as follows:
Generated Problem 1
TH
allows one to sum the investment cost that occurs at the beginning of
the planning horizon with the rate cost incurred by the entire network.
In order to determine the weights, there are some good approaches in the
literature, such as the analytical hierarchy process, the weighted
least square method, and the entropy method. However, determination of
the weights is not the focus of this study.
Solution Approach 2.
In this approach the weights of the objectives ( Z1, Z2 ) and preferred
DMs' service level are the same as in solution approach 1, but we
consider Z3 (TDELT objective) as a new constraint.
Generated Problem 2
In the payoff table we calculate optimum (or local optimum) values for the three objective functions. In this approach, to compare each objective function against the others, we use the performance Index as a compensation rate. Since objectives Z3 and Z2 are very interactive, it is important for the DMs to evaluate the impact of increasing γ % in total delivery time (TDELT) on the system costs (INV and TCOST). To generate new scenarios we calculate the γ parameter based on the DMs preferences.
Generated Problem 3:
To generate more scenarios we calculate η and γ parameters based on the DMs preferences.
Solution procedure
Step 1: Formulate the original stochastic MOSNLP model for the DPD problem.
Step 2: Obtain efficient extreme solutions (payoff values) used for constructing the right-hand side of the added constraints (first and third objective functions). If the DMs select one of them as a preferred solution, go to Step 10.
Step 3: Define upper and lower bounds of each objective functions from the payoff table.
Step 4: Formulate problems 1, 2 and 3.
Step 5: Ask the DMs if they want to modify the right-hand side of the newly-added constraints of problems 2 and 3.
Step 6: Introduce η and γ parameters to generate new scenarios - i.e., define a systematic rule for changing upper bound of Z1 and Z3
Step 7: Determine the values of the SC performance vector (W1 , W2, η, γ, K).
Step 8: Improve the generated scenarios with the performance vector determined in Step 7.
Step 9: Analyse outputs of generated scenarios and obtain non-dominated solutions. If the DMs select one of them as a preferred solution, go to Step 10; otherwise, go to Step 5.
Step 10: Stop.
Case Study
The case study presented here, with an example from
the food industry, illustrates the algorithm proposed in Section 4, as
well as the applicability and effectiveness of the model. This food
industry company is the leading producer of two main categories of
Iranian food and drink (rice and tea). The basic distribution data are
presented in the next sub-section.
Setup
A case study inspired by a food producer in Iran is presented to demonstrate the validity and practicality of the model and solution method. The company owns one production site and six potential DC/warehouse sites in the different customer zones (Figure 2). There are three types of products and twenty main retailers.
Lingo 8.0
optimisation software is used as the problem solver. All scenarios are
solved on a Pentium 4 (Core 2 Duo) with 1GB RAM and 4 GHz CPU.
Because
of confidentiality, the input data are randomly generated. However, the
generation process is done so that it will be close to the real data
available in the company. Without loss of generality and just to
simplify generation of the stochastic parameters, we apply the pattern
of a systematically normal distribution for our numerical test.
The
required throughput capacity of any warehouse for product ' is as
follows: s,= 2, s2= 5, s3= 4. Tables 2 and 3 list some of the other
basic distribution data.
Performance analysis
The interactive solution procedure using the proposed SGP method for the case study is as follows:
First,
formulate the original stochastic multi-objective DPD problem according
to equations (1)-(9). The goal of the model is to select the optimum
numbers, locations, and capacity levels of warehouses to deliver the
products to the retailers at the least cost, while satisfying the
desired service level of the retailers. The proposed model is
distinguished from the other models in this field in the modelling
approach. Because of the somewhat uncertain nature of retailers' demand
and DMs' aspiration levels for the goals, a stochastic modelling
approach is used. Additionally, a novel and generic SGP-based solution
approach is proposed to determine the preferred compromise solution.
Second,
obtain efficient extreme solutions for each of the objective functions.
These extreme solutions of the case study are presented in Table 4.
It
is assumed that the DMs do not choose any of the efficient extreme
solutions as the preferred compromise solution, and proceed to the next
step.
Considering the efficient extreme solutions given in Table
4, the lower and upper bounds of the objectives can be determined. In
our case, the corresponding minimum and maximum values of the efficient
extreme solutions are determined as the lower and upper bounds
respectively, as presented in Table 5.
After calculating the upper and lower bounds of each objective function, the next step is formulation of problems 1, 2 and 3. A summary of the results for the various scenarios is given in Tables 6, 9 and 11.
As
stated previously, the relative weights for the first and second
objective functions in problem 1 can be determined by DMs using various
methods. For the presented case study, DMs determine three weights for
the INV and TCOST objectives as follows: (0.7, 0.3), (0.5, 0.5) and
(0.3, 0.7). For this problem, no constraint on delivery time is included
and TH=1000 (planning horizon) hereafter. By fixing the values of W1
and W2, the solution given in Table 6 is obtained. In this table, for
three values of each objective function and three levels for the
customer service performance index (K), nine scenarios have been
generated.
In Table 6, the warehouse load ratio percentage (WRL)
column shows the efficiency of the opened warehouses. The average WRL in
approach 1 is 0.9865, and since Zp1 is a non-linear objective function,
the range of the CPU time for solving this problem is very wide, from 6
to 180 seconds.
Note that in scenarios 5 and 6, although the
customer service performance (90%, 75%) is lower than in the 4th
scenario (97.5%), the objective function is higher. Therefore these
scenarios are inferior and must be removed from the scenario list.
Figure 3 shows the results of equal weights for scenarios 1, 2 and 3,
comparing them with non-equal weights for scenarios 7, 8 and 9 in
approach 1. A comparison of the first and third scenarios in Table 7
shows that total cost is increased slightly from 6,988,496 to 7,111,742
(1.7%) when CSPI is increased from 75% to 97.5% (23%). This situation is
the same for scenarios 7, 8 and 9 in approach one and for the other
scenarios in the second and the third approaches (Tables 9 and 11). As
can be seen, the effect of customer service level decreasing on cost
improvement is negligible. This may support management's preference to
select K=97.5% because a large increase in CSPI results in a small cost
penalty. Selecting the first or the seventh scenario in this approach is
based on DMs' preferred objective weights.
To
solve problem 2, first the γ parameter must be calculated based on the
DMs' preferences for the right-hand side of the new constraint (TDELT).
Table 8 shows three preferred values for the delivery time performance
index ( γ ).
Based on three values for W1, W2 and γ, eighteen
scenarios have been generated. The results of these scenarios are
presented in Table 9. In approach 2, the WRL average (0.9644) is lower
than approach 1 (0.9865); and by considering the TDELT objective in
approach 2, this effect was predictable. Considering the sixth column in
Table 8, it can be determined that since Zp2 is a non-linear objective,
the range of the CPU time to solve this problem is very wide, from 32
to 1,693 seconds. Comparing the CPU times in Table 6 and 9 shows that
these times for problem 2 are significantly larger than those for
problem 1. Unfortunately, LINGO optimisation software could not solve
the 16th scenario in 180 minutes. The results presented in Table 9 are
illustrated graphically in Figures 4 and 5.
Table 10 shows the preferred values for η and γ in problem 3. For this problem, six scenarios are examined. The performance vectors and the other results are presented in Table 11, and illustrated graphically in Figure 6. It is interesting to note that in approach 3 the WRL average is 0.9878, and it is higher than the other approaches.
In summary, we make the following observations from our case analysis:
- Ten cases out of 33 scenarios are dominated by the other ones.
- The solution results indicate that the proposed model is not very
sensitive to CSPI, so the preferred value for this parameter is 97.5%.
It
can be concluded that the proposed SGP solution using approach 3 may
provide different and even more preferable results when compared with
approaches 1 and 2
Summary and Conclusion
This
study proposed a multi-objective, multi-commodity distribution planning
model that integrates location and inventory control decisions in a
multi-echelon supply chain network with multiple capacity centres in a
stochastic environment. An interactive stochastic goal programming
formulation for food production is developed. The goal of the model is
to select the optimum numbers, locations, and capacity levels of the
warehouses to deliver the products to the retailers at the least cost,
while satisfying the desired service level. The modelling approach of
this model is distinguished from the other models in this field by the
fact that DMs' imprecise aspiration levels for the goals, and retailers'
imprecise demand are incorporated into the model using a stochastic
modelling approach, which is otherwise not possible by conventional
mathematical programming methods.
This paper also contributes to
the literature by proposing a novel and generic SGP-based solution
approach that determines the preferred compromise solution for
multi-objective decision problems.
An Iranian food industry case
study was used to demonstrate the feasibility of the proposed method for
real distribution problems. Some realistic scenarios have been
investigated, based on the DMs' strategies. These strategies can be
compared by determining the performance vector for each strategy. The
proposed method yields an efficient solution and overall degree of DMs'
satisfaction with the determined objective values. Accordingly, the
proposed method is practically applicable to solving real-world
multi-objective DPD problems in an uncertain environment.