An integrated inventory optimisation model for facility location allocation problem

Full Terms & Conditions of access and use can be found at
http://www.tandfonline.com/action/journalInformation?journalCode=tprs20
Download by: [115.85.75.131] Date: 22 December 2015, At: 04:42
International Journal of Production Research
ISSN: 0020-7543 (Print) 1366-588X (Online) Journal homepage: http://www.tandfonline.com/loi/tprs20
An integrated inventory optimization model for
facility location-allocation problem
R.P. Manatkar, Kondapaneni Karthik, Sri Krishna Kumar & Manoj Kumar
Tiwari
To cite this article: R.P. Manatkar, Kondapaneni Karthik, Sri Krishna Kumar & Manoj Kumar
Tiwari (2015): An integrated inventory optimization model for facility location-allocation
problem, International Journal of Production Research, DOI: 10.1080/00207543.2015.1120903
To link to this article: http://dx.doi.org/10.1080/00207543.2015.1120903
Published online: 21 Dec 2015.
Submit your article to this journal
View related articles
View Crossmark data

An integrated inventory optimization model for facility location-allocation problem
R.P. Manatkar, Kondapaneni Karthik, Sri Krishna Kumar and Manoj Kumar Tiwari*
Department of Industrial and Systems Engineering, Indian Institute of Technology Kharagpur, Kharagpur, India
(Received 25 May 2015; accepted 6 November 2015)
This paper presents an integrated inventory distribution optimisation model for multiple products in a multi-echelon
supply chain environment. Inventory, transportation and location decisions are considered. The objective is to offer prac-
tical guideline to the steel retail supply chain practitioners in choosing the correct distribution centre, finding out inven-
tory level at individual inventory keeping points (retailers and distribution centres) point thereby helping them in
reducing overall distribution cost. The framework presented endorses systems approach and suggests near-optimal
approach to calculating inventory for an individual distributor and his retailers. Two algorithms are used to solve this
problem, a novel hybrid Multi-objective Self-learning particle swarm optimiser and Non-dominated sorting genetic algo-
rithm-II. The model and solution methods are tested on real data-sets obtained from organisations in the steel retail envi-
ronment. The actual data on inventory holding, ordering and transportation costs of distributors and retailers are used as
inputs. The decisions like choosing correct set of Distribution centres, keeping optimal regular and safety stock inventory
levels are arrived at by applying practical constraints in the supply chain. Model developed assists in effective and
efficient distribution of the products manufactured from the optimal location at minimal cost.
Keywords: inventory distribution; facility location-allocation problem; supply chain network design; multi-objective
optimisation; swarm optimisation
1. Introduction
Innovative strategies and technologies are being implemented in the field of Supply Chain Management by enterprises
to tackle new challenges faced due to changing demands of customers. Efficiency and responsiveness are the two new
strategies for supply chain that have emerged in recent years. Efficiency aims at delivering products at reduced costs,
while responsiveness, on the other hand, is the ability of the organisation to respond quickly to customer demands and
save costs (Liao and Hsieh 2010). Amongst other drivers, an organisation’s responsiveness depends upon management
of inventory at plant warehouses, regional warehouses, distributor warehouse, transportation and retailers. The comprised
inventories on hand can cost between 20 and 40% of the value. Therefore, careful management of inventory becomes
predominant for economic reasons. Despite many strategies such as just-in-time, quick response and collaborative prac-
tices are being practised by the manufacturers, distributors and retailers, still there is a big scope for improvement, since
question of interest to various players in the supply chain is How much inventory is the right inventory? Higher cus-
tomer service level demands large inventory thereby inflating the costs and lower inventory may result in sales loss.
Hence, to make decent profit margins and remain competitive, firms are looking for measures to reduce their costs by
maintaining desired service level through better inventory management practices.
Managing inventory in a distribution network consists of three critical tasks. Primarily, determine the location of
Distribution centres (DCs) (from here warehouses and distribution centres are considered same) from those locations.
Second, allocate group of customers to each DC and thirdly, determine the amount of inventory at each location in the
Supply chain. In a stochastic environment, inventory at any facility is categorised into two types – regular stock and
safety stock. Regular stock is carried by any facility to serve its nominal demand, while the safety stock is to mitigate
the risk of stock outs due to the unpredictable nature of supply and demand. Therefore, it is essential to maintain
adequate regular and safety stock at each facility of the supply chain network (SCN).
Supply chain consists of six drivers: location, transportation, information, sourcing, pricing and inventory. These dri-
vers have direct or indirect impact on each other as well as on the whole SCN. Any decision on these drivers will have
an overall impact on the performance and operational efficiency of the SCN (Bashiri and Tabrizi 2010). Determining
stock point location is common in supply chain practices; however, identifying the exact location based on integrated
*Corresponding author. Email: mkt09@hotmail.com
© 2015 Taylor & Francis
International Journal of Production Research, 2015
http://dx.doi.org/10.1080/00207543.2015.1120903
Downloadedby[115.85.75.131]at04:4222December2015

inventory model (integrating location sourcing and transportation decisions) is difficult. As the market is becoming more
and more competitive, companies need to be more aggressive in achieving their service goals at the lowest possible cost.
For that reason, firms ought to design the SCN considering their inventory, location, sourcing and transportation in a
holistic manner.
The model developed in this paper is on Multi-objective integrated allocation-inventory problem (MOIAIP).The
objective is to lessen inventory carrying and transportation cost. This includes inventory holding cost, transportation cost
and the decisions regarding maximum inventory to be kept at any echelon. Specifically, it is helping in making an
integrated decision regarding:
• The optimal assignment of a group of retailers to multiple DCs.
• The level of safety stock to be maintained at each facility by upholding given service level.
• The level of regular stock to be maintained at each facility.
• The maximum inventory at any echelon in the system.
2. Literature review
The literature on strategic issues like facility location and design of distribution network focuses on developing models
for spotting the best possible locations of the facilities, identify the optimal number of DCs and assignment of retailer’s
to DCs. This mainly includes fixed transportation and facility costs, but the shortage costs and operational inventory are
usually overlooked (Melkote and Daskin 2001). Daskin and Owen (2003) did the analysis on the facility location
modelling, while Daskin (1995) clearly depicted FLPs. FLPs can sometimes be modelled as SCN design problems.
Ambrosino and Scutellà (2005) have studied complex distribution network design problems, which involve warehous-
ing, facility location, inventory and transportation decisions. Facility location modelling comes with issues and
challenges in designing a SCN. These are clearly shown by Melnyk, Narasimhan, and DeCampos (2014).
Recently, several researchers have put their effort on integrating inventory management theory in the supply chain
design field. For multiple retailers, multiple supplier distribution networks a near optimal inventory policy was presented
by Ganeshan (1999). In their model, inventory at retailers and DC are synchronised to minimise the total logistics costs
taking into account the service level requirements. A linear safety stock function was formulated by Nozick and Turn-
quist (2001). They proposed Lagrangian-based scheme to solve uncapacitated FLP. On the similar line, a DC location
model was introduced by Daskin, Coullard, and Shen (2002) that incorporates safety stock and working inventory cost
at the DCs. Shen, Coullard, and Daskin (2003) as well as Daskin, Coullard, and Shen (2002) proposed a single-echelon
set-covering problem to build an optimal SCN. He demonstrated that if the demand faced by the DC is Poisson distribu-
tion, then the problem can be tackled effectively. Economic order quantity model and safety stock model are incorpo-
rated into FLP by Miranda and Garrido (2004). Gebennini, Gamberini, and Manzini (2009) had proposed a model for
safety stock optimisation and control service level for the dynamic location–allocation. Mizgier, Jüttner, and Wagner
(2013) proposed a new methodology to identify the bottlenecks in SCN design in order to make an informed decision
by the firms. Some of the researchers Shu (2010) and Li et al. (2013) proposed approximate algorithms to solve SCN
design problems.
In recent years, FLPs are extended to multi-echelon SCN design. Teo and Shu (2004) proposed a two-echelon DC-
retailer network for the set-covering model under deterministic demand. Later this model was extended to non-standard
demands by Shu, Teo, and Shen (2005). A two-echelon model that incorporates location-specific transportation costs
with location decisions was proposed by Romeijn, Edwin, and Teo (2007). This model determines the number of DCs
to open, location of the DCs and the way to serve from DCs to the retailers under a single-sourcing policy. Their work
put forward inventory policies to reduce the location, transportation costs for two-echelon inventory model. Askin,
Baffo, and Xia (2014) have modelled the distribution network of a logistics firm for multiple sources, multiple products
and multiple retailer environments. FLPs can be classified crisply into two types capacitated and uncapacitated. Many of
the research works in FLPs is uncapacitated, but without considering capacities at the entities in the supply chain the
problem is always incomplete. Silva and de la Figuera (2007) presented a capacitated FLP with constrained backlogging
probabilities taking stochastic nature of demand.
With the increased complexity with multi-modal and non-linear problems, approximate algorithms have been finding
increased usage for solving this kind of problems. Integration of inventory, safety stock, allocation decisions etc., into
FLPs makes these non-linear and complex. This led many researchers to find application of EA in FLPs. Guner and
Sevkli (2008) have proposed a discrete particle swarm optimisation (PSO) approach to solve an uncapacitated FLP. Lee,
Moon, and Park (2010) proposed a model integrating vehicle-routing decisions with FLP and solved it using a hybrid
EA approach based on the swarm intelligence to determine the optimal location of DCs. This paper used hybrid EA
2 R.P. Manatkar et al.

approach based on the fast elitist Non-dominated Sorting Genetic Algorithm-II (NSGA-II) to determine the optimal loca-
tion of DCs. Yapicioglu, Smith, and Dozier (2007) solved a semi-desirable FLP using a single-objective PSO and multi-
objective PSO and found that usage of multi-objective approach was efficient. On similar lines using a multi-objective
optimisation approach, Bashiri and Tabrizi (2010) tackled a problem of locating a DC in a single DC, multiple retailers
system. They have used PSO approach to finding the location of the DC.
In this paper, we adopted a similar research work proposed by Bashiri and Tabrizi (2010), however, they proposed
model for a single DC multiple retailer environments. We extended the model to multiple DCs, multiple products and
multiple retailer environments. Even though model developed by Bashiri and Tabrizi (2010) curtailed maximum inven-
tory at retailers, in our model, we are curtailing maximum inventory at DCs. Bashiri and Tabrizi (2010) solved their
multi-objective problem by integrating two objectives and solved it using basic PSO. In our extended model with r
retailers, d DCs and p products, the solution space is (d)rp
. When it comes to higher dimensional problems, the solution
space will be huge. With this huge solution space and non-linear nature of the problem, the algorithm will get struck in
an infinite loop making our problem NP hard. Our allocation problem is analogous to classic halting problem as shown
in Figure 1. To solve this type of NP-hard problem, we employed a novel hybrid EA, Multi-Objective Self-Learning
Particle Swarm Optimisation (MOSLPSO) to estimate the Pareto front.
The rest of this work is structured as follows. Section 3 describes the problem and gives the overview of the mathe-
matical model developed for MOIAIP. Section 4 discusses the MOSLPSO and NSGA-II algorithms with steps to solve
MOIAIP. Section 5 presents the industrial cases for implementing the feasibility of applying the proposed MOIAIP
model in the real situation and discusses the significant findings. Section 6 illustrates results and discussions. Section 7,
concludes, recommends and gives suggestions for future work.
3. Problem environment
Manufacturing plants, in general, route their product through DCs. We considered a problem of designing distribution
network which encompasses a two-echelon environment with single manufacturing plant, multiple DCs distributed
geographically across a territory distributing multiple products to a set of retailers. The product shipment between
manufacturing plants and retailers is facilitated by DCs acting as intermediate facilities, as depicted in Figure 2. These
types of network commonly exist for most of the manufacturing organisation in which positions of DCs are fixed.
Important aspects concerning this network architecture are the development of an ideal network in which all the retail-
ers are optimally allocated to the existing DCs in a way that inventory holding cost, total transportation cost is min-
imised and optimal amount of inventory at all places. The Distributor wishes to redesign their SCN in such a way
that it supports the inventory replenishment activities of its retailers under stochastic demand at specified service
levels and at the lowest possible cost. The nominal demands and standard deviation of the demands are available for
each retailer. The problem is MOIAIP, which deals with the modelling, redesigning, planning and controlling
two-echelon supply chains.
Figure 1. proof of NP – hard complexity.
International Journal of Production Research 3

3.1 Assumptions
The model is developed with following general basic assumptions:
(1) All products of each retailer will be served exactly by one warehouse.
(2) There is no pipeline inventory.
(3) DCs are uncapacitated and demands of all retailers are uncertain.
Based on assumptions and problem stated above, the proposed mathematical model with notations is illustrated
below in Table 1.
3.2 Multi-objective non-linear integer programming model
As described earlier, we are considering the situation of multiple DCs, multiple products and multiple retailers.
Objective functions:
Min Z
P
i2I
P
j2J
P
p2P
Crw
ijplr
ipdrw
ij Xrw
ijp þ
P
j2J
P
p2P
Cwm
jp dwm
j ð
P
i2I
lr
ipXrw
ijp Þ þ
P
i2I
Ar
i nr
i þ
P
j2J
Aw
j nw
j
þ
P
i2I
P
p2P
hr
iplr
ip
2nr
i
þ
P
j2J
P
p2P
P
i2I
hw
jplr
ipXrw
ijp
2nw
j
þ
P
i2I
P
j2J
P
p2P
hr
ipkr
iprr
ipXrw
ijp
ffiffiffiffiffiffiffiffiffiffiffiffi
cr
ipdrw
ij
p
þ
P
j2J
P
p2P
hw
jpkw
jp
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
cw
jpdwm
j
P
i2I
ðrr
ipÞ2
Xrw
ijp
r
8
>>>>><
>>>>>:
(1)
Min W (2)
Subjected to Constraints:
X
j2J
X
p2P
Xrw
ijp ¼ 1 8i 2 I (3)
X
i2I
X
p2P
lr
ipXrw
ijp
2nw
j
þ
X
p2P
kw
jp
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
cw
jpdw
j
X
i2I
ðrr
ipÞ2
Xrw
ijp
r
W 8 j 2 J (4)
Figure 2. Two-echelon distribution network problem.

Derivation of ordering frequency for DC:
The total annual operational cost (TC) of ordering inventory from the plant at the DC j is given by
TC ¼
X
p2P
X
j2J
Crw
ijpdrw
ij lr
ipXrw
ijp þ Ar
i nr
i þ
X
p2P
hr
iplr
ip
2nr
i
(5)
Ar
i À
X
p2P
hr
iplr
ip
2ðnr
i Þ2
¼ 0 (6)
Solving for nr
i , we obtain,
nr
i ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
X
p2P
hr
iplr
ip
2Ar
i
s
(7)
TC ¼
X
p2P
Cwm
jp dwm
j
X
i2I
lr
ipXrw
ijp þ Aw
j nw
j þ
X
p2P
X
i2I
hw
jplw
ipXrw
ijp
2nw
j
(8)
Where nr
i is the (unknown) number of orders per month for retailer i, and other parameters have their usual meaning as
described earlier. Equation (5) represents the total annual cost incurred by the DC to the retailers, which include trans-
portation cost, ordering cost and inventory carrying cost. In order to find the optimal number of orders annually, we take
the derivative of Equation (5) with respect to nr
i and equate it to zero, we obtain Equation (6). Solving the Equation (6),
Table 1. Notations for mathematical model.
Notation Definition of sets and indexes
I Set of retailers, indexed by i
J Set of warehouses, indexed by j
P Set of products manufactured by the plant, indexed by p
Definition of parameters
Ar
i Fixed administrative and handling cost of placing an order by retailer i for all the products required by him, ∀i ∊ I
Aw
j Fixed administrative and handling cost of placing an order by distributor for warehouse j for all the products required by
it, ∀j ∊ J
Cwr
ijp Cost of transporting one unit of product p from warehouse j to retailer i for one unit of distance,
8i 2 I; 8j 2 JðiÞ; 8p 2 PðiÞ PðjÞ
Cmw
jp Cost of transporting one unit of product p from plant to warehouse j for one unit of distance, 8j 2 J; p 2 PðjÞ
drw
ij Euclidean distance between retailer i and warehouse j, 8i 2 I; 8 j 2 JðiÞ
dwm
j Euclidean distance between the plant and warehouse j, ∀j ∊ J
hr
ip Inventory holding cost per unit per year at retailer i for product p, ∀i ∊ I, ∀p ∊ P(i)
hw
jp Inventory holding cost per unit per year at warehouse j for product p, ∀j ∊ J, ∀p ∊ P(j)
kr
ip Parameter corresponding to service level in standard normal distribution at retailer i for product p. 8 i 2 I; 8p 2 P
kw
jp Parameter corresponding to service level in standard normal distribution at warehouses j for product p.
8 i 2 IðiÞ; 8p 2 PðiÞ
nr
i Number of shipments per year from the warehouse to the retailer i for all the products required by him, ∀i ∊ I
nw
j Number of shipments per year from the plant to the warehouse j for all the products required by warehouse, ∀j ∊ J
Wd Number of working days in a month
cr
ip Parameter that facilitate the conversion of distance between the retailer i and the warehouse which serves it into LEAD
TIME in inventory per day. Here we use cr
ip ¼ lr
ip=Wd.8 i 2 IðiÞ; 8p 2 PðiÞ
cw
jp Parameter that facilitate the conversion of distance between the plant and the warehouse j to Lead time in inventory per
day. Here we use cw
jp ¼
P
i¼IðjÞIðpÞ lr
ipXrw
ijp =Wd,8j 2 J; 8p 2 PðjÞ
lr
ip Average demand at retailer i for product p, ∀i ∊ I, ∀p ∊ P(i)
rr
ip Standard deviation of demand at retailer i for product p, ∀i ∊ I, ∀p ∊ P(i)
Definition of variables
Xrw
ijp
1; if warehouse j supplies product p to retailer i
0; Otherwise

W Maximum amount Inventory at warehouses

we will get optimal number of orders for retailer i in Equation (7). On similar lines, we derive optimal number of orders
for warehouses by taking the derivative of total annual cost in Equation (8) with respect to nw
j . By taking the derivative
of Equation (8) and equating it to zero (Equation (9)) and solving it we get optimal number of orders for warehouse j
(Equation (10)). Equations (7) and (10) has decision variable Xrw
ijp and are now substituted in Objective function 1
(Equation (1)).
Aw
j À
P
p2P
P
i2I hw
jplr
ipXw
ijp
2ðnw
j Þ2
¼ 0 (9)
Solving for nw
j , we obtain,
nw
j ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP
p2P
P
i2I
hw
jplr
jpXrw
ijp
2Aw
j
v
u
u
t
(10)
The first two terms in Equation (1) are the cost of transportation i.e. transportation cost between the DC and retailers
and the cost from plant to DCs for each product. The individual component of this cost comprises of three parts:
amount of demand, distance shipped and decision variable which are then multiplied by unit cost of transportation from
origin to destination. Third term and fourth term represents ordering cost for retailers and DCs. Fifth and sixth term rep-
resents regular stock holding cost for retailers and DCs. The total ordering cost and holding cost for both retailers and
DCs of all the above costs are from the Roundy’s popular 98% formulation (Roundy 1986). The last two cost terms are
the modified inventory cost for safety stock (Romeijn, Edwin, and Teo 2007) for retailer and distributor. In fact, safety
stock mainly depends on the distance from the source for material replenishment. The same is not visible in the formu-
lation by Romeijn, Edwin, and Teo (2007) and Qi and Shen (2007). The amount of safety stock depends on the lead
time, and the lead time depends upon the distance between source and destination of the product. This process is shown
as modifications in safety stock. Equation (2) is the second objective function; W constraints the maximum inventory
carried by any DC in SCN under consideration. Equation (3) shows the constraint that retailer will have Single product
source; demand of all products of a retailer are served by a single warehouse. Equation (4) Constraints the inventory
level at each DC. The first term in Equation (4) is the average inventory to satisfy the mean demand at the warehouse.
The second term is the safety stock to maintain the required service level. Safety stock is the product of required service
level parameter (kw
jp) and the total variation in the demand at the warehouse.
4. Solution methodology
The problem modelled in Section 3 is a multi-objective formulation and due to the inclusion of safety stock, it has
become non-linear and complex. In situations like this, EAs have been found effective in finding near optimal solutions
in less time. This encouraged us to employ an EA technique to solve problem at hand.
Over the years, NSGA-II has become a benchmark algorithm to solve multi-objective problems. NSGA-II works on
the similar lines of GA and having an additional feature of non-dominated sorting to rank solution with multiple objec-
tives. But, it is well known that GA has the tendency to converge pre-maturely to local optima. This prompted authors
to replace the working procedure of GA in NSGA-II with Self-Learning Particle swarm optimisation (SLPSO) procedure
to develop a new hybrid multi-objective EA; MOSLPSO, to tackle this problem. Flow chart of MOSLPSO is shown in
Figure 3. In following parts of this section, MOSLPSO and NSGA-II procedures are clearly depicted.
4.1 Multi-objective self-learning particle swarm optimisation
By observing the communication of information in bird flocking and fish schooling, Eberhart and Kennedy (1995)
developed PSO. PSO is a population (swarm)-based stochastic search technique. PSO is an iterative improvement based
meta-heuristic, wherein each iteration, a set of possible candidate solutions called swarm updated using Equations (11)
and (12). Each of the candidate solutions in the swarm is called a particle which is represented by two vectors; position
(xp) and velocity (vp). Velocity of the particle is updated by guiding it partially towards its local best position (pbest)
and partially towards global best position (gbest) as in Equation (11). Position of the particle is updated by simply add-
ing velocity vector to old position vector as in Equation (12). ^xp and ^vp are updated position and velocity vectors
respectively.
^vp ¼ x Â vp þ g1 Â rp Â ðpbest À xpÞ þ g2 Â rp Â ðgbest À xpÞ: (11)

Figure 3. Flowchart of proposed MOSLPSO.

^xp ¼ xp þ vp (12)
In recent years, PSO has been extensively considered by researchers and applied to numerous practical complex prob-
lems which yielded challenging results, yet it suffers from some drawbacks. In general, PSO algorithms use a particular
learning style (Equation (11)) for all the particles that enforce all the particles to use only one kind of thinking. This
single-learning strategy may lack intelligence of a particular particle; consequently, PSO algorithm may tend to be
incapable to tackle problems with high complexity (Li, Yang, and Nguyen 2012). Furthermore, in many complex multi-
model problemata solving the algorithm tends to local optima with a large numbers of local optima (Liang et al. 2006).
Li, Yang, and Nguyen (2012) took away these drawbacks of PSO, and invented SLPSO algorithm by proposing four
learning sources produced by four operators that guide particle (Equation (13)) to learn from the particle’s current best
solution, (Equation (14)) jumping out of a local optimum, (Equation (15)) exploit a local optimum, and (Equation (16))
explore new promising areas.
SLPSO is based on swarm intelligence that helps to find the near optimal solution to the problem. Li, Yang, and
Nguyen (2012) proposed four operators to update the velocity and position vectors. The four operators correspond to
four learning equations, respectively, as follow.
Operator 1: involvement of particle best fitness (pbest) in a learning source
^vp ¼ x Â vp þ g Â rp Â ðpbest À xpÞ: (13)
Operator 2: learning from a random position nearby, jumping out of local optima.
^xp ¼ xp þ vavg Â Nð0; 1Þ: (14)
Operator 3: involvement of the best position of a random particle (better than its fitness) in a learning source
^vp ¼ x Â vp þ g Â rp Â ðpbestrand À xpÞ: (15)
Operator 4: the abest position (super particle) in a learning source
^vp ¼ x Â vp þ g Â rp Â ðabest À xpÞ (16)
In this subsection, subscript p represents the pth particle; ^xp and xp represent the current and the previous position vec-
tors of the particle p, respectively; ^vp and vp are the velocity vectors of the current and the previous iterations; ω ∊
(0, 1) is the inertia weight that decide velocity preservation criterion from previous iteration; η is the acceleration coeffi-
cient; rp is a random number generated uniformly from the interval [0, 1]; pbestp is the best position found by the indi-
vidual particle p so far; pbestrand is the pbest of a random particle that is better than to pbestp; abest is the archived
position of the global best (gbest) particle so far; N(0, 1) generates a random number from the standard normal distribu-
tion with mean 0 and variance 1; and vavg ¼
PN
p¼1 vp

=N is the average speed of all particles, where N is the popula-
tion size.
These four operators produce four different learning sources that independently increase the search efficiency of each
particle. Operator 1 exploits the local optimum solution, whereas Operator 2 is mutation operator that can be used in
escaping the local optima; Operator 3 enables a particle to explore the non-searched areas with high probability, and
Operator 4 enables particles converge to the current global best position (Li, Yang, and Nguyen 2012).
In this paper, we are extending SLPSO for Multi-objective optimisation; MOSLPSO, using the non-domination sort-
ing (see NSGA-II; Deb et al. 2002) operation from NSGA-II. We used crowding distance operator and ranking of fronts
in order to distinguish among solutions. For more detailed understanding of how final ranks of solutions are calculated
(see NSGA-II; Deb et al. 2002). The steps of Algorithm are shown below. Only non-dominating sorting procedure was
added to SLPSO to make it MOSLPSO. Rest of the steps of algorithm is same a SLPSO. In SLPSO, initially all the
four operators are given same percentage in roulette wheel. As the algorithm progresses, the probabilities of selecting
each operator will be updated. For more detailed understanding of SLPSO and updating of parameters, please see
(Li, Yang, and Nguyen 2012). Flowchart of MOSLPSO is shown in Figure 3.
Step 1: Initialize swarm of particles and establish parameters for each particle.
Step 2: Evaluate fitness for each particle p.
Step 3: Set current generation counter t = 1.
Step 4: For each particle p do
• Select one of the Operators using roulette wheel selection.
○ Update velocity using selected operator i.
○ Update position using selected operator i.

○ Evaluate fitness for particle p.
○ Update particle’s parameters.
- UpdateGk
i .
- Updategk
i .
- Update progress value Pp
i of ith Operator for pth particle.
- Update pbest of particle.
- Update abest of particle.
- Update mk.
○ If Update criterion for selection ratios is met.
- Update selection ratios.
- Set mk = 0.
- Set Gk
i = 0; gk
i = 0; Pp
i = 0;
Step 5: if iteration number is one, store first non-dominated front else update first Non-dominated front.
Step 6: Update current iteration counter t++.
Step 7: Update algorithm parameters.
Step 8: If a termination criterion is met stop the algorithm and take results, else go to Step 4.
Gk
i is the statistic that contains information regarding selection of Operator i for particle k.
gk
i is the statistic that contains information regarding successful learning times using Operator i for particle k.
mk is the number of successive unsuccessful leaning for particle k.
Pp
i is the progress value for particle p using Operator i.
Termination condition: Pre-specified maximum number of Generations or acceptable quality of solutions is attained.
4.2 Non-dominating sorting genetic algorithm
Over the years, NSGA-II is considered as one of the earliest multi-objective EA based on Genetic Algorithm. NSGA-II
is also a population-based search technique for multi-objective optimisation. In order to find multiple optimal solutions
for the multi-objective optimisation, which form a non-dominating front (Deb et al. 2002) developed NSGA-II. They
used crowding distance operator and ranking of fronts in order to distinguish among solutions.
The steps followed for getting the results through NSGA-II are as follow:
Step 1: Initialize population (i.e. population, P(0))
Step 2: Evaluate fitness of each chromosome in the population (P(0)).
Step 3: Assign fronts and crowding distance to each individual chromosome in the population(P(0)) (non-dominated
sorting).
Step 4: Update the population set (P(0)) as follows,
(a) Set iteration counter t → t + 1,
(b) Create a new population set (NP(t)).
(1) Select two individuals from the population (P(0)), parent 1(p1) and parent 2(p2).
(2) Combine these two parents (p1 and p2) using crossover operator and produce two children (c1 and c2).
(3) Apply mutation operator on each child (c1 and c2).
(4) Insert children (c1 and c2) into (NP(t)).
(5) Continue this process until size of new population (NP(t)) is equal to size of population(P(t)) .
(c) Combine old population (P(t – 1)) and new population (NP(t)) and do non-dominate sorting.
(d) Select the best half from the combined population based on fronts and crowding distance as the new regular
population set (P(t)).
Step 5: If termination criterion is not met go to step 4, else terminate program and extract required results.
Termination condition: Pre-specified maximum number of Generations or acceptable quality of solutions is attained.
5. Case study
Real-world data are considered from India’s leading manufacturer of GI wire, barbed wire and chain links. Barbed wire
and chain link is the product of GI wire. This company sells GI wire and its products through a network of independent
distributors. The problem faced by the distributors includes decisions such as number of DCs required, retailer allocation

to DC, optimal inventory at DC and optimal inventory at the retail outlets. Addressing all these issues will help the
distributor to reduce overall distribution cost including transportation and inventory holding cost (for safety and regular
stocks).
5.1 Case study with scenario 1
Case scenario 1 analyses the problem with the single distributor with three DCs. The data were taken from one of the
distributors of this organisation. To ensure the confidentiality of the data, a real distributor facing the problem is referred
as ‘D’. D purchases and distributes GI wires, barbed wire, chain link items to a total of over hundred and eighty-nine
customers (e.g. retailers including institutional customers and converters) across his territory. D presently holds three
DCs, represented as DC 1, DC 2 and DC 3. D currently faces stochastic demand of retailers. It maintains regular and
safety stocks at each of his DC and supplies products from any one of the DC to retailers based on judgement. D wants
to know which DCs to be maintained to serve retailers i.e. D wants to see the effect of various options, such as closing
or opening of the DC. Further, D wants to know which retailer should be served from which DC. Moreover, D wants to
find out what should be the optimal inventory (Regular stock and Safety stock) at each of his DC and also at each of
his retailer so that a given service level can be achieved. The decision on above will help him in minimising the sys-
tem-wide distribution and inventory cost.
The present system of D was optimised i.e. DC at three district places denoted as DC 1, 2 and 3 respectively and
studied the effect of possible alternative DC combinations. Other than 3 DC locations above, one more potential loca-
tion was provided by management team to explore as DC 4. The authors are unable to provide input data-set in this
paper because of the excessive size of Tables, but we are able to take part of the data-set to show a numerical example
in the following sections. Input data for the model include distance between retailers and DC locations, Inventory carry-
ing cost, service level, price, demand and standard deviation of each product at retailer.
5.2 Case study with scenario 2
Similar to the earlier case, the organisation under study has 30 distributor territories with more than 6000 retailers in the
retail chain. The problem size is much bigger as the retailers are spread across the geographical territories of India and
there exists a gap in retailer reach in individual territory. Challenge is to identify the gap and locate appropriate DC or
DC to tap the changing market demand to serve customers better and remain cost competitive.
The territory under study had six distributors with eight DCs. Other than these eight DC locations, another potential
location for DC was also taken for study as suggested by the top management. Each distributor has one or more than
one DC and is allotted a well-defined territory. The company manager wishes to divide the particular region into optimal
number of territories. The study was conducted at nine potential DC locations.
6. Results and discussions
The implementation of the evolutionary multi-objective optimisation technique involved coding and testing the inte-
grated allocation–inventory distribution network mathematical module and the MOSLPSO and NSGA-II algorithms
using MATLAB (2012b). Following process is adopted to solve the problem:
(1) Devising a mathematical model that is constrained with a single product source for each retailer with same
source for all other products and retailers demanded at that site.
(2) Setting the chromosome (NSGA-II) or particle MOSLPSO structure to represent a candidate solution of
(MOIAIP) for m customers and r DCs allocation problem. Each chromosome has (m + 1) genes. The first m
genes take values ranging from 1 to r. Each integer in the vector takes the value ranging from 1 to r. The integer
values in these genes represent DC allocated to that particular retailer. The (m + 1)th gene represent the
maximum inventory level at any facility. Table 2 illustrates the particle representation of MOIAIP with eight
customers served by two DCs.
Table 2. Schema of MOIAIP with two warehouses serving eight customers.
Customer 1 2 3 4 5 6 7 8 W (Inventory)
Ware House 2 2 1 2 1 1 1 2 102.6 (mt)

(3) Define the particle evolution criterion that is based on fitness value of all the objectives considered in the prob-
lem.
(4) Generate initial swarm of particles. The structure of particle is specified in step 2. This step employs velocity
updating operator and particle updating operator. The particle representation for MOIAIP is set of integers (DCs
id) for first m vectors and last vector is real coded. The particle and its velocity updating operators work as
described in Section 4.2. For NSGA-II, uniform crossover and random mutation for updating of chromosomes
(see Figure 4).
6.1 Results of case study scenario I
Coding for Case Study Scenario1 was done in MATLAB (2012b), Intel core duo processor, windows basic for proposed
MOSLPSO and NSGA-II algorithms by setting the parameters as shown in Table 3. The parameter values for both the
algorithms are obtained using trial and error method by means of incremental method (changing one parameter at a
time). Since, we are dealing with allocation problem therefore number of DCs must be known prior. Now for N-DCs,
there can be (2N
– 1 – N) ways of serving the customer. In this paper, we have ruled out the possibility of the single
DC. For a P-DCs situation, each retailer can be served in P ways; for Y number of retailers, there are PY
(P Â P Â P Â P Â P ðupto Y timesÞ). The number of ways customers are served is the total search space (complexity).
There are total of 11 possible combination of DCs are possible as shown in Table 4. Total search for each of 11(24
– 1
– 4) possible scenarios and the number of function evaluations for each of the DC combinations to get converged are
shown in Table 5. With the increased complexity of the problem, algorithm needs more function evaluations to reach
the near global optimum.
After doing experimentation with all the 11 DC combinations, final results are summarised in Table 6. Table 6
shows the points in the final near optimal Pareto front for case study 1. The occurrence of multiple points in the
Pareto front for each combination is due to slightly different allocation of retailers to DCs which resulted in higher
or lesser maximum inventory. Results indicate that only four combinations of DCs (DC1-DC4, DC1-DC2-DC4,
DC1-DC3-DC4 and DC1-DC2-DC3-DC4) are making into final Pareto front. One can perceive the importance of
DC4 as it is involved in all four DC combinations that are part of final Pareto front. The points in the Pareto front
help the Distributor to take strategic decisions like which DCs to keep and how much maximum storage capacity a
DC must be given. The question of which DCs to keep and which DCs must be removed is a strategic one. This
depends upon several external factors like how much storage capacity available at the new potential DCs? how
much additional storage capacity available at present DCs? Future market trends etc. This is the main reason
Figure 4. Crossover and mutation operations for NSGA-II.

behind considering a multi objective formulation. The solutions in the Pareto front (Figure 5) help the distributor to
explore different possibilities and find the optimal combination considering all the factors influencing it.
The solutions of the Pareto front also help us to explore the possibilities of decreasing or increasing the maximum
capacity of all the DCs in a selected combination. For example, the first and second points in the Pareto front belong to
same combination of DCs (DC1-DC4). We can see that with a slightly different allocation of retailers resulted in a
decrease of 18 tonnes of inventory (15.65% increase) with just an increase of just Rs. 24,558 in total operational cost
(2.88% increase).
At present, distributor of the organisation is maintaining DC1, DC2 and DC3. Distributor present total cost of
the SCN is Rs. 1,345,781. Once the network is optimised, all the points in Pareto front (Table 6) have better costs
with different varying maximum inventories. These solutions help in making an effective yet efficient decision. Min-
imum total operational cost for all experimental DCs combinations are given in Table 7. Present DC combination
(case id 7) is optimised but results show that other combinations give better results (Table 7). Actual cost of case
Id 10 before optimisation is (Rs. 1057,892). Optimised results show an improvement of Rs. 73,305 (6.93%
decrease).
In order to validate the performance of MOSLPSO, It has been compared NSGA-II which is considered as bench-
mark algorithm in multi-objective optimization. Figure 6 shows the comparison of Pareto fronts of both algorithms. The
Pareto front is obtained after 10 repeated experimental trails. It can be clearly seen that NSGA-II is able to generate
only part of the Pareto front obtained by MOSLPSO. In all the 10 trials experimented, solutions of MOSLPSO are
found to be non-dominated which made us to conclude that algorithm is converged.
Table 3. Parameter values of MOSLPSO and NSGA-II for case scenario 1.
Parameter values for MOSLPSO Parameter values for NSGA-II
Acceleration coefficient 0.55 Crossover probability 0.9
Probability 0.8 Mutation probability 0.1
Velocity preserver 0.5
Table 4. Experimented DC combinations for case study I.
Case Id DC combination Case Id DC combination
1 DC1-DC2 7 DC1-DC2-DC3
5 DC2-DC4 11 DC1-DC2-DC3-DC4
6 DC3-DC4
Table 5. Complexity of the cases and function evaluations required for scenario 1.
Case (warehouse
combination)
Complexity (total
search space)
Number of function
evaluations Case
Complexity (total
search space)
Number of function
evaluations
1-2 2196
30,000 1-2-3 3196
35,000
1-3 2196
30,000 1-2-4 3196
35,000
1-4 2196
30,000 1-3-4 3196
35,000
2-3 2196
30,000 2-3-4 3196
35,000
2-4 2196
30,000 1-2-
3-4
4196
45,000
3-4 2196
30,000

Table 6. Pareto front points of case study I.
Sr no. Total operational cost Maximum curtailed inventory DC combination
1 827638.7723 115 1-4
2 852196.8064 97 1-4
3 893473.5912 83 1-4
4 902289.9048 82 1-4
5 911225.3085 81 1-2-4
6 916957.2897 79 1-2-4
7 926566.3727 78 1-2-4
8 939950.1895 77 1-2-4
9 947675.3574 75 1-2-4
10 966853.1557 70 1-3-4
11 983786.9702 65 1-3-4
12 992439.6332 63 1-3-4
13 1012587.372 62 1-3-4
14 1017282.842 60 1-3-4
15 1030234.18 57 1-2-3-4
16 1041635.651 53 1-2-3-4
17 1052389.502 52 1-2-3-4
18 1075368.19 51 1-2-3-4
19 1078197.886 50 1-2-3-4
20 1112861.314 49 1-2-3-4
40
50
60
70
80
90
100
110
120
800000 850000 900000 950000 1000000 1050000 1100000 1150000
Maximumcurtailed
inventory(tons)
Total Operational cost (Rs)
Pareto front for case 1
Figure 5. Final Pareto front for case study I for MOSLPSO.
45
55
65
75
85
95
105
115
125
800000 850000 900000 950000 1000000 1050000 1100000 1150000
MaximumCurtailedInventory(t)
Total Operational cost (Rs.)
MOSLPSO
NSGA-II
Figure 6. Comparison of pareto fronts of MOSLPSO and NSGA-II.

An integrated inventory optimisation model for facility location allocation problem

Empfohlen

Empfohlen

Weitere ähnliche Inhalte

Was ist angesagt?

Was ist angesagt? (19)

Ähnlich wie An integrated inventory optimisation model for facility location allocation problem

Ähnlich wie An integrated inventory optimisation model for facility location allocation problem (20)

An integrated inventory optimisation model for facility location allocation problem