Optimisation of the Beer Distribution Game with complex customer demand patterns.
ABSTRACT This paper examines a simulation of the Beer Distribution Game and a number of optimisation approaches to this game. This well known game was developed at MIT in the 1960s and has been widely used to educate graduate students and business managers on the dynamics of supply chains. This game offers a complex simulation environment involving multidimensional constrained parameters. In this research we have examined a traditional genetic algorithm approach to optimising this game, while also for the first time examining a particle swarm optimisation approach. Optimisation is used to determine the best ordering policies across an entire supply chain. This paper will present experimental results for four complex customer demand patterns. We will examine the efficacy of our optimisation approaches and analyse the implications of the results on the Beer Distribution Game. Our experimental results clearly demonstrate the advantages of both genetic algorithm and particle swarm approaches to this complex problem. We will outline a direct comparison of these results, and present a series of conclusions relating to the Beer Distribution Game.

Conference Paper: Applying Adapted PSO Approach to Minimize Costs in the Beer Distribution Game Using Three Dynamic Demand Patterns
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ABSTRACT: In this paper an artificial intelligence's monoobjective algorithm known as Particle Swarm Optimization (PSO) was applied. This PSO was adapted to optimize the minimum costs in the Beer Distribution Game Problem. The PSO makes an adjustment of the policy's order of the participants in the supply chain (retailer, wholesaler, distributor and manufacturer) minimizing the costs involved into the inventory holding and resulting costs from the backlogs in orders. The supply chain was tested under three dynamic consumer demand patterns: One Step, Uniform and Cyclic. The test results were compared based on the NonParametric Wilcoxon's SignedRank Test, showing which is better to use equal or individual order policy for each participant in the chain.Electronics, Robotics and Automotive Mechanics Conference (CERMA), 2012 IEEE Ninth; 01/2012
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Optimisation of the Beer Distribution Game with Complex
Customer Demand Patterns
Hongliang Liu, Enda Howley and Jim Duggan
Abstract—This paper examines a simulation of the Beer
Distribution Game and a number of optimisation approaches to
this game. This well known game was developed at MIT in the
1960s and has been widely used to educate graduate students
and business managers on the dynamics of supply chains.
This game offers a complex simulation environment involving
multidimensional constrained parameters. In this research we
have examined a traditional genetic algorithm approach to
optimising this game, while also for the first time examining
a particle swarm optimisation approach. Optimisation is used
to determine the best ordering policies across an entire supply
chain. This paper will present experimental results for four com
plex customer demand patterns. We will examine the efficacy of
our optimisation approaches and analyse the implications of the
results on the Beer Distribution Game. Our experimental results
clearly demonstrate the advantages of both genetic algorithm
and particle swarm approaches to this complex problem. We
will outline a direct comparison of these results, and present a
series of conclusions relating to the Beer Distribution Game.
I. Introduction
This paper examines the well known Beer Distribution
Game (BDG), which is a commonly used supply chain
analysis tool. This simple game has been widely used for
almost five decades to illustrate human decision making and
the concepts of supply chain management [1], [2]. The tradi
tional game is normally played by four players, representing
four individuals, a retailer, a wholesaler, a distributor and
a manufacturer. Each individual faces a decision making
challenge involving how they manage their current stock
inventories. Each participant in the game seeks to minimise
their total cost by managing their inventories in the face of
uncertain demand. It has been shown that this simple game
provides complex and often nonlinear dynamics due to feed
backs and time delays. It has been shown through simulation
and also real life experiments that game participants find
it extremely difficult to perform well in this game. Their
decisions commonly result in large divergences which are
far from optimal behavior. These result in large oscillations,
deterministic chaos and other forms of complex behavior [3].
The primary goal of this paper is to examine optimal
solutions for the BDG. This becomes increasingly difficult
when more complex customer demand patterns are examined.
We define optimality as the best set of strategies for the
overall efficiency of the supply chain, thereby, minimising the
total costs for all participants of the game. Determining the
most optimal ordering policy is always a difficult problem for
Hongliang Liu, Enda Howley and Jim Duggan are with the De
partment of Information Technology, National University of Ireland,
Galway(email:h.liu1@nuigalway.ie,
jim.duggan@nuigalway.ie).
enda.howley@nuigalway.ie,and
any participant in the Beer Distribution Game. A number of
possible approaches will be examined in this paper. We will
present a particle swarm optimisation (PSO) approach. Such
PSO approaches have been successfully used in other similar
contexts to find optimal solutions, yet, this is the first time it
has been applied to the BDG. We will also examine optimal
solutions found using a genetic algorithm (GA) approach.
This paper differs from existing research which is primarily
based on very simple customer demand patterns [1], [4]. This
paper will address a number of important research questions:
1) How effective are PSO and GA approaches to optimis
ing the BDG?
2) What are the differences between our PSO approach
and other optimisation approaches to this problem?
3) What are the effects of more complex customer de
mand patterns on the results?
4) What are the differences between common and distinct
ordering policies among game participants?
These research questions will be referred to regularly
throughout this paper and answered directly in the Conclu
sions section. The following sections of this paper are struc
tured as follows. In Section II, we will discuss background
research and in Section III we will outline our experimental
setup. Section IV will provide a detailed examination of
our experimental results. In Section V we will outline our
conclusions, while finally in Section VI we will briefly
summarise the contributions of this paper and outline some
future work.
II. Background Research
Our interpretation of the BDG is based on the specification
outlined by Sterman [1]. Later sections will provide a de
tailed overview of Particle Swarm Optimisation, and Genetic
Algorithm search techniques. These sections will also outline
previous optimisation approaches to the BDG.
A. Introduction to the Beer Distribution Game
BDG is a classic supply chain optimisation problem [5].
It has been widely used in the domain of supply chain
management [5], [6], [7]. This game offers a simplified
implementation of common real world production and distri
bution systems. This system consists of four participants: Re
tailer, Distributor, Wholesaler and Manufacturer. As shown
in Fig. 1, each participant has control and responsibility for
its own inventory.
• Retailer: The retailer receives orders from customers
though the “customer demand”. Subsequently the Re
tailer must order beer from the Wholesaler to replenish
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Current
Inventory
Retailer
Current
Inventory
Wholesaler
Shipment
Current
Inventory
Distributor
Shipment
Current
Inventory
Manufacturer
Shipment
Customer
Demand
Production
Delay
OrderingOrderingOrdering
Fig. 1.The structure of the Beer Distribution Game
its inventory. These orders will then arrive after a
specified time delay and when the Wholesaler is capable
of fulfilling those orders.
• Wholesaler: The Wholesaler supplies beer from its
inventory to fulfill its orders received. The Wholesaler
orders and receives beer from the Distributor.
• Distributor: The Distributor supplies beer to the
Wholesaler and receives beer from the Manufacturer.
• Manufacturer: The Manufacturer brews beer in order
to maintain its own inventory and fulfill orders from the
Distributor.
These processes are effected by a number of delays. Delays
represent shipping delays and order receiving delays. These
involve the time it takes to receive, process, ship and deliver
orders. In the case of the manufacturer these delays represent
the amount of time required to replenish its inventory. These
delays are an intrinsic part of the BDG. Their presence
causes significant challenges in attempting to maintain opti
mal levels of inventory. Each week, every participant must
decide how much to order from its respective supplier or how
much to brew in order to meet current and future demands.
It is common for each participant to order more (or less)
than actually needed due to instabilities in the supply chain.
Furthermore, demand information is often distorted in the
supply chain from the end customer to the manufacturer.
This phenomenon is known as the bullwhip effect [8].
As we have stated previously, the objective of participants
is to minimise cumulative costs over a 150 week period
by keeping inventories as low as possible while avoiding
outofinventory conditions which cause backlogs. The BDG
commonly uses the following costs to penalise inventory
holding and backlogs. The cost of inventory holding is $0.5
for each case of beer per week and the cost of backlogs
is $2.0 for each case of beer per week. It is intuitive for
a player to order more beer when inventory falls below a
desired level. Similarly a player is likely to order less beer
when stocks begin to accumulate.
B. The Beer Distribution Game Ordering Policy
Since all the game participants experience time delays, it
becomes quite difficult to manage their inventory levels. To
deal with this challenge, Sterman proposes that the partici
pants adopt three ordering principles [1]. Sterman’s ordering
principles are as follows, and state that game participants
should place sufficient orders to:
• Satisfy Expected Demand: The participants should
order enough to satisfy the demand. However, to predict
the exact future customer demand is often a difficult
task. In this paper, we use an “adaptive expectations”
formulation (see Equation (3)) to model future customer
demand.
• Adjust Inventory Levels: There is a chance of predic
tion errors from the previous principle. Therefore, it is
necessary to adjust orders above or below the expected
orders. This serves to correct actual inventory levels in
line with desired inventory levels.
• Adjust for Orders Currently in the Supply Line:
Orders currently in the supply line should be factored
into future ordering decisions. Therefore, a participant
should be capable of ensuring a stable response to rapid
changes in customer demand. It is pointless to place
orders for items already ordered in a previous time step.
The following equations formalise an intuitive ordering
policy based on the principles discussed above. We will refer
to this as the “Stock Management Structure” (SMS). First,
orders must be nonnegative:
OPt= max(0,OP∗
t)(1)
In Equation (1), OPtrepresents the actual orders placed at
week t, whereas OP∗
the “ordering heuristic” at week t. This is defined as follows:
trepresents the orders calculated through
OP∗
t= EDt+ α(INV∗
t− β ∗ OSL) (2)
In Equation (2), EDt represents the expected demand at
week t (see Equation (3)), INV∗
between desired and actual inventory at week t (see Equation
(4)), OSL represents the orders in the supply line; α and
β represent the adjustment paraments for the inventory and
the supply line, respectively; The expected demand (EDt) in
Equation (2) is expressed as:
trepresents the discrepancy
EDt= θ ∗CDt−1+ (1 − θ) ∗ EDt−1
(3)
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In Equation (3), EDtand EDt−1are the expected demand
at week t and t − 1; CDt−1is the customer demand at week
t − 1, θ describes the rate at which the demand expectations
are updated. The parameter θ is typically set to 0.25 [1]. The
discrepancy between desired and actual inventory (INV∗
Equation (2) is formulated as follow:
t) in
INV∗
t= Q − INVt+ BLt
(4)
In Equation (4), Q represents the desired inventory level,
INVtrepresents the current inventory and BLtrepresents the
backlogs at week t. It is clear from Equation (2), that the
SMS comprises of three factors: expected demand, inventory
level and orders in the supply line. In the BDG, Equation (1)
is used to determine how much beer to produce in the case
of the manufacturer. Furthermore, in all other cases Equation
(1) is used to determine how much beer to order from one’s
respective supplier.
It has been proposed that two parameters are used by the
game participants to determine their orders [9].
(α) This represents the discrepancy between desired
and actual inventory ordered. This parameter is
usually represented in the range. (0 ? α ? 1)
(β)This represents the fraction of the supply line taken
into account. This parameter is usually represented
in the range. (0 ? β ? 1) If β = 1, the participant
factors in all orders in the supply line or conversely,
if β = 0, the participant factors in no orders in the
supply line.
The combination of the adjustment parameters (α,β) cor
responds to a set of behaviours for a given participant. These
parameters are fundamental to our analysis of the BDG. They
will provide the main basis for our PSO and GA optimisation
approaches. The following two experimental scenarios will
be examined in this paper.
• Scenario A: all participants in the supply chain use
the same ordering policy (the same α and β values).
Therefore there are the only two parameters involved.
We will refer to this scenario as Sa.
• Scenario B: each participant uses different ordering
policies (different α and β values). Therefore there are
eight parameters involved as there are four individuals
in the supply chain and each of these has two strategies
parameters (α and β). We will refer to this scenario as
Sb.
C. The Beer Distribution Game Objective Function
In the BDG, all the participants wish to minimize their
total cost. This objective can be formulated as:
totalCost =
m
?
i=1
(
n ?
j=1
(2.0 ∗ BLj
i) + 0.5INVj
i)(5)
In Equation (5), m is the total number of weeks and n is
the total number of participants, in our case n = 4. The cost
of holding inventory is 2.0 while the cost of maintaining a
backlog is 0.5. BLj
i; INVj
We use equation (5) as the objective function and employ
both our PSO and GA to determine the optimal α and β for
the entire supply chain.
iis the jthparticipant’s backlog at week
iis the jthparticipant’s inventory at week i.
D. Particle Swarm Optimization
Particle swarm optimization (PSO) was first described in
1995 by Kennedy and Eberhart [10], [11]. This approach
was first inspired by the phenomena of bird flocking and
fish schooling. It has been successfully applied to many
problems including the economic dispatch [12], and also
reactive power and voltage control [13]. However, PSO has
not been widely used in the case of supply chain problems.
For each particle {
Initialise it in the search space (it must satisfy
all the constraints);
}
Do {
For each particle {
Evaluate its position according the objective
function;
If the current position is better than the
previous personal best position (
update;
}
Choose the particle with the best position of all
the particles according the objective value as
the;
gBest
t), then
For each particle {
Calculate its velocity according the equation(6);
Update its position according the equation (7);
}
}while it does not reach a stop criteria(i.e.
maximum iterations)
pBes
pBest
Check if its position is still in the search space.
If not, reinitialise a new particle that satisfies
the constraints;
Fig. 2.The Modified PSO
In PSO, there is a population of solutions referred to
as particles. Particles fly around the Ddimensional solution
space, and are evaluated according to a fitness criteria after
each iteration. The ith particle’s position is represented
as Xi = (xi1, xi2,..., xid,..., xiD). The flying velocity for
a particle i is represented as Vi = (vi1,vi2,...,vid,...,viD).
In every iteration, each particle’s flying velocity is updated
according to the following two positions. The first one is the
position at which its best fitness has achieved so far. This
position is a “personal best” position and recorded as pBest.
The second position is the best position obtained so far by
all particles in the population (or by its local neighborhood,
in the local version of PSO). This position is a “global best
position” (or local best) and called gBest (or lBest). After
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updating velocity, each particle updates its position based on
its velocity. The Equations (6) and (7) formalise how each
particle’s velocity and position is updated. The inertia weight
w in (6) has been shown to provide improved performance
in a number of applications [14] and the c1 and c2 are
two positive constants, rand() and Rand() are two random
functions in the range [0,1].
vid= w ∗ vid+ c1∗ rand() ∗ (pBest − xid)
+c2∗ Rand() ∗ (gBest − xid)
xid= xid+ vid
(6)
(7)
E. The modified PSO
Particle Swarm Optimisation has been shown to be very
effective for many optimization problems. For the BDG
problem, however, the PSO needs to be modified to deal
with the game constraints.
Our modification involves reinitialising unfeasible parti
cles. Therefore in each iteration, once a particle fails to
satisfy the game constraints, it is reinitialised and positioned
randomly in the valid search space. Therefore, this particle
will again satisfy all the game constraints while also retaining
its memory of (a) its personal best position to date pBest,
and (b) the location of the global best position gBest. This
modification only concerns those particles which break the
game constraints, therefore all other particles are updated as
normal as in the traditional PSO.
This modification is similar to that proposed by Hu and
Eberhart [15] which was applied to constrained nonlinear
optimisation problems. Our modification benefits the overall
diversity of particles, and maintains a good search capability
since the total number of particles does not reduce. An
overview of the modified PSO is shown in Fig. 21.
F. Genetic Algorithms
Genetic Algorithms (GAs) are inspired by evolutionary
biology such as selection, crossover (also called recombina
tion) and mutation. GAs have been successfully applied to
optimisation problems like wire routing, scheduling, adaptive
control, supply chain management, etc [16].
An implementation of a GA begins with a population of
chromosomes that encode candidate solutions to a problem.
For each generation G, each chromosome is evaluated and
assigned a fitness value. After evaluation, Selection is ap
plied to the population according their fitness value and an
intermediate population is created. The algorithm applies
recombination and mutation to create the next population
in G + 1. Once evaluation, selection, recombination and
mutation is completed, this new population is then used and
the process repeats over successive generations. This process
continues for a predefined period of time (G), or until the
most optimal solutions are found.
Previous research has applied a GA to the BDG problem.
O’Donnell et al. [17] and Lu et al. [18] have successfully
used GAs to reduce bullwhip effect for the BDG. Strozzi
1Please note that our modification is shown in italic bold.
et al. [4] also used a GA to optimise the BDG ordering
policies for an entire supply chain. However, these papers
examined simple customer demand patterns. In this paper,
we will examine a series of more complex demand patterns
and investigate the implications of these demand patterns.
Furthermore, we will use a realcoded GA which Wright has
claimed should increase efficiency and precision [19]. The
previous GA approaches mentioned here use binary encoded
GAs which differs from our approach. Our realcoded GA
uses the Java Genetic Algorithms Package (JGAP) [20].
III. Experimental Setup
Our experiments examine a number of customer demand
patterns. These demand patterns use a common mean value
which is set to 8. Then we can compare the participants’
behaviour when they use the best policies obtained from
PSO or GA. As has been outlined by Yan and Woo, we
can model a number of alternative customer demand patterns
[21]. These demand patterns are also illustrated in Fig. 3, and
less than 150 data points are used for clarity purpose. These
demand patterns are as follows:
• One Step Demand (OSD): This demand changes only
once in the period of this simulation. It is typically set
that the customer demand is four cases of beer per week
until week 4 and then steps to eight cases of beer per
week.
• Stationary Demand (SD): The mean demand remains
constant over time. The distribution of the demand
conforms to a normal distribution. We use a mean of
8 and the standard deviation of 2.
• Uniform Demand (UD): This demand fluctuates ran
domly and is generated using a uniform distribution in
the range of [0,16].
• Cyclic Demand (CD): This pattern varies cyclically
over time, usually in response to some seasonal effect of
season or the standard business cycle. The mean value of
demand changes periodically. The cycle of the demand
is 50 weeks. In every cycle of the first 25 weeks, we
use normal distribution with the mean of 10 and the
deviation of 2; In the following 25 weeks, we use normal
distribution with the mean of 6 and the deviation of 2.
As explained we will examine four distinct customer
demand patterns. The One Step Demand pattern is the most
commonly used demand pattern when analysing the BDG.
These other three demand patterns are more realistic and
reflect a closer representation of real market dynamics. For
example, demand for goods such as rice normally follows
the SD pattern, demand for Christmas ornaments normally
follows the CD pattern. New product demand normally
follows the UD pattern.
As mentioned previously we will examine two distinct ex
perimental scenarios. Scenario A: Where all individuals use
the same α and β strategies. We will then examine Scenario
B: Where all individuals can use different α and β values.
These two distinct scenarios will be used to compare the
performance of our PSO and GA solutions. These scenarios
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0
2
4
6
8
10
12
14
16
0 20 40 60 80 100 120 140
Customer Demand
Time(week)
(a) One Step Demand
0
2
4
6
8
10
12
14
16
0 20 40 60 80 100 120 140
Customer Demand
Time(week)
(b) Stationary Demand
0
2
4
6
8
10
12
14
16
20 40 60 80 100 120 140
Customer Demand
Time(week)
(c) Uniform Demand
0
2
4
6
8
10
12
14
16
0 20 40 60 80 100 120 140
Customer Demand
Time(week)
(d) Cyclical Demand
Fig. 3.The Simulation Results of Demand Patterns
fundamentally change the optimisation problem involved. In
Scenario A, all individuals use the same ordering policy and
therefore only two parameters must be optimised. In Scenario
B, four individuals use different policies and therefore eight
parameters need to be optimised.
In the case of all these experiments, optimality is consid
ered as the collective optimal solution for the entire supply
chain. This differs from using individual optimality for an
individual participant in the BDG. For example, the most
optimal solution for the Distributor may well be different, if
we were to assume he is acting in a selfish manner. However,
in this case the collective performance of the supply chain
among all its participants is our fundamental concern. There
fore we define the most optimal solution as “those parameters
which provide the most optimal performance across the entire
supply chain”. This optimal policy is formally specified
through the minimum cost as calculated through Equation
(5).
A. Beer Distribution Game Parameters
The parameters involved in our BDG are as follows. The
total simulation length was 150 weeks. The α and β values
exist in the range of [0,1]. The θ value used was 0.25
(see Equation (3)). Delays equal 4 weeks and the desired
inventory levels, Q = 17.
B. PSO Parameters
Our PSO implementation is based on an existing PSO
implementation by Hu and Eberthart [15]. Therefore, we
adopt the following parameters which are specified by Hu
and Eberthart [15]. The population size used was N = 50 and
the total number of iterations was 500. The inertia weight
was w = 0.5 + rand()/2.0, while the maximum allowed
velocity Vmaxwas set equal to range of the dimension. Each
dimension was set to Vmax = 1, since the α and β values
were limited to the range of [0,1]. Finally the constants c1
and c2were set to 1.49445.
C. GA Parameters
As mentioned earlier, our realcoded GA was implemented
using JGAP [20]. The population size was set to N = 50 and
the total number of iterations (Generations) used was = 500.
The crossover rate was 0.8 and the mutation rate was 0.01.
TABLE I
Comparison Between PSO and GA in Scenario A
Customer
Demand
MethodWorstBest
μσ
OSD
PSO
GA
PSO
GA
PSO
GA
PSO
GA
1854.0
1896.0
2527.0
2765.0
7489.5
8041.5
13533.0
13702.5
1772.0
1791.0
2459.0
2459.0
7488.0
7488.0
12015.0
12020.0
1834.6
1847.0
2472.8
2549.19
7490.2
7605.4
12579.2
12946.9
27.5
28.6
15.0
61.5
0.5
147.7
700.4
727.2
SD
UD
CD
IV. Experimental Results
In this section we will provide a series of detailed ex
perimental results, involving the BDG, and our different
optimisation approaches.
Page 6
TABLE II
Comparison results with enumeration methods in Scenario A
Customer
Demand
OSD
SD
UD
CD
Enumeration
Method
1772.0
2459.0
7488.0
12015.0
PSOGANo.
PSO
8
17
43
18
No.
GA
0
4
6
0
1772.0
2459.0
7488.0
12015.0
1791.0
2459.0
7488.0
12020
TABLE III
The best α and β for all participants in Scenario A
Customer
Demand
Method
αβ
OSD
PSO
GA
PSO
GA
PSO
GA
PSO
GA
0.8835
0.8866
0.2039
0.2018
0.0296
0.0300
0.0249
0.0246
0.8478
0.8470
0.8132
0.8132
0.7274
0.7252
0.1349
0.1323
SD
UD
CD
A. Simulation Results in Scenario A
The data in Table I shows the results obtained by our
modified PSO and GA. These results are from 50 individual
runs of our PSO and GA. Data is shown representing the
maximum cost (Worst), the minimum cost (Best), the average
cost (μ) and standard deviation (σ) for 50 individual runs.
Compare the results obtained by PSO and GA, we observe
that the results from the PSO are marginally better than the
results from the GA. Only two best results are the same when
the participants face the SD and UD. So it seems that PSO
provides a better solution for the BDG in Sa.
In order to clarify our results from Sa, we compare the
results obtained from using our PSO and GA implementa
tions with the results we have obtained from a bruteforce
search of the α, β strategy space. This corresponds to an
enumeration method, and therefore provides us with the
optimal solutions for this problem. Here, we used a search
resolution of (0.0001) in Sa. This resolution is high enough
to obtain the global optimum due to the fact that there were
no further improvements when higher resolutions were used.
However, the bruteforce method was not feasible in Sbdue
to the size of the search space. Table II shows the optimal
results from the enumeration method, the best results from
the PSO and GA and the number of hits to the optimal in
50 individual runs of the PSO and GA which we record as
(No. PSO) and (No. GA). From Table II, we can see that
the PSO find the optimal results in all customer demands
while GA does not find all the optimal results. However,
the results obtained by GA are very close to the optimal
when the participants face the OSD and CD that GA does
not obtain the optimal results. Furthermore, the PSO hits the
optimal much more times than the GA. It should be noted
that these differences do not indicate a clear benefit for using
a PSO over a GA approach. Instead they clearly show the
differences in the specific case and for the specific parameters
used in our PSO and GA algorithms.
Our results in Table I and II show that the most opti
mal strategies differ significantly when alternative customer
demand patterns are used. This means that the demand
patterns have a significant effect on the overall performance
of the supply chain. Total costs vary significantly across
the various customer demand patterns. We believe this cor
relates strongly with the degree of complexity involved in
interpreting each of these demand patterns. It is plausible to
argue that where complexity is represented as C that COSD<
CSD< CUD. It appears that Cyclical Demand (CD) is more
complex, or more difficult to interpret than Uniform Demand
(UD). While the standard deviation of the actual customer
demand in each of these patterns would lead us to believe
the converse, the actual amplification effects on inventory
levels is the primary driver of this behaviour [21]. Further
more, this problem is influenced through our optimisation
approaches “overfitting” for a particular period of the cycle
in (CD). This does not occur when interpreting the Uniform
Demand customer demand. This effect is reduced when the
period used in Cyclical Demand is reduced and therefore
CD changes more regularly over shorter periods. We can
therefore conclude that CUD< CCD. The total cost becomes
larger as the demand pattern becomes more complex. So the
simulation results show that the more complex the demand
pattern, the more difficult to manage the inventory when the
participants use the same strategies.
Table III shows the best ordering parameters for all
participants obtained by the PSO and GA in 50 runs. The
results achieved using both PSO and GA approaches are quite
similar. It is clear that the α and β values do not follow
conventional wisdom as stated by Sterman [1] once more
complex customer demand patterns occur. From the values
shown in our table it is clear that as uncertainty increases
about the customer demand, game participants are better to
use lower α and β values. Thereby not factoring in all orders
in their supply chain, or trying to fully correct their inventory
levels. This is a symptom of individuals performing better by
not reacting to every small change in demand.
From the results shown and the our discussion above, we
have shown that both the PSO and GA approaches achieve
high quality solutions for the BDG in Sa.
B. Simulation Results in Scenario B
The data in Table IV shows the results obtained from using
our modified PSO and GA optimisation approaches. This
data represents the maximum cost (Worst), the minimum
cost (Best), the average cost (μ) and standard deviation (σ)
over 50 individual runs. The results indicate that the GA was
more successful than the PSO in its search for an optimum
solution. It should be noted that the difference between the
PSO and the GA approaches are not very significant and
could be due in part to some of the parameters used in the
two approaches. It is clear from analysing the results from
both optimisation approaches, that more optimal solutions
are achieved in Sbthan in Sa. Furthermore, it is clear from
Table IV that total costs increased as more complex customer
Page 7
demand patterns were simulated. This reinforces this same
observation which we discussed in Sa.
Table V shows the best ordering parameters for each
participant as obtained by our approaches in 50 attempts.
Clear differences are identifiable in the performance of the
two optimisation approaches. These stem from a number
of factors. Firstly, the search space is much larger, and
more complex due to numerous local minima. Secondly,
the specific parameters used by each approach may well
influence their ability to find the most optimal solutions in
the complex landscape. Apart from these specific issues, it
appears quite clear that Sb offers a much more expressive
landscape of solutions with greater potential for identifying
more optimal solutions than those available in Sa. This more
expressive landscape is also much more difficult to search,
and therefore this causes certain differences in the solutions
identified by our optimisation approaches.
C. Comparison Results in both Scenarios
In this section, we will outline the differences and simi
larities between the results shown from Saand Sb. Table VI
clearly shows a reduction in total cost in Sbcompared with
Sa. This stems from the increased α, β landscape. In Sa
these strategy values are constrained so that the same α,
β values are used by all participants in the supply chain.
Sb differs significantly, as all individuals are free to use
different α, β values to their peers on the supply chain. This
results in a much larger set of possible optimal solutions in
the game. This larger, more complex landscape is further
augmented when more complex customer demand patterns
are simulated. This results in a heightened degree of difficulty
when attempting to determine the most optimal α, β values.
This observation is reinforced in the results shown, where
it is clear that more complex customer demand patterns
result in ever increasing differences between Saand Sb. For
more complex demand patterns, Sb offers greater potential
to approximate to the global optimum than Sa.
TABLE IV
Comparison Between PSO and GA in Scenario B
Customer
Demand
MethodWorstBest
μσ
OSD
PSO
GA
PSO
GA
PSO
GA
PSO
GA
1664.5
1560.0
1866.0
1837.5
4874.5
4537.0
6976.5
6644.0
1422.0
1351.5
1651.0
1635.0
4378.0
4282.5
5640.5
5353.5
1525.0
1424.8
1742.4
1704.4
4562.0
4341.4
6233.4
5772.2
55.0
51.3
43.2
43.4
114.6
39.5
304.1
292.1
SD
UD
CD
V. Conclusions
The research outlined in this paper investigated optimised
ordering strategies in the BDG when a number of com
plex customer demand patterns are simulated. This research
holds particular significance for those interested in classic
supply chain problems. Earlier in this paper, we posed a
TABLE VI
The cost reduced in Sbcompared with Sa
Method
Customer
Demand
OSD
SD
UD
CD
OSD
SD
UD
CD
WorstBest
μ
PSO
10.2%
26.2%
34.9%
48.5%
17.7%
33.6%
43.6%
51.5%
19.8%
32.9%
41.5%
53.0%
24.6%
33.5%
42.8%
55.5%
16.8%
29.5%
39.1%
50.4%
22.9%
33.2%
42.9%
55.4%
GA
number of important research questions. In response to the
first research question, both optimisation approaches were
clearly very successful in achieving their task. The optimised
α and β values in Sa and Sb, indicate optimal ordering
strategies when more complex customer demand patterns
occur. While global optima were only found in a limited set
of circumstances, it is clear that both optimisation approaches
performed well despite the complexity of the initial game,
and the added complexity of the customer demand patters
examined in this paper. From examining previous research,
and the results outlined here we can confidently conclude
that both optimisation approaches performed well in their
optimisation tasks.
It is also worth noting that previous research examining
this optimisation problem has used genetic algorithm ap
proaches to this problem. Existing research has predomi
nantly examined much simpler customer demand patterns.
This paper has extended this existing research to include
much more complex customer demand patterns and also
introduces for the first time a PSO approach to this op
timisation problem. Our second research question refers
to this extension. This PSO approach offers a new and
alternative approach to these BDG optimisation problems,
while specifically in the case of Sa it has been shown to
perform very well. It found the optimum solution more often
than the GA approach (Table II).
Our third research question refers to the effects of more
complex customer demand patterns. It is apparent that as
these patterns become more complex the total cost increases
as game participants struggle to cope with the increased
difficulty. Furthermore, our results show that this increased
difficulty also resulted in higher variances across all results.
It is clear that this increased complexity posed significant
challenges for our optimisation approaches, as they struggled
to find optimal α and β values. The more complex customer
demand patterns used in this research demonstrates a more
realistic interpretation of the BDG. Furthermore, the results
also show the benefits of low α and β values when deter
mining an optimal solution. When customer demand patterns
become more complex and difficult to interpret, perhaps its
best to simply react rapidly without too much regard for
previous trends.
Our final research question involves the use of Sa and
Sbin our experimental simulations. As explained previously
Page 8
TABLE V
The best α and β for each participant in Scenario B
Customer
Demand
Method
Rα
Rβ
Wα
Wβ
Dα
Dβ
Mα
Mβ
OSD
PSO
GA
PSO
GA
PSO
GA
PSO
GA
0.8880
0.2017
0.2215
0.0267
0.0205
0.0113
0.0187
0.0168
0.8713
0.8925
0.7832
0.2753
0.4118
0.0196
0.2890
0.2248
0.1019
0.8848
0.0823
0.6076
0.0209
0.0270
0.0225
0.0210
0.8384
0.8587
0.7577
0.7578
0.4506
0.5961
0.0945
0.0406
0.9170
0.9976
0.1602
0.1693
0.9193
0.9832
0.3654
0.2967
0.7823
0.8294
0.7342
0.6992
0.6734
0.6356
0.5615
0.5051
0.9116
0.9975
0.8437
0.9726
0.9349
0.9463
0.9031
0.9963
0.7753
0.7310
0.7167
0.7391
0.4566
0.4606
0.4465
0.5141
SD
UD
CD
the strategy space in Sa is significantly limited since all
individuals must use the same α and β values. Conveniently
this provided us with a simplified strategy space which was
easier to analyse. Furthermore, this also made optimisation
much easier. Sb provided a much more complex strategy
search space where the α and β values were not fixed across
all individuals in the game. It is clear that this initially
simple game results in some very complex dynamics which
are inherently difficult to analyse. These are quite apparent
from our results outlined for Sb. This complexity is further
magnified when more complex customer demand patterns are
introduced. Despite this, we identified lower costs in Sbover
Sa across all customer demand patterns. This stems from
Sboffering greater freedom to individuals and their specific
strategy choice. This effect is further magnified when more
complex customer demand patterns are encountered.
VI. Summary and future research
This paper has examined the BDG, its optimisation and
the effects of complex demand patterns. A number of fun
damental factors influence this study. Firstly, the degree of
autonomy offered to game participants and the scope of
their individual rationality. This has significant implications
for this paper. Even in the limited case of this paper, we
have seen the effects of reduced and increased freedom to
determine α and β strategies. The second significant factor
influencing this study involves the complexity of customer
demand patterns. We have provided an initial study of these
factors, yet much more work is required.
Acknowledgment
The authors would like to gratefully acknowledge the
continued support of Science Foundation Ireland.
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