"Optimisation of closed loop supply chain decisions using integrated game theoretic particle swarm algorithm" Kalpit Patne, Visiting Fellow, SMART Infrastructure Facility presented a summary of his research as part of the SMART Seminar Series on 8 July 2016. For more information, visit the event page at: http://smart.uow.edu.au/events/UOW217694.

1. Presented by:
Kalpit Patne
4th Year UG Student,
Department of Industrial and Systems Engineering,
Indian Institute of Technology (IIT) Kharagpur, INDIA

2.  Introduction
 Problem Definition
 Solution Methodology
◦ Background of Key Concepts
◦ Improved Particle Swarm Optimisation
 Numerical Analysis
 Results and Comparison
 Future prospects

3. Source: Google images

4.  Closed-loop supply chains are designed and
managed to explicitly consider the reverse and the
forward supply chain activities over the entire life
cycle of the product.
Forward chain
Reverse Chain
New Product

5. Source: Google images

6. Source: Google images

7.  Key considerations in optimizing the Supply
Chain Network
 Location and distance
 Current and future demand
 Inventory level
 Size and frequency of shipment
 Warehousing costs
 Transportation costs
 Mode of transportation

8. The significant aspects we aim to address in
this study-
1. Retailer Facility Location
2. Customer Zone Allocation
3. Optimal Pricing Policy
4. Optimal Inventory Cycle Time

9. 1. Retailer Facility Location
 Customer zones and production centre locations are known
 We want to optimally locate retailer facilities
Source: Google images

10. 2. Customer Zone Allocation
 We want to know the customer zone allocation schema
Source: Google images

11. 3. Optimal Pricing Policy
 We need to decide on the selling price of the new
product and the rebate price (incentive) paid to the
customer for returning the used product, so that
the overall profit is maximized.
 Since the demand and willingness to return is
sensitive to selling and rebate price respectively,
there exists a binding constraint on their values.

12. 4. Optimal Inventory Cycle Time
 Deciding optimal inventory replenishment time
 This has to be done considering the return
shipments of used products from retailer facilities
to the production centre for remanufacturing.

13. Mathematical Representation of the Problem
Z= Max 𝑖 𝑗 𝑝 − 𝑐 𝑁𝑖 𝑒−𝑘𝑝 𝑥𝑖𝑗 𝛼𝑖𝑗--------------------------------I
+ 𝑖 𝑗(𝑠 − 𝑐 𝑟 − 𝑟)𝑁𝑖(1 − 𝑒−𝑔𝑟
)𝑥𝑖𝑗 𝛽𝑖𝑗------------------------II
− 𝑗 𝐹𝑗 𝑦𝑗---------------------------------------------III
− 𝑗
𝐴 𝑗
𝑇 𝑗
+ 𝑖 𝑁𝑖 𝑒−𝑘𝑝
𝑥𝑖𝑗 𝛼𝑖𝑗 𝑇𝑗
ℎ𝑛 𝑗
2
+ 𝑁𝑖 1 − 𝑒−𝑔𝑟
𝑥𝑖𝑗 𝛽𝑖𝑗 𝑇𝑗
ℎ𝑟 𝑗
2
𝑦𝑗----IV
I – Profit function for new products
II – Profit in terms of assets retrieved from customers
III – Establishment and Operational cost of the facility
IV – Ordering and Inventory holding costs

14. We have considered the proposed model as a
combination of two sub-problems:
 Location-Allocation Problem (LAP)
1. Retailer facility location
2. Customer zone allocation
 Pricing-Inventory Problem (PIP)
3. Optimal pricing policy
4. Optimal inventory cycle time

15. LAP solved
using modified
PSO
• No. of customer zones
• Their locations
• Production facility location
• No. of retailer facilities
• Retailer
locations
• Operational
facilities
• Customer
allocation
• Capacity
• Fixed cost
• Operational
cost
• Distance
parameters
PIP solved
with Gradient
search
• Optimal pricing
policy
• Optimal
inventory cycle
time

16. 1. Particle Swarm Optimization (PSO)
 Memory based
(personal and global best
position)
 Communication involved
(for searching the global best
position of the swarm)
Source: Google images

17. 1. Particle Swarm Optimization (PSO)
 Involves two main search variables- Position and
Velocity
 Updated in each iteration – fitness value for updated
position of particle keeps on increasing with iteration
 Drawback- If a particle discovers a local optimal, all the
other particles will move closer to it, then particles are
in the dilemma of local optimal point.
◦ This is known as premature convergence.

18. 2. Evolutionary game-based replicator dynamics
 It allows the fitness to incorporate the distribution of population
types rather than setting the fitness of a particular type constant.
 The general form of the evolutionary game consists of three key
components: Players, Strategies space and Payoff function.
Particles in a swarm ------- Players in a game
Research space ------- Strategies space
Velocity ------- Rate of proportion change
Fitness function ------- Payoff function
gBest ------- Evolutionary stable strategy
Based on: Liu, Wei-Bing, and Xian-Jia Wang. "An evolutionary game
based particle swarm optimization algorithm." Journal of Computational
and Applied Mathematics 214.1 (2008): 30-35.

19. 2. Evolutionary game based replicator dynamics
Where
.
xi – change rate of proportion of population of type i
xi - proportion of population opting for strategy i,
x=(x1,…,xn) - vector of the distribution of population types,
fi(x) - fitness of type i (which is dependent on the population),
ɸ(x) - average population fitness

20. 2. Evolutionary game based replicator dynamics
 We update the velocities of particles using the replicator
equation and then update the positions.
 This speeds up the convergence of our algorithm.

21.  To avoid premature convergence, we use the concept of
mutation.
 Mutation- To diversify the population when it starts
converging at a point.
 A larger research space is explored and chances of obtaining
a global optima increases
 Diversity index, D(t)- Variable responsible to trigger mutation
Based on: Lin, M. O., and Zheng Hua. "Improved PSO algorithm with adaptive
inertia weight and mutation." Computer Science and Information Engineering,
2009 WRI World Congress on. Vol. 4. IEEE, 2009.
Proposed Solution Approach

22. IPSO
 Adaptive Weight Update
 In our algorithm we have used adaptive weight update
method to make the updating process more dynamic.
 We use the rate of change of fitness at each iteration to
update the weight.
Proposed Solution Approach

23. Initialization
Iterations
If t > T
Weight update
(linear)
Weight Update
(adaptive)
Mutation
trigger
Probabilistic
mutation of some
particle positions
FalseTrue
Yes
Velocity and Position
Update (PSO)
Velocity and Position
update (Replicator
Dynamics)
Update pBesti
Update gBest
No

24.  We use a 100x100 2D space as our search space.
 We used real life data where possible; otherwise, realistic
assumptions were made
Parameter Value
# Customer Zones, M 20
# Retailer Facilities, N 10

25. This is a randomly generated market area showing the locations of
production facility and all the identified customer zones

26. Results obtained after implementation of the IPSO algorithm

27. Performance of IPSO algorithm

28.  Once the LAP problem is solved, we will have the
retailer locations and customer zone allocation
schema determined.
 We then tackle the PIP problem and try to maximize
the profit function Z (the mathematical model
developed earlier) using gradient search method.
 In our numerical example, we assumed:
◦ manufacturing cost ‘c’ for a new unit = $100;
◦ remanufacturing cost ‘cr’ = $15; and
◦ salvage value ‘s’ of the returned product = $60.

29.  The gradient search algorithm produces the optimal value
of selling price as p =127.57 for a new product and the
incentive r=18.89 for returned products.
 So, the maximum profit that the company can make by
selling one unit of new product is
(127.57 -100) = 27.57 per unit.
 If the company decides to sell the remanufactured product
at it’s salvage value, it will earn
(60-18.89-15) = 26.11 per unit

30.  The optimal inventory time Tj, is calculated from the standard
Economic Order Quantity (EOQ) model equation.
Tj =
2𝐴 𝑗
𝑖 [𝑁 𝑖 𝑒−𝑘𝑝 𝑥 𝑖𝑗 𝛼 𝑖𝑗 𝑇 𝑗ℎ𝑛 𝑗+𝑁𝑖 1−𝑒−𝑔𝑟 𝑥𝑖𝑗 𝛽𝑖𝑗 𝑇 𝑗ℎ𝑟 𝑗]
Retailer Facility (j)
Optimal Inventory
Cycle Time (Tj)
1 Inf
2 6.95
3 11.42
4 11.42
5 5.03
6 4.31
7 2.81
8 7.66
9 4.12
10 10.41

31.  We have compared the results produced by
our algorithm with three other algorithms:
PSO, simulated annealing (SA), genetic
algorithm (GA) to evaluate the
◦ Performance (e.g. solution quality); and
◦ Efficiency (e.g. convergence speed)
of the proposed algorithm.

32. 200
400
600
800
1000
1200
1400
1600
0 5000 10000 15000 20000 25000 30000 35000
IPSO
PSO
SA
GA
Number of function evaluations
Fitnessvalue
1. For M=20 and N=10

33. 250
270
290
310
330
350
370
390
410
430
0 5000 10000 15000 20000 25000 30000 35000
IPSO
PSO
SA
Number of function evaluations
Fitnessvalue
1. For M=20 and N=10
(excluding GA)

34. 2. For M=30 and N=15
400
600
800
1000
1200
1400
1600
1800
2000
2200
0 5000 10000 15000 20000 25000 30000 35000
IPSO
PSO
SA
GA
Number of function evaluations
Fitnessvalue

35. 450
470
490
510
530
550
570
590
0 5000 10000 15000 20000 25000 30000 35000
IPSO
PSO
SA
2. For M=30 and N=15
Fitnessvalue
Number of function evaluations
(excluding GA)

36. 3. For M=50 and N=20
0
500
1000
1500
2000
2500
3000
3500
4000
0 5000 10000 15000 20000 25000 30000 35000
IPSO
PSO
SA
GA
Fitnessvalue
Number of function evaluations

37. 3. For M=50 and N=20
780
800
820
840
860
880
900
920
940
0 5000 10000 15000 20000 25000 30000 35000
IPSO
PSO
SA
Number of function evaluations
Fitnessvalue

38.  Further sensitivity analysis to improve robustness
 Routing decision making can also be incorporated
into the model
 Salvage value of the returned products can be
differentiated based on the age of used product

39. SMART Infrastructure Facility
 Prof Pascal Perez
 Ms Tania Brown
 Dr Nagesh Shukla
School of MMM
 Prof Gursel Alici
 Prof Kiet Tieu
 Dr Senevi Kiridena
Prof Roger Lewis (ADR, EIS)