SMART Seminar Series: "Optimisation of closed loop supply chain decisions using integrated game theoretic particle swarm algorithm"

Closed-loop Supply Chain decisions using Game Theoretic Particle Swarm Optimisation

Presented by:

Kalpit Patne

4th Year UG Student,Department of Industrial and Systems Engineering,Indian Institute of Technology (IIT) Kharagpur, INDIA

Presentation Outline Introduction Problem Definition Solution Methodology

◦ Background of Key Concepts◦ Improved Particle Swarm Optimisation

Numerical Analysis Results and Comparison Future prospects

Introduction

Source: Google images

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

Introduction

New Product

Used ProductProducer Consumer

Fun Fact!


Introduction


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

Introduction

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

Problem Definition

1. Retailer Facility Location

Customer zones and production centre locations are known We want to optimally locate retailer facilities

Problem Definition


2. Customer Zone Allocation

We want to know the customer zone allocation schema

Problem Definition


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.

Problem Definition

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.

Problem Definition

Mathematical Representation of the Problem

Z= Max --------------------------------I

------------------------II

---------------------------------------------III

----IV

I – Profit function for new productsII – Profit in terms of assets retrieved from customersIII – Establishment and Operational cost of the facilityIV – Ordering and Inventory holding costs

Problem Definition

We have considered the proposed model as a combination of two sub-problems:

Location-Allocation Problem (LAP)1. Retailer facility location2. Customer zone allocation

Pricing-Inventory Problem (PIP)3. Optimal pricing policy4. Optimal inventory cycle time

Solution Methodology

Solution Methodology

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

1. Particle Swarm Optimization (PSO)

Memory based (personal and global bestposition)

Communication involved(for searching the global bestposition of the swarm)

Primary Algorithms used


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.

Primary Algorithms used

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 gameResearch space ------- Strategies spaceVelocity ------- Rate of proportion changeFitness function ------- Payoff functiongBest ------- Evolutionary stable strategy

Proposed Solution Approach

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.

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


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.


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

Improved PSO (IPSO) Algorithm

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.


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.


Initialization

Iterations

If t > TWeight 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

We use a 100x100 2D space as our search space.

We used real life data where possible; otherwise, realistic assumptions were made

Numerical Analysis

Parameter Value# Customer Zones, M 20# Retailer Facilities, N 10

Numerical Analysis

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

Results obtained after implementation of the IPSO algorithm

Numerical Analysis: Results

Performance of IPSO algorithm

Numerical Analysis : Results

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.


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


The optimal inventory time Tj, is calculated from the standard Economic Order Quantity (EOQ) model equation.

Tj =


Retailer Facility (j)

Optimal Inventory Cycle Time (Tj)

1 Inf2 6.953 11.424 11.425 5.036 4.317 2.818 7.669 4.12

10 10.41

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.

Results and Comparison


0 5000 10000 15000 20000 25000 30000 35000200

400

600

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1400

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IPSOPSOSAGA

Number of function evaluations

Fitne

ss v

alue

1. For M=20 and N=10


0 5000 10000 15000 20000 25000 30000 35000250

270

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430

IPSOPSOSA


Fitne

ss v

alue


(excluding GA)

Results and Comparison2. For M=30 and N=15

0 20000 40000 60000 80000 100000 120000400

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IPSOPSOSAGA


Fitne

ss v

alue


0 5000 10000 15000 20000 25000 30000 35000450

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IPSOPSOSA


Fitne

ss v

alue


(excluding GA)


0 5000 10000 15000 20000 25000 30000 350000

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IPSOPSOSAGAFit

ness

val

ue



0 5000 10000 15000 20000 25000 30000 35000760

780

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IPSOPSOSA


Fitne

ss v

alue

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

Further Improvements

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)

Acknowledgment

Q & A

Thank You!