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Multi-objective Analysis Of An Integrated Supply Chain Scheduling Problem

PRESENTED BY,

SREEPATHY R NAIK

IEM 2014-16

NIT CALICUT

2Introduction

Multi-objective optimization problems Best compromise solution and decision makers preference Pareto optimal solutions Cost vs. service levels Make to order industry Integration between production and distribution Importance in make to order business models

3Literature survey

Title Remarks Solution methodology

Integrated production and out bound distribution scheduling

Review paper: survey of existing models

Synchronizing scheduling of assembly and multi-destination air-transportation in a customer electronics supply chain

Optimizing total tardiness + total earliness + distribution costs

Orders with equal sizes

Simulated annealing variant

Scheduling a production-distribution system to optimize the tradeoff between delivery tardiness and distribution cost

Shows that an integrated approach yields significantly better results when compared with a sequential approachOrders with equal sizes

Dynamic programming algorithm

4Literature survey

Problem statement

Remarks Solution methodology

The coordination of scheduling and batch deliveries

A variant of scheduling costs is considered involving total weighted completion time, the maximum lateness, total weighted number of late jobs, and the total tardinessVehicle has Infinite capacity

Several algorithms are used

Supply chain scheduling: Conflict and cooperation in assembly systems

Conflict issues are discussed when each party has its own objectivesSuppliers: total completion timeManufacturers : minimum lateness

Various supply chains are studied ,no heuristics

5Nomenclature J set of jobs such that J {1,2,…,n} B set of vehicle trips such that B {1,2,…,n} set of job to trip pairs i, j job k, b trip vehicle capacity time required to perform the trip distribution cost =1 if job i precedes j; 0 otherwise =1 if trip is performed; 0 otherwise =1 if job j assigned to trip k; 0 otherwise

6Nomenclature the time at which job jJ finishes its required processing each trip delivery starting time time at which the job j is delivered total number of jobs ordered by customer associated with

trip b the size of the job(volume) j J time between completion of jobs i and j the weight of the job(priority) j J the due time of job j J TWT Total Weighted Tardiness TC Transportation Cost

7Problem description

Scheduling n orders in an integrated production and distribution system.

There exist an infinite number of vehicles. Whose capacity are finite

The objective is to optimize the trade-off between total weighted tardiness and total distribution costs.

Orders are received by a manufacturer, processed on a single production line, and delivered to customers by capacitated vehicles.

Only direct deliveries without any intermediate stops are allowed

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Customer 1

• J1,J2,J3,J4

• ISRO

Customer 2

• J5,J6,J7,J8

• DRDO

Customer

3

• J9,J10

• HAL

FactoryEg. Valeth ceramics

NB : Each order (job) is associated with a customer, weight (priority), processing time, due time, and size

9Model :objective functions

Problem is formulated as a mixed integer programming problem Tardiness of the job j J , = max

Objective functions: Minimize total weighted tardiness, [1] Minimize total transportation cost, [2]

10Model :constraints

Jobs are assigned to a single production line with unique predecessor and successor

[3]

[4] Job i completes time units before j

[5] Jobs are assigned to one of the available trips associated with

same customer

[6]

11Model :constraints (contd.)

Vehicle capacity cannot be exceeded

[7]

Vehicle cannot start delivery until all jobs to be delivered to corresponding customer have finished processing

[8]

is the sum of delivery start plus delivery time[9]

12Model :constraints (contd.)

Each possible trip should be performed if job is assigned to it

[10] Tardiness is calculated as

[11]

In the model above the objectives are aggregated into a single objective by summing up the weighted objectives

ie. αT+(1-α)C ,where α is a constant

The reduced problem is NP hard

13Solution methodology

1. IBM CPLEX

2. Heuristics

Different Genetic algorithm based approaches are used

Chromosomal coding:

Consider 3 customers with orders

fig.1 example chromosome representation

Job J1 J2 J3 J4 J5 J6 J7 J8 J9 J10

trip 8 6 4 8 5 7 1 3 4 9

14Solution methodology (contd.)

Trip assignments are performed by randomly selecting a trip from the set of pre-determined trips of the associated customer

For each customer, possible trips are pre-determined, one for each job ordered by that customer, and none of the possible trips overlap between customers.

jobs of customer 1 can only be assigned to a trip randomly chosen between 1 and 4, any trips numbered from 5 to 8 can deliver jobs to customer 2, and jobs of customer 3 can only be assigned to either trip 9 or 10

Fig.2 example chromosome modified representation

Customer 1 Customer 2 Customer 3

Job J1 J2 J3 J4 J5 J6 J7 J8 J9 J10

Trip 2 3 4 3 6 7 8 7 10 10

15Solution methodology (contd.)

Crossover :

Two parent chromosomes and a crossover point are selected randomly.

one-point crossover is applied with a probability of pc

Job assignments before the crossover point are copied from the first parent, and the rest are copied from the second parent. If crossover is not applied to the parent , they are copied directly to the offspring

Fig 3.Example of crossover operation

Customer 1 Customer 2 Customer 3

Parent 1

2 3 4 3 6 7 8 7 10 10

Parent 2

1 2 4 2 7 8 6 6 10 9

Customer 1 Customer 2 Customer 3

Child 1 2 3 4 3 6 8 6 6 10 9

Child 2 1 2 4 2 7 7 8 7 10 10

16Solution methodology (contd.)

Mutation :From each chromosome of the offspring a job is randomly selected and assigned to a random trip (batch) of the associated customer according to a pre-defined mutation probability, pm

Penalty function: Each infeasible solution is penalized by multiplying both objective values by an exponential function of the number of infeasible trips

Fig 4. example of mutation

CUSTOMER 1 CUSTOMER 2 CUSTOMER 3

CHILD1 2 3 4 3 6 7 8 7 10 10

CHILD1(M)

2 1 4 3 6 7 8 7 10 10

17Steps of NSGA IIStep 1:Initialization :generate n feasible starting solutions .Set these solutions as where

k=0

Step 2:offspring :using crossover and mutation operators, generate offspring population from parent population both of size N

Step 3:mating pool :combine parent and offspring population to create .sort based on non domination

Step 4:next generation: from 2N solutions in , select n best solutions to form

Step 5:generation update: kk+1. if kstop otherwise, go to step 2

18Fast non dominated sorting

For each individual solution i, identify the no of solutions that dominates i, n and create S, set of solutions that i dominates .and set k=1

Each unplaced solution I with n=0 is placed in the k th front For each solution i in ,find each individual j that is dominated

by i and reduce n by 1 If all solutions are placed in a front, stop. Otherwise, k k+1

and go to step 2

19Crowding distance

For each objective m, sort all solutions in particular front k, in non decreasing order of m th objective function value and calculate crowding distance of each individual i w.r.t. objective m as:

, where E is a very large no. Total crowding distance of each solution i in which M is the no of

objectives

20Sample problem

Jobs 1 2 3 4 5 6 7

Customer 1 1 1 1 2 2 2

Processing time

11 9 8 18 20 8 13

Due date 20 25 15 35 37 22 42

Weight (priority)

3 2 2 1 2 3 1

Volume 1 1 1 1 1 1 1

Table : sample problem data

• =3 ••

21Solution in excel sheet

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25Reference

Chen, Z.L., 2010. Integrated production and outbound distribution scheduling: Review and extensions. Operations Research, 58,130–148.

Li, K.P., Ganesan, V.K., and Sivakumar, A.I., 2005. Synchronizing scheduling of assembly and multi-destination air-transportation in a customer electronics supply chain. International Journal of Production Research, 43 (13), 2671–2685

Deb, K., et al., 2002. A fast and elitist multi-objective genetic algorithm: NSGA-II. IEEE Transactions on Evolutionary Computation, 6, 182–197.

Deb, K., 2001. Multi-objective optimization using evolutionary algorithms. Chichester, UK: Wiley.

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