Metaheuristics for Lot sizing and scheduling problem

CAPACITATED LOT SIZING AND
SCHEDULING PROBLEM
Rohit Voothaluru

Outline of the Presentation
 Review of the lot sizing problems

 AIS and SFL as alternative approaches

 Implementation

 Results and Scope for future work
Rohit Voothaluru, IIT Guwahati

Review of Lot sizing Problems
Characteristics used in defining lot sizing:

Planning Horizon- time interval on which the

Plan schedule extends into the future.
 No. of levels
 Resource constraints – capacitated or un-
capacitated.
 Deterioration of items.
 Demand.
 Inventory shortage.

Classifications & Approaches
Specialized Heuristics


 Lot sizing step
 Feasibility step
Feed-back mechanism

Look ahead mechanism

Improvement step


Mathematical-Programming based Heuristics


Metaheuristics


Assumptions
The demand is deterministic, varying with time


Shortages aren’t allowed


Replenishment lead time is zero


Size of the replenishment must be established for at least one

period

The item is treated as independent from other items,

replenishment in groups aren’t allowed

Parameters
Qj : Replenishment order quantity in the jth period(units)


A : Fixed cost component (independent of replenishment

quantity) incurred with each replenishment quantity

D (j) : Demand rate of the item in period j (j=1,2...N)


TRC (Q) : Total replenishment cost per unit time



Problem
Ij : Ending inventory in period j (units)


h : Inventory cost per unit ( $/unit)


 ( A (Q j )  hI j )
n
Minimize: Total replenishment cost :

i 1

Subject to:

Ij = Ij-1 + Qj − Dj ; j = 1, 2,…,N
Qj ≥ 0; j = 1, 2,…,N
Ij ≥ 0; j =1, 2,…,N
δ(Qj) = 0, if Qj =0
= 1, if Qj >0

Heuristic


Heuristics

The lot sizing and scheduling deals with two tasks


Finding the best replenishment procedure


The best possible schedule for the jobs on

specified machines

Heuristics

Lot sizing task is NP-Hard


Scheduling problem in this case is also NP-Hard


We need to solve these separately for best solution



Heuristics
NP-Hard implies no polynomial time


algorithm
Heuristics are used to suggest a possible


procedure
It may be correct, but may not be proven to


produce an optimal solution#

# Pearl, Judea (April 1984). Heuristics. Addison-Wesley Publication.

Heuristics
Fundamental goals of any polynomial time

algorithm:
Finding algorithms with good runtime
(i)

Finding algorithms to get optimum quality solution
(ii)

Heuristics abandon one or both of the above


Lack proof; But, backed by good results over the


past few decades

Proposed approach


Proposed Approach
Artificial Immune Systems strategy


Performance on other NP-Hard problems


Application of AIS in previous works


prompted our decision to explore its
ability on CLSP IIT Guwahati
Rohit Voothaluru,

Artificial Immune Systems
An antigen is used to represent the

programming problem to be addressed

A potential solution is called an antibody


Generating an antibody set



Affinity is the attraction between the antigen and the

antibody (receptor cells)

Analogous to the shape-complementary structures in

biological systems

The affinity function is defined as


Affinity = 1/ (objective function)


Affinity criterion is used to determine


Fate of the antibody

Completion of the algorithm


When the antibody set has not yielded affinity

relating to algorithm completion, individual
antibodies are replaced, cloned or hypermutated

Operative Mechanisms

The operative mechanisms of immune system


Clonal Selection


 Affinity Maturation

These mechanisms form the basis for the AIS


strategy

Cloning
Initial Set


Initial population
TRC Affinity (1/TRC)
1–0–1–0–0–1–1–0–0–1–0 500 0.00200
1–1–0–1–0–0–0–1–1–0–0 580 0.00172
1–0–0–1–1–0–0–0–1–0–1 430 0.00232
1–1–1–0–0–0–0–1–0–1–1 610 0.00164
1–1–1–1–1–0–0–0–0–0–1 730 0.00137

Average Value of Affinity = 0.00181

Cloning
New Population


Cloned Generation
TRC Affinity (1/TRC)
1–0–0–1–1–0–0–0–1–0–1 430 0.00232
1–0–0–1–1–0–0–0–1–0–1 430 0.00232
1–0–1–0–0–1–1–0–0–1–0 500 0.00200
1–0–1–0–0–1–1–0–0–1–0 500 0.00200
1–1–0–1–0–0–0–1–1–0–0 580 0.00172

Average Value of Affinity = 0.00207

Affinity Maturation

The process of mutation and selection of

antibodies that better recognize the antigen

Basic mechanisms

1) Hypermutation

 2) Receptor Editing


Mutation
Two phase mutation procedure has been

adopted in the present algorithm for lot
sizing problem

They are


 Inverse
 Pair-wise interchange

Artificial Immune Systems-Mutation
Inverse Mutation:


Sequence between two points ‘i’ and ‘j’ is
inversed in the antibody

Eg.:
Clone: 1 – 0 – 1 – 1 – 1 – 0 – 0 – 1 – 0
New: 1 – 0 – 1 – 1 – 0 – 0 – 1 – 1 – 0

Artificial Immune Systems-Mutation

Pair-wise interchange mutation


‘i’ and ‘j’ positions are selected randomly and
interchanged to obtain a new antibody

Eg.:
Clone: 1 – 0 – 1 – 1 – 1 – 0 – 0 – 1 – 0
New: 1 – 0 – 1 – 1 – 0 – 0 – 1 – 1 – 0

Representation

Suitable for the problem


Close interaction between encoding and


affinity function

Satisfy the problem at hand


Representation

Replenishment is done at the beginning of each period


Best strategy must involve quantities that serve for an

integer number of periods

Binary encoding with N bits


N is the number of periods in planning horizon



Representation
The replenishment quantity in any period i,Q i is given

by i T
Qi   D( j )
i

j 1

Where Ti is the number of bits from ith bit to the first bit

on the right, which has value 1

If ith bit has a value =1 then, we need to replenish at the

beginning of that period

Representation - Illustration
Let this be a potential solution


1 0 0 1 0 0 0 1 0 1 0 1

First replenishment is at first period, i=1, Ti = 2

Q1 = D1 + D2 + D3
Q4 = D4 + D5 + D6 + D7 ; i=4, Ti = 3
Q8 = D8 + D9 ; i=8, Ti = 1
Q10= D10 + D11 + D12 ; i=10, Ti = 2

This scheme is proposed to handle the problem using


Evaluation
Total replenishment cost

T T
TRC   kA  h QCk
k 1 k 1

Tk
QCk   ( j  1) D j
j 1
T = number of replenishments


QCk = carrying units corresponding to kth replenishment


Tk = number of ‘0’ bits between kth and (k+1)th period


Algorithm
1: Generate an antibody set (solution population)


2: Determine the affinity of these antibodies


3: Cloning according to affinities


4: For generated strings:


a) Inverse Mutation

b) Decode and evaluate the total replenishment cost

c) if TRC(new string) < TRC(clone), clone = new string

else go to d)


Algorithm
d) Pairwise interchange mutation

e) Decode and evaluate the total replenishment cost

f) if TRC(new string) < TRC(clone), clone = new string

else, clone=clone; antibody=clone


5. New antibody population


6. Receptor editing


7. If no. of iterations=Max or affinity criterion is

satisfied: Stop,
else, go to Step 2


Scheduling phase


Scheduling

Follows the replenishment phase


Assignment of orders to work centers


Relative priorities of the jobs



Scheduling

Encountered in any shop floor with ‘m’


machines and ‘n’ jobs
Allocation of tasks to time intervals on

machines
Minimizing the makespan



Scheduling

Each job consists of sequence of tasks


Hard to find optimal solution


Several heuristics were employed



Scheduling

The problem has two constraints:


 (i) Sequence constraints
 (ii) Resource constraints


Scheduling

Sequence constraint: Two operations cannot


be processed at the same time

Resource constraint: No more than one job can


be handled on one machine at the same time

Problem
n m
Z   ( qimk ( X ik  pik ))
Minimize:
i 1 k 1

Subject to :
m m

q ( X ik  pik )   qi ( j 1) k X ik
i)Sequence constraint imk
k 1 k 1

X hk  X ik  pik  ( H  pik )(1  Yihk )
ii)Resource constraints:

X ik  X hk  phk  ( H  phk )Yihk
where, pik is the processing time of job i on machine k, Xik be the starting/waiting time
of job i on machine k ,Yihk = 1 of i precedes h on machine k or else 0; qijk is 1 if
operation j of job i requires processing on machine k; H is a very large number

Scheduling
AIS developed can be modified for use in

scheduling case

The objective function differs between the two


We also propose a memetic heuristic for

comprehensive study

Proposed strategies
Development of a Shuffled Frog Leaping

algorithm

Shuffled Frog Leaping has not been explored to a

great extent in case of the lot sizing problems

We intend to provide a new way of solving the

problem along with our existing solution

Proposed strategies
Why shuffled frog leaping only?

 PSOs were successful with scheduling
 Memetic algorithms were also successful to an
extent

SFLA combines the benefits of genetic based

MAs and the social behavior based PSOs

Notifications
Notifications


Actual SFLA
Solutions Frogs

Subset of
Memeplexes
solutions

Comparison
AIS Shuffled Frog Leaping Algorithm

Qualities can be transferred Information can be
 
only from one chromosome to Transmitted between any two
its clone individuals
Improved idea can be Improved idea can be
 
incorporated after full incorporated as and when it is
generation is replenished found
Improvement by cloning is Number of individuals that
 
limited to the number of can take over from single
clones based upon affinity entity does not have a limit


Advantages
Progressive improvement of ideas held by the


frogs (potential solutions)
Ideas are passed between all individuals in the


population
Unlike parent sibling relation in other AI


techniques


Shuffled Frog Leaping
Goal of the frogs is to find the stone with maximum amount of food as quickly
as possible by improving their memes

Passing information in same culture


Different Cultures interact among themselves and leap


Exchange of information by communicating the best local position and
adjusting leap step size

Quick achievement of final goal due to local and global interaction and
adjustment of leap size accordingly


A sample of virtual frogs constitutes the

population

Partition into memeplexes


Our SFLA considers discrete variables as opposed

to PSO and Shuffled Computing Evolution


Defined number of memetic evolution steps


Information is passed by shuffling


Enhances solution quality due to exchange in

information from different sources


Shuffling ensures that evolution is free from bias


The process is repeated


Local search and shuffling repeat until

convergence criterion is satisfied

Number of frogs (solutions)


Number of memeplexes


Number of generations before

Main
shuffling
parameters

Max. Number of shuffling iterations


Maximum step size for leaping



The algorithm
1. Generate the population

2. Choose the number of memeplexes

3. Select the number of steps to be completed in a memeplex before shuffling

4. Divide the population into subsets (memeplexes)

5. Determine the best and worst frog in each memeplex

6. Improve the worst frog position

The algorithm
7. Repeat for a specific number of iterations

8. Combine the evolved memeplexes

9. Sort the population in decreasing order of their fitness and check for termination
If true, End


Transformation
SFL requires transformation from permutation

space to search space
Greatest Value Priority is employed for

transformation
Condition to be satisfied by the transformation

function f
For any memetic vector in search space there must be

one and only one permutation corresponding to it

Transformation
For arbitrary position in space,

X = {x1, x2, …, xn}

where xi ε { -P_min,-P_max}


for i = { 1, 2, …, n}

The only permutation that corresponds to X

is A = { a1, a2, … , an} which represents the
solution

Transformation
For a component xi,

n

 if ( xj  xi ).1, else.0
k=1+

j 1

Then, ak = i


In GVP the maximum quantity in Xi is first

chosen out and its index number becomes
the value of the first element a1 in A

Representation
The velocity function shall be similar to that in

PSO
Vi I 1  Vi I  C1 * Rand () * ( X bI  X w )  C2 * Rand () * ( X g  X w )
I I I

X w1  X w  Vi I 1
I I

Where C1, C2 are constants and Rand()

generates random number between 0 and 1

Results

Fixed setup cost = 200 units


Holding cost = 20 per unit in inventory


Number of periods is taken as a parameter


The algorithm was run on C platform on a


1GHz Pentium Dual Core computer


Results
S. No. No. of periods SM solution AIS solution % Improvement
1 10 1400 1400 0.00
2 12 2650 2650 0.00
3 15 3450 3450 0.00
4 20 5350 5100 0.04
5 25 7050 6950 1.44
6 28 14350 13000 10.38
7 30 13100 12350 6.07
8 35 38250 37950 0.07

Results
S. No. No. of periods SM solution AIS solution % Improvement
9 40 39400 35200 11.93
10 45 89050 87550 1.71
11 50 47450 46400 2.26
12 52 65150 62650 3.99
13 55 48050 47650 0.84
14 60 64500 64300 0.31
15 65 114950 105550 8.91
16 100 203550 199950 1.80

Lot sizing problem

2.5e+5

2.0e+5
AIS value and SM value

1.5e+5

1.0e+5

5.0e+4

0.0

0 20 40 60 80 100 120

No. of periods
SM value vs No. of periods
AIS value Vs No. of periods.

Results
Algorithm was tested on 10 and 12 period

problems

Per unit inventory holding cost = 0.4 units


With varying demands for each period proposed

by Hindi9 as 10, 62, 12, 130, 154, 129, 88, 124, 160,
238, 41, 52

Results

No No. of periods Hindi TS solution Proposed soln. Improvement
KS1 10 679.20 679.20 0.00
KS2 12 550.80 550.80 0.00
KS3 12 430.80 430.80 0.00
KS4 12 692.00 692.00 0.00
KS5 12 855.20 852.80 2.81


Results
Tested the AIS and SFL algorithms for the


second phase
The algorithms were tested on problem


instances from OR-library contributed by Dirk
Mattfield and Rob Vassens
The results are as shown in the following table



Results
Problem n m SFL AIS
ABZ5 10 10 1234 1234
ABZ6 10 10 943 943
ABZ7 20 15 666 666
ABZ8 20 15 669 678
ABZ9 20 15 684 693
ORB1 10 10 1062 1064
ORB2 10 10 891 890

Summary
The algorithms worked well for most of the

instances
AIS algorithm was particularly successful in lot

sizing decisions involving larger number of
periods
For fewer periods the results obtained were on

par with the existing solutions

Summary
AIS algorithm proposed can be employed for

both phases
Results obtained showed that SFL worked

better in case of certain problems for the
second phase
We can thus employ the AIS for evaluating

TRC and SFL for the scheduling phase

Scope for future work
The AIS algorithm suggested can be coupled

with other metaheuristics to develop a hybrid
algorithm

The solutions can be further improved by

employing different representation schemes in
SFL

Scope for future work
Owing to the simply constructed nature of the

algorithms they can be tweaked to
accommodate new constraints

The algorithms can be successfully employed

for solving the huge number of variants of lot
sizing problems

Metaheuristics for Lot sizing and scheduling problem

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