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 workRohit 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.

Rohit Voothaluru, IIT Guwahati

Classifications & Approaches

Specialized Heuristics

Lot sizing step

Feasibility step Feed-back mechanism

Look ahead mechanism

Improvement step

Mathematical-Programming based Heuristics

MetaheuristicsRohit Voothaluru, IIT Guwahati

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)

Minimize: Total replenishment cost :

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

n

i jj hIQA1

))((


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.Rohit Voothaluru, IIT Guwahati

Heuristics

Fundamental goals of any polynomial time algorithm:

(i) Finding algorithms with good runtime

(ii) Finding algorithms to get optimum quality solution

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


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 setRohit Voothaluru, IIT Guwahati


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

1 – 0 – 1 – 0 – 0 – 1 – 1 – 0 – 0 – 1 – 0

1 – 1 – 0 – 1 – 0 – 0 – 0 – 1 – 1 – 0 – 0

1 – 0 – 0 – 1 – 1 – 0 – 0 – 0 – 1 – 0 – 1

1 – 1 – 1 – 0 – 0 – 0 – 0 – 1 – 0 – 1 – 1

1 – 1 – 1 – 1 – 1 – 0 – 0 – 0 – 0 – 0 – 1

TRC Affinity (1/TRC)

500 0.00200

580 0.00172

430 0.00232

610 0.00164

730 0.00137

Average Value of Affinity = 0.00181Rohit Voothaluru, IIT Guwahati

Cloning New Population

Cloned Generation

1 – 0 – 0 – 1 – 1 – 0 – 0 – 0 – 1 – 0 – 1

1 – 0 – 0 – 1 – 1 – 0 – 0 – 0 – 1 – 0 – 1

1 – 0 – 1 – 0 – 0 – 1 – 1 – 0 – 0 – 1 – 0

1 – 0 – 1 – 0 – 0 – 1 – 1 – 0 – 0 – 1 – 0

1 – 1 – 0 – 1 – 0 – 0 – 0 – 1 – 1 – 0 – 0

TRC Affinity (1/TRC)

430 0.00232

430 0.00232

500 0.00200

500 0.00200

580 0.00172

Average Value of Affinity = 0.00207Rohit Voothaluru, IIT Guwahati

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 interchangeRohit Voothaluru, IIT Guwahati

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 – 0Rohit Voothaluru, IIT Guwahati

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 handRohit Voothaluru, IIT Guwahati

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, is given by

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

iTi

j

i jDQ1

)(

iQ


Representation - Illustration

Let this be a potential solution

First replenishment is at first period, i=1, Ti = 2= D1 + D2 + D3

= D4 + D5 + D6 + D7 ; i=4, Ti = 3= D8 + D9 ; i=8, Ti = 1= D10 + D11 + D12 ; i=10, Ti = 2

This scheme is proposed to handle the problem using Artificial Immune Systems

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

1Q

4Q

8Q

10Q

Evaluation

Total replenishment cost

T = number of replenishments

QCk = carrying units corresponding to kth replenishment

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

T

k

T

k

kQChkATRC1 1

kT

j

jk DjQC1

)1(


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

i

m

k

ikikimk pXqZ1 1

))((

m

k

m

k

ikkjiikikimk XqpXq1 1

)1()(

)1)(( ihkikikikhk YpHpXX

ihkhkhkhkik YpHpXX )(

Minimize:

Subject to :

i)Sequence constraint

ii)Resource constraints:

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

Solutions

Subset of solutions

SFLA

Frogs

Memeplexes


Comparison

Qualities can be transferred only from one chromosome to its clone

Improved idea can be incorporated after full generation is replenished

Improvement by cloning is limited to the number of clones based upon affinity

Information can be Transmitted between any two individuals

Improved idea can be incorporated as and when it is found

Number of individuals that can take over from single entity does not have a limit

AIS Shuffled Frog Leaping Algorithm


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


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

Shuffled Frog Leaping


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



Main parameters

Number of frogs (solutions)

Number of memeplexes

Number of generations before shuffling

Max. Number of shuffling iterations

Maximum step size for leaping


The algorithm

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

2. Choose the number of memeplexes

1. Generate the population

6. Improve the worst frog position

5. Determine the best and worst frog in each memeplex

4. Divide the population into subsets (memeplexes)

The algorithm

9. Sort the population in decreasing order of their fitness and check for termination

If true, End

8. Combine the evolved memeplexes

7. Repeat for a specific number of iterations


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,

k = 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

n

j

elsexixjif1

0.,1).(

Representation

The velocity function shall be similar to that in PSO

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

11

21

1 )(*()*)(*()*

I

i

I

w

I

w

I

w

I

g

I

w

I

b

I

i

I

i

VXX

XXRandCXXRandCVV


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

No. of periods

0 20 40 60 80 100 120

AIS

valu

e an

d SM

val

ue

0.0

5.0e+4

1.0e+5

1.5e+5

2.0e+5

2.5e+5

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


THANK YOU

metaheuristics for lot sizing and scheduling problem

Technology

shuffled frog

artificial

total replenishment

period rohit

rohit voothaluru

replenishment

operative

finding algorithms