2. 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
3. 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
5. 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
Rohit Voothaluru, IIT Guwahati
6. 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
Rohit Voothaluru, IIT Guwahati
7. 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
Rohit Voothaluru, IIT Guwahati
9. 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
Rohit Voothaluru, IIT Guwahati
10. 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
Rohit Voothaluru, IIT Guwahati
11. 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#
Rohit Voothaluru, IIT Guwahati
# Pearl, Judea (April 1984). Heuristics. Addison-Wesley Publication.
12. 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
Rohit Voothaluru, IIT Guwahati
14. 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,
15. 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
Rohit Voothaluru, IIT Guwahati
16. Artificial Immune Systems
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)
Rohit Voothaluru, IIT Guwahati
17. Artificial Immune Systems
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
Rohit Voothaluru, IIT Guwahati
18. Operative Mechanisms
The operative mechanisms of immune system
Clonal Selection
Affinity Maturation
These mechanisms form the basis for the AIS
strategy
Rohit Voothaluru, IIT Guwahati
19. 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
Rohit Voothaluru, IIT Guwahati
20. 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
Rohit Voothaluru, IIT Guwahati
21. Affinity Maturation
The process of mutation and selection of
antibodies that better recognize the antigen
Basic mechanisms
1) Hypermutation
2) Receptor Editing
Rohit Voothaluru, IIT Guwahati
22. Mutation
Two phase mutation procedure has been
adopted in the present algorithm for lot
sizing problem
They are
Inverse
Pair-wise interchange
Rohit Voothaluru, IIT Guwahati
23. 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
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25. Representation
Suitable for the problem
Close interaction between encoding and
affinity function
Satisfy the problem at hand
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26. 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
Rohit Voothaluru, IIT Guwahati
27. 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
Rohit Voothaluru, IIT Guwahati
28. 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
Artificial Immune Systems
29. 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
Rohit Voothaluru, IIT Guwahati
30. 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)
Rohit Voothaluru, IIT Guwahati
31. 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
33. Scheduling
Follows the replenishment phase
Assignment of orders to work centers
Relative priorities of the jobs
Rohit Voothaluru, IIT Guwahati
34. Scheduling
Encountered in any shop floor with ‘m’
machines and ‘n’ jobs
Allocation of tasks to time intervals on
machines
Minimizing the makespan
Rohit Voothaluru, IIT Guwahati
35. Scheduling
Each job consists of sequence of tasks
Hard to find optimal solution
Several heuristics were employed
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36. Scheduling
The problem has two constraints:
(i) Sequence constraints
(ii) Resource constraints
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37. 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
Rohit Voothaluru, IIT Guwahati
38. 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
39. 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
Rohit Voothaluru, IIT Guwahati
40. 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
Rohit Voothaluru, IIT Guwahati
41. 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
Rohit Voothaluru, IIT Guwahati
42. Notifications
Notifications
Actual SFLA
Solutions Frogs
Subset of
Memeplexes
solutions
Rohit Voothaluru, IIT Guwahati
43. 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
Rohit Voothaluru, IIT Guwahati
44. 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
Rohit Voothaluru, IIT Guwahati
45. 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
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48. Shuffled Frog Leaping
Exchange of information by communicating the best local position and
adjusting leap step size
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49. Shuffled Frog Leaping
Quick achievement of final goal due to local and global interaction and
adjustment of leap size accordingly
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50. Shuffled Frog Leaping
A sample of virtual frogs constitutes the
population
Partition into memeplexes
Our SFLA considers discrete variables as opposed
to PSO and Shuffled Computing Evolution
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51. Shuffled Frog Leaping
Defined number of memetic evolution steps
Information is passed by shuffling
Enhances solution quality due to exchange in
information from different sources
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52. Shuffled Frog Leaping
Shuffling ensures that evolution is free from bias
The process is repeated
Local search and shuffling repeat until
convergence criterion is satisfied
Rohit Voothaluru, IIT Guwahati
53. Shuffled Frog Leaping
Number of frogs (solutions)
Number of memeplexes
Number of generations before
Main
shuffling
parameters
Max. Number of shuffling iterations
Maximum step size for leaping
Rohit Voothaluru, IIT Guwahati
54. 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
55. 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
Rohit Voothaluru, IIT Guwahati
56. 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
Rohit Voothaluru, IIT Guwahati
57. 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
58. 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
59. 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 w1 X w Vi I 1
I I
Where C1, C2 are constants and Rand()
generates random number between 0 and 1
Rohit Voothaluru, IIT Guwahati
60. 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
Rohit Voothaluru, IIT Guwahati
63. 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.
64. 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
Rohit Voothaluru, IIT Guwahati
66. 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
Rohit Voothaluru, IIT Guwahati
68. 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
Rohit Voothaluru, IIT Guwahati
69. 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
Rohit Voothaluru, IIT Guwahati
70. 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
Rohit Voothaluru, IIT Guwahati
71. 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
Rohit Voothaluru, IIT Guwahati