CAPACITATED LOT SIZING AND
SCHEDULING PROBLEM
                   Rohit Voothaluru
Outline of the Presentation
 Review     of the lot sizing problems

 AIS   and SFL as alternative approaches

 Implemen...
Review of Lot sizing Problems
    Characteristics used in defining lot sizing:

      Planning Horizon- time interval on ...
Classifications & Approaches
    Specialized Heuristics


     Lot sizing step
     Feasibility step
            Feed-b...
Assumptions
    The demand is deterministic, varying with time




    Shortages aren’t allowed




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



    A : Fixed cost component (independent o...
Problem
    Ij : Ending inventory in period j (units)


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




                 ...
Heuristic




  Rohit Voothaluru, IIT Guwahati
Heuristics

    The lot sizing and scheduling deals with two tasks




    Finding the best replenishment procedure




...
Heuristics

    Lot sizing task is NP-Hard




    Scheduling problem in this case is also NP-Hard




    We need to so...
Heuristics
           NP-Hard implies no polynomial time
    

           algorithm
           Heuristics are used to sug...
Heuristics
    Fundamental goals of any polynomial time

    algorithm:
            Finding algorithms with good runtime
...
Proposed approach




  Rohit Voothaluru, IIT Guwahati
Proposed Approach
    Artificial Immune Systems strategy




    Performance on other NP-Hard problems




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

    A potenti...
Artificial Immune Systems

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

   ...
Artificial Immune Systems

    Affinity criterion is used to determine


        Fate of the antibody
    
        Compl...
Operative Mechanisms

    The operative mechanisms of immune system


      Clonal Selection
    

     Affinity Matura...
Cloning
    Initial Set




          Initial population
                                              TRC        Affinit...
Cloning
    New Population




        Cloned Generation
                                            TRC        Affinity ...
Affinity Maturation

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

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

    ...
Artificial Immune Systems-Mutation
    Inverse Mutation:




    Sequence between two points ‘i’ and ‘j’ is
    inversed ...
Artificial Immune Systems-Mutation

    Pair-wise interchange mutation




    ‘i’ and ‘j’ positions are selected randoml...
Representation

    Suitable for the problem




    Close interaction between encoding and


    affinity function

   ...
Representation

    Replenishment is done at the beginning of each period




    Best strategy must involve quantities t...
Representation
    The replenishment quantity in any period i,Q i is given

    by             i T
               Qi  ...
Representation - Illustration
    Let this be a potential solution


        1    0    0   1    0    0      0   1   0   1...
Evaluation
    Total replenishment cost

                     T               T
         TRC   kA  h QCk
            ...
Algorithm
    1: Generate an antibody set (solution population)


    2: Determine the affinity of these antibodies


  ...
Algorithm
        d) Pairwise interchange mutation
    
        e) Decode and evaluate the total replenishment cost
    ...
Scheduling phase




  Rohit Voothaluru, IIT Guwahati
Scheduling

    Follows the replenishment phase




    Assignment of orders to work centers




    Relative priorities...
Scheduling

    Encountered in any shop floor with ‘m’


    machines and ‘n’ jobs
    Allocation of tasks to time interv...
Scheduling

    Each job consists of sequence of tasks




    Hard to find optimal solution




    Several heuristics ...
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: ...
Problem
                                           n      m
                                  Z   ( qimk ( X ik  pik )...
Scheduling
    AIS developed can be modified for use in

    scheduling case

    The objective function differs between ...
Proposed strategies
    Development of a Shuffled Frog Leaping

    algorithm

    Shuffled Frog Leaping has not been exp...
Proposed strategies
    Why shuffled frog leaping only?

     PSOs  were successful with scheduling
     Memetic algori...
Notifications
    Notifications



         Actual                              SFLA
         Solutions                  ...
Comparison
               AIS                           Shuffled Frog Leaping Algorithm

    Qualities can be transferred ...
Advantages
    Progressive improvement of ideas held by the


    frogs (potential solutions)
    Ideas are passed betwee...
Shuffled Frog Leaping
Goal of the frogs is to find the stone with maximum amount of food as quickly
as possible by improvi...
Shuffled Frog Leaping
Passing information in same culture

         Rohit Voothaluru, IIT Guwahati
Shuffled Frog Leaping
Different Cultures interact among themselves and leap

         Rohit Voothaluru, IIT Guwahati
Shuffled Frog Leaping
Exchange of information by communicating the best local position and
adjusting leap step size
      ...
Shuffled Frog Leaping
Quick achievement of final goal due to local and global interaction and
adjustment of leap size acco...
Shuffled Frog Leaping

    A sample of virtual frogs constitutes the

    population

    Partition into memeplexes




...
Shuffled Frog Leaping

    Defined number of memetic evolution steps




    Information is passed by shuffling




    ...
Shuffled Frog Leaping

    Shuffling ensures that evolution is free from bias




    The process is repeated




    Lo...
Shuffled Frog Leaping
                     Number of frogs (solutions)
                 

                     Number of ...
The algorithm
                           1. Generate the population


                      2. Choose the number of memepl...
The algorithm
                     7. Repeat for a specific number of iterations




                        8. Combine th...
Transformation
    SFL requires transformation from permutation

    space to search space
    Greatest Value Priority is...
Transformation
    For arbitrary position in space,

    X = {x1, x2, …, xn}

    where xi ε { -P_min,-P_max}


    for ...
Transformation
    For a component xi,

             n

            if ( xj  xi ).1, else.0
    k=1+

            j 1...
Representation
    The velocity function shall be similar to that in

    PSO
     Vi I 1  Vi I  C1 * Rand () * ( X bI...
Results

    Fixed setup cost = 200 units


    Holding cost = 20 per unit in inventory


    Number of periods is taken...
Results
S. No.   No. of periods   SM solution   AIS solution   % Improvement
  1           10            1400           14...
Results
S. No.   No. of periods   SM solution   AIS solution   % Improvement
  9           40            39400         352...
Lot sizing problem


                         2.5e+5



                         2.0e+5
AIS value and SM value




       ...
Results
    Algorithm was tested on 10 and 12 period

    problems

    Per unit inventory holding cost = 0.4 units




...
Results

No    No. of periods         Hindi TS solution Proposed soln.   Improvement
KS1   10                     679.20  ...
Results
    Tested the AIS and SFL algorithms for the


    second phase
    The algorithms were tested on problem


   ...
Results
 Problem             n                  m    SFL    AIS
  ABZ5              10                  10   1234   1234
 ...
Summary
    The algorithms worked well for most of the

    instances
    AIS algorithm was particularly successful in lo...
Summary
    AIS algorithm proposed can be employed for

    both phases
    Results obtained showed that SFL worked

   ...
Scope for future work
    The AIS algorithm suggested can be coupled

    with other metaheuristics to develop a hybrid
 ...
Scope for future work
    Owing to the simply constructed nature of the

    algorithms they can be tweaked to
    accomm...
THANK YOU
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Metaheuristics for Lot sizing and scheduling problem

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Metaheuristics for Lot sizing and scheduling problem

  1. 1. CAPACITATED LOT SIZING AND SCHEDULING PROBLEM Rohit Voothaluru
  2. 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. 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
  4. 4. Classifications & Approaches Specialized Heuristics   Lot sizing step  Feasibility step Feed-back mechanism  Look ahead mechanism  Improvement step  Mathematical-Programming based Heuristics  Metaheuristics  Rohit Voothaluru, IIT Guwahati
  5. 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. 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. 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
  8. 8. Heuristic Rohit Voothaluru, IIT Guwahati
  9. 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. 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. 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. 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
  13. 13. Proposed approach Rohit Voothaluru, IIT Guwahati
  14. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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 Rohit Voothaluru, IIT Guwahati
  24. 24. 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 Rohit Voothaluru, IIT Guwahati
  25. 25. Representation Suitable for the problem  Close interaction between encoding and  affinity function Satisfy the problem at hand  Rohit Voothaluru, IIT Guwahati
  26. 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. 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. 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. 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. 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. 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 
  32. 32. Scheduling phase Rohit Voothaluru, IIT Guwahati
  33. 33. Scheduling Follows the replenishment phase  Assignment of orders to work centers  Relative priorities of the jobs  Rohit Voothaluru, IIT Guwahati
  34. 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. 35. Scheduling Each job consists of sequence of tasks  Hard to find optimal solution  Several heuristics were employed  Rohit Voothaluru, IIT Guwahati
  36. 36. Scheduling The problem has two constraints:   (i) Sequence constraints  (ii) Resource constraints Rohit Voothaluru, IIT Guwahati
  37. 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. 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. 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. 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. 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. 42. Notifications Notifications  Actual SFLA Solutions Frogs Subset of Memeplexes solutions Rohit Voothaluru, IIT Guwahati
  43. 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. 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. 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 Rohit Voothaluru, IIT Guwahati
  46. 46. Shuffled Frog Leaping Passing information in same culture Rohit Voothaluru, IIT Guwahati
  47. 47. Shuffled Frog Leaping Different Cultures interact among themselves and leap Rohit Voothaluru, IIT Guwahati
  48. 48. Shuffled Frog Leaping Exchange of information by communicating the best local position and adjusting leap step size Rohit Voothaluru, IIT Guwahati
  49. 49. Shuffled Frog Leaping Quick achievement of final goal due to local and global interaction and adjustment of leap size accordingly Rohit Voothaluru, IIT Guwahati
  50. 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 Rohit Voothaluru, IIT Guwahati
  51. 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 Rohit Voothaluru, IIT Guwahati
  52. 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. 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. 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. 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. 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. 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. 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. 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 w1  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. 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
  61. 61. 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
  62. 62. 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
  63. 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. 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
  65. 65. 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 Rohit Voothaluru, IIT Guwahati
  66. 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
  67. 67. 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 Rohit Voothaluru, IIT Guwahati
  68. 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. 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. 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. 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
  72. 72. THANK YOU

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