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- 1. CAPACITATED LOT SIZING AND SCHEDULING PROBLEM Rohit Voothaluru
- 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
- 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. 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
- 8. Heuristic 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
- 13. Proposed approach 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 Rohit Voothaluru, IIT Guwahati
- 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. Representation Suitable for the problem Close interaction between encoding and affinity function Satisfy the problem at hand Rohit Voothaluru, IIT Guwahati
- 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
- 32. Scheduling phase Rohit Voothaluru, IIT Guwahati
- 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 Rohit Voothaluru, IIT Guwahati
- 36. Scheduling The problem has two constraints: (i) Sequence constraints (ii) Resource constraints Rohit Voothaluru, IIT Guwahati
- 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 Rohit Voothaluru, IIT Guwahati
- 46. Shuffled Frog Leaping Passing information in same culture Rohit Voothaluru, IIT Guwahati
- 47. Shuffled Frog Leaping Different Cultures interact among themselves and leap Rohit Voothaluru, IIT Guwahati
- 48. Shuffled Frog Leaping Exchange of information by communicating the best local position and adjusting leap step size Rohit Voothaluru, IIT Guwahati
- 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. 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. 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. 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
- 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. 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. 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
- 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. 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. 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. 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
- 72. THANK YOU

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