TRENDS IN DYNAMIC LOT SIZING RESEARCH - a tutorial -

1. TRENDS IN DYNAMIC LOT SIZING RESEARCH- a tutorial - Albert P.M. Wagelmans Econometric Institute Erasmus School of Economics July 1, 2010

2. Turkish football supporters in Holland (2008)

3. Management Science, October 1958, Volume 5, Number 1, pp. 89-96

4. 592 citations

7. Search for “dynamic/economic lot sizing/size” yields more than 6200 results

9. Overview Wagner-Whitin model Solution approaches Extensions

10. Wagner-Whitin model • Single product • T consecutive time periods • For period t{1,2,…,T}: • dt : known demand in t • ft : fixed setup cost in t • pt : unit production cost in t • ht : unit holding cost in t • Objective: find production plan that satisfies all demand at minimal total cost

11. Wagner-Whitin model • MIP formulation • Variables for period t: • xt : number of items produced • yt : binary variable to indicate a setup (=1) or not (=0) • It : ending inventory level

12. Wagner-Whitin model MIP formulation:

13. Wagner-Whitin model • Notes: • The original W-W model had stationary costs (and therefore no unit production costs) • The inventory variables can be substituted out of the model, resulting in a model without holding costs, but with modified unit production costs rt

14. Solutions approaches • Dynamic programming • Heuristics • Mixed integer linear programming

15. Dynamic programming • Crucial observation: there exists an optimal production plan such that for every period t: It-1 = 0 or xt = 0 (zero-inventory property) • This means that an optimal production plan is completely determined by its set of production periods • F(t): optimal value of the lot sizing problem for the first t periods

16. Dynamic programming Complexity: O(T2) Refinements to bring down the (practical) running time Planning horizon theorem (W&W): when the cost coefficients are stationary, then the optimal last production period for the first t+1 periods is not earlier than the optimal last production period for the first t periods

17. Dynamic programming • Improved DP approaches (Federgruen & Tzur, Wagelmans et al., Aggarwal & Park, early 90s): O(T log T); linear for special cases (incl. stationary costs) • Geometric technique:

18. Heuristics • Motivation: • W&W algorithm too complicated • W&W algorithm leads to nervousness when implemented in a rolling horizon context • Heuristics may outperform W&W algorithm in a rolling horizon context • Assume stationary costs (ignore unit production costs)

19. Heuristics • Example Silver-Meal: • Let C(t) denote the cost of producing in period 1 for the first t periods, i.e., • Let t* be the first period for which • Produce in period 1 for the first t* periods • Continue in a similar way with period t*+1; and so on

20. Heuristics • Numerous other heuristics: Least unit cost, Part period balancing, Lot for lot,… • Single pass or with look-back/look-ahead feature to marginally adjust tentative lot sizes • Many computational studies • Approximation results (Axsäter, Bitran et al., early 80s)

21. Heuristics • Examples of approximation results: • Worst case ratio of Part period balancing is 2 • Worst case performance of the Silver-Meal heuristic can be arbitrarily bad • Axsäter, 1985: for a large class of lot sizing heuristics, including all well-known single pass heuristics, have worst case ratio at least 2

22. Heuristics Van den Heuvel & Wagelmans, 2010: any online heuristic has worst case ratio at least 2 Result also holds if we allow look-back/look-ahead for at most a fixed number of periods

23. Heuristics W&W algorithm may perform badly in a rolling horizon context because end-of-horizon effects (IT= 0) Stadtler, 2000: forecast demand beyond T Fisher et al., 2001: ending inventory evaluation Van den Heuvel & Wagelmans, 2005: straightforward application of W&W algorithm to extended horizon with demand forecasts

24. Heuristics Average percentage deviation from optimality AR: average rank

25. Mixed integer programming • Natural MIP formulation: • May have fractional LP relaxation; often large integrality gap

26. Mixed integer programming Simple plant location formulation: disaggregate xt into variables xti denoting the amount produced in period t to satisfy demand in periods i ≥ t The LP relaxation of this formulation has an integer optimal solution (Krarup & Bilde, 1977)

27. Mixed integer programming LP relaxation of the natural MIP formulation has an integer optmal solution if we add the (m,S) inequalities (Barany et al., 1984): In other words, we have obtained a complete linear (polyhedral) description of the convex hull of feasible solutions

28. Mixed integer programming Shortest path network for T = 4 Mij: cost of producing in period i for period i through j Shortest path (flow) formulation:

29. Extensions • Backlogging • Multi-echelon • Production capacities • Price dependent demand • Product returns • Also: lost sales, minimum order quantities, set-up time, bounds on inventory, different cost functions, perishable goods, product substitution, time windows, multiple items, etcetera, etcetera, ...

30. Backlogging • Zangwill, 1966/69: concave costs; optimal solutions correspond to extreme flows in certain networks

31. Backlogging • Dynamic programming based on the property that every production period t produces exactly Dij for some i and j with i ≤ t ≤ j • O(T2) running time for general concave costs; O(T log T) for setup + linear production costs and linear holding and backlogging costs

32. Production capacities • Florian & Klein, 1971: concave costs; between any two consecutive periods with zero inventory, there is at most one production period which produces below its capacity • Florian et al., 1980: lot-sizing with production capacities is NP-hard, in general • Stationary capacities: O(T4) DP algorithm • Stationary capacities and linear holding costs: O(T3) DP algorithm (Van Hoesel & W, 1996)

33. Production capacities • Solution approaches for non-stationary capacities: • DP algorithms • MIP • Approximation

34. Production capacities • DP algorithms: • Florian et al., 1980: straightforward DP for general cost functions with variables Ft(It) denoting the minimal cost in the first t periods to attain ending inventory It; O(C1TD1T)running time, where C1T denotes total capacity over the horizon • Shaw &W, 1998: O(qD1T) algorithm for piecewise linear production costs and general holding costs, where q is the total number of pieces in the production cost functions

35. Production capacities • DP algorithms (cont’d): • Chen et al., 1994: piecewise linear costs (geometric techniques)

36. Production capacities • MIP approaches; tight LP bounds through: • Extending the formulation (simple plant location formulation) • Reformulation (shortest path, Eppen & Martin, 1987) • Polyhedral description (Wolsey and others)

37. 2006

38. Management Science, December 2002, Volume 48 ,  Issue 12,  pp. 1587 - 1602 

39. Production capacities • Approximation: • Van Hoesel & W, 2001: FPTAS for capacitated problem with backlogging and general monotone cost functions • Based on DP approach with “dual” variables Gt(b): maximum value of It , which can be achieved by production in the first t periods if the total cost incurred in these periods is at most b • Chubanov et al, 2006: FPTAS based on standard DP approach

40. Multi-echelon • Zangwill, 1969: concave costs, no capacities; O(LT4) DP algorithm

41. Multi-echelon • Van Hoesel et al., 2005: concave costs, stationary production capacities; O(LT2L+3) DP algorithm; better running times for more specific cost functions

42. Price dependent demand • Kunreuther & Schrage, 1973: where αt,βt≥ 0 en δ(p) non-increasing in p • Find p* between pl and pu that maximizes profit • K&S give a local improvement procedure that is not guaranteed to find p* (but often does) • Van den Heuvel & W, 2006: polynomial method to find p* • Geunes et al, 2009: generalization to stationary capacities

43. Product returns • Teunter et al., 2006: dynamic lot sizing with product returns and remanufacturing; stationary costs • Minimize the total cost composed of holding cost for returns and (re)manufactured products and set-up costs • Distinguish between joint setup cost for manufacturing and remanufacturing and separate set-up costs (single vs. dedicated production lines) • O(T4) DP algorithm for joint setup cost case • Conjecture NP hardness of separate setup cost case

44. Product returns • Retel Helmrich et al, 2010: same model, but with non-stationary costs • Both cases are NP-hard; seperate setup case is even NP-hard with stationary costs • Comparison of different MIP formulations

45. Product returns

46. Product returns

47. Product returns

48. Concluding remarks • After more than half a century, dynamic lot sizing is still a thriving research area • Academic research has lead to practical methods for real-life problems • THANK YOU!