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TRENDS IN DYNAMIC LOT SIZING RESEARCH - a tutorial -. Albert P.M. Wagelmans Econometric Institute Erasmus School of Economics July 1, 2010. Turkish football supporters in Holland (2008). Management Science , October 1958, Volume 5, Number 1, pp. 89-96. 592 citations.

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trends in dynamic lot sizing research a tutorial


Albert P.M. Wagelmans

Econometric Institute

Erasmus School of Economics

July 1, 2010


Wagner-Whitin model

Solution approaches


wagner whitin model
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
wagner whitin model11
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
wagner whitin model12
Wagner-Whitin model

MIP formulation:

wagner whitin model13
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
solutions approaches
Solutions approaches
  • Dynamic programming
  • Heuristics
  • Mixed integer linear programming
dynamic programming
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
dynamic programming16
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

dynamic programming17
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:
  • 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)
  • 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
  • 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)
  • 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

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


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


Average percentage deviation from optimality

AR: average rank

mixed integer programming
Mixed integer programming
  • Natural MIP formulation:
  • May have fractional LP relaxation; often large integrality gap
mixed integer programming26
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)

mixed integer programming27
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

mixed integer programming28
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:

  • 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, ...
  • Zangwill, 1966/69: concave costs; optimal solutions correspond to extreme flows in certain networks
  • 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
production capacities
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)
production capacities33
Production capacities
  • Solution approaches for non-stationary capacities:
    • DP algorithms
    • MIP
    • Approximation
production capacities34
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
production capacities35
Production capacities
  • DP algorithms (cont’d):
    • Chen et al., 1994: piecewise linear costs (geometric techniques)
production capacities36
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)

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

production capacities39
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
multi echelon
  • Zangwill, 1969: concave costs, no capacities; O(LT4) DP algorithm
multi echelon41
  • Van Hoesel et al., 2005: concave costs, stationary production capacities; O(LT2L+3) DP algorithm; better running times for more specific cost functions
price dependent demand
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
product returns
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
product returns44
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
concluding remarks
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