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## TRENDS IN DYNAMIC LOT SIZING RESEARCH - a tutorial -

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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

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

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 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 model

MIP formulation:

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

- Dynamic programming
- Heuristics
- Mixed integer linear 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 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 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:

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)

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

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)

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

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

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

Mixed integer programming

- Natural MIP formulation:
- May have fractional LP relaxation; often large integrality gap

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 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 programming

Shortest path network for T = 4

Mij: cost of producing in period i for period i through j

Shortest path (flow) formulation:

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, ...

Backlogging

- Zangwill, 1966/69: concave costs; optimal solutions correspond to extreme flows in certain networks

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

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 capacities

- Solution approaches for non-stationary capacities:
- DP algorithms
- MIP
- Approximation

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 capacities

- DP algorithms (cont’d):
- Chen et al., 1994: piecewise linear costs (geometric techniques)

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 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-echelon

- 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

- 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

- 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 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

- 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!

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