Synchronized Base Stock Policies

Transcription

1 Synchronized Base Stock Policies Ton de Kok Department of Technology Management Technische Universiteit Eindhoven

2 Contents Value Network Model Network transformation principles Divergent structures and allocation rules Optimality conditions on base stock levels Computational aspects Comparison against LP rolling schedule Managerial insights Conclusions and further research 2

3 Value Network Model 3

4 4

5 Multi-item multi-echelon inventory systems Arbitrary BOM no alternative recipees, no multi-sourcing Stochastic end-item demand I.i.d. periodic Linear holding costs Customer service objectives End-item linear shortage costs End-item service level constraints Item order lead times Constant (planned) Unsatisfied end-item demand is backordered 5

6 Network transformation principles 6

7 Basic principle: Divergent node structures At time 0 item(s) with longest cumulative lead time has (have) been ordered At time t items m C 0 have been ordered, of which some at time t At time t+l another set of items C is ordered End-items in P(C) use items in C ( ) PC ( ) PC ( ) 0 c L Immediate excess coverage P( C 0 ) C 0 C 0 ( ) P( C ) P C ( ) P C 0 0 C 0 C C 0 C P(C P(C ) = ) P(C P(C 0 ) 0 ) 7

8 Product structure requirements for SBSpolicies Acyclic BOM If two components i and i 2 are common to items j and j 2 then a a ij 2 i j aij = = c a i j ii 2 Condition allows for elimination of i 2 with respect to (j,j 2 ) a D + a D ij j ij j 2 2 a D + a D i j j i j j = c ii 2 mutually non-zero elements of rows i and i 2 are dependent 8

9 Elimination of different consumption ratios Create versions of items, whereby sum of consumption rates of versions satisfy the original product structure constraint P={3} 3 P={3} P={3,4} C={,2,5} C={3} C={2} P={4} 4 P={4} C={,2} C={4} 9

10 Elimination of cycles in BOM If item i is used at n different levels in the BOM structure, then create n different 3 versions of item i. P={} P={4} P={4} P={4} P={4} 5 P={4} C={9}={7} C={8}={6} C={2,3} 3 3 C={2,3,7} 2 C={2,3,6} P={5} C={4} P={5} C={,2} C={5} 3 3 0

11 Natural hierarchy BOM and lead time structures determine a natural hierarchy defined by divergent structurs Divergent structures relate sets of end-items to sets of items Divergent structures determine forecasting requirements An order of a set of items associated with a node can be released based on the forecast of demand for the set of end-items associated with this node over its cumulative lead time (plus one period) Safety stock requirements over the cumulative lead time

12 Dead stocks average stock in divergent structures [ ] ˆ PC ( ) dead i C i E X = E X + E X { Ci C} Due to synchronization mechanism not all combinations of average stocks of items are possible If average inventories are taken from real life data and SBS policies are used, dead stock may result net stock of an item cannot get below some positive number Computation of performance for given average stock levels requires determination of dead stocks there is no unique solution to this minimize value of dead stock 2

13 Divergent structures and allocation rules 3

14 ME divergent networks W i s u p p li 0 U i i k E i c u st o m er e r s s 4

15 Definitions h i p k U i W i E i Y i X i I ki added value created at (holding item i cost) at item i penalty cost per unit short of end-item k set of predecessors of i, i.e. all child items set of all items in the echelon of i set of all end-items in the echelon of i echelon inventory position of i net stock of item i net stock of end-item k in a system with root node i 5

16 Base stock policies and linear allocation rules Shortfall of i ( ] ( ) Y () t = S q Z ( t L) + D t L, t j j j i i i i j i ( ) S = S + q i j j j j V End-item demand during lead time of i + Parameter that determines stock kept at i on behalf of j j-specific eliminates the effect of zero-cost successors on non-zero-cost successors: no stock of item i ( ) [ ] = ( 0, ] E X q E Z D L + i j j i i i j Vi 6

17 Allocation fractions Eppen and Schrage (98): equal stockout probability q j σ j = σ m V i m Van der Heijden et al (997): minimal imbalance σ µ 2 2 j j q j = σ m 2 µ m j Vi j Vi 7

18 Optimization problem Given allocation fractions, determine base stock levels for all items Expressions for costs and performance are similar to expressions for serial systems Imbalance should be sufficiently low to ensure validity of analytical results in case imbalance too high, base stock levels should be such that average stocks are increased at upstream stages with high imbalance requires experimental research 8

19 Optimality conditions on base stock levels 9

20 Optimal order-up-to-policies for divergent MEIS: Generalized Newsvendor equations theorem for periodic review systems without setup costs Optimal order-up-to-policies and allocation functions under balance assumption satisfy generalized Newsvendor equations: Probability of non-stockout at downstream stockpoint k in the divergent subsystem with stockpoint i as most upstream stage equals or equivalently { 0} P I ki = p k + j U k k j j U i h p + h + h j p + h = p + h + h P I j Ui j Uk k j k k j ki k { 0} 20

21 Sample path condition on control policies Assume that each node i is controlled according to a policy, which is defined by a parameter ξ i and possibly other parameters and functions Sample path condition If ξ j is increased by E j ε for all j W j, and there is upstream availability then a one-time additional flow of ε units is created towards each k E j 2

22 Main theorem If sample path condition holds then i I p + h = p + h + h P I k j k k j ki k E j U \ W k E j U i k i i k { 0} Proof based on echelon costs and similar to proof of single-item Newsvendor equation Relationship can be used as heuristic 22

23 Computational aspects 23

24 Finite horizon ruin probabilities Non-stockout probabilities P{I ki 0} can be written as finite horizon ruin probabilities j P{ Iki 0 } = P Xk ξ j, j =,..., i k = Expressions can be accurately approximated recursively j Gi( ξ,..., ξi) = P Xk ξ j, j =,..., i k = 24

25 Recursive computations Define random variables Y i { } = { } PY x P X x i Gi( ξ,..., ξi, x) =, Gi ( ξ,..., ξi ) 0, =2,...,. { } P Y x x i N Theorem { } { i + i, i ξi } PY { ξ } P X Y x Y PY x =, i= 2,..., N i i i 25

26 Two-moment recursion Fit mixture of Erlang distributions on E[Y i ] and σ 2 (Y i ) [ i] [ i] [ i i ξi ] ( ) ( ) ( ) EY = E X + EY Y, i= 2,..., N- σ Y = σ X + σ Y Y ξ, i = 2,..., N - i i i i i Compute P{I ki 0} = G i (ξ,..., ξ ) ι recursively G( ξ) = P{ Y ξ} G ( ξ,..., ξ ) = P{ Y ξ } G ( ξ,..., ξ ), i = 2,..., N i i i i i i 26

27 Computational efficiency Solving cost-balance equations or generalized Newsvendor equations is equivalent to solving a one-dimensional bisection scheme to find a target value in (0,) Finding base stock levels for N-item divergent structure requires solution of N bisection schemes As efficient as solving N single-echelon systems Finding base stock levels for N-item general systems requires solution of at most 3N bisection schemes 27

28 Comparison against LP rolling schedule 28

29 LP-based rolling schedule approach Define objective function linear holding and penalty costs incorporate safety stocks by goal programming formulation Feasibility constraints inventory balance equations at item level material availability constraints w.r.t. dependent requirements No lot sizing constraints Solve LP problem Implement immediate decisions 29

30 Experimental comparison of SCOP paradigms: LP vs. SBS for uncapacitated systems Example product structure 4 final products E[D] = 00 P = 95% added value $0 assembly lead time 4 specific items value $0 2 semi-specific items value $30 common item value $50 30

31 An -item system L s 5 L sc 9 L f (c,α ) L s 6 L f 2 (c 2,α 2 ) L c L s 7 L sc 0 L f 3 (c 3,α 3 ) L s 8 L f 4 (c 4,α 4 ) 3

32 Results simulation study 2 Supply Chain Inventory capital Customer service c i (L f, L s,l sc,l c ) SBS ana SBS sim LP % P,SBS P,LP 0.25 (,,2,4) % 95% 95% 0.25 (,4,2,) % 95% 95% 0.25 (,,4,2) % 95% 95% 0.5 (,,2,4) % 95% 95% 0.5 (,4,2,) % 95% 95% 0.5 (,,4,2) % 95% 95% (,,2,4) % 95% 95% (,4,2,) % 95% 95% (,,4,2) % 95% 95% 2 (,,2,4) % 94% 95% 2 (,4,2,) % 95% 95% 2 (,,4,2) % 94% 95% 32

33 Impact of final product cost on safety stocks, LP (c 2, c 2 2, c 3 2, c 4 2 ) (h, h 2, h 3, h 4 ) (L f,l s,l sc,l c ) LP-based safety stocks P P 2 ss ss 2 ss 3 ss 4 (0.25, 0.25, 0.25, 0.25) (20, 20, 20, 0) (,,4,2) (0.25, 0.25, 0.25, 0.25) (20, 20, 20, 0) (,,2,4) (0.50, 0.50, 0.50, 0.50) (20, 20, 20, 0) (,,2,4) (0.50, 0.50, 0.50, 0.50) (20, 20, 20, 0) (,,4,2) (,,, ) (20, 20, 20, 0) (,,2,4) (,,, ) (20, 20, 20, 0) (,,4,2) (2, 2, 2, 2) (20, 20, 20, 0) (,,2,4) (2, 2, 2, 2) (20, 20, 0, 0) (,,4,2) (0.25, 0.25, 0.50, 0.50) (0, 0, 0, 0) (,,4,2)

34 Impact of final product cost on safety stocks, SBS (c 2, c 2 2, c 3 2, c 4 2 ) (h, h 2, h 3, h 4 ) (L f,l s,l sc,l c ) SBS-based safety stocks P P 2 ss Ss 2 ss 3 ss 4 (0.25, 0.25, 0.25, 0.25) (20, 20, 20, 0) (,,4,2) (0.25, 0.25, 0.25, 0.25) (20, 20, 20, 0) (,,2,4) (0.50, 0.50, 0.50, 0.50) (20, 20, 20, 0) (,,2,4) (0.50, 0.50, 0.50, 0.50) (20, 20, 20, 0) (,,4,2) (,,, ) (20, 20, 20, 0) (,,2,4) (,,, ) (20, 20, 20, 0) (,,4,2) (2, 2, 2, 2) (20, 20, 20, 0) (,,2,4) (2, 2, 2, 2) (20, 20, 0, 0) (,,4,2) (0.25, 0.25, 0.50, 0.50) (0, 0, 0, 0) (,,4,2)

35 Managerial insights Most of inventory capital tied up downstream informally speaking optimal upstream stock never exceeds 0% of overall supply chain inventory capital Flexibility only pays off downstream lead time reduction replenishment frequency Focus on efficiency upstream Postponement should come almost for free Shortages and overages should be shared among parent items Moving CODP upstream yields substantial savings Control parameters, i.e. safety stocks, depend on the operational planning logic applied 35

36 Inventory capital allocation CW lead time inventory investm ent 00% 90% 80% 70% 60% 50% 40% 30% 20% 0% 0% 0% 20% 30% 40% 50% 60% 70% 80% 90% 00% value at CW central local 36

37 Inventory capital allocation CW lead time 5 inventory investment 00% 90% 80% 70% 60% 50% 40% 30% 20% 0% 0% 0% 20% 30% 40% 50% 60% 70% 80% 90% 00% value at CW central local 37

38 Inventory capital allocation CW lead time 0 inventory investment 00% 90% 80% 70% 60% 50% 40% 30% 20% 0% 0% 0% 20% 30% 40% 50% 60% 70% 80% 90% 00% value at CW central local 38

39 Lot sizing and allocation rules R i review period of item i Ri rj =, rj, j Vi R V it θ jt j set of successors of item i that order at time t the last ordering moment of successor j at or before time t Allocation policy should be such that net stock of i non-negative Linear allocation policies defined before in general do not satisfy this constraint 39

40 Lot sizing and allocation rules ( ) X () t = q W() t i j j i j V i + Then we can recursively define the random variables W(t), ( ] W( L) = Z (0) + D 0, L i i i i Wt () = Zi( t Li) + Di( t Lt i, ] Dj( θ jt, t qw j ( θ jt) q j j Vit j Vit j V it The allocation policy follows from ( ) Y () t = S q W() t, if j orders at time t j j j j + 40

41 Conclusions SBS policies allow for control of general multiitem multi-echelon networks Close-to-optimal SBS policies can be efficiently computed SBS policies perform well compared with standard rolling scheduling SBS policies can be implemented for operational control requires forecast-dependent base stock levels 4

42 Challenges for further research Extension of SBS with (period order) lot sizing Incorporation of multi-level forecasting future detailed forecasts are conditional on today's aggregate forecast multi-level forecast structure should be aligned with value network structure Finite capacity Should finite capacity be modelled explicitly through constraints or explicitly through model of transformation process or implicitly through planned lead time, i.e. loosely coupling SCOPlevel and shopfloor control level Empirical research to test various modelling options 42