Introduction Optimization Model Solution approaches Numerical Results Conclusion

Dynamic uncapacitated lot sizing with random demand under a fillrate constraint Horst Tempelmeier and Sascha Herpers Seminar f¨ ur SCM und Produktion Universit¨ at zu K¨ oln EURO Conference 2009

Bonn, July 2009

SCMP c Prof. Dr. Horst Tempelmeier — www.scmp.uni-koeln.de

1/30

Introduction Optimization Model Solution approaches Numerical Results Conclusion

Agenda 1

Introduction The Problem Solution Approaches

2

Optimization Model Formulation

3

Solution approaches Exact solution Heuristic solution

4

Numerical Results Experiment 1 Experiment 2

5

Conclusion

c Prof. Dr. Horst Tempelmeier — www.scmp.uni-koeln.de

2/30

Introduction Optimization Model Solution approaches Numerical Results Conclusion

1

Introduction The Problem Solution Approaches

2

Optimization Model Formulation

3

Solution approaches Exact solution Heuristic solution

4

Numerical Results Experiment 1 Experiment 2

5

Conclusion

c Prof. Dr. Horst Tempelmeier — www.scmp.uni-koeln.de

The Problem Solution Approaches

3/30

Introduction Optimization Model Solution approaches Numerical Results Conclusion

The Problem Solution Approaches

Planning situation Dynamic and Random Demand

Demand (forecasted averages and variations) Period t µt σt

25/2009 ... ...

26/2009 ... ...

... ... ...

35/2009 ... ...

Holding costs Setup costs Service level

c Prof. Dr. Horst Tempelmeier — www.scmp.uni-koeln.de

4/30

Introduction Optimization Model Solution approaches Numerical Results Conclusion

The Problem Solution Approaches

Alternatives

Common sense approach (MRP, APS) Compute safety stocks and add to forecasted demand (st , qt )-policy, (rt , St )-policy Use a stationary inventory policy with dynamic adjustment of parameters ”Static-dynamic uncertainty” strategy Fix replenishment periods in advance, adjust production quantity ”Static uncertainty” strategy Fix replenishment periods and quantities in advance

c Prof. Dr. Horst Tempelmeier — www.scmp.uni-koeln.de

5/30

Introduction Optimization Model Solution approaches Numerical Results Conclusion

1

Introduction The Problem Solution Approaches

2

Optimization Model Formulation

3

Solution approaches Exact solution Heuristic solution

4

Numerical Results Experiment 1 Experiment 2

5

Conclusion

c Prof. Dr. Horst Tempelmeier — www.scmp.uni-koeln.de

Formulation

6/30

Introduction Optimization Model Solution approaches Numerical Results Conclusion

Formulation

Model Formulation I Model SSIULSPqβc : Minimize Z =

T X t=1

s. t.


 s · γt + h · E [It ]+

(1)

It−1 + qt − Dt = It

t = 1, 2, . . . , T

(2)

qt − M · γt ≤ 0

t = 1, 2, . . . , T

(3)

Itf ,prod = − [It−1 + qt ]−

t = 1, 2, . . . , T

(4)

Itf ,end = − [It ]−

t = 1, 2, . . . , T

(5)

Ft = Itf ,end − Itf ,prod

t = 1, 2, . . . , T

(6)

c Prof. Dr. Horst Tempelmeier — www.scmp.uni-koeln.de

7/30

Introduction Optimization Model Solution approaches Numerical Results Conclusion

Formulation

Model Formulation II lt = (lt−1 + 1) · (1 − γt )

t = 1, 2, . . . , T

l0 = −1

(8) t = 1, 2, . . . , T − 1

ωt = γt+1 ωT = 1 ( E

1−

E

(

(7)

(9) (10)

t X

j=t−lt t X

j=t−lt

Fj

Dj

)

) ≥ βc⋆

c Prof. Dr. Horst Tempelmeier — www.scmp.uni-koeln.de

t ∈ {t | ωt = 1}

(11)

8/30

Introduction Optimization Model Solution approaches Numerical Results Conclusion

Formulation

Symbols used I βc⋆ Dt Ft γt h It Itf ,end Itf ,prod lt M

target fillrate demand in period t (random variable) backorder in period t (random variable) binary setup indicator in period t inventory holding cost net inventory at the end of period t (random variable) backlog at the end of period t (random variable) backlog immediately after production in period t (random variable) number of periods since the last setup prior to period t large number

c Prof. Dr. Horst Tempelmeier — www.scmp.uni-koeln.de

9/30

Introduction Optimization Model Solution approaches Numerical Results Conclusion

Formulation

Symbols used II

ωt

qt s T [x]+ [x]−

indicator variable: ωt = 1, if production takes place in period t + 1; ωt = 0, otherwise production quantity in period t setup cost length of planning horizon = max{0, x} = min{0, x}

c Prof. Dr. Horst Tempelmeier — www.scmp.uni-koeln.de

10/30

Introduction Optimization Model Solution approaches Numerical Results Conclusion

Formulation

Expected Inventory

E {Itp }

=

Z

Q (t) 0 (t)

=Q

(Q (t) − y ) · fY (t) (y ) · dy

− E {Y (t) } + GY1 (t) (Q (t) )

t = 1, 2, . . . (14)

Q (t) – cumulated production quantity from period 0 to t Y (t) – cumulated demand from period 0 to t

c Prof. Dr. Horst Tempelmeier — www.scmp.uni-koeln.de

11/30

Introduction Optimization Model Solution approaches Numerical Results Conclusion

1

Introduction The Problem Solution Approaches

2

Optimization Model Formulation

3

Solution approaches Exact solution Heuristic solution

4

Numerical Results Experiment 1 Experiment 2

5

Conclusion

c Prof. Dr. Horst Tempelmeier — www.scmp.uni-koeln.de

Exact solution Heuristic solution

12/30

Introduction Optimization Model Solution approaches Numerical Results Conclusion

Exact solution Heuristic solution

Shortest-Path Network E{C14 } E{C24 (P2 )}

1

E{C12 }

2

E{C23 (P2 )}

3

E{C34 (P3 )}

4

E{C13 }

E {Cτ t } = s + h ·

t−1 X ℓ=τ

" #+  ℓ   X E Iτ −1 (Pτ ) + qτ⋆t − Di  

c Prof. Dr. Horst Tempelmeier — www.scmp.uni-koeln.de

(15)

i =τ

13/30

Introduction Optimization Model Solution approaches Numerical Results Conclusion

Exact solution Heuristic solution

Shortest-Path Network

1 1 1 1 1 1 1 .. .

Setup 2 0 1 1 1 1 0 .. .

in period 3 4 0 0 0 0 1 0 1 1 1 1 1 0 .. .. . .

5 0 0 0 0 1 0 .. .

On hand inventory E {I5p } 79.61 81.14 87.01 103.89 118.95 86.95 .. .

Table: Expected on-hand inventory at the end of period 5 as a function of the setup pattern

c Prof. Dr. Horst Tempelmeier — www.scmp.uni-koeln.de

14/30

Introduction Optimization Model Solution approaches Numerical Results Conclusion

Exact solution Heuristic solution

Solution Procedure 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 17: 18: 19: 20:

M := U := {2, 3, . . . , T } E := {(τ, t)|τ = 1, 2, . . . , T ; t = τ + 1, τ + 2, . . . , T − 1} for all ((0, t) ∈ E with t ∈ U) do Predecessor(t):= 1; C (t) := E{C1t } end for while (M = 6 ∅) do Select τ ∈ M with minimum C (τ ) M := M \ τ ; U := U \ τ if (τ = T ) then end else for all ((τ, t) ∈ E with t ∈ U) do M := M ∪ t if (C (τ ) + E{Cτ t } < C (t)) then Predecessor(t):= τ C (t) := C (τ ) + E{Cτ t } end if end for end if end while

c Prof. Dr. Horst Tempelmeier — www.scmp.uni-koeln.de

15/30

Introduction Optimization Model Solution approaches Numerical Results Conclusion

Exact solution Heuristic solution

Dynamic Lot Sizing Heuristic 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12:

τ := 1 while (τ < T ) do t := τ while (t < T ) do if (Cτ t ≤ Cτ,t+1 ) then t := t + 1 else Make current lotsize for period τ permanent. τ := t + 1 end if end while end while

c Prof. Dr. Horst Tempelmeier — www.scmp.uni-koeln.de

16/30

Introduction Optimization Model Solution approaches Numerical Results Conclusion

Exact solution Heuristic solution

Silver-Meal Rule

s +h· E {Cτ t } =

t X ℓ=τ

" #+  ℓ   X E Iτ −1 (Pτ −1 ) + qτ∗t − Di   i =τ

t −τ +1

c Prof. Dr. Horst Tempelmeier — www.scmp.uni-koeln.de

(16)

17/30

Introduction Optimization Model Solution approaches Numerical Results Conclusion

Exact solution Heuristic solution

Least-Unit-Cost rule

E {Cτ t } = E

 " #+  t ℓ X X        s +h· Iτ −1 (Pτ −1 ) + qτ∗t − Di      i =τ

ℓ=τ

     

c Prof. Dr. Horst Tempelmeier — www.scmp.uni-koeln.de

t P

i =τ

Di

(17)

     

18/30

Introduction Optimization Model Solution approaches Numerical Results Conclusion

Exact solution Heuristic solution

Least-Total-Cost rule

E {Cτ t } = E

  

s +h·

t X ℓ=τ

"

Iτ −1 (Pτ −1 ) + qτ∗t −

c Prof. Dr. Horst Tempelmeier — www.scmp.uni-koeln.de

ℓ X i =τ

Di

#+  

(18)



19/30

Introduction Optimization Model Solution approaches Numerical Results Conclusion

1

Introduction The Problem Solution Approaches

2

Optimization Model Formulation

3

Solution approaches Exact solution Heuristic solution

4

Numerical Results Experiment 1 Experiment 2

5

Conclusion

c Prof. Dr. Horst Tempelmeier — www.scmp.uni-koeln.de

Experiment 1 Experiment 2

20/30

Introduction Optimization Model Solution approaches Numerical Results Conclusion

Experiment 1 Experiment 2

Expected Demands

Series # 1 2 3 4

E {Dt } 92 80 50 10

92 100 80 10

92 125 180 15

92 100 80 20

92 92 92 92 92 50 50 100 125 125 0 0 180 150 10 70 180 250 270 230

c Prof. Dr. Horst Tempelmeier — www.scmp.uni-koeln.de

92 92 93 100 50 100 100 180 95 40 0 10

21/30

Introduction Optimization Model Solution approaches Numerical Results Conclusion

Experiment 1 Experiment 2

Parameters

Series # T 1 2 3 4

12 12 12 12

s

TBO

CVD

βc⋆

500 500 500 500

1–12 1–12 1–12 1–12

{0.1, 0.2, 0.3, 0.4} {0.1, 0.2, 0.3, 0.4} {0.1, 0.2, 0.3, 0.4} {0.1, 0.2, 0.3, 0.4}

{0.5, 0.525, 0.05, . . . , 0.975} {0.5, 0.525, 0.05, . . . , 0.975} {0.5, 0.525, 0.05, . . . , 0.975} {0.5, 0.525, 0.05, . . . , 0.975}

c Prof. Dr. Horst Tempelmeier — www.scmp.uni-koeln.de

22/30

Introduction Optimization Model Solution approaches Numerical Results Conclusion

Experiment 1 Experiment 2

Results

1

LUC SM LTC PPA AC Groff

Average cost increase (%) 4.9 5.0 5.3 5.8 6.1 24.5

2

SM LTC LUC PPA AC Groff

5.9 7.2 7.5 7.9 7.9 24.8

Series # Heuristic

c Prof. Dr. Horst Tempelmeier — www.scmp.uni-koeln.de

Maximum cost increase (%) 30.4 30.4 29.1 43.5 30.1 52.7 32.7 28.7 31.3 49.9 32.9 60.2

% Optimal % Worst 56.6 56.0 48.6 57.8 48.6 3.9

3.76 3.97 7.00 19.23 7.52 75.24

42.8 41.5 39.9 51.4 38.3 0.4

4.79 5.31 5.83 17.50 5.31 69.90 23/30

Introduction Optimization Model Solution approaches Numerical Results Conclusion

Experiment 1 Experiment 2

Results

3

AC PPA SM LTC Groff LUC

Average cost increase (%) 9.7 10.2 10.7 11.5 23.2 29.0

4

SM AC PPA LTC Groff LUC

6.0 13.2 15.3 15.8 17.1 39.1

Series # Heuristic

c Prof. Dr. Horst Tempelmeier — www.scmp.uni-koeln.de

Maximum cost increase (%) 37.0 55.8 36.6 41.1 58.9 67.9 30.9 50.3 59.0 52.3 47.6 61.2

% Optimal % Worst 35.4 46.0 29.5 33.2 10.3 1.8

3.02 3.96 1.04 5.83 33.13 54.69

49.9 27.8 34.2 22.5 14.3 3.4

1.88 0.73 14.58 2.19 7.81 74.48 24/30

Introduction Optimization Model Solution approaches Numerical Results Conclusion

Experiment 1 Experiment 2

Results for Demand Series 1 40

Averagde cost increase (%)

35

Groff

30 25 20 15 10

AC

PPA

LTC

5

Silver-Meal

0

LUC

0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

Fillrate

c Prof. Dr. Horst Tempelmeier — www.scmp.uni-koeln.de

25/30

Introduction Optimization Model Solution approaches Numerical Results Conclusion

Experiment 1 Experiment 2

Parameters

E {Dt } ≃ Uniform(0, 100) Series # T s TBO CVD 5 5 500 1–5 {0.1, . . . , 0.4} 6 10 500 1–5 {0.1, . . . , 0.4} 7 15 500 1–15 {0.1, . . . , 0.4} 8 20 500 1–15 {0.1, . . . , 0.4}

c Prof. Dr. Horst Tempelmeier — www.scmp.uni-koeln.de

βc⋆ {0.5, 0.525, 0.05, . . . , 0.975} {0.5, 0.525, 0.05, . . . , 0.975} {0.5, 0.525, 0.05, . . . , 0.975} {0.5, 0.525, 0.05, . . . , 0.975}

26/30

Introduction Optimization Model Solution approaches Numerical Results Conclusion

Experiment 1 Experiment 2

Results

5

SM AC LTC LUC Groff PPA

Average cost increase (%) 4.7 5.6 5.9 10.5 13.8 15.1

6

SM AC LTC Groff LUC PPA

4.2 10.2 11.5 13.8 15.4 19.9

Series # Heuristic

c Prof. Dr. Horst Tempelmeier — www.scmp.uni-koeln.de

Maximum cost increase (%) 35.4 39.7 63.4 64.4 60.5 56.7 32.7 41.2 44.7 70.5 55.3 60.4

% Optimal % Worst 65.4 61.2 62.6 50.6 34.7 38.6

14.47 13.07 11.16 25.25 36.77 42.72

43.7 21.6 16.5 13.9 12.6 12.2

3.75 10.05 13.35 27.95 21.25 43.78 27/30

Introduction Optimization Model Solution approaches Numerical Results Conclusion

Experiment 1 Experiment 2

Results

7

LTC AC PPA LUC SM Groff

Average cost increase (%) 8.5 8.6 9.1 11.0 12.1 30.3

8

SM AC LTC PPA LUC Groff

8.8 10.7 11.6 12.5 13.9 29.9

Series # Heuristic

c Prof. Dr. Horst Tempelmeier — www.scmp.uni-koeln.de

Maximum cost increase (%) 42.8 44.8 64.9 53.9 40.3 64.4 38.8 46.5 46.2 66.2 49.4 64.6

% Optimal % Worst 40.4 39.5 53.1 35.3 25.6 2.3

3.14 2.26 15.04 8.18 5.96 68.47

21.1 17.9 17.0 32.5 14.7 1.0

1.54 1.48 3.64 16.42 9.33 69.41 28/30

Introduction Optimization Model Solution approaches Numerical Results Conclusion

1

Introduction The Problem Solution Approaches

2

Optimization Model Formulation

3

Solution approaches Exact solution Heuristic solution

4

Numerical Results Experiment 1 Experiment 2

5

Conclusion

c Prof. Dr. Horst Tempelmeier — www.scmp.uni-koeln.de

29/30

Introduction Optimization Model Solution approaches Numerical Results Conclusion

Conclusion

Exact solution for the stochastic Wagner-Whitin problem Adjusted cost criteria used in standard dynamic lot sizing heuristics Silver-Meal rule superior to Groff rule Directly applicable in ERP/AP systems Static uncertainty strategy: no nervousness, no bullwhip effect Target service level (instead of backorder costs) Possible extension: Capacities (done)

c Prof. Dr. Horst Tempelmeier — www.scmp.uni-koeln.de

30/30