1

HEURISTICS FOR DYNAMIC SCHEDULING OF

MULTI-CLASS BASE-STOCK CONTROLLED SYSTEMS 

Bora KAT and Zeynep Müge AVŞAR

Department of Industrial Engineering Middle East Technical University

Ankara TURKEY

2

OUTLINE

Two-class base-stock controlled systems

Related work in the literature

Analysis Model to maximize aggregate fill rate Solution approach Structure of the optimal dynamic (state-dependent) scheduling policy

Heuristics to approximate the optimal policy

Numerical Results Symmetric case (equal demand rates) Asymmetric case

Optimality of the policy to minimize inventory investment subject to aggregate fill rate constraintto maximize aggregate fill rate under budget constraint

Conclusion and future work

3

SYSTEM CHARACTERISTICS

Single facility to process items

Exponential service times

Poisson demand arrivals

No set-up time

Backordering case

Preemption allowed

Each item has its own queue managed by base-stock policies

Performance measure: aggregate fill rate over infinite horizon

4

TWO-CLASS BASE-STOCK CONTROLLED SYSTEM

2 ,1 iforkSnn iiii

::::::

i

i

i

i

i

Sknn items of type i in process

backorders of type ibase-stock level for type idemand rate for type iservice rate

items of type i in stock1

2

5

RELATED WORK IN THE LITERATURE

•Zheng and Zipkin, 1990. A queuing model to analyze the value of centralized inventory information.

•Ha, 1997. Optimal dynamic scheduling policy for a make-to-stock production system.  

•van Houtum, Adan and van der Wal, 1997. The symmetric longest queue system.  

•Pena-Perez and Zipkin, 1997. Dynamic scheduling rules for a multi-product make-to-stock queue.  

•Veatch and Wein, 1996. Scheduling a make-to-stock queue: Index policies and hedging points.  •de Vericourt, Karaesmen and Dallery, 2000. Dynamic scheduling in a make-to-stock system: A partial charact. of optimal policies.  •Wein, 1992.

Dynamic scheduling of a multi-class make-to-stock queue.  •Zipkin, 1995.

Perf. analysis of a multi-item production-inventory system under alternative policies.•Bertsimas and Paschalidis, 2001.

Probabilistic service level guarantees in make-to-stock manufacturing systems.  •Glasserman, 1996.

Allocating production capacity among multiple products.  •Veatch and de Vericourt, 2003.

Zero Inventory Policy for a Two-Part-Type Make-To-Stock Production System.

6

RELATED WORK IN THE LITERATURE

Zheng and Zipkin, 1990. A queuing model to analyze the value of centralized inventory information.

• symmetric case: identical demand (Poisson) and service (exponential) rates, identical inventory holding and backordering costs

• base-stock policy employed, preemption allowed

• main results on the LQ (longest queue) system• closed form steady-state distribution of the difference between the two queue lengths• closed form formulas for the first two moments of the marginal queue lengths• a recursive scheme to calculate joint and marginal distributions of the queue lengths • it is analytically shown that LQ is better than FCFS discipline under the long-run average payoff criterion

• alternative policy: specify (2S-1) as the maximum total inventory to stop producing imposing a maximum of S on each individual inventory

• -policy as an extension of LQ for the asymmetric case, extension of the recursive scheme to calculate steady-state probabilities

7

RELATED WORK IN THE LITERATURE

Ha, 1997. Optimal dynamic scheduling policy for a make-to-stock production system.

• two item types allowing preemption,demand (Poisson) and service (exponential) rates, and

inventory holding and backordering costs to be different

• perf. criterion: expected discounted cost over infinite horizon

• equal service rates: characterizing the optimal policy by two switching curves (base-stock policy, together with a switching curve, for a subset of initial inv. levels)

• different service rates: optimal to process the item with the larger index when both types are backordered

• heuristics:

static priority () rule

dynamic priority ( modified and switching) rules

8

RELATED WORK IN THE LITERATURE

van Houtum et al., 1997.The symmetric longest queue system.

• symmetric multi-item case: identical demand (Poisson) and service (exponential) rates, base-stock policy employed, not preemptive

• performance measure is fill rate (the cost formulation used is the same)

• investigating (approximating) the performance of the LQ policy with two variants: threshold rejection and threshold addition (to find bounds)

9

MODEL

.1

0,

1,,,1min1,,1,,

21

210

2112112112

2111

2121

where

nnf

nnfnnfnnfnnfnncnnf mmmmm

rates fill of average ted weigh:horizon timeoflength

cost total1run -long

221111),(:cost 2121 SnSn wwnnc ? :weight

21

iiw

10

SOLUTION APPROACH

• Value iteration

to solve for long-run avg. payoff

• No truncation

rates fill of average weighted

,,lim1 21121 nnfnnf mmm

22121

221211

221210

N0,1,..., ),( ,

1N0,1,..., ),( ,

N0,1,..., ),( 0,

nnfornnf

mnnfornnf

mnnfornnf

m

n2

n1N

N

N+mN+m-1

N+m-1

N+m

11

OPTIMAL SCHEDULING POLICY: Symmetric, Finite-Horizon Case

S1 n1

n2

S2

S1 n1

n2

S2

S2

n2

S1 n1

n2

S2

S1 n1

12

OPTIMAL SCHEDULING POLICY: Symmetric, Infinite-Horizon Case

S1 n1

n2

S2

S1 n1

n2

S2S2

n2

S1 n1

A(n1)

B(n1)

process type 1

process type 2

n2

S2

S1 n1

13

OPTIMAL SCHEDULING POLICY: Symmetric Case

S1=S2=9

S1=S2=9

A(n1)

S1 n1

B(n1)

PROCESS TYPE 1

S2

n2

PROCESS TYPE 1

PROCESS TYPE 2

PROCESS TYPE 2

n2

S2

PROCESS TYPE 2

PROCESS TYPE 2

S1 n1

PROCESS TYPE 1

PROCESS TYPE 1

B(n1)

A(n1)

14

STRUCTURE OF THE OPTIMAL POLICY: Symmetric Case

Cost: c(n1,n2)

15

STRUCTURE OF THE OPTIMAL POLICY: Symmetric Case

Region Iand11 Sn 22 Sn

Region III

22 Sn 11 Sn and

Region IV

11 Sn 22 Sn and

Region II

22 Sn 11 Sn and

none in stockoutLQ to avoid stockout for the item with higher risk

both types in stockoutSQ to eliminate stockout for more promising type

type 1 in stockoutB(n1): threshold to be away from region III and to reach region I

type 2 in stockoutB(n1): threshold to be away from region III and to reach region I

S1 n1

n2

IV III

S2

I II

n1

S2

n2

S1

16

HEURISTICS TO APPROXIMATE OPTIMAL POLICY: Symmetric Case(to approximate curve B)

S1 n1

(n1,n2)LQ

S2-n2

n1-(S1-1)

SQ

S2

n2

Best performance by heuristic 2.

Heuristics 1 and 2 perform almost equally well.

Heuristic 1

process type 1 2

22

1

11 1

nSSn

process type 2

Heuristic 2

process type 1 2

2211 1

nSSn

process type 2

17

HEURISTICS TO APPROXIMATE OPTIMAL POLICY: Symmetric Case

S1 n1

n2

SQ

S2

LQ

SQ

n2

2S

n12S

LQ

Heuristic 3 Heuristic 4 Heuristic 5

n2

S2

SQ

S1 n1

LQ

performs better than heuristics 3 and 4

for large .

steady-state probability distribution

by Zheng-Zipkin’s algorithm in LQ region

and then proceeding recursively in SQ region.

18

NUMERICAL RESULTS: Symmetric Case

Fill Rate (%)

1 2 3 4 6 8 11 15Optimal 53.72 76.26 87.73 93.75 98.43 99.61 99.95 100.00

LQ 43.81 70.25 84.70 92.26 98.07 99.53 99.94 100.00FCFS 46.15 71.01 84.39 91.59 97.56 99.29 99.89 99.99

Heuristic 1 53.72 76.06 87.60 93.67 98.40 99.61 99.95 100.00Heuristic 2 53.72 76.26 87.72 93.74 98.43 99.61 99.95 100.00Heuristic 3 53.72 75.03 87.03 93.40 98.35 99.59 99.95 100.00Heuristic 4 53.72 74.68 85.11 91.09 96.89 98.98 99.83 99.99Heuristic 5 53.72 76.06 87.43 93.48 98.31 99.57 99.95 100.00

Optimal 44.97 66.00 78.14 85.86 94.12 97.58 99.36 99.89LQ 30.81 53.95 69.88 80.48 91.91 96.67 99.13 99.85

FCFS 33.33 55.56 70.37 80.25 91.22 96.10 98.84 99.77Heuristic 1 44.97 65.92 77.95 85.69 93.98 97.49 99.34 99.89Heuristic 2 44.97 66.00 78.14 85.85 94.12 97.58 99.36 99.89Heuristic 3 44.97 62.94 75.62 84.15 93.41 97.29 99.29 99.88Heuristic 4 44.97 64.74 75.57 82.52 90.94 95.41 98.43 99.65Heuristic 5 44.97 65.92 77.95 85.57 93.83 97.39 99.29 99.88

Optimal 35.46 53.20 63.86 71.24 81.26 87.71 93.47 97.19LQ 16.23 31.12 43.80 54.31 69.93 80.26 89.50 95.48

FCFS 18.18 33.06 45.23 55.19 70.00 79.92 89.00 95.07Heuristic 1 35.46 53.20 63.79 71.08 80.97 87.37 93.17 97.01Heuristic 2 35.46 53.17 63.74 71.13 81.19 87.66 93.44 97.17Heuristic 3 35.46 46.65 56.36 64.48 76.61 84.64 91.83 96.48Heuristic 4 35.46 52.48 62.11 68.50 77.24 83.38 89.74 94.72Heuristic 5 35.46 53.20 63.79 71.08 80.90 87.25 93.05 96.92

0.8

0.9

S

0.7

19

NUMERICAL RESULTS: Symmetric Case

0.0010.0020.0030.0040.0050.0060.0070.0080.0090.00

100.00

1 2 3 4 6 8 11 15

S

Fill Rat

e (%

)

Optimal

LQ

FCFS

Heuristic 1

Heuristic 2

Heuristic 3

Heuristic 4

Heuristic 5

20

NUMERICAL RESULTS: Symmetric Case

0.0010.0020.0030.0040.0050.0060.0070.0080.0090.00

100.00

1 2 3 4 6 8 11 15

S

Fill R

ate

(%)

OptimalLQFCFSHeuristic 1Heuristic 2Heuristic 3Heuristic 4Heuristic 5

21

THREE TYPES OF ITEMS: Symmetric Case

S FCFS1 50.000 46.323 46.366 60.735 60.7522 75.000 74.281 74.322 81.520 81.5463 87.500 88.324 88.354 91.476 91.4974 93.750 94.849 94.873 96.199 96.2166 98.438 99.037 99.048 99.282 99.2948 99.609 99.824 99.828 99.870 99.87311 99.951 99.986 99.987 99.990 99.991

Fill Rate (%), =0.75LQ Heuristic 2

S FCFS1 25.000 21.359 21.421 47.887 47.9202 43.750 40.856 40.955 66.429 66.4603 57.813 56.293 56.406 76.568 76.6004 68.359 67.943 68.049 83.138 83.1866 82.202 82.901 82.987 91.045 91.0978 89.989 90.906 90.975 95.249 95.27811 95.776 96.460 96.519 98.147 98.171

Fill Rate (%), =0.90LQ Heuristic 2

Simulation results for LQ policy and Heuristic 2

22

THE OPTIMAL SCHEDULING POLICY: Asymmetric Case

PROCESS TYPE 1

A(n1)

S2

n1

PROCESS TYPE 2

PROCESS TYPE 1

B(n1)

S1

PROCESS TYPE 2

n2n2

S2

PROCESS TYPE 1

PROCESS TYPE 2

PROCESS TYPE 2 B(n1)

A(n1)

S1 n1

PROCESS TYPE 1

S1=S2=8S1=S2=8

21

21 2ww

23

HEURISTIC 1 TO APPROXIMATE OPTIMAL POLICY: Asymmetric Case (to approximate curves A and B)

(n1,n2)

(n1,n2)

n1-(S1-1)

S2-n2S1-n1

S2-n2

(n1,n2)

S2

S1 n1

n2

n2-(S2-1)

III

n1-(S1-1)

III

IV

Region I

Region II

process type 1 2

22

1

11 1

nSSn

process type 2

Region III

process type 12

22

1

11

nSnS

process type 2

process type 1 2

22

1

11 11

SnSn

process type 2

21

21

ww

Curve A approximated for regions I and III is the diagonal when 1=2.

24

HEURISTIC 2 TO APPROXIMATE OPTIMAL POLICY: Asymmetric Case

Region I

Region II

process type 1 2

2211 1

nSSn

process type 2

Region III

process type 12

22

1

11

nSnS

process type 2

process type 1 2

22

1

11 11

SnSn

process type 2

(n1,n2)

(n1,n2)

n1-(S1-1)

S2-n2S1-n1

S2-n2

(n1,n2)

S2

S1 n1

n2

n2-(S2-1)

III

n1-(S1-1)

III

IV

21

21

ww

25

NUMERICAL RESULTS: Asymmetric Case

Fill Rate (%)

122 1 2 3 4 6 8 11 15

Optimal 56.21 77.12 87.94 93.72 98.37 99.59 99.95 100.00FCFS 47.69 71.90 84.54 91.30 97.11 98.99 99.78 99.97Heuristic 1 56.16 77.00 87.71 93.57 98.31 99.57 99.95 100.00Heuristic 2 56.16 76.90 87.71 93.60 98.32 99.58 99.95 100.00Delta 1 45.11 70.09 84.38 91.99 97.94 99.48 99.93 100.00Delta 2 46.77 70.19 83.64 91.46 97.81 99.45 99.93 100.00Delta 3 47.77 71.14 83.47 90.77 97.52 99.38 99.92 100.00Delta 5 48.78 72.12 84.18 90.72 96.74 99.09 99.88 99.99Delta 8 49.31 72.66 84.58 90.98 96.62 98.62 99.75 99.98Optimal 48.76 67.89 78.93 86.17 94.19 97.60 99.37 99.90FCFS 35.06 57.23 71.44 80.68 90.86 95.52 98.39 99.57Heuristic 1 48.72 67.65 78.51 85.81 93.93 97.46 99.33 99.89Heuristic 2 48.72 67.54 78.54 85.93 94.06 97.54 99.35 99.89Delta 1 32.05 53.97 69.68 80.26 91.74 96.57 99.09 99.85Delta 2 33.86 54.49 69.05 79.63 91.47 96.48 99.07 99.84Delta 3 35.16 55.99 69.29 78.91 90.93 96.24 99.01 99.83Delta 5 36.83 57.96 71.04 79.51 89.55 95.37 98.75 99.79Delta 8 38.12 59.52 72.44 80.63 89.71 94.16 98.02 99.65Optimal 41.12 56.93 65.78 72.25 81.72 88.00 93.64 97.27FCFS 19.64 35.14 47.42 57.19 71.27 80.43 88.71 94.38Heuristic 1 41.07 56.76 65.31 71.54 80.88 87.22 93.07 96.96Heuristic 2 41.07 56.45 65.20 71.78 81.40 87.78 93.51 97.21Delta 1 17.07 31.26 43.77 54.23 69.83 80.17 89.44 95.45Delta 2 18.48 32.00 43.57 53.85 69.55 80.01 89.37 95.43Delta 3 19.64 33.59 44.30 53.61 69.06 79.65 89.19 95.35Delta 5 21.48 36.16 47.02 55.40 68.09 78.49 88.48 95.04Delta 8 23.45 38.97 50.02 58.27 69.76 77.35 86.72 94.17

S

7

8

n2

1

S2

0

S1 n1

21

21 2ww

Instead of LQ policy,delta policy (Zheng-Zipkin): for 1>2, process type 2 when n2-n1>.

26

0.00

20.00

40.00

60.00

80.00

100.00

1 2 3 4 6 8 11 15S

Fill R

ates

(%)

optimal

fcfs

h1

h2

delta3

NUMERICAL RESULTS: Asymmetric Case21

21 2ww

27

0.00

20.00

40.00

60.00

80.00

100.00

1 2 3 4 6 8 11 15S

Fill R

ates

(%)

optimalfcfs

h1

h2

delta3

NUMERICAL RESULTS: Asymmetric Case21

21 2ww

28

NUMERICAL RESULTS: Asymmetric Case Weighted Cost Function

21

21 2ww

Fill Rate (%)

S1 n1

n2

1

S2

01w

2w

Indices used for the heuristics

are multiplied by the respective wi.

Heur-mult: index1 is multiplied

also by (1-).

(adjustment for high )

21

iiw

S 1 2 3 4 6 8 11 15optimal 51.66 74.98 86.96 93.26 98.24 99.56 99.95 100.00FCFS 44.84 68.92 82.17 89.63 96.38 98.70 99.71 99.96h1 51.54 74.41 86.63 92.98 98.11 99.49 99.93 99.99h1-mult 50.98 74.71 86.86 93.09 98.14 99.49 99.93 99.99h2 51.54 74.52 86.66 93.00 98.13 99.50 99.93 100.00h2-mult 50.98 74.92 86.95 93.15 98.16 99.51 99.93 100.00delta1 40.08 66.37 82.18 90.77 97.59 99.38 99.92 100.00delta2 41.18 65.81 80.86 89.92 97.39 99.34 99.92 100.00delta3 41.84 66.44 80.25 88.80 96.96 99.23 99.91 99.99delta5 42.52 67.10 80.73 88.44 95.83 98.82 99.85 99.99delta8 42.87 67.46 80.99 88.61 95.61 98.18 99.67 99.98optimal 42.69 64.62 77.84 85.89 94.20 97.62 99.38 99.90FCFS 32.47 53.85 68.14 77.80 88.97 94.40 97.92 99.43h1 42.50 63.39 76.67 84.99 93.71 97.32 99.25 99.86h1-mult 41.80 64.46 77.75 85.74 93.99 97.42 99.27 99.87h2 42.50 63.36 76.71 85.04 93.76 97.37 99.28 99.87h2-mult 41.80 64.52 77.84 85.80 94.06 97.48 99.30 99.88delta1 28.03 50.30 66.99 78.40 90.92 96.22 98.99 99.83delta2 29.24 50.02 65.53 77.18 90.41 96.04 98.95 99.82delta3 30.11 51.02 65.08 75.71 89.46 95.63 98.85 99.81delta5 31.22 52.33 66.25 75.62 87.22 94.29 98.46 99.74delta8 32.08 53.37 67.18 76.37 87.03 92.49 97.43 99.54optimal 32.95 52.01 64.79 73.24 83.56 89.48 94.48 97.64FCFS 17.86 32.27 43.95 53.47 67.63 77.24 86.36 92.95h1 32.73 48.98 60.70 69.59 81.19 87.90 93.52 97.12h1-mult 32.22 51.79 64.42 72.82 83.17 89.08 94.04 97.31h2 32.74 49.29 61.30 70.10 81.41 88.03 93.65 97.22h2-mult 32.22 52.01 64.78 73.21 83.48 89.30 94.26 97.46delta1 14.72 28.66 41.44 52.25 68.48 79.26 88.96 95.24delta2 15.65 28.70 40.40 51.14 67.72 78.80 88.73 95.15delta3 16.43 29.76 40.37 49.96 66.50 77.96 88.29 94.96delta5 17.65 31.47 42.18 50.68 63.98 75.59 86.90 94.36delta8 18.97 33.34 44.18 52.59 64.72 73.01 83.98 92.95

0.7

0.8

0.9

29

OPTIMAL POLICY for ,

),...,( min1

11

I

iIii

I

iii SSFRwSc

BudgetScSSFRwI

iii

I

iIii

111 ),...,(max

Lower expected inventory levels under heuristic policies when is not too small.

Minimum Base-Stock Levels to satisfy Target Fill Rate

S1 S2 FR(%) exp.inv. S1 S2 FR(%) exp.inv. S1 S2 FR(%) exp.inv. S1 S2 FR(%) exp.inv.0.25 2 1 91.837 1.347 2 1 91.807 1.346 2 1 92.462 1.346 2 1 92.462 1.3460.50 3 2 92.593 2.037 3 2 92.936 2.030 2 2 90.139 1.562 2 2 90.070 1.5550.60 3 3 92.128 2.309 3 3 92.681 2.296 3 3 93.616 2.313 3 3 93.493 2.3040.75 5 5 92.224 3.617 5 4 90.259 3.127 4 4 90.458 2.745 5 4 92.136 3.1380.90 12 11 90.001 7.450 12 11 90.502 7.405 9 9 90.000 5.545 11 10 90.924 6.5230.95 24 23 90.470 14.905 23 23 90.284 14.399 18 18 90.858 10.844 21 21 90.897 12.6320.25 2 2 97.959 1.837 2 2 98.085 1.836 2 2 98.115 1.836 2 2 98.115 1.8360.50 3 3 96.296 2.519 3 3 96.741 2.513 3 3 96.989 2.517 3 3 96.971 2.5160.60 4 4 96.626 3.275 4 4 97.167 3.267 4 3 95.837 2.789 4 3 95.764 2.7840.75 6 6 95.334 4.570 6 6 95.990 4.552 6 5 95.864 4.108 6 5 95.532 4.0780.90 15 15 95.071 10.722 15 15 95.482 10.693 13 12 95.214 8.533 14 13 95.177 9.2780.95 30 30 95.034 20.972 30 29 95.006 20.462 24 24 95.075 15.850 27 27 95.081 18.1120.25 3 3 99.708 2.834 3 3 99.755 2.834 3 2 99.025 2.335 3 2 99.025 2.3350.50 5 4 99.177 4.004 4 4 99.052 3.504 4 4 99.113 3.505 4 4 99.109 3.5040.60 6 6 99.380 5.255 6 5 99.258 4.754 5 5 99.042 4.259 5 5 99.026 4.2570.75 10 9 99.194 8.012 9 9 99.278 7.509 8 8 99.007 6.526 9 8 99.194 7.0140.90 23 23 99.010 18.545 23 22 99.065 18.040 20 20 99.014 15.617 21 21 99.007 16.5570.95 47 46 99.046 37.091 46 45 99.033 36.090 40 40 99.073 30.810 43 43 99.047 33.619

Heuristic 3

0.90

0.95

0.99

FCFS LQ Heuristic 2

30

COMPARISON OF THE POLICIES in terms of fill rate, exp. backorders and inventories

S FCFS LQ Heur.2 FCFS LQ Heur.2 FCFS LQ Heur.21 0.40000 0.37500 0.49435 0.90000 0.87500 0.99435 0.40000 0.37500 0.494352 0.64000 0.62813 0.71433 0.54000 0.50312 0.66484 1.04000 1.00313 1.164843 0.78400 0.78380 0.83442 0.32400 0.28692 0.40970 1.82400 1.78692 1.909704 0.87040 0.87589 0.90458 0.19440 0.16281 0.24512 2.69440 2.66281 2.745126 0.95334 0.95990 0.96898 0.06998 0.05200 0.08156 4.56998 4.55200 4.581568 0.98320 0.98720 0.99007 0.02519 0.01652 0.02632 6.52519 6.51652 6.5263211 0.99637 0.99771 0.99822 0.00544 0.00295 0.00472 9.50544 9.50295 9.5047215 0.99953 0.99977 0.99982 0.00071 0.00030 0.00047 13.50071 13.50030 13.50047

=0.75Fill Rate (%) Expected Backorders Expected Inventory

S FCFS LQ Heur.2 FCFS LQ Heur.2 FCFS LQ Heur.21 0.18182 0.16227 0.35459 3.68182 3.66223 3.85455 0.18182 0.16227 0.354592 0.33058 0.31117 0.53169 3.01240 2.97341 3.35688 0.51240 0.47344 0.856913 0.45229 0.43797 0.63745 2.46469 2.41139 2.90969 0.96469 0.91141 1.409704 0.55187 0.54307 0.71125 2.01656 1.95446 2.50083 1.51656 1.45448 2.000856 0.70002 0.69933 0.81191 1.34993 1.28298 1.79612 2.84993 2.78299 3.296148 0.79918 0.80257 0.87658 0.90367 0.84188 1.25659 4.40367 4.34190 4.7566011 0.89001 0.89505 0.93436 0.49495 0.44743 0.71384 6.99495 6.94744 7.2138515 0.95071 0.95482 0.97173 0.22180 0.19260 0.32378 10.72180 10.69262 10.82379

=0.90Fill Rate (%) Expected Backorders Expected Inventory

31

COMPARISON OF THE POLICIES in terms of fill rate, exp. backorders and inventories

Fill Rate (%), =0.90

0.300.350.400.450.500.550.600.650.700.750.800.850.900.951.00

1 2 3 4 6 8 11 15S

FCFSLQHeur.2

Expected Backorders, =0.90

0.00

1.00

2.00

3.00

4.00

1 2 3 4 6 8 11 15S

FCFSLQHeur.2

Expected Inventory, =0.90

0.001.002.003.004.005.006.007.008.009.00

10.0011.00

1 2 3 4 6 8 11 15S

FCFSLQHeur.2

32

MODEL TO INCORPORATE SET-UP TIME

f : minimum cost while processing items of type 1g : minimum cost while processing items of type 2sf : minimum cost while setting up the facility for type 1sg : minimum cost while setting up the facility for type 2set-up rate

21

211211

2112112

2112111

2121

, ,1,,1min

1,,1,min ,1,,1min

,,

nnfnnsgnnf

nnsgnnfnnsgnnf

nncnnf

m

mm

mm

mm

m

33

CONCLUSION

Summarymulti-class base-stock controlled systemsnumerical investigation of the structure of the optimal policy

for maximizing the weighted average of fill ratesoptimal policy for

smaller than LQ (symmetric case), (asymmetric case), FCFS policies give smaller expected inventory when is not too small

accurate heuristics adapted for extensions: asymmetric case, more than two types of itemsdisadvantage: not that easy to implement compared to LQ, and FCFS policies

Future Workoptimizing base-stock levelsset-up timetype-dependent processing time

instead of working with aggregate fill rate

how to determine the values of ?

),...,( min1

11

I

iIii

I

iii SSFRwSc

iSSFRSc iIi

I

iii ),...,( min 1

1

iw

I

iIii SSFRw

11 ),...,(

),...,( 1 ISS