5th Annual IEEE Conference on Automation Science and Engineering
Bangalore, India, August 22-25, 2009
Stability Considerations and Service Level Measures in Production Inventory Systems: a Simulation Study
D. Bijulal, Jayendran Venkateswaran, and N. Hemachandra
Abstract— This paper analyzes the ability of the general
production-inventory replenishment policy to maintain desired
level of service. Key control parameters that affect stability of
the replenishment policy are the fractional rates of adjustments
of discrepancies in WIP and inventory. The effect of these
control parameters on the service level (order fill rate) of
the production-inventory system which face independent and
identically distributed (i.i.d.) random demand is investigated.
A replenishment policy that explicitly accounts for the safety
stock, and thus improves the service levels, has been proposed.
The range of parameter settings that assures the desired service
level without affecting system stability has been determined.
I. INTRODUCTION
Past research in production-inventory systems addressed
the problems associated with refinement of production ordering policies, cost optimization, stability analysis, comparison
of production ordering policies, bullwhip reduction, etc.
The structure of production-inventory systems has flows
of information, materials, their accumulations and decision
rules, and delays involved in processing information and
in production. The decision rules, the parameters, and the
delays involved are decisive in the system behavior. The
general replenishment rules are based on the base stock level
and rules defined for achieving the base stock to meet the
customer demand. Studies related to variance amplification
in production inventory systems - also called bullwhip effect,
revealed that the gaps in the desired values of the work-inprocess (W IP ) and inventory are to be adjusted at fractional
rates to control the variance amplification [1].
The dynamic behavior of production inventory systems
were well addressed by research [2]. It is shown by previous research that System Dynamics (SD) simulations have
a role to play in supply chain design [3]. SD modeling
technique has been adopted in research [4] for modeling
supply chain systems. The dynamic behavior of supply
chains and production-inventory systems were analyzed by
control theoretic modeling. For dealing with time varying
nature of the supply chain systems, the use of control theory
has been well explained in [5]. Control system theory has
lessened the efforts required in the modeling and analysis of
the dynamic structure of production-inventory systems with
the use of frequency domain calculation. The application of
SD modeling and control theoretic analysis together became
The authors are with the interdisciplinary programme in Industrial Engineering and Operations Research, Indian Institute of Technology Bombay,
Mumbai, INDIA.
D.
Bijulal
<[email protected]>,
Jayendran
Venkateswaran
<[email protected]>, N. Hemachandra <[email protected]>
978-1-4244-4579-0/09/$25.00 ©2009 IEEE
instrumental in supply chain and production inventory systems analysis, especially in stability and bullwhip related
research [1][6][7][8].
A. Related Literature
A common structure of the production-inventory system
analyzed in literature is from the inventory and order based
production control system (IOBPCS) family of control policies [9], the latest among them being the automatic pipeline,
variable inventory and order based production control system
(APVIOBPCS)[1]. The control system structure of this policy has three tuning parameters. Two of them are the delays
in adjusting discrepancies in production pipeline (WIP) and
inventory, and the third is the smoothing constant selected
for demand forecast [1]. Most of the past research [1][6][10]
reported the analysis of variance amplification in orders and
in inventory, stability conditions of models, etc.
A three stage periodic review production system has been
analyzed as discrete time system in the research in [7].
The importance of the system parameters and the effect of
update frequency on the dynamic stability of the productioninventory system has been established by that research. They
utilized the z-transform technique for deriving the stability
conditions in terms of the tuning parameters. In production
stages, they introduced higher order delays in place of fixed
pipeline delays. This modification helped in avoiding the
approximations, and to get exact solutions. This modeling
methodology has been followed in [11] for getting the
conditions of stability for a closed-loop system.
An important aspect not considered in the past research
is the service level constraints in the production-inventory
models, especially while considering the system stability.
Production-inventory systems work in stochastic demand
condition [8]. Inventory systems become unable to meet customer demand due to the demand variability during a period.
An important performance measure of an inventory system
is the service level. It is the measure of the productioninventory system’s performance which shows the degree
to which the customer demand is satisfied [12]. The past
research so far addressed the question of getting a desired
service level in an inventory system when the customer
demand is uncertain.
Some measures of performance accepted as service level
are: average out standing back orders, average stock outs per
unit time, average number of lost sales occurrences in unit
time, fraction of demand lost, etc. [13]. A most common
measure of service level is the probability that an order
is fully satisfied from stock [14]. This measure is termed
489
A. System Model
The system dynamics structure of the system model is
adopted from a previous work [11]. The control theoretic
model of the system is shown in Fig. 1. The accumulations
are represented by the integration of the difference in the
inflow and outflow signals. The variable W IP is the accumulation of the difference between the inflow (P REL) to the
production process, and the outflow production completion
rate (P CR) from the production process. Inventory in the
system is the accumulation of the difference between P CR
and customer demand (CD). Each production stage has a
delay of (Lp /Q), where Lp is the Production Lead time and
Q is the Number of Stages. We make use of higher (Qth )
order delay modeled by cascading Q number of I st order
exponential delays.
Forecasted
Demand
PREL
PCR
Prodn Process
K Ts
z-1
K Ts
II. INTRODUCTION TO THE PROBLEM
A three stage production system with generalized orderup-to (G OUT) replenishment policy for production orders is
analyzed for its ability to meet the required service levels.
The model analyzed in this study is similar to the model
analyzed for stability in [7] and [11]. G OUT structure
of production systems uses two negative feedback loops
to control the production order releases to the plant. The
feedbacks are based on the discrepancies in the desired
values of W IP and IN V respectively.
The system model represents a manufacturer receiving
orders (demand) from the distributor, and producing goods
based on the replenishment policy. The replenishment policy
targets to maintain a base by keeping sufficient work-inprocess inventory. The gap between desired WIP (DW IP )
and W IP is adjusted at a fractional rate α. The gap in base
stock (BS) and IN V is corrected at a fractional rate β. α
and β are the control parameters in the system. The question
to be answered is that: whether keeping the system stable is
enough for meeting the service level requirements? If the
cost optimization is also involved, what other considerations
are required in the parameter setting? It is also aimed to
study the variation of the performance measure (i.e., service
level) within the stable region.
This research analyzes the effect of the control parameters
on the production-inventory system performance in terms of
stability as well as service levels. The system faces a random
i.i.d. demand of the form N (µ, σ 2 ). The average is estimated
as the demand forecast and the standard deviation is assumed
to be known. The objective is that: for a desired OF R,
examine how the system performs at different (α, β) pairs,
inside the stability region. The changes in cost requirements
are also observed to get a feel of the investment comparison
with different parameter settings.
Sales &'
Demand Forecasting
z-1
Base Stock
Calculation
as the Order Fill Rate (OFR), which is also known as
α-Service level or S1 -Service level. In certain cases the
customer accepts partial supply against their orders. The
balance quantity in the order can become either lost or back
ordered [12]. In such cases, the ratio of the available stock to
the demand quantity is called the Item Fill Rate (IFR) or Fill
Rate, which is also known as β-Service level or S2 -Service
level [14]. Service levels measured as fill rate in productioninventory systems while reducing the sum of bullwhip and
inventory variance is analyzed in [8]. The research content
in [8] is based on the parameter setting α = β, with different
demand patterns. This parameter setting is known as DezielEilon line or D-E line [15], where the discrepancies in WIP
and inventory (IN V ) are corrected with same amount of
delay (delay = 1/f ractional rate).
It will be insightful if we can predict the service level the
system can achieve, while the tuning parameters α and β are
selected within the entire stability region. A research on this
direction which addresses system stability and service level
requirements as well as average system costs are presented
in this paper.
WIP
Alpha
DWIP
Lp
INVENTORY
Beta
BS
Fig. 1.
The model structure
B. Assumptions
The basic assumption made in the system model is that it
is a periodic review system with simulation time step kept
as one period. The system state at the beginning of period
is kept constant during the period and updated at the end
of period. To keep the system linear, it is assumed that the
capacities of storage and processing stages are infinite. In
each period there is only one order from the customer side.
The model uses G OUT policy for replenishment. If an order
is not fully met from the system inventory, the unmet quantity
become back ordered.
C. OFR Modeling
The performance parameters selected in this analysis are
OFR and average system cost. OFR is measured as the ratio
of the number of orders completely filled from inventory to
the total number of orders.
OFR in this study is taken as the long run ratio of the
number of orders fully filled from the stock to the total
number of orders. It is modeled as follows. OFR has to keep
track of the number of orders completely satisfied from the
available stock. This is captured in each period as orders
filled (OF ), the value of which is binary. It is defined as
function of the stock and the customer demand. i.e.,
490
(
OF (n) =
1;
0;
IN V (n) ≥ CD(n)
otherwise
(1)
III. SYSTEM EQUATIONS AND STABILITY
CONDITIONS
The cumulative sum of this variable value over n periods1
gives the total number of orders filled (T OF ) from stock.
T OF (n) =
n
X
OF (i)
(2)
i=1
Therefore order fill rate in the system becomes:
OF R(n) =
T OF (n)
n
(3)
The coefficient of variation of the i.i.d. random demand is
kept at very low value so that the probability of occurrence
of negative demand become zero. However the simulation
model take care of such instances and avoid counting zero
and negative demands in T OF and total orders2 .
D. Average cost modeling
The system has costs associated with the quantity back
ordered in a period: the back order cost (b1 per item) and
associated with the quantity stored in the system in a period
(h per item). The total back orders in a period is observed
for calculating the back order cost. It will include the back
orders which are carried from previous periods also. The
variable BO, which is the accumulation of the net effect of
inflow back order rate (BOR) and outflow back order fill
rate (BOF R), which is modeled as follows.
BO(n) = BO(n − 1) + BOR(n − 1) − BOF R(n − 1)
(4)
(
CD(n) − IN V (n), CD(n) > IN V (n);
BOR(n) =
(5)
0,
otherwise.
"
#
8
>
<M in BO(n),
, IN V (n) > CD(n);
BOF R(n) =
IN V (n) − CD(n)
>
:
0,
otherwise.
(6)
The model variables and their interactions are represented
by difference equations in time [11]. These equations are
transformed using z-transformation and the stability of the
system is analyzed against the exogenous variable CD. The
stability is defined in terms of system parameters, fractional
rate of adjustment for WIP discrepancy (α) and fractional
rate of adjustment for inventory discrepancy (β).
The exponentially smoothed demand forecast (F Dn ) for
period ‘n’ is represented by (8), with ρ as the smoothing
constant. Inventory (IN Vn ) at period n is represented by
(9), showing the accumulation of net effect of P CR and
CD with the initial value. The system WIP in period n is the
accumulation of net effect of P REL and P CR, represented
by (10). The production release, which is the sum of the
base stock, and adjustments for the discrepancies in WIP
and INV, is represented in (11). The adjustments for WIP and
inventory are shown in (12) and (13) respectively. Base stock
is taken as the demand forecast (14). WIP in j th production
stage, in period n is represented by (15). The production
completion rate (P CR) is represented by (16).
F D(n) = F D(n − 1) + ρ(CD(n − 1) − F D(n − 1))
W IP ADJ(n) = α (Lp BS(n) − W IP (n))
IN V ADJ(n) = β (BS(n) − IN V (n))
p
P CR(n) = W IPQ (n) × (Q/Lp )
(7)
OF R(n) and Avg. Cost(n) at the end of the simulation
are assumed to be better approximation of OFR and Average
cost in the system. These variables are part of the system
response which are not fed back to the system for decision
making and do not affect the stability. Therefore this information is not part of the stability analysis and is omitted in
the discussions on stability.
(12)
(13)
BS(n) = F D(n)
(14)
” !
“
8
Q
>
W IP1 (n − 1) 1 − L
>
>
p
; j = 1;
>
>
< +P REL(n − 1)
1
0
“
”
W IPj (n) =
Q
>
>
>@ W IPj (n − 1) 1 − Lp A ; j ∈ {2, · · · , Q}
>
>
Q
:
+W IPj−1 (n − 1) L
The quantity BO(n) will incur back order cost per period,
BO(n) × b1 . IN V (n), in period n will incur a carrying cost
per period, IN V (n)×h. The total of these costs accumulated
till period n gives the total cost. The per period average cost
is calculated as the ratio of this accumulated costs to n.
n
1X
(IN V (i) × h + BO(i) × b1 )
Avg. Cost(n) =
n i=1
(8)
IN V (n) = IN V (n − 1) + (P CR(n − 1) − CD(n − 1))
(9)
W IP (n) = W IP (n − 1) + (P REL(n − 1) − P CR(n − 1)) (10)
P REL(n) = F D(n) + W IP ADJ(n) + IN V ADJ(n)
(11)
(15)
(16)
The z-transforms of the equations (8) to (14) are shown
in equations (17) to (23) respectively.
1 Since
it is assumed that in each period there is one order, number of
orders and number of periods will be equal.
2 The occurrence of negative demand in an order is captured by N eg Sale
which is a function of CD.
(
1; CD(n) ≤ 0
N eg Sale(n) =
0; otherwise
ρ
CD(z)
z+ρ−1
1
IN V (z) =
(P CR(z) − CD(z))
z−1
1
W IP (z) =
(P REL(z) − P CR(z))
z−1
F D(z) =
P REL(z) = F D(z) + W IP ADJ(z) + IN V ADJ(z)
491
(18)
(19)
(20)
W IP ADJ(z) = α (Lp BS(z) − W IP (z))
(21)
IN V ADJ(z) = β (BS(z) − IN V (z))
(22)
BS(z) = F D(z)
N eg Sale is then subtracted from T OF and Total Orders, n.
(17)
(23)
Recursive solution of (15) yield the z-transform of WIP in
the last production stage as (24) and (16) becomes (25):
(Q/Lp )(Q−1)
W IPQ (z) =
z − 1 + Q/Lp
(24)
P CR(z) = W IPQ (z) × (Q/Lp )
(25)
The above simultaneous algebraic equations ((17)· · · (25))
are solved to get the transfer function between the input variable CD and the output variable P REL. These equations
are solved using Mathematicar 6.0, assuming forecasting
constant ρ = 1. Thus:
«
“
”Q „
(z − 1)(1 + αLp )
(z − 1) z + LQ − 1
p
+2zβ − β
P REL(z)
1
0 “
=
(26)
”Q
CD(z)
Q
z
+
−
1
(z
+
α
−
1)
B
C
Lp
“ ”Q
z(z − 1) @
A
−(α − β) LQ
p
This is the general expression of the transfer function for
any number of stages and any value of production delay.
Since the model in this analysis assumes three stages, (Q =
3), and total production lead time Lp = 3 (i.e., Lp = Q = 3),
this equation reduces to:
z 5 (1 + 3α + 2β) − z 4 (2 + 6α + 3β)
+z 3 (1 + 3α + β)
P REL(z)
=
CD(z)
(z − 1) {z 5 + z 4 (α − 1) + z(β − α)}
(27)
System stability is usually tested against a bounded input.
The system (27) is Bounded Input Bounded Output (BIBO)
stable, if the roots of the denominator polynomial (poles)
lie within the unit circle in complex plane. The stability
conditions in terms of α & β has been obtained by solving
the denominator of (27) and the stability region is plotted in
Fig. 2.
3
Fractional rate of
adjustment of WIP (α)
Unstable region
D-E
Lin
e
2
Stable region
1
α>β
α<β
Unstable region
0
0
1
2
Fractional rate of
adjustment of inventory (β)
3
Fig. 2. Stability boundary of the system defined by the system parameters
The (α, β) region is divided to stable and unstable regions
by a closed critically stable boundary. The selection of (α, β)
pairs within the stable region makes the system stable against
any change in the input. The output parameter will eventually
converge to a steady value with or without initial oscillations.
The selection along the boundary makes the system to
continue sustained oscillations and selection of points outside
the boundary will cause the system variables to continue
oscillations with exponentially increasing amplitude.
A. OFR for i.i.d. Normal demand
The above discussion on stability applies to pulse and
step inputs. For i.i.d. Normal, N (µ, σ 2 ) demand, a better
approximation of future demand is µ itself. Exponentially
smoothed demand forecast can approximate µ. Therefore,
there is 50% probability that the demand is greater than the
stock if the G OUT policy is used without modifications.
This leads to OFR values as low as 50%. To overcome
this difficulty, the inventory systems make use of the safety
stocks. In this analysis also a safety stock is assumed and
added to the expected demand, FD. Thus the base stock
changes to F D+safety stock, where, safety stock= kσ and
k = Φ−1 (DOF R) represents the safety factor [12]. DOF R
is the target service level, desired OFR. Here we take an
estimator for σ as the product of F D(n) and a constant
‘ν’, as in [8]. With this modification in base stock, (14) get
modified to (28), where, a = (1 + ν)k, a constant. Therefore
the system transfer function (27) get modified to (29):
BS(n) = aF D(n)
z 5 (1
P REL(z)
=
CD(z)
(28)
z 3 (1
+ 3aα + β(1 + a)) +
+ 3aα + aβ)
−z 4 (2 + 6aα + β(1 + 2a))
(z − 1) {z 5 + z 4 (α − 1) + z(β − α)}
(29)
Comparing (27) and (29), it is evident that the denominator
polynomials do not change with this modification. Therefore
the BIBO stability of the system do not get affected by
modifying the base stock to account for DOFR as in (28).
However the effect of modification is visible in the numerator, which controls the amplitude of the system response. It
is noted that at a = 1, (29) reduces to (27).
IV. SUB-REGIONS WITHIN STABILITY REGION
The stability region can be divided into two by D-E line
(α = β line) as α > β and α < β. Another division of the
region can be α < 1 and β < 1, where the points will always
show only fractional rates of adjustments (corresponds to
delays greater than 1 period). A discussion on the expected
system response for different combinations of parameters in
stability regions is made below.
D-E line shows the situation where the adjustments for
inventory and WIP discrepancy are done at the same rate.
α = β = 1 is typical, where the discrepancies are completely
accounted for corrections. At this setting the replenishment
policy is called the pure Order-up-to (pure OUT) policy.
Along D-E line, if the discrepancies are adjusted by a small
fraction, the parameter setting become α = β, and α < 1
and β < 1.
A. Relation between Bullwhip and system performance
It is shown by [1] and [8], that the bullwhip in the replenishment orders will be eliminated if the control parameters
setting is α < 1 and β < 1. Once the variance amplification
in the production orders are reduced/eliminated, the variance
in inventory will also be reduced, which in turn can help
smooth inventory build up and less holding costs as well as
492
2.5
Frac. Rate of adj. for WIP - a
better service level, compared to pure OUT or aggressive
ordering (α > 1, & β > 1) policies.
Aggressive ordering policy make the system over react to
the changes and the order variance get amplified. This variance amplification can be carried to inventory, and cause the
service level to reduce because of fluctuations in inventory.
Also the cost can increase because of increased number of
stock out occasions.
B. Parameter selection and variation in cost and service
level
The system model is simulated at two DOFR values
with the base stock fixed as in (28). The variation of the
performance parameters are plotted as contour plots inside
the stability region, which are shown in Fig. 3 to Fig. 6.
The OFR and average cost plots show different behaviors
in relation to the D-E Line. Along the D-E line, however,
1.5
75
1.0
95
90
80
85
0.5
0.0
0.5
Fig. 3.
1.0
1.5
2.0
2.5
Frac. Rate of adj. for inventory - b
3.0
Contour plot of OFR with 95% DOFR
Frac. Rate of adj. for WIP - a
2.5
2.0
125
115
112
1.5
110
108
1.0
140
107
0.5
0.0
0.0
Fig. 4.
0.5
1.0
1.5
2.0
2.5
Frac. Rate of adj. for inventory - b
3.0
Contour plot of Average System Cost for 95% DOFR
Frac. Rate of adj. for WIP - a
2.5
V. SIMULATION EXPERIMENTS
A. Results and discussions
55
65
0.0
The parameter selection can be on either side of the D-E
line, i.e., either (α > β) or (α < β).
• Case I α > β: Here the delay in adjusting the WIP
discrepancy is less than that for inventory discrepancy.
In the present setting, the quantity of discrepancy in
WIP will always be more than the discrepancy in
inventory. Therefore a correction for a higher quantity
(WIP discrepancy) is made at a higher rate than the
correction for a lesser quantity (inventory discrepancy).
This can cause the WIP and inventory to build up at a
faster rate, but with higher variability. The service level
can improve because of increased average inventory and
at the same time holding cost can also increase.
• Case II α < β: Here the delay in adjusting the inventory
discrepancy is less than that for WIP discrepancy. The
inventory build up in this case can be expected to be
slower than in the previous case leading to less average
inventory and also with less variability. This reduced
variability can influence the cost associated with the
system to become lesser compared to the previous case.
However, it can adversely affect the service level.
The simulation experiments of the system model prepared
in Powersimr 2.5 are conducted to observe the behavior
of the system. The back order cost b1 and holding cost h
are taken as 2 and 1 per period per item respectively. The
demand distribution is assumed iid N (100, 1). The DOFR
are selected to be 95% and 99% and the corresponding
values of safety factor k becomes 1.64 and 2.33 respectively.
Two set of experiments are conducted to obtain the system
performance for the two DOFR values with a wide range
of (α, β) pairs selected over the entire stable region. The
total number of (α, β) pairs selected for the simulation study
is around 270, spread uniformly within the stability region
(Fig. 2). The replication length of simulation is kept as
3650 days (10 years), with the update period as one day,
which is assumed to be sufficient for this study.
2.0
2.0
60
70
1.5
80
95
1.0
90
99
0.5
0.0
0.0
Fig. 5.
0.5
1.0
1.5
2.0
2.5
Frac. Rate of adj. for inventory - b
3.0
Contour plot of OFR with 99% DOFR
there is intersection of the two conflicting objectives. The
selection of (α, β) pairs for higher OFR always lie above the
D-E line, while the selection of parameters for lesser average
cost lie below the D-E line. This confirms the arguments
made in Section IV-B.
The zeros (roots of numerator polynomial) of a transfer
function decide the amplitude of oscillations of the system
response. If any one of the zeros is positioned near one on
the real axis, the amplitude and overshoot increase. From the
system transfer function (27), it is seen that when α → ∞
or β → 0 one of the zeros shifts towards 1. Therefore the
relative positioning of the parameter pairs about the DE line
will generate different amplitudes and settling times. It is
493
Frac. Rate of adj. for WIP - a
2.5
2.0
affecting the service level, (Fig. 3, & Fig. 5). These results
give insights into the criteria for selection of the parameter
pairs for tuning the system and achieving the performance
measures at the desired levels. The results of this study
provide a clear picture of the service level variation within
the entire region of system stability. This helps in selecting
the parameter pairs, (α, β) based on required service level
and cost constraints.
125
115
112
1.5
110
109
140
1.0
0.5
VI. OBSERVATIONS AND CONCLUSIONS
A. Future Work
Selection of forecasting constant ρ as one is a limitation
in this study. The influence of ρ in service level and its sensitivity are to be analyzed in detail. The delays in production
stages also has significant role in the dynamic behavior of
the system, which has to be explored in detail. Selection of
the safety factor, k < Φ−1 (DOF R) can help reduce the
system costs. However the scale of reductions possible in
the average cost, safety factor, and the boundary for (α, β)
pairs to maintain service level can be determined only with
more experiments. A generalization scheme for the parameter
selection will be much more helpful from practical point of
view. These aspects will be dealt with in detail in the future
work.
The system can achieve desired order fill rate only when
the value of fractional rate of adjustment for inventory (β) is
set ≤ 0.7. However, the value of fractional rate of adjustment
for inventory (α) can increase up to 1.2. The D-E Line
(α = β line) forms a lower bound for the parameter setting
to maintain the desired order fill rate. This is equivalent
to saying that the inventory discrepancy can be accounted
up to a maximum of 70% to maintain the service level
(Fig. 3 & Fig. 5). Even though β can be set as low as 0
for α ≤ 1, α = 0 is possible only for β ≤ 4/9. This shows
the inevitability of considering the supply line while making
replenishment decisions.
For any value of desired OFR proper selection of the
parameters will enable achieving OFR above the desired
value. This analysis also shows that for achieving DOFR,
the general replenishment policy, α < 1 and β < 1, is better
than the pure OUT replenishment policy, i.e., α = β = 1.
Interestingly, the average cost plots, Fig. (4) & Fig. (6),
show that the parameters are to be selected below the D-E
line to assure minimal costs. This is in agreement with the
intuitive discussions made in Section IV-B. However, it can
be observed that the same service level can be achieved with
different average costs and safety factor. Eg. for OF R =
95%, (α, β) = (1, 0.5) will be a parameter selection (Fig. 3)
with an average cost ≈ 112 (Fig. 4). If less cost is desired,
(α, β) = (0.5, 0.5) will be a better option with average cost
≈ 107. If safety factor can be fixed for DOF R = 99%,
95% OFR can be achieved by setting (α, β) = (1, 1) with
average cost ≈ 110 (Fig. 5 & Fig. 6).
A proper selection of (α, β) pairs can offer lower cost
as well as higher service levels. It is observed that the D-E
line forms an upper bound for the parameter setting to keep
the average system cost at lower levels. This will also help
in selecting the safety factor, k < Φ−1 (DOF R), without
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0.0
0.0
Fig. 6.
0.5
1.0
1.5
2.0
2.5
Frac. Rate of adj. for inventory - b
3.0
Contour plot of Average System Cost for 99% DOFR
observed that α > β will always increase the amplitude of
oscillation. This setting increases buffer inventory and thus
improves the service level with associated cost (due to higher
inventory levels). The α < β setting reduces the overshoot
and causes smoother inventory build up, resulting in lesser
costs but lower service levels.
R EFERENCES
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