International Journal of Production Research, 2017

Vol. 55, No. 14, 4053–4067, https://doi.org/10.1080/00207543.2016.1245884

Exploring nonlinear supply chains: the dynamics of capacity constraints

Borja Pontea*, Xun Wangb, David de la Fuentea and Stephen M. Disneyb
a
Department of Business Administration, Polytechnic School of Engineering, University of Oviedo, Gijón, Spain; bLogistics Systems
Dynamics Group, Cardiff Business School, Cardiff University, Cardiff, UK
(Received 7 March 2016; accepted 28 September 2016)

While most supply chain models assume linearity, real production and distribution systems often operate in constrained
contexts. This article aims to analyse the consequences of capacity limits in the order-up-to replenishment policy with
minimum mean squared error forecasting under independently and identically distributed random demand. Our study
shows that the impact of this nonlinearity is often significant and should not be ignored. In this regard, we introduce the
concept of a settling capacity, which informs when our knowledge from a linear analysis is a reasonable approximation
in a nonlinear context. If the available capacity is less than the settling capacity, the nonlinear effects can have a
significant impact. We compare the Bullwhip Effect and Fill Rate in constrained contexts to well-established results for
linear supply chains. We reveal the capacity limit acts as a production smoothing mechanism, at the expense of
increasing inventory variability. We proceed to analyse the economic consequences of the capacity constraint and show
that it can actually reduce costs. We provide an approximate solution for determining the optimal capacity depending on
the demand, the unit costs and the lead time.
Keywords: bullwhip effect; capacity planning; fill rate; order-up-to policy; supply chain dynamics

1. Introduction
The Bullwhip Effect is a major source of supply chain inefficiencies due to its harmful consequences including storage,
shortage, labour, obsolescence and transport costs (Lee, Padmanabhan, and Whang 1997). Its strategic importance has
led to a large amount of research over the last two decades. In this regard, the amplification of the variability of orders
has become a widely used metric to assess internal supply chain performance. At the same time, the Fill Rate is a key
measure of customer satisfaction within production and distribution systems (Song 1998). It is popular in high-volume
industries as it measures the demand fulfilment experienced by the customer. That is, the Fill Rate measures the external
supply chain performance. The major challenge for supply chains is to provide high customer service level at low oper-
ating cost. Thus, the complementary indicators, the Bullwhip Effect and the Fill Rate, are popular metrics in the analysis
of supply chains.
Control engineering has been a useful technique in supply chain studies (Dejonckheere et al. 2003). Multiple works
have used this methodology to analyse and improve the dynamic supply chain performance (e.g. John, Naim, and Towill
1994; Disney and Towill 2002; Warburton and Disney 2007; Jaipuria and Mahapatra 2014). This approach is based on
the assumption of system linearity, where supply chains are represented by a set of linear difference or differential equa-
tions.
Linear supply chain models are based on some key assumptions in order to: (1) provide a meaning for the negative
values of variables (e.g. inventories can become negative indicating backlogs or orders can become negative indicating
free returns to suppliers); (2) avoid saturation (e.g. unconstrained capacities); and (3) consider the system has no process
uncertainty (e.g. orders are always fulfilled completely, on time, and with no quality loss or over production). Nonethe-
less, these assumptions could be deemed unrealistic in many practical settings (Spiegler, Naim, and Winker 2012).
Hence, our research objective is to understand the consequences of nonlinear features in production and distribution
systems1.
Due to the complexity involved in nonlinear modelling, this article focuses only on capacity constraints and its
impact on supply chain dynamics via the following research questions (RQ):

RQ1. How do capacity constraints affect the well-established Bullwhip Effect and the Fill Rate supply chain measures?

*Corresponding author. Email: ponteborja@uniovi.es

© 2016 Informa UK Limited, trading as Taylor & Francis Group

4054 B. Ponte et al.

RQ2. Can we use the same key performance metrics in both linear and constrained models?

RQ3. Is there an optimal capacity limit? If so, what does it depend on?

RQ4. What level of capacity would allow managers operating in constrained environments to safely use the results from a
linear analysis?

With this aim, we employ simulation methods supported by statistical techniques. Sterman (2002) highlights the
importance of computer-based simulation by arguing that a fundamental limit of human cognition is our inability to
mentally simulate the dynamics of complex nonlinear systems. When linearity assumptions are removed, complex
dynamic behaviours are revealed (Wang, Disney, and Wang 2014), and the nonlinear effects may even play the
dominant role (Nagatani and Helbing 2004). In this context, mathematical analysis becomes so complex that resorting
to simulation to gain insight has become an essential alternative. Nonetheless, we note that Spiegler et al. (2016a,
2016b) have recently studied the impact of supply chain nonlinearities by developing a methodology, built upon nonlin-
ear control theory, based on (1) simplifying the model and (2) applying advanced linearisation techniques.
Our study is concerned with the classic order-up-to (OUT) replenishment policy (Chen, Ryan, and Simchi-Levi
2000) for three main reasons. First, it is a widely used replenishment method in both academic studies and in the indus-
try (Disney et al. 2013). Second, it is optimal at minimising local inventory costs (Karlin 1960). Third, this policy is rel-
atively simple and operates on a discrete time periodic basis, reducing the complexity of our analysis. In order to gain a
thorough understanding of the nonlinear effects induced by capacity constraints, we study only a single supply chain
echelon.
The structure of this paper is as follows. Section 2 reviews the literature and outlines our contribution. Section 3
summarises a linear supply chain, while Section 4 highlights the key performance indicators and previously known
results. Section 5 presents our nonlinear simulation model, and Section 6 considers the dynamic consequences when fac-
ing a step input and explores both the operational and the financial impact of capacity constraints under stochastic
demand. Finally, Section 7 concludes by revisiting our research questions and suggesting a future research agenda.

2. Literature review and our contribution

Capacity issues have been explored in the literature from different perspectives. Cachon and Lariviere (1999) investi-
gated the relationship between the ordering decisions of the retailer and the capacity choice of the supplier, exploring
both manipulable and truth-inducing mechanisms for capacity allocation. Olhager, Rudberg, and Wikner (2001) argued
that managers need to determine appropriate capacity levels required to support their production plan. In this regard,
two main capacity strategies were identified by Chou et al. (2007): reactive and conservative. Chou et al. (2007) found
that although reactively adjusting the capacity according to a demand forecast was the dominant alternative, the conser-
vative strategy – which entails making capacity decisions only after the organisation is running at full capacity as a con-
sequence of an actual increase in demand – worked better in some circumstances. Cheng et al. (2012) proposed a
production-inventory model for manufacturing systems with capacity constraints. Facing a seasonal demand, they find
an optimal trade-off between inventory cost and capacity utilisation. In this field, the concept of ‘demand-supply balanc-
ing’ has emerged and acquired strategic importance. Coker and Helo (2016) have recently reviewed this notion and its
key role in operations management, as well as successful practical methodologies for balancing demand and supply.
However, works which consider the implications of capacity limits on the dynamic supply chain performance are
rather scarce. Of those that do exist, the questions we have raised have not been addressed, but these works do offer
some insight and background to our problem.
Evans and Naim (1994) explored capacity-constrained supply chains and noticed that while capacity limits tend to
decrease inventory service levels, capacity limits generally led to improved dynamic performance. Using Multi-Attribute
Utility Theory, they accounted for eight different indicators, concluding that the unconstrained system did not always pro-
duce the best response. This intriguing finding warrants further investigation. Since then, several works have assessed the
impact of limited capacity on supply chain performance, often using discrete event simulation, which we now review.
Helo (2000) highlighted the damaging consequences of capacity constraints on supply chain agility. Suwanruji and
Enns (2006) compared Materials Requirement Planning, Kanban and the OUT policies, examining how capacity con-
straints affected their performance. They found that the relative performance ranking depends on the capacity limit.
Studying a model with finite capacity, Wikner, Naim, and Rudberg (2007) showed that production planning and control
systems should proactively regulate backorders. Wilson (2007) studied a short duration capacity loss due to transportation
disruption highlighting a negative impact on the Fill Rate, although a dynamic improvement was sometimes generated.
 International Journal of Production Research 4055

Although these previous works did not mention the concept directly, the dynamic improvement induced by capacity
constraints can be explained by a mitigation in the Bullwhip Effect. Cannella, Ciancimino, and Márquez (2008) showed
that supply chain performance can be enhanced as capacity restrictions limit the ability to overproduce and have a
dampening effect on order variability. They studied six different capacity levels revealing that – in the presence of infor-
mation distortions – increasing the capacity can lead to higher costs. Chen and Lee (2012) and Li et al. (2016) also
highlighted that a finite capacity had a smoothing effect on the order variance, as the receipts tend to be less variable
than the original orders. Nevertheless, Nepal, Murat, and Chinnam (2012) and Hussain, Khan, and Sabir (2016) con-
cluded that capacity limits increased the variability of net stock, reducing Fill Rates. Yang et al. (2014) analysed inven-
tory models with set-up costs and concluded that the optimal policy depended on the capacity restriction.
In summary, the literature has shown that constraining capacity allows the supply chain to mitigate the Bullwhip
phenomenon (increasing internal performance) but at the expense of reducing inventory service levels (decreasing exter-
nal performance). This insight hints at the existence of an optimal, finite, capacity limit. However, little previous work
has been directed towards this idea, leading us to pose the four aforementioned, insufficiently explored, research ques-
tions.
The contribution of this paper is to increase our understanding of nonlinear supply chains by analysing the relation-
ship between the system performance and its capacity. We confirm the existence of an optimal capacity limit from a
financial perspective and identify it, which does not appear to have been previously obtained. We introduce the concept
of a settling capacity to indicate when the knowledge derived from a linear analysis can be applied in a constrained
environment. We also question if the same metrics that have proven to be so useful in linear models, such as the convex
weighted sum of the square root of the orders and inventory variances (see Section 4.2), can still be used in a nonlinear
setting.
Having briefly summarised the pertinent literature on the subject and highlighted our research aims, before consider-
ing the capacity-constrained supply, we will next define the linear model in Section 3 and review the known linear
results in Section 4.

3. Description of a linear supply chain and its main assumptions

The sequence of events shown in Figure 1 describes the discrete time operation of a supply chain (Disney et al. 2016).
First, both product (material flow) and demand (information flow) are received from the upstream and downstream eche-
lons, respectively. Then demand is satisfied and both the net stock and the work-in-progress (WIP) levels are updated.
Finally, demand is forecasted and orders are placed. Figure 1 focuses on a single supply chain echelon, but multiple
echelons can be linked together to form an entire supply chain. Notice we highlight the time lag between shipping and
receiving the product. If this lead time is zero, the product will be received the moment after the order is placed (at the
beginning of the next period) and that receipt will influence the order placed in the next period.
A common replenishment algorithm is the OUT policy (Disney and Lambrecht 2008). This inventory model can be
expressed in words as ‘let the production targets be equal to the sum of a forecast of demand ( D b t ), plus the discrepancy
between target (TNS) and actual (NSt ) levels of net stock (on-hand inventory), plus the difference between the target
(Wb t ), and actual (W t ) WIP (on-order inventory)’. Formally, the order quantity at time t (Ot ) is given by

Figure 1. The sequence of events in the supply chain model.

4056 B. Ponte et al.
 
b t þ ðTNS  NSt Þ þ W
Ot ¼ D b t  Wt : (1)

The receipts (Rt ) are the orders that have been placed T P þ 1 periods ago, where T P is the lead time,
Rt ¼ OtTP 1 : (2)
This linear model assumes that orders are always fulfilled after a fixed lead time (key assumption I). Both orders and
receipts are real numbers. When orders are negative, products are returned to the supplier without cost (key assumption
II). Moreover, orders are unconstrained. That is, unlimited production and distribution capacities are assumed (key
assumption III).
The actual net stock (NSt ) is the accumulated difference between receipts and demand. The inventory balance
equation is given by
NSt ¼ NSt1 þ Rt  Dt : (3)
The inventory can be either positive or negative. The former refers to storage, while the latter refers to back orders.
Hence, demand that is not fulfilled in a period is backlogged and will be fulfilled as soon as on-hand inventory becomes
available (key assumption IV). In addition, unlimited storage capacity is assumed (key assumption V), and the model
does not consider defective products, quality loss or random yields (key assumption VI).
The WIP (W t ) represents the inventory in-transit between echelons and is the accumulated sum of the difference
between the previous order and the current receipt. With (2), W t can also be modelled as the sum of the orders that
have been placed but not yet received via
X
TP
Wt ¼ Oti ¼ Wt1 þ Ot1  Rt : (4)
i¼1

Note that when T P ¼ 0, items are received before the next order is generated, hence the WIP is zero.
Regarding the target levels, it is usual to consider that a constant safety stock, a TNS and a variable target WIP exist
(John, Naim, and Towill 1994). Under this approach, the TNS is a decision variable to be optimised, while the desired
WIP ( Wb t ) is a forecast of future consumption over the lead time given by

bt ¼ D
W b t  TP : (5)
It is a frequent practice in inventory management literature (e.g. Schneeweiss 1974; Costas et al. 2015; Disney et al.
2016) to consider demand (Dt ) to be an independent and identically distributed (i.i.d.) random variable following a nor-
mal distribution with a mean of lD and a standard deviation of rD (Dt  N ðlD ; rD Þ). In this case, the minimum mean
b tþx ) is given by its conditional expectation (Disney
square error (MMSE) forecast of the demand for all future periods ( D
et al. 2016):
b tþx ¼ lD :
8x; D (6)
Having described a linear supply chain model as a base case, we will now review its known performance in Section 4
in preparation for our nonlinear supply chain study in Sections 5 and 6.

4. Performance indicators and background in the linear supply chain

In this section, we describe the main operational and financial indicators used to measure supply chain performance. In
addition, we summarise the main known results for the linear model as a preliminary step to analyse the nonlinear
system.

4.1 Operational metrics

Wang and Disney (2016) provide a recent review of Bullwhip literature. They assert that this phenomenon is usually
measured through the Bullwhip (BW) ratio defined as the variance of the orders divided by the variance of demand,
r2O
BW ¼ : (7)
r2D
The net stock amplification (NSAmp) measure is the ratio of the variance of net stock to the variance of the demand,
 International Journal of Production Research 4057

r2NS
NSAmp ¼ : (8)
r2D
Order variability mainly contributes to production costs, while inventory variability determines the echelon’s ability to
meet demand in a cost-effective manner. This is the key trade-off faced by supply chain managers (Disney et al. 2006),
and the reason why both the BW and the NSAmp ratios are commonly used to measure supply chain performance.
Under i.i.d. demand, the OUT policy with MMSE forecast acts as a ‘pass-on-orders’ strategy, similar to the Kanban
system often advocated by the lean production community (Disney et al. 2016). That is, the orders in each period are
simply the current demand (Ot ¼ Dt ). Consequently, BW ¼ 1 and NSAmp ¼ 1 þ T P .
The NSAmp ratio is commonly used as a p proxy for the ffiinventory service level in Bullwhip studies, as in linear sys-
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
tems inventory availability is proportional to NSAmp  r2D (Chen and Disney 2007). Furthermore, the Fill Rate, b, is
decreasing in NSAmp. This is a popular metric in high-volume industries as it expresses the proportion of demand that
is immediately fulfilled from net stock (Disney et al. 2015). The fulfilled demand (F t ) is calculated by
8
> 0 if Dt  0;
<
0 if Rt þ NSt1  0;
Ft ¼ (9)
: Rt þ NSt1 if 0  Rt þ NSt1  Dt ;
>
Dt if Rt þ NSt1 [ Dt :
Note that Rt þ NSt1 is the inventory available after the deliveries have been received from the upstream node that can
be used to fulfil current demand. Negative demand means that net returns from customers are larger than those deliv-
ered. Thus, b is the ratio of mean fulfilled demand to the mean of the demand that can be satisfied, ðDt Þþ :
EðFt Þ
b¼ : (10)
EððDt Þþ Þ
Here EðÞ is the expectation operator and ðxÞþ ¼ maxð0; xÞ is the maximum operator. The calculation of b is complex
as it is influenced by a number of variables (the lead time, safety stock, mean and standard deviation of the demand,
demand correlation, and the correlation between demand and inventory) and is often only studied in linear systems
(Sobel 2004; Disney et al. 2015).

4.2 Financial metrics

Assuming the main goal of a firm is to maximise financial gain, many authors have used economic measures to assess
supply chain performance. In the literature, inventory and order costs are often considered jointly in order to evaluate
the performance of a replenishment strategy (Disney and Grubbström 2004).
The classic approach to modelling inventory costs (IC) is to assign a unit inventory holding cost (h) and a unit
backlog cost (b) (Kahn 1987). Note that a backlog is incurred when the net stock is negative; otherwise, a holding cost
is incurred. This inventory cost model can be summarised as
IC ¼ h  EððNSt Þþ Þ þ b  EððNSt Þþ Þ: (11)
As the demand is assumed to be normally distributed, and a linear system exists, the inventory levels will also be nor-
mally distributed and the inventory costs will be governed by the TNS and minimised when
 
b
TNS ¼ rNS  U1 (12)
bþh
(Hosoda and Disney 2009). In (12), U1 ðÞ is the inverse of the cumulative distribution function of the standard normal
distribution.
There is a wide range of costs functions that can be used to assign order costs (OC), see Disney, Gaalman, and
Hosoda (2012) for a review. A common order cost function considers a certain guaranteed capacity (GC) is available in
each period. If less capacity than GC is required, labour stands idle for a proportion of the period, hence an opportunity
cost is incurred (n per unit). If more than GC is needed, labour works overtime at a higher unit cost (of p per unit). This
model can be described by
OC ¼ n  EððGC  Ot Þþ Þ þ p  EððOt  GCÞþ Þ: (13)
First-order conditions reveal that to minimise (13), we should set GC to
4058 B. Ponte et al.
 
 1 p
GC ¼ EðOt Þ þ rO  U : (14)
pþn
The minimised total costs (TC), i.e. the minimised sum of inventory and order costs, per period are linearly related to
the standard deviation of orders and inventory (Kahn 1987; Disney, Gaalman, and Hosoda 2012) according to
minðTCÞ ¼ KNS  rNS þ KO  rO ; (15)

ffiffiffiffi  eerf ðbþh1Þ and K O ¼ g2 ðn; pÞ ¼ p ffiffiffiffi  eerf ð1nþpÞ : Here erf 1 ðÞ is the inverse error
1 2b 2 1 2
2n
where K NS ¼ g1 ðb; hÞ ¼ pbþh
2p
nþp
2p
function. Thus, the objective function J can be formulated as a convex weighted sum of the square root of the BW and
the NSAmp ratios by
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi
J ¼ KNSA  NSAmp þ KBW  BW; (16)
where K NSA ¼ K NSKþK
NS
O
and K BW ¼ 1  K NSA are used to express the weight (relative importance) of each indicator. If J
is minimised, total costs are minimised in the linear model.
Having defined the performance measures commonly used in linear models, we will now define our capacitated sup-
ply chain model, before analysing its performance in Section 6.

5. The capacity-constrained supply chain: simulation model

In this section, we consider the impact of constrained capacity, one of the key nonlinearities highlighted in Section 2. In
this case, a decision variable must be added – the capacity limit (CL), defined as the maximum quantity of product that
the firm can receive in each period2. To incorporate this nonlinearity into the OUT policy, the most common approach
is to place a limit on the order quantity (Cannella, Ciancimino, and Márquez 2008), by replacing (1) with
n   o
Ot ¼ min D b t þ ðTNS  NSt Þ þ W b t  Wt ; CL : (17)

With the aim of studying the impact of the capacity limit on the supply chain, we have developed a nonlinear simula-
tion model, where (2)–(6) and (17) represent the system equations following the sequence of events in Figure 1, (7)–
(10) have been implemented as operational metrics, and (11), (13), (15) and (16) have been used as financial indicators.
We have employed MATLAB R2014b to construct a simulation model, whose parametric space is formed by 10 fac-
tors that can be categorised into: (a) Three external parameters: mean and standard deviation of the demand, and back-
log unit cost, (b) Four internal parameters: lead time; and holding, opportunity and over-time unit costs and (c) Three
decision variables: target net stock, guaranteed capacity and the capacity limit.
Hence, the general problem can be defined as
½BW; NSAmp; b; J ; TC ¼ f ðlD ; rD ; TP ; TNS; CL; GC; b; h; n; pÞ þ n; (18)
where n is the unexplained part of the response.

6. Analysis of the capacity-constrained supply chain

Having defined our capacitated supply chain model, we now consider its dynamic performance in Section 6.1, its opera-
tional (stochastic) performance in Section 6.2 and its financial performance in Section 6.3.

6.1 Dynamic analysis of the step response

This section compares the step response of the capacity-constrained supply chain to the linear supply chain in order to
expose the differences in the two scenarios. This classic rich picture provides one with a firm understanding of dynamic
behaviour (Disney and Towill 2003). Note that in this step demand case, the forecast D b t , as this exaggerates the differ-
ences in the dynamic performance, facilitating our discussion.
Figure 2 depicts the inventory and order response of both the linear and nonlinear systems when facing a 10-unit
step in demand at time t ¼ 0. The outputs have been obtained for TNS ¼ 10, while the lead time has been set to
T P ¼ 1 and T P ¼ 4. In each case, we have used two different values of the capacity limit in order to study the effect of
this variable. The inventory and order response of the linear OUT policy quickly return to steady state at the expense of
 International Journal of Production Research 4059

Figure 2. Step response (net stock and orders) for the linear and the capacitated system. (a) Net stock for Tp = 1. (b) Orders for
Tp = 1. (c) Net stock for Tp = 4. (d) Orders for Tp = 4.

generating the Bullwhip Effect. Notice for T P ¼ 1 the largest order is 30, while for T P ¼ 4 it is 60. For this reason, the
capacity limit has been set to 15 and 25 when T P ¼ 1, and 20 and 50 when T P ¼ 4, in the nonlinear system.
It can be seen from Figure 2(b) and (d) that the difference between the linear orders and the capacity limit at t = 0
is added to the nonlinear orders in subsequent periods. Under these circumstances, we can imagine the BW ratio (and
order costs) would be desirably reduced if demand was instead a random variable rather than a step input.
Figure 2(a) and (c) shows the negative impact of capacity constraints on the inventory. The inventory level takes
much longer time to return to the steady state in the constrained supply chain. As the inventory level is lower in this
transient phase, we expect the supply chain is more vulnerable to demand shocks and may suffer from reduced Fill
Rates.
We conclude that the capacity constraint acts to reduce the BW ratio at the expense of increasing the NSAmp ratio
and decreasing the Fill Rate. The more severe the capacity constraint, the more accentuated these effects become (how-
ever, CL [ lD is required to ensure that the system is stable). This insight is consistent with prior research (Cannella,
Ciancimino, and Márquez 2008; Nepal, Murat, and Chinnam 2012, Lin, Jiang, and Wang 2014). In the following sub-
sections, we aim to further explore the impact of the capacity constraints via the system response to a random demand
and an economic analysis.
4060 B. Ponte et al.

6.2 The operational impact of capacity constraints

This section analyses the consequences of the capacity limit via the Bullwhip Effect and the Fill Rate3. Note, since the
cost parameters and the guaranteed capacity do not have an impact on these indicators, the simulation problem in (18)
can be simplified to ½BW; NSAmp; b ¼ f ðlD ; rD ; T P ; TNS; CLÞ þ n.
In the linear system, for the OUT policy under i.i.d. demand with MMSE forecasting, BW ¼ 1 and
(NSAmp  T P Þ ¼ 1 (Disney et al. 2016). That is, these indicators do not depend on the variables (lD ; r; T P ; TNS). In
order to explore the behaviour of these indicators, we first wonder whether these relationships hold true in the nonlinear
environment. Preliminary tests revealed the following hypothesis for the capacitated model:

Hypothesis 1. TP and TNS have no significant effect on BW and (NSAmp − TP).

To investigate this premise, we used a full factorial design of experiments (DoE) with two levels per factor
(lD ¼ f100; 200g; rD ¼ f20; 80g; T P ¼ f1; 4g; TNS ¼ f10; 50g; CL ¼ f210; 300g). We chose this parameter set as it
covers a wide range of supply chain settings (e.g. the coefficient of variation CoV ¼ rD =lD varies between 10% and
80%, and the coefficient of capacity CoC ¼ CL=lD varies between 1.05 and 3).
Statistically analysing the 32 runs reveals the p-value associated with both variables (T P and TNS) in the parameter
estimates of a least-squares fit is significantly higher than 0.05 (even close to 1) for both outputs, see Table 1. This
means that the impact of T P and TNS on BW and ðNSAmp  T p Þ can be ignored.
This allows us to study BW and ðNSAmp  T P Þ as a function of the remaining three factors (lD ; r; CL). Since both
the standard deviation and the capacity limit can be related to the mean demand – by means of the coefficient of varia-
tion (CoV ¼ rD =lD ) and the coefficient of capacity (CoC ¼ CL=lD ) – further experimentation led us to consider if
these could be combined to reduce the problem to two dimensions. We developed the following hypothesis:

Hypothesis 2. BW and (NSAmp − TP) can be expressed as a function of CoC and CoV.

To study this premise, we followed the same methodology we used to test Hypothesis 1. We laid out a full factorial
DoE with three factors and three levels (lD ¼ f100; 200; 400g; CoC ¼ f1:1; 1:2; 2g; CoV ¼ f0:1; 0:2; 0:4g). In this case,
the p-value of lD obtained for both indicators is greater than 0.05, see Table 1, leading us to fail to reject H2 at 5% sig-
nificance. This shows that the impact of lD is not significant when the ratios CoC and CoV are considered.
In line with RQ1, the results from H1 and H2 reveal that, in the capacity constrained supply chain, BW and
ðNSAmp  T P Þ can be considered to be a function of CoC and CoV, i.e. BW ¼ f 1 ðCoC; CoVÞ and
ðNSAmp  T P Þ ¼ f 2 ðCoC; CoVÞ. Thus, from this point on we investigate the nonlinear supply chain for different val-
ues of CoC and CoV. Figure 3(a) and (b) displays the BW and NSAmp as functions of both coefficients4. In each
graph, the abscissa represents the CoC (CoC  1 is required for stability). We have plotted the curves for three different
values of the CoV: 10, 20 and 40%, as this parameter is typically less than 50% in practice (Dejonckheere et al. 2003).
Figure 3a reveals that the nonlinear model response tends to that of the linear model – that is, BW = 1 and
ðNSAmp  T P Þ ¼ 1 – when the capacity limit becomes sufficiently large. If the capacity frequently constrains the

Table 1. Statistical analysis of the results of the DoEs performed to verify Hypotheses 1 and 2.

Contrast Factor F (BW) p–v (BW) F (NSAmp*) p–v (NSAmp*)

Hypothesis 1 Intercept 5.677 0.025** 1.041 0.317

μD 36.406 0.000** 7.669 0.010**
σD 7.348 0.012** 6.215 0.019**
TP 0.000 0.997 0.001 0.973
TNS 0.000 0.994 0.003 0.956
CL 33.140 0.000** 7.632 0.010**
Overall model 15.379 0.000** 4.304 0.006**
Hypothesis 2 Intercept 1.676 0.208 16.060 0.001**
μD 0.000 0.985 0.006 0.938
CoC 43.173 0.000** 17.164 0.000**
CoV 104.049 0.000** 16.862 0.000**
Overall model 49.074 0.000** 11.494 0.000**

*We refer to the difference between the NSAmp ratio and TP, as in the linear model.
**The significant factors at the confidence level 95% are highlighted.
 International Journal of Production Research 4061

Figure 3. BW and NSAmp functions in the capacity constrained system. (a) BW ratio versus CoC and CoV, with dashed lines in
BW = {0.95, 0.99, 1}. (b) (NSAmp−TP) versus CoC and CoV, with dashed lines in (NSAmp−T P ) = 1.

orders, Bullwhip is reduced. Figure 3(a) also shows that the BW ratio approaches zero when CL ¼ lD and increases
exponentially in the CoC. Notice, larger CoV leads to greater reductions in Bullwhip.
Figure 3(b) shows that the capacity limit reduces the order variance, but increases the inventory variability (the
greater the CoV, the greater the increase in inventory variance), which threatens the inventory service level. The trade-
off between the BW and the NSAmp ratios suggests that an optimal capacity level exists, which we will explore eco-
nomically in Section 6.3.
As seen, when the capacity does not constrain the system, the linear assumption can be maintained. Thus, in order
to investigate RQ4, we introduce the concept of a settling capacity, defined as the least capacity that allows us to
assume the behaviour of the nonlinear system can be predicted by the results from linear systems with a certain level of
confidence (95 or 99%). Figure 3(a) suggests that the settling capacity is approximately linear in the CoV leading to the
following expression for the settling capacity, CLs,
CLs ð95%Þ lD þ 1:86  rD ; (19)

CLs ð99%Þ lD þ 2:41  rD ; (20)

Figure 4. Fill Rate in the capacitated system for Tp = 1 and TNS = 0. The vertical dotted lines represent the settling capacity (95%
level of confidence) for the three CoVs.
4062 B. Ponte et al.

for normal i.i.d. demand with MMSE forecasting5. If we use linear knowledge in a constrained supply chain, where the
available capacity is lower than the settling capacity, we may incur significant approximation errors.
Having studied the internal performance, we now focus on the external impact of capacity constraints by evaluating
the Fill Rate. Figure 4 shows the consequences of the capacity limits on the Fill Rate. The Fill Rate is highly influenced
by T P and TNS in the linear system and we have selected T P ¼ 1 and TNS ¼ 0 in these graphs.
Completing the analysis of RQ1, Figure 4 shows how capacity constraints reduce Fill Rates revealing that when the
CoV increases, the Fill Rate decreases and becomes more sensitive to the capacity limit. It can be seen that when
CL [ CLs the linear assumption can be maintained. However, when CL\CLs , the impact of the nonlinearity must be
considered as the Fill Rate can be seriously affected.
The impact of the capacity constraints on the Fill Rate is consistent with its impact on the NSAmp ratio. Nonethe-
less, we must statistically confirm whether (similar to the linear system) reducing the NSAmp ratio also leads to
increased Fill Rates in this nonlinear system. Thus, the following hypothesis has been developed for RQ2:

Figure 5. Optimal CoC in terms of J in the capacity constrained system. (a). For Tp = 1. (b). For Tp = 4.

Figure 6. The relationship between the optimal CoC in terms of J and in terms of TC in the capacity constrained system.
 International Journal of Production Research 4063

Hypothesis 3. There is a negative correlation between NSAmp and the Fill Rate.

To validate this hypothesis, as the normality of the continuous inventory variable cannot be verified, a Spearman corre-
lation test was carried out. The p-value was found to be 0.000, hence the idea that the correlation is due to random sam-
pling can be rejected: we can assume a correlation exists. Spearman’s rho (−0.810) shows that this correlation is
negative. That is, in the capacity-constrained supply chain where backlogs are allowed, reducing the NSAmp ratio also
increases the Fill Rate.

6.3 The economic impact of capacity constraints

This section explores the existence of an optimal capacity – the focus of RQ3. When using the optimal values for TNS
and GC from (12) to (14), Figure 5(a) (for T P ¼ 1) and 5(b) (for T P ¼ 4) represent the CoC that minimises the objec-
tive function J in the capacitated system over the whole range of permissible K BW ¼ 1  K NSA 2 ½0; 1 for the three dif-
ferent values of the CoV6.
As expected, the BW ratio (J when K BW ¼ 1) is minimised when CoC = 1, i.e. CL ¼ lD . On the other hand, the
NSAmp ratio (K BW ¼ 0Þis minimised when CL ¼∞. Between these points, the optimal CoC can be found, which is
decreasing in K BW , increasing in CoV and decreasing in T P , see Figure 5. For example, if the supply chain demand
follows a normal distribution with rD ¼ 100; lD ¼ 20 (CoV ¼ 20%) and K BW ¼ 0:3, the optimal capacity limit is
approximately 123 (CoC 1:23 from Figure 5(a)) if T P ¼ 1 and 118 (CoC 1:18 from Figure 5(b)) when T P ¼ 4.
Under these circumstances, they key question is: Does minimising J lead to minimising the economic metric TC in
the capacitated system? To understand this, we conducted another simulation study focused on supply chain costs. This
leads to the following hypothesis, in line with RQ2, being revealed:

Hypothesis 4. Optimising J leads to minimising TC.

We have explored different scenarios based on sets of values of the relative weight of the BW ratio (K BW , which
depends on the four unit costs), the lead time (T P ) and the coefficient of variation (CoV, by varying the standard devia-
tion of the demand while lD ¼ 100). With the aim of analysing this premise, we developed a full factorial DoE with
three levels per factor (K BW ¼ f0:25; 0:5; 0:75g; T P ¼ f1; 2; 4g; CoV ¼ f10%; 20%; 40%g)7, which we assume ade-
quately covers the parameter space.
We have simulated the resultant 27 different scenarios to capture both the CoC that minimises J and the CoC that
minimises TC. With these data, a Wilcoxon signed-rank test (as normality is not necessary for the dependent variables)
was performed to evaluate the difference between these paired observations. The p-value obtained was 0.000 in both
cases, so we clearly reject the null hypothesis (equality of means). Therefore, we cannot assume that, as in the linear
system, the CoC that minimises J also minimises TC. Nevertheless, the confidence interval (95%) for the difference
between both values is (0.0062, 0.0187). Thus, even though we have not found the same relationship that exists in the
linear system, the difference between both CoCs is rather small. Figure 6 represents the 27 scenarios in a scatter plot
that represents both CoCs (note that the dash-dotted line shows the equality). This situation allows us to consider the
optimal CoC from Figure 5 to be a good approximation to minimise the sum of inventory and order costs depending on
the demand, the unit costs and the lead time.
As an example, Figure 7(a), (c) and (e) plot J and TC obtained from simulation of three scenarios defined by:
ðK BW ¼ 0:25; T P ¼ 1; CoV ¼ 20%Þ, ðK BW ¼ 0:5; T P ¼ 2; CoV ¼ 40%Þ, and ðK BW ¼ 0:75; T P ¼ 4; CoV ¼ 10%Þ8. As
previously demonstrated, the CoC that minimises J tends to be slightly lower than the CoC that minimises TC. Notice
that the corresponding TNS and GC have also been provided in Figure 7(b), (d) and (f).
Importantly Figure 7(a), (c) and (e) – together with the remaining scenarios – confirm the existence of an optimal
capacity limit in the capacitated supply chain. That is, selecting an appropriate capacity reduces total costs and the
unconstrained system is bettered (recall the performance of the linear system can be ascertained by letting CL ! 1).
For example, in Figure 7(e) the costs in the linear supply chain are approximately 57, in the nonlinear system these can
be reduced to approximately 42.5, a 25% decrease.
These results suggest that capacity limitations stop unnecessarily large orders being issued and this has some eco-
nomic value. The decrease in the Fill Rate can be outweighed by the consequences of improving the operational perfor-
mance of the supply chain. Furthermore, higher values of K BW lead to greater percentage reductions in costs as the
main advantage of constraining orders is to alleviate the Bullwhip phenomenon.
4064 B. Ponte et al.

Figure 7. Economics of the capacitated system. (a). J and TC for KBW = 0.25. (b). TNS and GC for KBW = 0.25. (c). J and TC for
KBW = 0.5. (d). TNS and GC for KBW = 0.5. (e). J and TC for KBW = 0.75. (f). TNS and GC for KBW = 0.75.
 International Journal of Production Research 4065

7. Conclusions and managerial implications

We have considered the impact of capacity constraints on supply chain performance, a problem largely untouched by
the literature due to the complexity of the analysis involved. The supply chain we explored used the OUT replenishment
policy and MMSE forecasting to satisfy an i.i.d. demand drawn from a normal distribution. For the four research ques-
tions we posed in the introduction, we found the following answers:

RQ1. Capacity constraints help to reduce the Bullwhip Effect as they have a smoothing effect on the orders. However, they
increase inventory variability and reduce the achieved Fill Rate. In the supply chain scenario analysed, both the BW and
(NSAmp – TP) can be expressed as functions depending exclusively on the CoV and on the CoC; i.e. they are largely unaf-
fected by the lead time, safety stock, and the mean demand. We have provided a graphical relationship for these dependencies.

RQ2. Similar to the linear supply chain, we have verified that reducing the NSAmp ratio leads to improving the Fill Rate in
the capacitated system. The BW ratio has also proven to be a powerful indicator of the economic consequences of order vari-
ability. However, in contrast to a linear system, an optimisation based on a convex weighted sum of the standard deviation of
order and inventory does not result in minimal costs in the capacity constrained system. Nonetheless, the difference between
the optimum capacity in terms of both convex functions has proven to be small.

RQ3. We have demonstrated that there is an optimal capacity limit that balances Bullwhip (capacity) and inventory costs. For
random demand, the optimal capacity limit depends on the first and second moments of demand, a function of the inventory
holding and backlog costs, and the normal and overtime production costs. We have provided an approximation for the optimal
coefficient of capacity that accounts for the lead time and the coefficient of variation.

RQ4. The estimated settling capacity gives an indication of when managers can safely use the results from a linear analysis
with a certain degree of confidence.

Managerially, we have revealed some practical benefits to capacity constraints exist. Capacity limits can prevent unnec-
essarily large orders being issued, mitigate the Bullwhip phenomenon and reduce order costs. This reduction in order
variability occurs at the expense of incrementing the net stock variance; consequently, decreasing the Fill Rate and
increasing inventory costs. In this regard, our investigation revealed the existence of an optimal capacity in the capaci-
tated system which may significantly outperform the linear system. Thus, supply chain managers can enjoy significant
savings by viewing the capacity as a decision variable. We also provided an approximate solution for the optimal capac-
ity depending upon the unit costs, the lead-time and the demand variability. This optimal capacity has shown to be
decrease in the relative importance of the variance of orders in comparison with the variance of inventory, increase in
the demand coefficient of variation and decrease in the lead time.
Further research could be conducted to explore this nonlinear system since the results from linear models can be per-
ilously inaccurate in some settings, perhaps with the nonlinear control theory techniques that have recently been applied
to supply chains, see Spiegler et al. (2016a, 2016b), or with Markovian decision processes, see Li, He, and Wu (2016).
Another line of future work could consider the impact of capacity constraints when the proportional OUT policy (Dis-
ney et al. 2016) is used to place replenishment orders. Finally, the study of other nonlinearities, such as non-negative
orders (forbidden returns) and non-negative inventories (lost sales), may be worthy of further research. Our experience
suggests that inventory restrictions lead to rich dynamic behaviours.

Disclosure statement
No potential conflict of interest was reported by the authors.

Funding
This work was supported by Government of the Principality of Asturias [grant number BP13011].

Notes
1. We use the term ‘nonlinear supply chain models’ to refer to supply chain models that include at least one nonlinearity. Herein we
specifically studied the nonlinearity introduced by placing an upper limit on the order quantity.
2. Note the difference between the decisions variables CL and GC. CL is a capacity limit that cannot be surpassed by the order rate
at any time (i.e. ∀t, CL ≥ Ot ). GC is the guaranteed capacity which can be exceeded by working overtime or by using a subcon-
tractor at a higher cost. Naturally, CL ≥ GC.
4066 B. Ponte et al.

3. Sections 6.2 and 6.3 are derived from simulating 201,000 time periods. The first 1000 periods were discarded in order to avoid
any possible impact of initial conditions. Stability of the response and consistency of the results were checked via individual and
range control charts and Levene’s test for equality of variance after running two repetitions of each test. The model has been vali-
dated using known results from the literature for the linear model (i.e. the nonlinear model when CL = ∞).
4. In Figure 3(a), (b) and 4 CoC increases in increments of 0.02.
5. To obtain (19) and (20), we used regression to approximate the relationship between BW and CoV (BW was chosen as we
noticed this was the most sensitive of the three measures, see Figures 3(a), (b) and 4). After testing several alternatives, we found
that BW 1  elCoCa ð1 þ sinðu  CoCa Þ, where CoCa = 1 - CoC, and l ¼ f19:6271; 3:70803; 1:66262g and
u ¼ f8:76281; 2:42235; 1:32266g when CoV ¼ f10%; 20%; 40%g; respectively. We then set BW ¼ f0:95; 0:99g and solved
for CoCa for each of the three CoV levels. Using regression again, we then found the following linear relationships were a good
approximation: CoC(0.95) ≈ 1 + 1.85905 ⋅ CoV, and CoC(0.99) ≈ 1 + 2.41286 ⋅ CoV. Finally, multiplying CoC throughout by μD
provides (19) and (20).
6. The convex function J has been obtained in the capacity-constrained system via simulation.
7. We have employed the following combinations of costs: for KBW=0.25, {b = 6, h = 3, n = 1, p = 2}; for KBW = 0.5, {b = 4,
h = 2, n = 2, p = 4}; and for KBW=0.75, {b = 2, h = 1, n = 3, p = 6}.
8. Figures 6a, 6b, 6c, 6d, 6e, and 6f span from CL=μD to CL=CLS(95%). These graphs have been obtained via simulation, where
CL increases in increments of 0.25. Each point plotted is the average of ten simulation runs.

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