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Horizontal recursion in soft OR
Abstract
In operational research (OR), the concept of recursion explains particular relationships
between modelled systems. It clarifies how the same system properties are replicated vertically
across hierarchically interdependent units, meaning these units should be amenable to the same
analytical conventions. OR views recursion as hierarchical and therefore does not consider these
properties in a horizontal sense. This paper uses theory from other disciplines to develop criteria
that define recursive modelling for soft OR as vertical or horizontal. Empirical data was captured
using WASAN to improve efficiency in a police force customer contact department. Four units were
modelled using WASAN, and additional analysis using recursion was conducted to understand the
horizontal interdependence across these four units. Feedback from participants suggests the
horizontal recursion analysis provided valuable insights beyond that of individual models.
Keywords: Problem structuring; Recursion; Horizontal recursion; Soft OR; Meta-system modelling
Introduction
When applying soft OR modelling approaches, the complexities and interdependencies often
associated with a problem context can make it unclear how far to model and what level of detail is
required. To address this, Ackoff (1979a) advocated the principle of expansionism—that is, to better
understand the analysed system (the system-in-focus), one should model the surrounding systems.
Expansionism can be achieved by operationalising the concept of recursion, which helps modellers
explain the relationship between complex, hierarchically interdependent units and represent them
across multiple models. The relationship between recursive models is usually understood as vertical,
in that a model is related to higher-level or lower-level equivalents. For example, Figure 1 shows 12
vertical levels of recursion within the Chilean Government (Hoverstadt, 2008). At each recursive
level in Figure 1 the same structural and analytical rules apply, so a model can be built at any level
using the same conventions with linkages established across models at different levels. In this paper,
we expand the understanding of recursion to include both vertical and horizontal recursion. In
horizontal recursion, a recursive relationship can exist between models on the same hierarchical
plane. The aim of horizontally recursive modelling is to provide a new way for modellers to explain
and represent relationships between systems, thereby giving decision makers more insight into
problem situation and the cascading effects of possible solutions.
Although there are some examples of recursive modelling in soft OR, they are not
recognised as such, and thus the field has not fully exploited the potential of this approach. The
strongest presence of recursion in OR is in the viable systems model (VSM) (Beer, 1981), for which
the benefits of vertical recursion in modelling are well documented. First, building vertically
1
recursive models “allows [the] elegant representations of organization” (Jackson, 2003, p. 87); when
systems are linked recursively using the same rules and structure, visual representations that
illustrate these connections are more appealing and effective than modelling each system in
isolation, e.g., Figure 1 represents more complexity than a single “Whole Nation” model. A single
large model such as the “Whole Nation” model can encompass the meta-system of the lower level
models, but emergent properties of the individual systems in that model would be harder to identify
(Tejeida-padilla and Badillo-pin, 2010). Second, Beer (1984) claims that understanding vertical
relationships between systems is critical on the basis of “Hegel's Axiom of Internal Relations: the
relations by which terms [or in this case, recursions] are related are an integral part of the terms [or
recursions] they relate” (p. 16). A system behaves as it does because it is linked to other systems in
the ways that it is. So, studying the relationships between systems (the notion of holism) provides
more insight than independent analyses of those units. Recursion allows a bounded system to be re-
opened (Beer, 1999), allowing the representation of interdependence of different systems (Jackson,
2003). As Beer (1989) states, “you cannot have a successful solution to a systemic problem that
does not take its embedments into account … I advocate study of the metasystemic embedment
precisely because it enables a social system to understand and accept its own responsibilities” (p.
275-276). This aligns with the concept of expansionism (Ackoff, 1979b) by defining systems through
the systems they contribute to, not the systems they contain. These benefits may also apply to other
soft OR approaches, and this paper reconceptualises what recursion means in soft OR by exploring
how units can be modelled to develop outcomes that are horizontally coordinated across a meta-
system and therefore yield better results. Thus, this paper addresses the following research
questions:
RQ1. How can recursive modelling be broadened to include both vertical and horizontal
recursion?
RQ2. What benefits could recursive modelling have for soft OR?
Figure 1 – Twelve levels of recursion in the Chilean Government (Hoverstadt, 2008)
2
We first identify the theoretical principles of recursion in a literature review from which we
derive three criteria for recursion. Second, we present the methodological considerations of this
study. Third, we show horizontally recursive modelling in action through a case study of a police
force’s customer contact department using the WASAN OR approach (Shaw and Blundell, 2010).
Fourth, we discuss the implications of our findings for soft OR. Finally, we conclude the paper with its
limitations and future research opportunities.
Theoretical Context
To introduce recursion, we briefly consider its meaning in four contexts where this concept
is well understood: geometry, linguistics, computer science, and VSM. Key principles are extracted to
establish a definition of recursion. These criteria offer us a way to judge whether a modelling
approach has employed recursion. We then consider the difference between horizontal and vertical
recursion. Finally, we apply the identified criteria for recursion to recursive model building in soft
OR.
Recursion in four contexts
In geometry, shapes can be defined recursively. An example the von Koch (1906) Curve. It is
described by Falconer (2003) as follows: “We let E0 be a line segment of unit length. The set E1
consists of the four segments obtained by removing the middle third of E0 and replacing it by the
other two sides of the equilateral triangle based on the removed segment. We construct E2 by
applying the same procedure to each of the segments in E1, and so on. Thus Ek comes from
replacing the middle third of each straight line segment of Ek−1 by the other two sides of an
equilateral triangle” (p. xviii-xix) (Figure 2). Because these shapes are too irregular to be described by
Euclidian (traditional) geometry, they are defined recursively.
In linguistics, Chomsky (1959) demonstrates the presence of recursion in sentences that are
embedded within one another to provide unlimited context (Hauser, Chomsky, and Fitch, 2002).
Sauerland and Trotzke (2011) describe a recursive sentence structure as one where string B is
preceded and followed by two non-trivial strings (A and C). Originally called self-embedding,
Chomsky (1959) and provides unlimited context, which differentiates human language from that of
animals (Hauser et al, 2002).
In computer science, a recursive routine includes a repeated function where the output of
each repetition is the input of the next repetition until a stopping criterion is reached, providing the
final solution (Seitman, 1991). Recursive programs deconstruct a large problem into sub-problems
and tend to be easier to program and use less program code (Seitman, 1991). A simple recursive
program can calculate a factorial where the factorial ‘n !’ is the product of all positive integers less
than or equal to n, as in the example below.
3
IF N= 0 //This checks for the stopping criterion of the recursion
THEN FACTORIAL = 1 //This is the answer for the stopping criterion: FACTORIAL(0) = 1
ELSE //Any case other than the stopping criterion
FACTORIAL = N*FACTORIAL (N- 1) //The recursion calculation e.g. FACTORIAL(5) = 5 * FACTORIAL(4)
END //End of the conditional statement
Figure 2 – Adapted von Koch Curve (Falconer, 2003)
In VSM, recursion is hierarchical, as “any viable system contains, and is contained in, a
viable system” (Beer, 1979, p. 118). Due to the difficulty of representing the separation of multiple
systems, a VSM model represents one level, including its connections and interactions with higher
and lower recursive levels (Tejeida-padilla and Badillo-pin, 2010) (see Figure 1). Around each (sub-
system is a boundary depicting what is included within the recursive level.
The recursive systems theorem states, “if a viable system contains a viable system then the
organisational structure must be recursive” (Beer, 1981, p. 228). Leonard (1999) explains this as the
structure of the whole model being replicated in each of its parts and the relationships between
parts, i.e., as “self-similarity” (Jackson, 2003, p. 118). When analysing recursive levels, the same
modelling approach is applied at each level. The consistent recursive structure allows comparison
across an organisation to eliminate inconsistencies (Leonard, 1992), and recursion demands a
replication of the structure in each case (Beer, 1984). The structure of the VSM identifies how
sustainable organisations should be configured. It comprises five sub-systems: S1, Operations; S2,
Co-ordination; S3, Control/Monitoring; S4, External Environment; and S5, Identity (Beer, 1981). All
sub-systems exist in any viable organisation and should be in balance. This is the second proposition
of VSM: that the viability, cohesion, and self-organization of an enterprise depends upon the
specified units operating recursively at all levels (Schwaninger, 2004). For example, S1 (Operations)
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are expected to be “viable” (Bustard et al, 2007) and therefore possess the same five sub-systems to
ensure this.
Thus, it is theoretically possible to build infinite models (Beer, 1984), each becoming
progressively larger or smaller and reflecting S1-S5 and the meta-system (a wider system beyond the
bounds of the system-in–focus). However, pragmatically, a stopping criterion is applied when further
models become unhelpful for understanding the system. Jackson (2003) suggests a stopping
criterion of three recursive levels: the “system-in-focus” (Level 1) (the primary model); the meta-
system (Level 0); and sub-systems of the primary model (Level 2). Level 2 is a model of the S1
(Operations) units in Level 1, which itself is a model of the S1 units in Level 0. The three recursive
levels allow an understanding of the meta-system and the sub-systems. To manage the detail of the
models in Levels 0 and 2, Beer (1979) introduced the use of a black box, from which emerges
linkages to Level 1 to reduce overwhelming detail. He identifies two regulatory aphorisms: it is not
necessary to model the black box to understand the nature of the function it performs (Beer, 1979,
p. 40); and it is not necessary to model the black box to calculate the variety that it may generate (p.
47). Thus, black boxes ensure that we do not need to model all recursion levels with the same depth
to understand the system.
Criteria to define recursion
From all four contexts, we can identify three conditions to determine whether a recursive
relationship exists. First, there must be consistent replication—for example, repetition of process to
draw a shape (in geometry), in a sentence structure (in linguistics), in a function (in computer
science), or in the modelling of five sub-systems in Levels 2, 1, and 0 (in VSM). Second, recursion
must be self-referencing or self-generating. In geometry, the recursive shape (Sn) references a
shape (Sn−1) as a starting point for the process. A recursive linguistic structure is naturally generated
by a recursive program. In computer science, the recursive program calls (references) itself until a
stopping rule is reached. A VSM Level 1 model references Level 2 models through the S1
(Operations) units. Third, the recursive operation must provide greater understanding of a problem
than a single iteration of the recursion. In geometry, the shape is not generated without the
recursive steps, i.e., we cannot derive von Koch’s curve S5 without the previous four iterations. In
linguistics, recursion provides context, i.e., answers to questions do not require infinite words. In
computer science, only by completing the recursive program can you obtain the answer. In VSM,
building a more detailed model of the S1 operational unit uncovers additional information about the
sub-system and how it behaves within the system-in-focus (Level 2 model). Modelling the meta-
system increases the breadth of understanding (Level 0 model). These criteria are now used to
consider horizontal recursion.
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Horizontal Recursion – Definition
The vertical recursion in Figure 1 does not suitably explain all problem contexts that soft OR
may address, e.g., it misses the interrelationships between systems on the same hierarchical plane.
For example,1 assume Firm (F) comprises three sub-systems, called Departments (D), that
collectively ensure F is viable, namely Marketing (MK), Finance (FI), and Operations (OP). These
Departments can be represented as vertically recursive under SystemF as per Figure 3.
Figure 3 – Recursion between Firm and Departments
The relationship between SystemMK, SystemFI and SystemOP is horizontal because SystemFI
and SystemOP are in the environment of SystemMK. To represent these relationships, ModelMK can use
black boxes (Beer, 1979). When SystemFI and SystemOP are referenced as black boxes (Figure 4b) in
ModelMK, we can use the same approach to model these external systems as in ModelMK. Aggregating
ModelMK, ModelFI, and ModelOP on the same hierarchical plane, and explaining the linkages between
them, builds a model that provides insight to managers in Marketing ,Finance ,∧Operations , as
well as the Firm, as a whole (Figure 4a). This is not the same as building a single ModelF at the higher
vertical level of recursion, which could miss the depth of individual sub-systems and their
interconnections because (for the sub-systems) it would only include elements that pertain to the
Firm. Note the boundary of analysis is much narrower in Figure 4b than in Figure 4a.
Figure 4a – Horizontal recursion between
DepartmentsFigure 4b – Relationship between Marketing and
other systems as black boxes.
1 To clarify our definition of a model and modelling we distinguish between a unit, a system, and a model. A unit
is a real-world entity that could be modelled if desired, e.g., a switchboard where calls are received from the public and
routed into an organisation. A system is a conceptual tool used to think about the unit, i.e., “a particular way of describing
the world. It does not tell us what the world is…it may only be described as a system” (Checkland, 1983, p. 671) (here the
switchboard system, abbreviated to SystemSB). A model is a representation of a system using systems concepts and a
coding scheme, e.g., for the switchboard this would be ModelSB. Therefore, SystemSB refers to the switchboard conceptually
and not the real-world entity. When we model the Switchboard in ModelSB we model SystemSB based on the experiences of
participants included in the modelling process. We cannot detach the modelling conventions from systems concepts; these
are applied to the entity as understood through the experiences of participants.
6
Recursion in soft OR
Beyond VSM, recursive modelling is not associated with soft OR; however, some of the
principles identified above are present in other soft OR approaches, e.g., the limited use of nesting
models in soft systems methodology (SSM). For vertical recursion, Wilson (1990) builds conceptual
models in SSM and describes different resolutions to ensure the models are not too complex when
defining a vertical hierarchy of systems (Figure 5). Using a root definition to build each “resolution”
replicates the same methodology: criterion 1; the outer system of “Second resolution model for
activity X” in Figure 5 can be considered a black box, which can be modelled; this constitutes self-
referencing, criterion 2; displaying the system models side-by-side as in Figure 5 shows data
aggregation, criterion 3.
Figure 5 – Vertical recursion in SSM (Wilson, 1990, p. 34)
Figure 6 – Horizontal recursion in SSM (Wilson, 1990, p. 219)
There are also examples of horizontal recursion within SSM. Figure 6 is an SSM conceptual
model that shows the interaction between horizontally recursive systems, e.g., the “Planning
System” and “Marketing System”. The dotted lines constitute Level 1 system boundaries and contain
the Level 2 systems. Interactions (depicted with solid arrows) are evident between processes across
Level 1 boundaries, e.g., “Negotiate business” from the “Marketing System” interacts with “Decide
what facilities need to be developed to meet long-term requirement” from the “Resource
Development System”. Other horizontally recursive black box systems are not modelled (shown with
arrows pointing off the model such as “The technology”, top right). This model provides more
information than individual Level 1 models, detailing interactions across systems. Each model was
built replicating the same methodology (meeting criterion 1). Recursive systems are referenced;
7
some are modelled, and others are black boxes (meeting criterion 2). The aggregated model
provides information beyond a single model (meeting criterion 3).
These examples are not recognised as recursive model building by SSM and are not widely
used within SSM, meaning the benefits of recursion, particularly better understanding of
interdependence between modelled systems, cannot be fully exploited. We now introduce the
methodological considerations of this study and explore the two research questions.
Methodology
We conceptualise horizontal recursion through a case study that sought to provide knowledge to a
client regarding a particular problem. The research project followed an action research framework
(Hult and Lennung, 1980), “a cyclical process that involved formulating a definitive plan of action,
fact-finding in accordance with that plan, reformulation of the plan on the bases of research results
and implementing the next action plan to meet the goals of the revised plan” (Cunningham, 1976, p.
217). Thus, we developed the same analytical methodology over several learning loops by modelling
different units. The final learning loop combined analysis from four modelled units using horizontal
recursion. To explain the methodological considerations in this project we introduce the following:
the case context, selection of analytical approach, data collection, and the recursive modelling
approach.
The case context
The case study was a UK police force that aimed to maintain service delivery despite
reduced funding. Our focus was the Customer Contact department, which is comprised of four units:
Switchboard (SB), which receives non-emergency calls from the public and resolves them or routes
them to another unit and is staffed by eight full-time equivalent (FTE) employees; Call Handling
(CH), which receives emergency and non-emergency calls, logs information, and assesses a call’s risk
to decide whether to deploy police (52 FTE); the Crime Desk (CD), which takes crime reports from
the public over the phone and conducts low-level investigations (16 FTE); and, Crime Admin (CA),
which inputs crime reports into databases, corresponds with victims, and manages information
requests (12 FTE).
The department managers wanted to identify process improvements. A soft OR approach
was taken to model the complex sub-system interrelationships involving internal stakeholders
through bottom-up identification of problems and solutions. The project’s aims were as follows: to
remove “wasted” time to improve the management of customer contact; to reduce “wasted” time
from units interacting unproductively; to reduce processing time for customers; to involve internal
stakeholders; to maintain service performance; and to build an audit trail of recommendations. To
fulfil these aims, we analysed each unit’s performance and its impact on other units. As the focus for
8
analysis, these four units occupy recursive Level 1; thus, Customer Contact occupies Level 0, and
individual staff roles occupy Level 2, as shown in Figure 9. Constructing four Level 1 models would
miss the systemic complexity arising from the units’ interactions. Furthermore, building a single
Level 0 Customer Contact model would include elements irrelevant to our project and could
potentially cloud issues and miss emergent properties of the individual units. Thus, horizontal
recursion was used to preserve the integrity of the four individual models (to understand each unit)
whilst building models that could be joined (to understand interactions between units).
Selection of WASAN as analytical approach
The selection of WASAN was based upon the project aims identified above. WASAN met
these criteria by focusing on waste reduction and linking upstream and downstream systems to
explore how waste moves through the expanded system. WASAN structures participants’ knowledge
to understand where, when, and why waste is created within a system. By understanding waste and
its sources, modelling helps stakeholders identify actions to minimise avoidable waste. WASAN
clarifies how units are affected by up/downstream operations through perturbations that result in
waste and identifies how units can minimise waste. WASAN has four stages (Figure 7) (for a fuller
description of WASAN see Shaw and Blundell, 2010):
Figure 7 – WASAN stages (Shaw and Blundell, 2010)
A. Define the system boundary: Agree the scope of analysis, e.g., the process, wastes;
B. Analyse operations: Identify concerning issues and candidate solutions through 1. Analysing
internal operations by exploring waste production inside the unit and 2. Analysing external
operations by exploring the impact of up/downstream processes on waste production in a unit;
C. Evaluate actions: Evaluate and select candidate actions to implement;
D. Program deliverables: Consider candidate actions in the wider workplan.
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Data collection
For two years the first author was embedded in the police force providing analytical support
within the Customer Contact department. Rich understanding of the context and culture was built
through two years of daily ethnographic observations (Van Maanen, 2011), which culminated in the
implementation of WASAN. Each of the four modelled units comprised a separate action research
learning loop; thus, each unit was modelled independently from the others. In a single loop,
individuals were interviewed in Stages A and B to build unit models. Stage C developed an
amalgamated model that involved all participants in focus group settings. Stage D involved a meeting
with up to two managerial level participants. The process was supported by the second author who
had experience of WASAN. Each interview/focus group was audio recorded, and a summary of key
points was sent to the participant(s) for validation. In total, 34 people informed the model
development (see Table 1).
UnitUnique
ParticipantsWASAN
Stage
Data Collection Method
Number of Participants
Total Time (minutes)
A Interview 1 27B Interviews 5 190C Focus Group 4 41D Focus Group 2 30A Interview 1 37B Interviews 11 638C Focus Group 6 90D Focus Group 2 60A Interview 1 7B Interviews 4 224C Focus Group 4 41D Interview 1 30A Interview 1 8B Interviews 4 140C Focus Group 3 29D Interview 1 30
Total 34 1622
Crime Admin 6
Total
Switchboard 7
Call Handling 15
Crime Desk 6
Table 1 – Collected data
Below, we apply WASAN to Call Handling, showing its application in one action research
learning loop. Stage A captured the Customer Contact Manager’s descriptions of the system and
modelled them into one definition of the system properties, the output being an agreed system
boundary. Also, the waste to be examined was identified as “time”—if a Call Handler spends 10
minutes processing a phone call, these minutes can never be reused. If only four minutes were
needed, the remaining six minutes were wasted. Thus, time can be split into useful time and wasted
time. We applied WASAN to identify wasteful aspects and improve efficiency.
10
In Stage B, we interviewed 11 stakeholders (Call Handlers, Control Room Supervisors, Quality
Assurance Officer, a Police Inspector, and a Police Chief Inspector). Because the interviewees
performed emergency roles, staff members were interviewed individually using a semi-structured
interview schedule with each interview informing a Level 2 model. Interviewees were given the
boundary definition from Stage A to define the scope of analysis. They then discussed instances
where they felt time was wasted, either because calls took longer than expected or were not
appropriate for them to answer; these conversations identified sources of waste. Then, a keyword
analysis was conducted on these wastes, interviewees were asked how to “avoid” and “minimise”
each waste, with their answers forming the audit trail. The Level 2 models were aggregated to form
a composite Level 1 WASANCH model. The output for Stage B was the identification of sources of
waste and candidate corrective actions. These models were text-based network diagrams
representing the movement and flow of avoidable waste through the meta-system. See Figure 10 for
an example.
Decreased overall riskNeutral overall risk
Minor increased overall risk Major increased overall risk
Decreased overall riskNeutral overall risk
Minor increased overall risk Major increased overall risk
Decreased overall riskNeutral overall risk
Minor increased overall risk Major increased overall risk
Decreased overall riskNeutral overall risk
Minor increased overall risk Major increased overall risk
Decreased overall riskNeutral overall risk
Minor increased overall risk Major increased overall risk
Decreased overall riskNeutral overall risk
Minor increased overall risk Major increased overall risk
Decreased overall riskNeutral overall risk
Minor increased overall risk Major increased overall risk
Decreased overall riskNeutral overall risk
Minor increased overall risk Major increased overall risk
Decreased overall riskNeutral overall risk
Minor increased overall risk Major increased overall risk
Medium correlation with the blueprint –
timeframe and ethos
High savings compared to investment
Medium savings compared to investment
Low savings compared to investment
Low correlation with the blueprint – timeframe
and ethos
High savings compared to investment
Medium savings compared to investment
Low savings compared to investment
Alignment with long term blueprint
Savings to investment ratio
Risk to the public, staff and officers
High correlation with the blueprint –
timeframe and ethos
High savings compared to investment
Medium savings compared to investment
Low savings compared to investment
Figure 8 – Action evaluation grid
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Stage C involved interviews and a focus group. An action evaluation grid (AEG) (Figure 8) was
used to rate actions according to the following criteria chosen by the Customer Contact Manager:
alignment with the long-term vision; savings-to-investment ratio; and risk to the public, staff, and
officers. The AEG was initially completed by interviewing the Superintendent, who was the senior
participant. The six-person focus group worked systematically through candidate actions, rating
them on each criterion. Each row in the AEG has a relative priority level shown in the right-hand
column of Figure 8 (high [white cells], medium [light grey cells], and low [dark grey cells]). To avoid
the influence of priority levels on the ratings, they were decided by the Customer Contact Manager
one week after the focus group was conducted.
The Stage D focus group involved the Customer Contact Manager and a Control Room
Supervisor deciding a work programme to implement. All low priority actions were discarded to
focus on high and medium priorities. Some actions were discarded because of existing work streams,
ownership, and precedence. A final list was agreed.
The three other units within Customer Contact were independently modelled through
Stages A-D.
Recursive modelling
The final learning loop of the research project reviewed learning across all four units, as
inefficiencies arose from interactions across them. However the modelling approach taken with
WASAN in learning loops 1-4 did not allow for analysis across models. Hence, we used horizontal
recursion to expand the scope of the project and identify wastes through relationships between the
models.
First individual level models across all units were searched for interaction between Level 1
units. This was done by creating a black box for each model that logged information not relevant
within ght model boundary, but that a participant stated had influence on (or was influence by)
other units. For example, Call Handling identified Switchboard as a source of waste because this
unit receives “incorrect calls from Switchboard”. Therefore, this issue was considered in both Call
Handling and Switchboard.
Next black boxes on interactions between units were aggregated to create a meta-model
containing information from across the four models. Using Jackson's (2003) naming convention for
vertical recursion systems horizontally, the models built for police included: Level 2 models from
each interviewee; four Level 1 models (one for each unit) that aggregated the Level 2 models; and a
Level 0 model that aggregated the Level 1 models through their black boxes. Establishing the
principles of horizontal recursion through the combination of the four units’ Level 1 models is the
focus of the findings below.
12
Findings
To consider the evidence for RQ1 “How can recursive modelling be broadened to include
both vertical and horizontal recursion?” we explore whether the three recursion criteria (consistent
replication; self-referencing; recursive operations adding understanding) are present in the four
Level 1 models (and their black boxes). To consider the evidence for RQ2 “What benefits could
recursive modelling have for soft OR” we reflect upon participants’ statements to understand how
they perceived the benefits of modelling the four Level 1 systems recursively.
Consistent replication in WASAN
For models to be recursive they must follow the same analytical conventions to eliminate
inconsistencies when aggregated. Through the case study we identified that replication can be
considered in two ways: methodological and contextual. In recursive modelling the same
methodological approach must be used to build each model. All approaches will have modelling
conventions (Stages A-D in WASAN) that ensure methodological replication and thus comparability.
However, methodological replication alone did not create models that were contextually consistent
enough to be horizontally recursive. Two contextual factors required replication: the definition of
waste and the system boundary. In this police force case study, which used time as the waste, the
models could be horizontally recursive. If waste is not replicated across models, its effects cannot be
traced; thus, analysing the horizontal systems recursively would be unnecessary. The second
contextual factor in WASAN is the system boundary. Recursive modelling shows interactions
between different systems; therefore, recursive models must share a system boundary through
which they interact. At Level 1, the four systems do not replicate the same system boundaries, but
they do replicate (or share) a boundary because they are part of Level 0 and thus could be
horizontally recursive (Figure 9).
In our case, replication is understood as both methodological (respecting the principles of
the analytical approach) and contextual (with consistent contextual factors across models).
Figure 9 – Replicated Level 0 system boundary
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Self-referencing in WASAN
WASAN models use black boxes to limit the analytical boundary. Figure 10 shows the
WASANCA with “misrouted calls” as a waste source from a black box, . This source is created by the
black box misrouting calls to Crime Admin. These negative effects could be passed to downstream
systems, represented as a second black box. The boundary of analysis is shown as a dotted line. The
analysis focuses on the channels (the arrows) and the Crime Admin System (the grey box).
In recursive modelling, the boundary of analysis is expanded beyond a single system. This is
done by modelling the black boxes using the same modelling conventions and context outlined
above. Self-referencing in the models provides a route map for moving from one model to another.
Figure 10 – Upstream system as a black box
During Stage B of WASANCA a participant identified Switchboard as the source of misrouted
calls: “We get mixed up with Crime Desk sometimes…we can just put those back through to
Switchboard”. Thus, we can move from WASANCA to model the upstream black box at SystemSB to
understand why call misrouting occurs; this provides a meta-system for understanding the problem
identified in WASANCA. In the interviews, referencing other systems was widespread, participants
from all four units identified other systems that caused waste within their own units. Table 2
presents instances where a model or interviewee referenced an up/downstream system. The x-axis
shows from which model the issue was raised, and the y-axis shows the system being discussed.
Thus, the first row presents views on Switchboard raised by interviewees from Call Handling, the
Crime Desk, and Crime Admin. A description and context of the waste is provided [underlined] along
with a quote [in italics] from the participant. This table shows how misrouted calls are a systemic
problem affecting all units within Customer Contact. A single model (Figure 10) does not convey the
breadth of system-wide issues that must be assessed to eliminate misrouted calls within Customer
Contact. Thus, the meta-systemic analysis of data shown in Table 2 is necessary. Below, we
aggregate this data (and the WASAN models) to build a horizontally recursive model of misrouted
calls.
Recursive operations aid understanding in WASAN
The aggregation of recursive operations involves combining models using the links where
one model references another modelled system. Two types of aggregated models were identified:
14
(1) waste chains, where an action from an up/downstream black box caused a waste to be explored
by modelling the black box and (2) waste transference, where a decision from a Level 0 system
transferred waste across Level 1 systems. To illustrate waste chains and how recursive modelling
clarifies the situation (beyond showing them in isolation) we further investigate misrouted calls from
the Switchboard (Figure 10). Figure 11 expands WASANCA to include the upstream WASANSB as the
source of misrouted calls and includes the other systems that the Switchboard misroutes calls to.
Also, the impact of waste is traced downstream from WASANCA as calls are sent back to the
Switchboard or as Call Handlers are unavailable to answer incoming calls. This illustrates the
importance of identifying how a problem affects a meta-system because addressing this problem in
WASANCA alone would not fix the wider problem.
Figure 11 – Misrouted calls waste chain
The second example is when a change in policy transfers waste between two recursive Level
1 systems. Here, failure to consider the meta-system could foster decisions that
benefit/disadvantage a particular Level 1 system. Horizontal recursion helps avoid this. To illustrate,
using Figure 12, the WASANCD model identified ‘Sending callers to Action Fraud’ as a waste (Action
Fraud investigates some types of fraud). In WASANCD an action was included for the ‘Switchboard
system to send these types of callers directly to Action Fraud’ bypassing the Crime Desk. This would
reduce the number of calls to the Crime Desk but would increase processing time at the Switchboard
and require staff training. If implemented by the Crime Desk, this policy would shift the impact of
waste between horizontally recursive Level 1 systems (so, in Figure 12, the source of waste would be
transferred from WASANCD to WASANSB). This may benefit the Crime Desk but not the overall
system. Horizontal recursion enables decision makers at Level 0 to make consistent comparisons
between units. By identifying systemic effects, decision makers can balance the benefits and
drawbacks of a policy for the meta-system.
15
System Referenced
Where the issue was raisedWASANSB WASANCH WASANCD WASANCA
SystemSB
Calls put through to Call Handling by Switchboard that should not and result
in other calls not being answered
"Some calls go through that shouldn’t go through … really it should have gone to another department … but while we
are dealing with that, calls that are aimed more for us are not being
answered."
Misrouted calls from Switchboard that get redirected back to Switchboard
"Misdirected calls are quite prevalent … I think Crime Desk is the easy option as we have to answer the phone … I tend to put them back to Switchboard and
tell them they need to go too [pause]."
Crime Admin’s responsibilities mixed up with Crime Desk’s
"We get mixed up with Crime Desk sometimes … we can just put those
back through to Switchboard … we can [route them directly to Crime Desk] but
its more to highlight to Switchboard that we are not Crime Desk."
SystemCH
Calls bounced back to Switchboard from Call Handling
"There is some waste when callers inevitably end up coming back to the Switchboard because they have been put through to the wrong place and
they don’t always know where to put them through too."
Call Handlers taking details of historic crime for Crime Desk to follow up
Comms Centre [Call Handlers] sit and reproduce all that in a STORM incident
… they will then switch it through to the Crime Desk it’s up to the Crime Desk
then to chase the victim to manage the crime report.”
WASANCA did not reference SystemCD
SystemCD
Calls bounced back to Switchboard from Crime Desk
"Switchboard don’t always know what department deals with particular types
of enquiries."
Call Handlers tied up issuing crime numbers
I’ve got calls for service for fights and things that are going on but I’ve got
Call Handlers tied up with doing Crime Desk’s job.
Crime reports from Crime Desk with missing information
"We waste a lot of time with stuff coming through to us that hasn’t been
completed properly, the C1’s [crime reports]"
SystemCA
Calls bounced back to Switchboard from Crime Admin
"Switchboard don’t always know what department deals with particular types
of enquiries."
WASANCH did not reference SystemCA WASANCD did not reference SystemCA
Table 2 – Quotes referencing other units
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Figure 12 – Change in policy leading to movement of waste
Feedback from system owners
System owners received the recommendations from the WASAN modelling of individual
units more than a month before receiving the recursive modelling results. This allows us to
understand the impact of the additional recursive analysis on their understanding of the problem.
Recursive modelling did not provide additional recommendations; however, the utility of
modelling did have important indirect effects. First, it was transformative in system owners’ thinking
about the problem of waste within their units. Without recursion, they perceived a link between
units but did not fully understand how interconnected waste issues were or how their unit impacted
others. By viewing recursive models (such as Figure 11), the system owners developed more
ambitious and wide-reaching plans to address the identified waste issues. We do not have consent
to share exact recommendations, but in general terms, plans were reconsidered to provide “a more
joined up view of the interactions between [the units]”. Examples include revised training packages
and procedural changes.
Second, the recursive modelling fostered consensus across system owners and engendered
closer working relationships. It showed how interlinked each unit was with its meta-system and
therefore demonstrated a need for managers across that meta-system to work together more
closely. Feedback from system owners focused on how horizontal recursion pinpointed problems
that originated in upstream units. When discussing benefits, the Chief Inspector for Customer
Contact said, “[it] is useful looking at the front end, you know there is stuff we perhaps don’t have
control of, but can we have an impact on it”. An enhanced understanding of the situation allowed
strategic discussions between these managers to reduce waste across the modelled units. One
Control Room Supervisor remarked, “[the recursive analysis] brought it all together…it’s never
brought together and presented in a way forward…that was good”.
Discussion
This paper identified recursion as a horizontal property of systems in the case study. This
discussion addresses the two research questions by reflecting upon the findings with respect to the
three criteria for recursion.
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Research Question 1: How can recursive modelling be broadened to include both vertical and
horizontal recursion?
To answer RQ1, we first consider horizontal recursion within VSM and then consider
broadening the use of recursion to other soft OR approaches. This paper introduces recursion as a
horizontal property of modelling to analyse units across different models on the same horizontal
plane. To establish the three criteria for recursion, the paper drew heavily on vertical recursion in
VSM. We use these same three criteria to understand whether VSM can address horizontal
interdependence through modelling using horizontal recursion. Figure 13 shows the VSM structure
of two vertical levels of recursion adapted from Beer (1981). At Level 1 we have a detailed VSM with
the external environment on the left, the management functions (S3, S4, and S5) on the top right,
three operations units (S1) in the bottom right, each contained in a dotted oval with S2 providing the
link between the S1 operations units. The three operations units sit at vertical recursion Level 2 and
replicate the structure of the Level 1 unit with the S3, S4, and S5 units represented in boxes with S2
in the triangle and S1 represented as circles. Figure 13 is arranged to show the vertical recursion of
systems, but it also shows the horizontal recursion for VSM. When considering the three Level 2
systems in Figure 13, we can see self-similarity, where the same analytical method can be used to
represent the three units (criterion 1). In terms of self-referencing (criterion 2), Figure 13 clearly
shows links between horizontally recursive systems at Level 2, as illustrated by the double parallel
lines between each Level 2 system. Regarding the aggregation of models to provide deeper
understanding of a problem (criterion 3), using WASAN at the UK police force we identified waste
transference (Figure 12), where a policy decision shifts waste between two horizontally recursive
systems. Looking at the structure of the VSM in Figure 13 we can see how a policy decision could be
made by the S2 unit of the Level 1 system and then be implemented by the Level 2 systems. This is
supported by Beer (1981), who categorises S2 as “acting vis-à-vis System Three very much as the
input synapse on the horizontal command axis acts vis-à-vis System One” (p. 175).
VSM has the properties required for modelling systems using horizontal recursion, offering
an alternative or supplementary approach to vertical recursion. Horizontal recursion allows the
modelling of systems surrounding a system-in-focus without having to move up a vertically recursive
level and the modelling of the management functions of S3, S4, and S5. Thus, where appropriate,
horizontal recursion can be used to model parallel units using S1-S5 and then track interactions
across these units. This analysis may lead to a deeper understanding of a situation that is not
possible with vertical recursion alone. This approach should be tested empirically, and we suggest
this as a direction of future research.
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We briefly consider how recursion may be considered in another soft OR approach. Journey
making (Eden and Ackermann, 2004) links strategic issues using word/arrow diagrams with an action
orientation to produce a cognitive map arranged in a hierarchical teardrop (see Figure 14). The
hierarchy places goals at the top, strategies to realise those goals in the middle, and actions required
to implement those strategies at the base. Multiple Level 2 “participant maps” can build a composite
group Level 1 model (from Participants A and B in Figure 14). Each Level 2 model replicates the same
data analysis technique, meeting criterion 1. Overlaps between Level 2 models are used to stitch
them together to form the Level 1 model (see Nodes a, d, and g in Figure 14), constituting self-
referencing between Level 2, which meets criterion 2. Additional understanding can be gained from
the composite Level 1 model, which meets criterion 3 (e.g., the additional information from linking
nodes b & c and e & f). The process of moving from individual to group maps exemplifies vertical
recursion, while the relationship between individual participant models is horizontally recursive.
Figure 13 – Two levels of recursion (adapted from Beer, 1981)
Figure 14 – Vertical teardrop (adapted from Eden and Ackermann, 1998)
In this case study, we have shown how the criteria for recursion can be applied to WASAN
when horizontal recursion is employed. The discussion has shown the broader applicability of
horizontal recursion in VSM and recursion in journey making. Further research can exploit the
potential for thinking about recursion in these OR approaches. Figures 5, 6, 13, and 14 show that
vertical and horizontal recursion can be used within soft OR; however, currently neither is widely
employed. When modelling multiple linked systems with the same soft OR approach, recursion can
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lead to better understanding of vertical and horizontal interdependency among modelled units.
Further empirical work is required to understand how horizontal recursion can be employed within
other soft OR approaches beyond WASAN. Next, we consider the benefits of recursion for soft OR.
Research Question 2: What benefits could recursive modelling have for soft OR?
Feedback from the case study participants suggests the utility of additional information from
recursive modelling. Below, we discuss how the additional analysis improved their outcomes. The
Customer Contact Manager and Force Crime Registrar used the results from recursive modelling to
negotiate additional training across all Call Handler, Switchboard, Crime Desk, Crime Admin, and
Control Room supervisors. This training used these results (such as Figure 11) (similar to Lane,
Munro, and Husemann, 2016) to show how waste was permeating across the four units and to
demonstrate the benefits of changing behaviour. Additionally, when software changes were
proposed to reduce identified waste, these were negotiated on the basis of the effect of waste
across the four units, rather than on one single unit. The recursive models also helped achieve buy-in
from managers across all four units by showing the potential benefits for each of them rather than
presenting resource allocation as a zero-sum game within a single unit.
The introduction identified two benefits of vertical recursion in VSM: the elegant
representation of organisations (Jackson, 2003) and better understanding of internal relations
between units and system behaviours (Beer, 1984, 1999). These also apply to horizontal recursion.
Elegant representation refers to reducing the visual complexity of a model, thus removing irrelevant
information and highlighting important factors or relationships to users. Modelling different units in
separate models and then analysing the horizontal relationships between those models recursively
allows for a full examination of each individual unit without the models becoming too complex.
Furthermore, a second level of analysis across the meta-system considers horizontal relationships.
Each of these models are less visually complex than a larger single model trying to accomplish both
sets of analysis.
For a better understanding of relationships between units, we return to Beer's (1989) initial
justification for vertical recursion: “you cannot have a successful solution to a systemic problem that
does not take its embedments into account” (p. 275). Although Beer focuses on vertical
interdependency across embedded systems, we suggest this view could be enhanced to account for
the importance of the horizontal relationships across the metasystem. For example, the waste chain
in Figure 11 shows how misrouted calls are embedded in all four units more effectively than
independent analyses of each unit in isolation (Figure 10). Appraising the system without horizontal
recursion would lead to decisions that may seem sensible at the individual unit level but may not be
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as effective as those that take into account the interconnected nature of the problem by assessing
the meta-system using horizontal recursion.
Furthermore, this paper has shown how system owners felt the additional analysis of
recursive modelling provided a better understanding of avoidable waste created by, and moving
through, units on the same hierarchical plane. This allowed them to make decisions based on more
information than if only considering their unit in isolation. Therefore, we suggest that like vertical
recursion horizontal recursion allows for a better understanding of internal relations between units.
Conclusion
This paper suggests both the theoretical and modelling contribution of horizontal recursion
to soft OR. We outline a broad definition of recursion using literature beyond soft OR and conclude
that recursive models should: 1. have the same structures, allowing consistent replication with
models amenable to the same form of analysis; 2. reference each other; and 3. be aggregated to
provide understanding of a meta-system. The paper shows how recursion can be applied in soft OR
approaches through a case study of WASAN that demonstrates horizontally recursive modelling.
Despite its strengths, this work also has several limitations. First, we only collected empirical
data using one soft OR approach—WASAN. We aimed to show the relevance of horizontal recursion
across a range of soft OR approaches beyond WASAN, but this was based only on the literature and
our theoretical understanding of the approach. Second, due to the nature of the case study, we
were not able to compare the utility of recursive modelling in more experimental conditions. Future
work may consider whether horizontal recursion is applicable to new soft OR approaches, e.g., DPSIR
(Gregory et al, 2013). We are also interested in collecting empirical data on horizontal recursion
within the VSM to show how different Level 1 systems interact without having to model the S3-S5
sub-systems of the Level 0 system. The first criterion for recursion is how methodological factors are
replicated across applications while contextual factors alter to reflect the problem – and additional
applications are needed to consider how replication is considered in different methodologies and
contexts.
Soft OR has enjoyed a resurgence through new approaches to complement established
methods. Horizontal and vertical recursion can inform the application of existing techniques, as well
as refresh and add value to soft OR. This paper addresses a limitation inherent to all soft OR
approaches, which is the failure to fully exploit the potential of modelling across systems; it thus
provides a novel contribution to soft OR. We have shown the added value of modelling horizontally
in contexts where problems permeate beyond an individual system, where meta-systemic
understanding is required, where multiple linked systems can be modelled using the same
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conventions, and where links between models can be identified. Under these conditions horizontal
recursion can begin to contribute to soft OR as exemplified through the case study.
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