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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

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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)

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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.

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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.

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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;

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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

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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.

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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

















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




Alignment with long term blueprint

Savings to investment ratio

Risk to the public, staff and officers

High correlation with the blueprint –

timeframe and ethos




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.

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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:

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(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.

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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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