Governing the Filtration Process of Contractors to Insolvent Point in Construction Projects with Nmorcha (Threshold) Operator

Abstract

There are recurrent literatures to show that contractors in the construction industry are going insolvent. As a crippling process, there are early signs leading to contractor’s insolvency. Conflicting literatures have documented causes of insolvency in construction project in an irregular calibration. This paper argues theoretically that those causes can be stabilized calibrated with a view to establishing a threshold point or the limiting point beyond which a contractor becomes insolvent. A conceptual theory was built on spherical harmonic polynomial for which a sector of the sphere representing an entire economic sphere was taken to represent the construction industry sector. The sectors’ investigation precipitated five (5) main causes of insolvency from five (5) significant unethical practices. These variables were subjected to an orthogonal 5 × 5 square matrix to derive a pseudodifferential operator matrix with Fredholm properties narrated in Athanasiadis et al. and Claeys’ boundary integral problems. The derived determinant was used to operate as Nmorcha operator (∇N) on the population variable that assumed a chaotic attribute under a normal distribution curve at the critical stressed boundary plane. The Nmorcha operator(∇N) as a determinant operand at the boundary point of the critical region fittered variables to the skew of accepted and reject domains. The filtration operation at the boundary point presented a value of e2i4 representing falsification of projects financial status (e2) arising from poor project cost control (i4). Beyond this point in the array of causes in the calibrated insolvency scale, the contractor becomes insolvent.

Definition of Terms

Nmorcha operator (∇N): A theoretical operator derived from the characteristics property of a sectorial polynomial (m=n) matrix of spherical harmonic function that operates filtration on functions with indeterminate or chaotic state!

Threshold operation: An operation performed on operand to obtain the value of a threshold function.

Threshold gate: A logic plane where a threshold operation is performed.

Threshold operator: A logic element that performs a threshold operation.

Threshold: The starting point for a new state or experience.

Boundary operator: a certain element that permits transition by gradient profiling at boundary plane of a space to either divides of a domain.

Determinant: The characteristic value of a group of variables that ascribes peculiarity and representative attribute to the variables.

Keywords

Insolvency, Boundary filtration, Nmorcha operator, Threshold gate, spherical polynomial, Determinant

Introduction

Contractors in the construction business are gliding into insolvency or emphatically becoming insolvent. What remains indeterminate is; at what point in life of the project do they become insolvent? ASHURST sounded a tocsin that contractors’ insolvencies will increase from ASHURST [1] and advocated that contractors should devise an organizational metric to indicate the early warning signs of insolvency. Nevertheless, contractors 100% alertness may not be a rescue option to the inevitable because, combating insolvency in the construction industry is almost as a helpless combat of man with climate reclamation! The reason for this is that the world economic recession in the recent past came with global financial warming/depletion that necessarily will and still interfere with any remediation measure to combat insolvency in the real sector of the construction industry [2].

In the last 10 years, research efforts have been directed towards the investigation of the causes of insolvency amongst construction contractors. Typically, Arain and Low proposed measures of minimizing impacts of variations on construction project as a way of avoiding insolvency [3,4]. With the same objective, Bahram offered some legal claim protection methods that can mitigate insolvency on risk transfer basis by insertion of express legal clauses in contract documents. Also, the thought that insolvency may arise from some foundational ethical reasons inspired Wood to prescribe a friendly partnership relational model in construction projects [5]. More the same, the indicators given by ASHURST on early signs of contractor’s insolvency are very leading to the expiration of insolvency in construction contract business should contractors adhere. Assuredly, this paper opines that a threshold value from the ladder (hierarchy) of causes of insolvency in construction project contracts based on calibration will certainly give scientific credence to combating insolvency occurrence and friction the gliding into that economic state.

A Survey of Insolvency Literature in the Construction Industry

The concept of insolvency is not peculiar to construction sector of any economy. As a matter of fact, insolvency is an economic state of an organisation that reflects the financial health status of the organisation. The reason for its study is in an attempt to master its forms, its causes, its sources and its prevention. Arain did a classical investigation on the causes of insolvency and unethical practices of contractors in Pakistan construction industry [6]. The investigation was conducted vide a questionnaire survey administered to 90 contractors. Thirty (30) questionnaires were returned and analysed. The findings showed nineteen (19) speculative causes but with five (5) main causes of insolvency. The five significant causes identified in Arain were absence of barriers, cash flow problems poor financial control, knock-on effect and onerous conditions of contract. The study also noted that these five (5) significant causes of insolvency are also intrinsically linked to unethical practices of contractors with the exception of absence of barriers. This outcome collaborates earlier knowledge on the subject by [5-10] Arain, Assaf and Low, Newman, Dalla- Costa, Lowe and Low. A significant outcome of Arain study was that construction industry should pay attention to ethical behaviour and good practices in order not to jeopardize the financial stability of contractors in the supply chain. ASHURST before the Arain study proposed a pre-emptive measure on decimating insolvency, rather than studying the symptoms. The underlying preposition of the ASHURST literature is that contractors are to be vigilant to certain indicators of early warning signs of insolvency that could lead to financial embarrassment. According to the study, the following are early indicators of insolvency; sub-contractors making demands for payments directly, contractors requesting for early/advancement to cover the cost of sub-contractors or materials, scarcity of work on site which is a sign of possible cash flow in the future, slow progress of work and missed date, employees apathy arising from none payment of wages couple with reduced labour force, disappearance of materials, persistent rumours of declining financial status, application for spurious claims, ailing contractor’s account, diversion of income, court judgments that conveys orders of mandamus on the contractor and whether if the contractor’s parent company is showing any of these symptomatic signs.

However, symptoms and treatment of insolvency in construction projects need not be a laboratory science. As a cripping process, reminiscence of pathological, metamorphosis, its dynamics requires empirical study before the process converges to a literacy zero (0) i.e. contractor’s oblivion. A complementary study by Turner and Townsend proposed [11] a four (4) way test to show that a contractor is gliding to insolvency in consonance with Section 132 of the UK Insolvency Act of 1986. These include failure to pay a statutory demand over a threshold, execution on a judgment is unsatisfied, inability to pay debts arising from cash flow deficit and value of assets are less than liabilities. Bahram attempted a legal prescription to insolvency in construction projects with a review of the Perar BV vs. General Surety & Guarantee Co. Ltd. British Law Report (BLR) 72, Case. The outcome of that study was that insolvency is not a breach of contract as default, unless it is expressly provided in contract documents. Further, the study reported that it is neither to the benefit of the contractor nor the client to resort to termination of contract arising from insolvent situation [12-18]. In the same study, it averred that there is plethora of recovery measures towards diffusing insolvency in contractual arrangements. Specifically, it mentioned administrative receivership; Company Voluntary Arrangement (CVA) and scheme of arrangement that is subject to satisfactory compliance with statutory requirements. Other research efforts in unison offered some legal protection approach but with a caveat that Employer’s termination notice must necessarily show compliance with contractual requirements otherwise the situation may turn beneficial by way of compensation to the contractor. As a legal rescue mechanism, contractors and employers should turn to the prescribed procedure of resolving termination of contract due to insolvency. Appreciably, the JCT, under Clause 8 was very benevolent to both parties with Clause 8.2 conditioning that contractor’s termination shall not be issued unreasonably or vexatiously [19-22]. Contrary to NEC Clause 90.2 that permits employer to terminate for any reason he deemed fit. On the basis of clustered research reaction, this paper argues that, there are events that culminates to insolvency and seeks to investigate the threshold or the limiting point beyond which a contractor becomes insolvent.

Theoretical Setup

The Arain study, precipitated an outcome of six (6) unethical practices correlating to five (5) significant causes of insolvency on a shifted like scale of 1-6 mean score. Indicating a probability space between most likely and most unlikely as shown below (Table 1).

Table 1. Outcome of Arain (2013) study on source and cause of insolvency.

ϕ_i1 represents unethical practice, ϕ_e represents causes of insolvency such that absence of barriers (ϕ_i1) as a cause of insolvency is likely to be originated by the unethical practice of failure to reconcile with sub-contractor’s concern (ϕ_e5) = 2.81. Generally, the Arain source and cause correlation of 6 × 5 matrix is indicated to read (Table 2).

Table 2. Insolvency causes.

Conceptually, two variables such as source (unethical practices (ϕ_ei) and causes (of insolvency), ϕ_ej are said to be orthogonal if their scalar product vanishes. A family of variables is an orthogonal system on the nterval (a, b) with the weight function ω(x) (or distribution a(x)), if for any two distinct members of the family say (ϕ_eϕ_i) = 0. From the Arain results, the correlates can form a system of orthogonal family as;

Consequently, this paper idealizes a threshold estimation function for contractor’s insolvency which is a variable that takes a limiting value say “1” if a specified function of the argument exceeds a given threshold and ‘0’ if otherwise. A threshold operation when performed on Equation (1) as an operation performed on operands say ϕ₁ϕ₁, ϕ₁ϕ₂……….. ϕ₂ϕ₄, ϕ₂ϕ₅……… ϕ₅ϕ₁……… ϕ₆ϕ₅, in order to obtain the value of a threshold function are therefore consistent with the vanishing properties of an orthogonal function [23-31]. In doing so, the space between ϕ₁ϕ₁…… ϕ₆ϕ₅,is considered a uadratically integrable space and therefore separable. Since the space consist of finite elements or denumerable vectors been the length of space between each scalar combination, this paper deduces a threshold value as the limiting point of lim (ϕ_ek,ϕ_ik) = 0.

The Arain study on causes of insolvency and unethical practices using a cross-assessment table of mean item score, precipitated a 6 × 5 matrix with an overwhelming response determinant of delayed payment and knock-on effect as the resultant combinatorial cause of insolvency in Pakistan from the five (5) main identified causes of insolvency and six (6) unethical practices value (Table 3).

Table 3. Restructured arain (2013) matrix in tabular form variable of sources and unethical practices.

Following this calibration, this paper deduce a semblance of the scalar combination of unethical practices and insolvency causes to an orthogonal system of a finite sequence of ϕ_i,ϕ_e (i = 1, 2, …n, e = 1, 2, … n) or shortly as {ϕ_n (x)} with orthogonality property of (ϕ_hϕ_k) = 0 , h^4r-1 k noting that the zeros of the orthogonal polynomials are simple and located in the interior of the interval (a, b) From the Arain matrix, an rthogonal space of (ϕ_i1,ϕ_i2,ϕ₅,ϕ_e6) wherein the threshold element for insolvency is located in the interior of the interval of (a, b) is consequently idealized.

Foundationally, this paper made recourse to Maxwell’s theory of poles in acknowledgement of the Bateman’s spherical harmonic polynomial description in a reduced form with; λ₁, λ₂, λ₃ ‘to be independent variables, such that

(1)

and a differential operator defined as (2)

In consonance with Laplace’s equation and homogenous polynomial of degree n = a + b + c multiplied by r^-2n-1 wherein, for every polynomial (H_n) of degree n, the following are equivalently true;

Thus, if n = a + b + c, then,

(3)

From this relationship, it becomes fundamental that to every homogenous polynomial of degree n, with three quantities, for which,

(4)

There corresponds a harmonic polynomial of λ₁, λ₂, λ₃ of degree n and which in line with Hobson’s theory can be shown that,

(5)

and λ₁ =r cos θ, λ₂ = r sin θ cos , λ₂ = r sin θ sin

Consequently, for geometric interpretation, of Equatiom (5) surface harmonic function reduction becomes zonal if m =0, sectorial if m = n and tesseral if 1≥ m ≥ n

From Maxwell’s result, this paper, took η= (α_k, β_k, γ_k), k =1,2,…..,n as unit vectors which defines points on the unit-sphere as poles-such that surface harmonic of degree n with poles η_k defined by;

(6)

For the special case of ρ = 2, h (n, ρ) = (n + 1)²

This paper assumed to be a vector with four components as and

The following vectors are introduced

U= (i – its, -its – is, -t+s, 1 = ts)

Y = (i-T^σ ,-iT – I^σ , -T+^σ , 1 + T^σ )

So that it becomes;

(7)

From Equation (7), it shows that all (n + 1)² polynomials, defined by

(8)

are harmonic polynomial is degree n.

Consequently, this paper note that

and therefore the surface harmonics,

Form an orthogonal set of h (n, 2) = (n+^{1)2 of linearly independent surface harmonics, where,}

Accordingly from Eqn (8), this paper reduces the polynomial conditionalities to

To derive the components of this paper introduced the following units,

So that, consistently from Eqn (8), this paper derive that,

And

(9)

By substituting,

σ = -s(bc – ad) /bd,

So that expressing a, b, c, d in terms of yi yields

(10)

Arising therefrom, the following precipitates;

(11)

Similarly,

(12)

Ditto;

(13)

where denotes a purely Jacobi polynomial

Derivatively, if this paper associate cause of insolvency (ϕ_e) and unethical practices (ϕ_i) to two vectors with four components each and letting be a vector with the components;

(14)

where Y_iz_i = ϕ_eϕ_i

With the introduction of quaternions, into Eqn (14) it reduces to w₄ + iw₂ + iw₃ + kw₁ = (z₄ + iz₂ + iz₃ + kz₁)(y₄ + iy₂ + iy₃ + ky₁) where 1, i, j, k are fundamental coefficients. Then from the addition theorem;

(15)

The matrix of the output of Eqn (14) becomes

where k denotes the rows and I denotes the columns for which k = m, l = n and m = n in this study and recursively gives the determinant;`

which is in tandem with Eqn (2) ditto Zi’s with characteristic roots where and the roots of the association given by

Study Objective

To identify a threshold point on a calibrated insolvency scale by threshold operation been an operation performed on operands in order to obtain the value of a threshold point through a threshold gate. This point on the calibration scale is the point beyond which a contractor turn’s insolvent.

Methods of Research

This paper reviewed the existing literatures on contractors’ insolvency in construction projects and zeroed the investigation to the causes and sources of insolvency. This study mirrored the Arain investigation on causes of insolvency arising from some unethical practices amongst contractors in the Pakistan construction industry as a control “experiment”. The Arain study identified 5-causes of insolvency originating from 6-unethical practices. As replication logic to test the validity of that outcome, a similar investigation was conducted on the Nigerian construction industry sector in this paper. The investigation was done using a survey design method with 108 questionnaire soliciting information on causes of insolvency arising from some certain sharp or unethical practices of contractors. The respondents (sample size) was derived from a population of 150, made up of professional Civil Engineers, M &E Engineers, Builders, Quantity Surveyors and Architects in public, private and contracting organizations using a sample size formula of;

S = sample size being sought.

x² = Table value for chi-square at 1 degree of freedom at the desired alpha level 0.05 = 3.841; 0.01 = 6.64

N = Population size.

ρ = The population proportion usually 0.05 for maximum sample.

∂ = Degree of accuracy designed, expressed as a proportion (usually 0.05).

81 questionnaires were returned with 67 suitable for analysis indicating a response rate of 75%. The questionnaire was structured in sections in relation to the objective of the study. An open ended system of questioning was used in achieving this aim reflecting a 5-point likert scale with the following ordinate value

5: Very high, 4: High, 3: Moderate, 2: Low, 1 : Very low

The mean score method of analysis which usually represents the average value of all factors associated to a specific inquiry was employed. The justification for the use of mean item score instrument for analysis was in consonance with the opinion of Egwunatum, citing Atsar and Ogunsemi that the instrument is prevalent amongst construction management researchers and reported its use by Kulugara, Ling and Akintoye. Usually arithmetic mean is employed unless otherwise a condition for application is giving. A pool of several types of mean item score exists viz; Arithmetic mean, Geometric mean, Harmonic mean and weighted mean. This study applied the weighted mean score, which involves assigning numerical weight (i.e. ordinal value) to respondents’ ratings of factors with respect to severity. The weighted mean was computed from Rogers as;

represents factors under evaluation.

w₁, w₂, … w_n represents the weightings of the factors that translate to

w₁ = number of respondents who answered Very Low.

w₂ = number of respondents who answered Low.

w₃ = number of respondents who answered Moderate.

w₄ = number of respondents who answered High.

w₅ = number of respondents who answered Very High.

The mean score values of five (5) main causes of insolvency originating from five (5) main unethical practices that might have been the source of the cause of insolvency in Nigerian construction industry was precipitated in matrix form to show a multinomial scalar combination in Table 2, and a calibrated scale by ranking shown in Table 3. This was against the Arain study of the Pakistan construction industry that gave six (6) unethical practices from five (5) significant causes. This paper resorted to transition operator (see Equations 1 and 2 for recourse) that skews distribution variables to regions permissible only to them by theorizing a characteristic value for the distribution, been the distribution determinant from the ϕ_eϕ_i distribution matrix as the skew Eigen vector. The transition from a critical point (i.e. a threshold to a rejection region) requires that an operand operates at the boundary domain to govern the filtration process. Krall had demonstrated that such process exist with stieltjes differential – boundary operators using the uniform limit of Picard-like approximations [μ_λ(s, t)] that can give the limit of an integration of say an m = n or 5 × 5 matrix illustrated in:

This context as;

Arising from Table 2, a matrix determinant value was computed from Eqn 16, that corresponds to an operator been the characteristic property of the matrix responsible for the skew distributing gradient of ϕ_eϕ_i distribution. This property shows symmetric relation to the coercivity properties of combined boundary integral investigation carried out by Spence on high-frequency scattering of variables in distributive fields. This paper considered the ϕ_eϕ_i values as normally distributed having been subjected to a normality test and a chaotic state occasioned by their transition attempts to accept or reject domain. An investigation as to why certain values are restricted at the critical plane of a normal distribution made a logical appeal that bears semblance to a threshold gate. On this basis, the adaptation of the boundary point/critical plane of a geometrically concave normal distribution as threshold gate shows congruency to the Feischl, Feischl, Werndland, and Kita and Fausmann conditions of Adaptive Boundary Element Methods which ab initio restrains variables transition to regions. Most values of ϕ_eϕ_i will be impeded by operator filtration from entering the rejection region, except exceeding the transient threshold of the transition gradient operator that performs the threshold operation of filtration at the threshold gate as a replica of mixed impedance transmission boundary. Congruency of this preposition has been demonstrated in Costabel on Boundary integral operators on Lipschitz Domains and Bernardi, Costabel, Dauge and Girault on continuity properties of the Inf-Sup Constants for divergence in conservative spaces (Figure 1).

Accordingly, this paper deduce a threshold operation performed on ϕ_eϕ_i distribution in order to obtain a threshold value at the boundary of a critical plane using the distribution characteristic value (determinant) of ϕ_eϕ_i computed from Equatio (16) as an operand that permits the transition of variables by gradient filtration at the critical plane of the normal curve when operated on at point been the threshold gate/boundary. This process shows congruency with typical mixed impedance transmission problems which has been given exhaustive narration and applicability in Athanasiadis and obtained an adaptive logic license for it with recourse to Claeys “Quasi-local multitrace boundary integral formulation” process. This filtration process is affirmed to be consistent with the replication logic of the “Maxwell’s demon” which is an imaginary creature to whom Maxwell assigned the task of operating a door in a partition dividing a volume containing gas at uniform temperature. The door is usually opened by the demon to enable fast molecules to move (say) from left to right through a partition. In this way, without expenditure of external work, the gas on the right could be made hotter than before and that on the left made cooler. The probable value of the threshold, det (∇_*), operator on as a boundary point correlated to the introduction of quaternion in Equation (14, 15) and fitted to the calibration on Table 4 to deduce the threshold value of insolvency as an insolvent point (Figure 2).

Table 4. Summary of demographic information of respondents.

Paper Structure

Section 1, gave an insight and exposed the issue and problems of insolvency in construction projects

Section 2, identified the trend of research efforts towards resolving or mitigating insolvency in construction projects and the attendant gaps created by the research

For instance;

1. ASHURT proposed organizational sensitivity to early warning signs and alarms.

2. Turner and Townsend proposed a 4-way test of identifying insolvent situation.

3. Bahram proposed legal recovery measures and claim protection methods towards diffusing insolvency to mitigate insolvency on risk transfer basis.

4. Arain and Low proposed minimizing variation orders to stern the tide of insolvency

5. Wood prescribed friendly relational model in the form of mutual commitment to offset insolvency.

6. Arain identified several causes of insolvency amounting from some unethical practices of contractors.

In line with the objective of this paper, no effort has been made in terms of research to identify a contractor’s insolvent point amid the series of causes.

Section 3, attempts a logical description of the method employed to achieve the objective of identifying the point at which a contractor becomes insolvent. This was done by treating the economic sphere of Nigeria as a spherical harmonic polynomial and took a sector of the sphere as construction industry sector. Did a field investigation on the sector thereafter subjected the result to a rank order and performed a threshold operation on the variable to identify the threshold point of becoming insolvent.

Section 4, presented the results of the field survey and showed contrast or correlation with existing knowledge.

Section 5, presented the findings from the analysis of the field survey with respect to the theoretical framework.

Basic Assumption of Variables Chaotic State

This paper typifies its construct of a chaotic state to the Vieru general definitions of chaos for continuous and discrete-time processes. Vieru posited that let be a family of continuous functions, mapping a metric space (Θ, d₁) into a metric space (Λ,d₂) and depending on a parameter a, whose values are called initial conditions. Let, (Ω, d₀) be the metric space of initial conditions. . This paper will more often assume Θ and Λ are either connected sets or everywhere dense parts of connected sets. In discrete processes, this paper will only assume Θ has at least one cluster point (usually ∞). This paper will consider continuous processes and more often assume (Θ, d₁) is a path-connected metric space. However, our definitions still hold for metric spaces that are not path-connected. The restrictions contained in our definitions drop by themselves if the set of paths in the metric space (or in its considered subsets) is empty.

This paper will call the elements of Ω points, while the elements of the domain of the functions ψa will be called moments, even if Θ may not necessarily be a subset of R.

In this article, this paper will not consider cases in which Θ and Ωare sets of fractal dimension. Since there is no possible confusion, this paper will designate d₀ (x, y), d₁(z, t) and d₂ (u, v) by |x – y|, |z – t| and |u – v|. Regarding the normal distribution continuum, this paper subjected the entities of the distribution been ϕ_eϕ_i variables to the Vieru chaotic state for continuous field by idealizing a mapping metric space by an Nmorcha operator as:

Results and Discussion

The outcome of the result showing a multinomial combination values of ϕ_i unethical practices (ϕ_i) and causes of insolvency (ϕ_e) as ϕ_eϕ_i

i₅ = cash flow problem

i₄ = Poor cost control

i₃ = Onerous conditions of contract

i₂ = Knock-on effects

i₁ = Absence of barriers

e_i = payment delays

e₂ = falsification of financial status

e₃ = Abuse of resources

e₄ = Inaccurate estimating practices

e₅ = Irreconcilable contractor’s concern

On the basis of Eqn (16) our value read as det (∇_*) yielded 0.5252.

Recursive applications

In this study, this paper associated more appropriately the Nigerian economy as a closed sphere to a spherical harmonic polynomial and from Maxwell’s theory of poles on spherical polynomial, a geometric interpolation showed that spherical sections from Equation (5) reduces the sphere to a sector when m = n, zonal when m = 0 and tesseral when 1 ≥ m ≥ n (see Equation 5). The Nigerian construction industry sector became a consequence of Equation (5) from the spherical economy. The sector (construction industry) is burdened with having to determine the cause(s) of contractor’s insolvency emanating from some unethical practices. Field survey result from Table 5, yielded five (5) significant causes of insolvency arising from five (5) unethical practices i.e.(ϕ_eϕ_i)= (5, 5). The sector’s result by status of Equation (5) conditionalities yield a scalar combinatorial value of ϕ_eϕ_i in matrix form of order 5 × 5 that precipitated 25 combinatorial variables (Table 6).

Table 5. Mean score value on causes of insolvency arising from unethical practices of contractors) with m = n rank (m).