upon substituting z = 1 in (18) above. Finally, the lemma holds if (20) is rearranged and simplified. □

Corollary 6 From the numerical results (Tables 1-6 in the appendix), it is clear that

1) $\rho \to \rho \left(c\right)$ .

2) $\left\{N\left(\rho \right)\right\}\to \left\{N\left(\rho \mathrm{,}c\right)\right\}$ .

Lemma 7 (First Criterion) A maximizer of the group $\left\{N\right\}$ is the solution for the policy map-constraint-occupation rate problem $G\left(\mathrm{.}\right)$ such that

$G\left({\partial}_{T}N,{\partial}_{\rho}N,{\partial}_{c}N,N,T,c,\rho \right)=0.$ (21)

Proof. We seek a solution $N\left(c\mathrm{,}\rho \mathrm{,}T\right)$ for $G\left(\mathrm{.}\right)$ such that

$G(.)={\partial}_{c}N+F\left({\partial}_{T}N,{\partial}_{\rho}N,N,T,c,\rho \right).$ (22)

where $F\left(\mathrm{.}\right)$ is a real valued semi-linear continuous function with respect to all its arguments. (22) is equivalent to

$G(.)={\partial}_{c}N+{\partial}_{T}N+h\left(T,c,\rho ,N\right){\partial}_{\rho}N+\stackrel{\xaf}{h}\left(T,c,\rho ,N\right).$ (23)

By Lemma 1, T is necessarily compact. Let T be a fixed point of G(.). Then (23) reduces to

${\partial}_{c}N=-h\left(c,\rho ,N\right){\partial}_{\rho}N-\stackrel{\xaf}{h}\left(c,\rho ,N\right).$ (24)

Consider a differentiable arc $B\mathrm{:}\rho \to \rho \left(c\right)$ in the $\left(\rho \mathrm{,}c\right)$ plane such that points on $B\to \left(\rho \left(c\right)\mathrm{,}c\right)$ .

By multivariate chain rule on the left hand side of (24), we have

$\frac{\text{d}}{\text{d}c}N\left(\rho \left(c\right),c\right)={\partial}_{c}N\left(\rho \left(c\right),c\right)+{\partial}_{\rho}N\left(\rho \left(c\right),c\right)\frac{\text{d}\rho \left(c\right)}{\text{d}c}.$ (25)

Assuming that $N\left(\rho \mathrm{,}c\right)$ solves (21) and combining (24) and (25) and rearranging, we have

$\frac{\text{d}}{\text{d}c}N\left(\rho \left(c\right),c\right)=\left(-h\left(\mathrm{..}\right)+\frac{\text{d}\rho \left(c\right)}{\text{d}c}\right){\partial}_{\rho}N\left(\rho \left(c\right),c\right)-\stackrel{\xaf}{h}\left(\mathrm{..}\right)$ (26)

So^{5} that the couple differential equations

$\frac{\text{d}}{\text{d}\rho}N\left(\rho \left(c\right),c\right)=h\left(\rho \left(c\right),c,N\left(\rho \left(c\right),c\right)\right)$ (27)

and

$\frac{\text{d}}{\text{d}c}N\left(\rho \left(c\right),c\right)=\left(-\stackrel{\xaf}{h}\left(\rho \left(c\right),c,N\left(\rho \left(c\right),c\right)\right)\right)$ (28)

constitute in general a solution for $G\left(\mathrm{.}\right)$ for $N\left(\rho \left(c\right)\mathrm{,}c\right)$ when T is a fixed point of $G\left(\mathrm{.}\right)$ . □

Lemma 8 (Second Criterion) Any continuously differentiable solution $N\left(\rho \mathrm{,}c\right)$ for $G\left(\mathrm{.}\right)$ satisfying the first criterion above must coincide with the original solution $\stackrel{\u02dc}{N}\left(\stackrel{\u02dc}{\rho}\left(c\right)\mathrm{,}c\right)$ along the base curve $\rho =\rho \left(c\right)$ .

Proof. Since (27) and (28) are coupled systems, only in rare cases analytic solution exists in closed form. However, if we specify an initial value for ${c}_{0}$ , ${\rho}_{0}$ and ${N}_{0}$ , then the existence of a unique solution pair $\stackrel{\u02dc}{\rho}\left(c\right)$ and $\stackrel{\u02dc}{N}\left(\stackrel{\u02dc}{\rho}\left(c\right)\mathrm{,}c\right)$ is guaranteed. A solution $\stackrel{^}{N}\left(\stackrel{\u02dc}{\rho}\left(c\right)\mathrm{,}c\right)$ that did not pass through the origin leading to $\stackrel{\u02dc}{N}\left(\stackrel{\u02dc}{\rho}\left(c\right)\mathrm{,}c\right)$ cannot be a solution for $G\left(\mathrm{.}\right)$ since it is nowhere differentiable around $G\left(\mathrm{.}\right)$ . □

Lemma 9 (Optimality Criterion) Suppose $\Vert T\Vert \to \Vert {T}_{\mathrm{max}}\Vert $ so that the constraint dependent occupation rate ${\rho}_{c}\to {\rho}_{\mathrm{max}}\left(c\right)\in \left(0,1\right)$ . A solution $N\left(\rho \left(c\right),c\right)$ that passes through the origin $\stackrel{\u02dc}{N}\left(\stackrel{\u02dc}{\rho}\left(c\right)\mathrm{,}c\right)$ for $G\left(\mathrm{.}\right)$ is optimal a.s.

Proof. Given that $\Vert T\Vert \to \Vert {T}_{\mathrm{max}}\Vert $ , we have ${\rho}_{c}\to {\rho}_{\mathrm{max}}\left(c\right)\in \left(0,1\right)$ . By the numerical approximation (Tables 1-6 in the appendix) $N\left(\rho \left(c\right),c\right)\to {N}_{\mathrm{max}}\left(\rho \left(c\right),c\right)$ . Given that $G\left(\mathrm{.}\right)$ is $N\left(\rho \left(c\right),c\right)$ dependent, it is then trivial. □

3. Scope for Future Work

^{5
$\left(\mathrm{..}\right)=\left(\rho \left(c\right)\mathrm{,}c\mathrm{,}N\left(\rho \left(c\right)\mathrm{,}c\right)\right)\mathrm{.}$ }

There is a scope in extending our results to some special cases of the problem solved in this work. For instance, when the function $h\left(\mathrm{..}\right)$ is independent of N or when $h\left(\mathrm{..}\right)$ and $\stackrel{\xaf}{h}\left(\mathrm{..}\right)$ are linear or even non-linear combination of $\rho $ and T and N. The author are grateful to all literature sources used.

Competing Interest

There is no competing interest of any kind within the authorship of this work.

Authors Contribution

SS: Drafted the entire manuscript, provided the introductory chapter (Section 1) and proved Lemmas 1, 2, 3, 5, 7, 8 and 9 together with Corollary 6.

HM: Participated in the sequence alignment of the manuscript and provided the numerical simulations.

MLM: Participated in the design of the manuscript and proved lemma 4 under my supervision.

All authors read and approved the final manuscript.

Acknowledgements

The authors are grateful to Dr. Babangida A. Albaba; the current Rector of the Katsina State Institute of Technology and Management (KSITM) for spearing time to go through the entire thesis leading to this manuscript and for making valuable suggestions.

Appendix

For a numerical approximation, we study the model in (11) under various sizes of constraint numbers c and varying occupation rate $\rho $ for $E\left[N\right]$ . The following numerical results are obtained.

Table 1. E[N] when c = 0.

Table 2. E[N] when c = 10._{ }

Table 3. E[N] when c = 23._{ }

Table 4. E[N] when r = 0.5._{ }

Table 5. E[N] when r = 0.75._{ }

Table 6. E[N] when r = 0.9.

Conflicts of Interest

The authors declare no conflicts of interest.

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