Model predictive control based integration of pricing and production planning

Messaoud Bounkhel; Lotfi Tadj; Messaoud Bounkhel; Lotfi Tadj

doi:10.3934/math.2024113

AIMS Mathematics

2024, Volume 9, Issue 1: 2282-2307. doi: 10.3934/math.2024113

Previous Article Next Article

Research article Special Issues

Model predictive control based integration of pricing and production planning

Messaoud Bounkhel ^{1
,
,},
Lotfi Tadj ²

1.
Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
2.
Department of Industrial Engineering, Alfaisal University, Riyadh 12714, Saudi Arabia; ltadj@alfaisal.edu

Received: 02 September 2023 Revised: 09 December 2023 Accepted: 18 December 2023 Published: 22 December 2023
MSC : 49J15, 90B30

We witness an increased emphasis on the integration of different areas of manufacturing firms with the intention of avoiding suboptimal solutions. In particular, due to new technologies which make it easy to change the price of a product in real time, the integration of pricing and production planning may be garnering the most interest. We are proposing in this paper a way to model the dynamics of the price. Thus, the price and the inventory level are considered as state variables whereas the supply (production) rate is the control variable. The demand rate is dynamic and state-dependent. Using a model predictive control approach, the optimal supply rate, and thus the optimal price and inventory level, are obtained. Different examples are provided under different scenarios for the supply rate and for the demand rate.

Keywords:

Citation: Messaoud Bounkhel, Lotfi Tadj. Model predictive control based integration of pricing and production planning[J]. AIMS Mathematics, 2024, 9(1): 2282-2307. doi: 10.3934/math.2024113

Related Papers:

[1]	Junhong Li, Ning Cui . A hyperchaos generated from Rabinovich system. AIMS Mathematics, 2023, 8(1): 1410-1426. doi: 10.3934/math.2023071
[2]	A. E. Matouk . Chaos and hidden chaos in a 4D dynamical system using the fractal-fractional operators. AIMS Mathematics, 2025, 10(3): 6233-6257. doi: 10.3934/math.2025284
[3]	Hany A. Hosham, Thoraya N. Alharthi . Bifurcation and chaos in simple discontinuous systems separated by a hypersurface. AIMS Mathematics, 2024, 9(7): 17025-17038. doi: 10.3934/math.2024826
[4]	Ning Cui, Junhong Li . A new 4D hyperchaotic system and its control. AIMS Mathematics, 2023, 8(1): 905-923. doi: 10.3934/math.2023044
[5]	Rahat Zarin, Abdur Raouf, Amir Khan, Aeshah A. Raezah, Usa Wannasingha Humphries . Computational modeling of financial crime population dynamics under different fractional operators. AIMS Mathematics, 2023, 8(9): 20755-20789. doi: 10.3934/math.20231058
[6]	Michael Precious Ineh, Umar Ishtiaq, Jackson Efiong Ante, Mubariz Garayev, Ioan-Lucian Popa . A robust uniform practical stability approach for Caputo fractional hybrid systems. AIMS Mathematics, 2025, 10(3): 7001-7021. doi: 10.3934/math.2025320
[7]	Kareem Alanazi, Omar Naifar, Raouf Fakhfakh, Abdellatif Ben Makhlouf . Innovative observer design for nonlinear systems using Caputo fractional derivative with respect to another function. AIMS Mathematics, 2024, 9(12): 35533-35550. doi: 10.3934/math.20241686
[8]	Chatthai Thaiprayoon, Jutarat Kongson, Weerawat Sudsutad . Dynamics of a fractal-fractional mathematical model for the management of waste plastic in the ocean with four different numerical approaches. AIMS Mathematics, 2025, 10(4): 8827-8872. doi: 10.3934/math.2025405
[9]	A. M. Alqahtani, Shivani Sharma, Arun Chaudhary, Aditya Sharma . Application of Caputo-Fabrizio derivative in circuit realization. AIMS Mathematics, 2025, 10(2): 2415-2443. doi: 10.3934/math.2025113
[10]	Michael Precious Ineh, Edet Peter Akpan, Hossam A. Nabwey . A novel approach to Lyapunov stability of Caputo fractional dynamic equations on time scale using a new generalized derivative. AIMS Mathematics, 2024, 9(12): 34406-34434. doi: 10.3934/math.20241639

Abstract

1. Introduction

A graph labeling is an assignment of integers to the vertices or edges, or both, subject to certain conditions. Gallian ^[3] has written a dynamic survey of graph labeling. MacDougall et al. ^[5] introduced the notion of a vertex magic total labeling of graphs. Let $G$ be a graph of order $n$ and size $m$ . A vertex magic total labeling of $G$ is defined as a one-to-one function

$f: V(G) \cup E(G) \rightarrow \{1, 2, \cdots, n+m\}$

with the property that for each vertex $u$ of $G$ ,

$f(u)+\sum\limits_{\substack{v\in N(u)}}f(uv) = k$

for some constant $k$ where $N(u)$ is the neighborhood of $u$ . The constant $k$ is called the magic constant for $f$ . The vertex-magic total labelings of wheels and related graphs were studied in ^[6], and later in ^[11]. The properties of the general graphs such as cycles, paths, complete graphs, wheels, bipartite graphs and trees, which satisfy the vertex magic total labelings, were studied in ^[10]. MacDougall et al. ^[4] introduced the concept of a super vertex magic total labeling. They defined a vertex magic total labeling to be super if

$f[V(G)] = \{1, 2, \cdots, n\}.$

In 2017, Nagaraj et al. ^[7] introduced the concept of an even vertex magic total labeling. They called a vertex magic total labeling as even if

$f[V(G)] = \{2, 4, \cdots, 2n\}.$

A graph $G$ is called an even vertex magic if there exists an even vertex magic total labeling of $G$ . We note that if $G$ is an even vertex magic, then $n \leq m$ . The following results, which appeared in ^[7], are useful to us.

Theorem 1.1. ^[7] Let $G$ be a nontrivial graph of order $n$ and size $m$ . If $G$ is an even vertex magic, then magic constant $k$ is given by the following:

${k = \frac{m^2+2mn+m}{n}}.$

A wheel $W_n$ , $n \geq 3$ , is a graph of order $n+1$ that contains a cycle $C_n$ , for which every vertex in the cycle $C_n$ is connected to one other vertex known as the hub. The edges of the wheel which are incident to the hub are called spokes. The vertices and edges of the cycle $C_n$ in $W_n$ are called rim vertices and rim edges, respectively. It was shown in ^[7] that a wheel $W_n$ has no even vertex magic total labeling, as we state next.

Theorem 1.2. ^[7] A wheel $W_n$ is not even vertex magic.

In this paper, the labeling problem is related to the work in ^[1]. In addition to the aforementioned vertex labeling by even numbers $2, 4, \cdots, 2n$ , they studied vertex labelings by using three consecutive numbers $0, 1, 2$ with some specific properties. These labelings were referred to as a weak Roman dominating function and a perfect Roman dominating function.

From the studies in ^[8,9,12], there exist graphs with the same order and size that are even vertex magics. Moreover, the wheel related graphs, namely fans, cycles and suns, having the even vertex magic total labelings were established in ^[7]. However, since these graphs have the same order and size, it is interesting and challenging to study wheel related graphs when the size is greater than the order, which have an even vertex magic total labeling.

The $t$ -fold wheel $W_{n, t}$ , $n \geq 3$ , $t \geq 1$ , is a wheel related graph derived from a wheel $W_n$ by duplicating the $t$ hubs, each adjacent to all rim vertices, and not adjacent to each other. It is observed that the $t$ -fold wheel $W_{n, t}$ has a size $nt+n$ that exceeds its order $n+t$ . The goal of this paper is to study conditions for an even vertex magic $W_{n, t}$ in terms of $n$ and $t$ . Furthermore, we also determine an even vertex magic total labeling of some $t$ -fold wheel $W_{n, t}$ .

2. The conditions for an even vertex magic $W_{n, t}$

Since the $1$ -fold wheel $W_{n, 1}$ is isomorphic to the wheel $W_n$ and by Theorem 1.1, $W_n$ is not an even vertex magic. In this section, we consider the $t$ -fold wheel $W_{n, t}$ , where $n$ and $t$ are integers with $n\geq 3$ and $t\geq2$ .

In order to present the conditions for an even vertex magic $W_{n, t}$ , we initially explore the magic constant of the $t$ -fold wheel $W_{n, t}$ of order $n+t$ and size $nt+n$ by employing Theorem 1.1.

Proposition 2.1. Let $n$ and $t$ be integers with $n\geq 3$ and $t\geq2$ . If the $t$ -fold wheel $W_{n, t}$ is an even vertex magic, then the magic constant is defined as follows:

$k = 2nt+3n+\frac{n^2t^2+2n^2t+n}{n+t}.$

We are able to show the bound of an integer $t$ for the $t$ -fold wheel having an even vertex magic total labeling as follows.

Proposition 2.2. Let $n$ and $t$ be integers with $n\geq 3$ and $t\geq2$ . If the $t$ -fold wheel $W_{n, t}$ is an even vertex magic, then $2 \leq t \leq n$ .

Proof. Suppose that the $t$ -fold wheel $W_{n, t}$ is an even vertex magic with magic constant $k$ . By Proposition 2.1, we obtain the following:

$k = 2nt+3n+\frac{n^2t^2+2n^2t+n}{n+t}.$

On the contrary, assume that $t > n$ . Let $t = n+r$ , for some $r \geq 1$ . Then,

$n^2t^2+2n^2t+n = n^4+2n^3r+n^2r^2+2n^3+2n^2r+n$

and

$n+t = n+(n+r) = 2n+r.$

Let

$P(n) = n^4+2n^3r+n^2r^2+2n^3+2n^2r+n.$

By using the remainder theorem, the remainder when $P(n)$ is divided by $2n+r$ is as follows:

$P(-\frac{r}{2}) = \frac{r^4+4r^3-8r}{16}.$

$P(-\frac{r}{2}) = 0,$

then $r = -2$ , which is a contradiction. Thus,

$P(-\frac{r}{2}) \neq 0.$

Specifically, $n^2t^2+2n^2t+n$ is not divisible by $n+t$ . Thus, $k$ is not an integer, which is a contradiction. Therefore, $2 \leq t \leq n$ . □

According to Proposition 2.2, the $t$ -fold wheel $W_{3, t}$ is not an even vertex magic, where $t \geq 4$ . shows the even vertex magics $W_{3, 2}$ and $W_{3, 3}$ with magic constants $k = 36$ and $k = 50$ , respectively, where their vertices and edges are labeled by the even vertex magic total labelings. We present an even vertex magic total labeling of the $t$ -fold wheel $W_{n, t}$ by considering only the integer $n$ as the following results.

Figure 1. Even vertex magic total labelings of

$W_{3, 2}$ and

$W_{3, 3}$ .

DownLoad: Full-Size Img PowerPoint

Proposition 2.3. For every integer $n\geq 3$ , if the $n$ -fold wheel $W_{n, n}$ is an even vertex magic, then $n$ is odd.

Proof. Let $n$ be an integer with $n\geq 3$ . Suppose that the $n$ -fold wheel $W_{n, n}$ is an even vertex magic with a magic constant $k$ . On the contrary, assume that $n$ is even. There exists an integer $q$ such that $n = 2q$ . By Proposition 2.1,

$k = 2n^2+3n+\frac{n^3+2n^2+1}{2}.$

Since

$n^3+2n^2+1 = 8q^3+8q^2+1$

is odd, $n^3+2n^2+1$ is not divisible by $2$ . Thus, $k$ is not an integer, which is a contradiction. Therefore, $n$ is odd. □

As we have seen in , the $3$ -fold wheel $W_{3, 3}$ is an even vertex magic, as indicated by Proposition 2.3. By an argument similar to the one used in the proof of Proposition 2.3, we obtain the condition for an even vertex magic $W_{n, n-2}$ , as we now show.

Proposition 2.4. For every integer $n\geq 4$ , if the $(n-2)$ -fold wheel $W_{n, n-2}$ is an even vertex magic, then $n$ is even.

The even vertex magic total labeling of the $2$ -fold wheel $W_{4, 2}$ with a magic constant $k = 50$ is shown in Figure 2.

Figure 2. Even vertex magic total labeling of

$W_{4, 2}$ .

DownLoad: Full-Size Img PowerPoint

In order to deduce an even vertex magic total labeling of the $t$ -fold wheel for achieving the main result, we need some additional notation for the $t$ -fold wheel $W_{n, t}$ . For every pair of integers $n\geq 3$ and $t\geq2$ , let

$V(W_{n, t}) = \{u_1, u_2, \ldots, u_n, v_1, v_2, \ldots, v_t\}$

and

$E(W_{n, t}) = \{u_iu_{i+1} | 1 \leq i \leq n-1\}\cup\{u_nu_1\}\cup\{u_iv_j | 1 \leq i \leq n, 1 \leq j \leq t\}.$

Suppose the $t$ -fold wheel $W_{n, t}$ is an even vertex magic. Then, for any even vertex magic total labeling $f$ of $W_{n, t}$ , let

$S_{rv} = \sum\limits_{i = 1}^{n}f(u_i), \ \ \ S_{re} = \sum\limits_{i = 1}^{n-1}f(u_iu_{i+1})+f(u_nu_1)$

and

$S_{h} = \sum\limits_{j = 1}^{t}f(v_j), \ \ \ S_{s} = \sum\limits_{j = 1}^{t} \sum\limits_{i = 1}^{n}f(u_iv_j).$

Next, we present the following lemma to show the necessary condition for an even vertex magic $W_{n, t}$ with the following magic constant:

$k = 2nt+3n+\frac{n^2t^2+2n^2t+n}{n+t}.$

Note that

$S_{rv}+2S_{re}-S_{h} = (n-t)k.$

Lemma 2.5. Let $n$ and $t$ be integers where $n\geq 3$ and $t\geq2$ . If the $t$ -fold wheel $W_{n, t}$ is an even vertex magic, then

$S_{rv}+2S_{re}-S_{h} = \\ (t^2+4t+3)n^2+(-2t^3-6t^2-3t+1)n+\frac{(t^3+2t^2-1)(2nt)}{n+t}.$

With the aid of Lemma 2.5 and Proposition 2.2, the necessary condition for an even vertex magic total labeling of the $t$ -fold wheel $W_{n, t}$ can also be given in terms of $n$ and $t$ .

Proposition 2.6. Let $n$ and $t$ be integers where $n\geq 3$ and $t\geq2$ . If the $t$ -fold wheel $W_{n, t}$ is an even vertex magic, then

$(-t^2-2t+1)n^2+(2t^3+6t^2+7t+1)n-(t^2+t)-\frac{(t^3+2t^2-1)(2nt)}{n+t} \geq 0.$

Proof. Suppose that the $t$ -fold wheel $W_{n, t}$ is an even vertex magic. By Lemma 2.5,

$S_{rv}+2S_{re}-S_{h} = (t^2+4t+3)n^2+(-2t^3-6t^2-3t+1)n+\frac{(t^3+2t^2-1)(2nt)}{n+t}.$

Next, we consider the maximum of $(S_{rv}+2S_{re}-S_{h})$ .

By Proposition 2.2, $2\leq t \leq n$ , and then $2n+2t < nt+n+t+1$ . The maximum of

$\begin{array}{l} (S_{rv}+2S_{re}-S_{h}) & = & \sum\limits_{\substack{2t+2\leq i\leq 2n+2t\\ i\; \text{is even}}}i+2\sum\limits_{i = nt+n+t+1}^{nt+2n+t}i-\sum\limits_{\substack{2\leq i\leq 2t\\ i\; \text{is even}}}i \\ & = & \left(\sum\limits_{\substack{2\leq i\leq 2n+2t\\ i\; \text{is even}}}i-\sum\limits_{\substack{2\leq i\leq 2t\\ i\; \text{is even}}}i\right)+2\left(\sum\limits_{i = 1}^{nt+2n+t}i-\sum\limits_{i = 1}^{nt+n+t}i\right)-\sum\limits_{\substack{2\leq i\leq 2t\\ i\; \text{is even}}}i \\ & = & \left(\frac{(2n+2t)(2n+2t+2)}{4}-\frac{(2t)(2t+2)}{4}\right) \\ && +2\left(\frac{(nt+2n+t)(nt+2n+t+1)}{2}-\frac{(nt+n+t)(nt+n+t+1)}{2}\right) -\frac{(2t)(2t+2)}{4} \\ & = & 2n^2t+4n^2+4nt-t^2+2n-t \, . \end{array}$

Since $S_{rv}+2S_{re}-S_{h}$ does not exceed the maximum of $(S_{rv}+2S_{re}-S_{h})$ , the maximum of

$(S_{rv}+2S_{re}-S_{h}) - (S_{rv}+2S_{re}-S_{h}) \geq 0.$

Therefore,

$(-t^2-2t+1)n^2+(2t^3+6t^2+7t+1)n-(t^2+t)-\frac{(t^3+2t^2-1)(2nt)}{n+t} \geq 0.$

□

Now, we investigate the sufficient condition for a labeling $f$ that can be an even vertex magic total labeling of $W_{n, n}$ when $n$ is odd.

Theorem 2.7. Let $n$ be an odd integer where $n \geq 3$ . For every $n$ -fold wheel $W_{n, n}$ , let

$f: V(W_{n, n}) \cup E(W_{n, n}) \rightarrow \{1, 2, \ldots, n^2+3n\}$

be defined by the following:

$\begin{array}{l} f(u_i) ~ = ~ 2i, ~~ \mathit{\mbox{if }}~~ 1 \leq i \leq n , \\ f(v_j) ~ = ~ 2n+2j, ~~ \mathit{\mbox{if }}~~ 1 \leq j \leq n , \\ f(u_iu_{i+1}) ~ = ~ 2n+1-2i, ~~ \mathit{\mbox{if}}~~ 1 \leq i \leq n-1 , \\ f(u_nu_1) ~ = ~ 1, ~~ \\ f(u_{n+1-j}v_j) ~ = ~ n^2+3n+1-2j, ~~ \mathit{\mbox{if }}~~ 1 \leq j \leq n , \\ f[EH]-f[\{u_{n+1-j}v_j| 1 \leq j \leq n\}] ~ = ~ \{2n+1, 2n+3, \ldots, n^2+n-1\} ~~ \\ ~~ \cup\{4n+2, 4n+4, \ldots, n^2+3n\}, ~~ \mathit{\mbox{if}}~~ EH = \{u_iv_j| 1 \leq i, j \leq n\} \, . \end{array}$

$\sum\limits_{j = 1}^{n-1}f(u_1v_j) = \frac{n^3+4n^2-5}{2},$

then $f$ can be an even vertex magic total labeling of $W_{n, n}$ .

Proof. Assume that

$\sum\limits_{j = 1}^{n-1}f(u_1v_j) = \frac{n^3+4n^2-5}{2}.$

We have that

$\begin{array}{l} S_{s}-\sum\limits_{j = 1}^{n}f(u_{n+1-j}v_j) & = & \sum\limits_{\substack{2n+1\leq i\leq n^2+n-1 \\ i\; \text{is odd}}}i + \sum\limits_{\substack{4n+2\leq i\leq n^2+3n \\ i\; \text{is even}}}i \\ & = & \left(\sum\limits_{\substack{1\leq i\leq n^2+n-1\\ i\; \text{is odd}}}i - \sum\limits_{\substack{1\leq i\leq 2n-1\\ i\; \text{is odd}}}i\right) + \left(\sum\limits_{\substack{2\leq i\leq n^2+3n\\ i\; \text{is even}}}i - \sum\limits_{\substack{2\leq i\leq 4n\\ i\; \text{is even}}}i\right) \\ & = & \left(\frac{(n^2+n)^2}{4} - \frac{(2n)^2}{4}\right) + \left(\frac{(n^2+3n)(n^2+3n+2)}{4} - \frac{4n(4n+2)}{4}\right) \\ & = & \frac{n^4+4n^3-4n^2-n}{2}, \end{array}$

and then,

$\left(S_{s}-\sum\limits_{j = 1}^{n}f(u_{n+1-j}v_j)\right)-\sum\limits_{j = 1}^{n-1}f(u_1v_j) = \frac{n^4+3n^3-8n^2-n+5}{2}\, .$

Next, we consider the sum of the label of each vertex and the labels of all edges incident to this vertex. By the assumption, for $1 \leq j\leq n$ ,

$\begin{array}{l} f(v_j)+\sum\limits_{i = 1}^{n}f(u_iv_j) & = & f(v_j)+f(u_{n+1-j}v_j)+\sum\limits_{\substack{1\leq i\leq n\\ i \neq n+1-j}}f(u_iv_j) \\ & = & f(v_j)+f(u_{n+1-j}v_j)+\frac{S_{s}-\sum\limits_{j = 1}^{n}f(u_{n+1-j}v_j)}{n} \\ & = & (2n+2j)+(n^2+3n+1-2j)+\frac{n^3+4n^2-4n-1}{2}\\ & = & \frac{n^3+6n^2+6n+1}{2}, \end{array}$

$\begin{array}{l} f(u_1)+f(u_1u_2)&+&f(u_nu_1)+\sum\limits_{j = 1}^{n}f(u_1v_j)\\ & = & f(u_1)+f(u_1u_2)+f(u_nu_1)+f(u_1v_n)+\sum\limits_{j = 1}^{n-1}f(u_1v_j)\\ & = & 2+(2n+1-2)+1+(n^2+n-1+2)+\frac{n^3+4n^2-5}{2} \\ & = & \frac{n^3+6n^2+6n+1}{2} \, . \end{array}$

For $2 \leq i\leq n,$

$\begin{array}{l} f(u_i)+f(u_iu_{i+1})&+&f(u_{i-1}u_i)+\sum\limits_{j = 1}^{n}f(u_iv_j)\\ & = & f(u_i)+f(u_iu_{i+1})+f(u_{i-1}u_i)+f(u_iv_{n+1-i})+\sum\limits_{\substack{1\leq j\leq n\\ j \neq n+1-i}}f(u_iv_j)\\ & = & f(u_i)+f(u_iu_{i+1})+f(u_{i-1}u_i)+f(u_iv_{n+1-i})+\frac{ \left(S_{s}-\sum\limits_{j = 1}^{n}f(u_{n+1-j}v_j)\right)-\sum\limits_{j = 1}^{n-1}f(u_1v_j)}{n-1}\\ & = & 2i+(2n+1-2i)+(2n+1-2i+2)+(n^2+n-1+2i)+\frac{n^3+4n^2-4n-5}{2}\\ & = & \frac{n^3+6n^2+6n+1}{2} \, . \end{array}$

Therefore, $f$ can be an even vertex magic total labeling of $W_{n, n}$ with a magic constant

$k = \frac{n^3+6n^2+6n+1}{2}.$

□

Now, we investigate the sufficient condition for a labeling $f$ that can be an even vertex magic total labeling of $W_{n, n-2}$ when $n$ is even.

Theorem 2.8. Let $n$ be an even integer with $n \geq 4$ . For every $(n-2)$ -fold wheel $W_{n, n-2}$ , let

$f: V(W_{n, n-2}) \cup E(W_{n, n-2}) \rightarrow \{1, 2, \ldots, n^2+n-2\}$

be defined by the following:

$\begin{array}{rcll} f(u_i) & = & 2i, &\mathit{\mbox{if }}~~ 1 \leq i \leq n , \\ f(v_j) & = & 2n+2j, &\mathit{\mbox{if}}~~ 1 \leq j \leq n-2 , \\ f[EC] & = & \{a_1, a_2, \cdots, a_n\}, & \mathit{\mbox{if}}~~ EC = \{u_iu_{i+1}, u_nu_1 | 1 \leq i \leq n-1 \}, \\ f[EH] & = & \{1, 2, \cdots, n^2+n-2\} &\\&& -\{2, 4, \cdots, 2n+4, a_1, a_2, \cdots, a_n\}, &\mathit{\mbox{if}}~~ EH = \{u_iv_j| 1 \leq i \leq n, 1 \leq j \leq n-2\} . \end{array}$

$S_{s} = \frac{n^4+n^3-15n^2+20n-4}{2},$

then $f$ can be an even vertex magic total labeling of $W_{n, n-2}$ .

Proof. Assume that

$S_{s} = \frac{n^4+n^3-15n^2+20n-4}{2}.$

It suffices to show that for each vertex $u$ of $W_{n, n-2}$ ,

$f(u)+\sum\limits_{\substack{v\in N(u)}}f(uv) = k,$

where

$k = \frac{n^3+3n^2-3n}{2}.$

To do this, we consider the relevant sums, as follows.

Since the sum of the labels of all rim edges is equal to the sum of the labels of all vertices and the labels of all edges subtracted by the sum of the labels of all vertices and the labels of all spokes, it follows that

$\begin{array}{l} S_{re} & = & \sum\limits_{i = 1}^{n^2+n-2}i - \sum\limits_{\substack{2\leq i\leq 4n-4 \\ i\; \text{is even}}}i -S_{s}\\ & = & \frac{(n^2+n-2)(n^2+n-1)}{2} - \frac{(4n-4)(4n-2)}{4}-\frac{n^4+n^3-15n^2+20n-4}{2} \\ & = & \frac{n^3+5n^2-11n+2}{2}. \end{array}$

Since the sum of the labels of all hubs is equal to the sum of even integers from $2n+2$ to $4n-4$ ,

$\begin{array}{l} S_{h}+S_{s} & = & \sum\limits_{\substack{2n+2\leq i\leq 4n-4 \\ i\; \text{is even}}}i +S_{s}\\ & = & \left(\sum\limits_{\substack{2\leq i\leq 4n-4 \\ i\; \text{is even}}}i-\sum\limits_{\substack{2\leq i\leq 2n \\ i\; \text{is even}}}i\right) +S_{s}\\ & = & \frac{(4n-4)(4n-2)}{4} - \frac{(2n)(2n+2)}{4}+\frac{n^4+n^3-15n^2+20n-4}{2}\\ & = & \frac{n^4+n^3-9n^2+6n}{2}. \end{array}$

Since the sum of the labels of all rim vertices is equal to the sum of even integers from $2$ to $2n$ ,

$\begin{array}{l} S_{rv}+2S_{re}+S_{s} & = & \sum\limits_{\substack{2\leq i\leq 2n \\ i\; \text{is even}}}i+2S_{re}+S_{s} \\ & = &\frac{(2n)(2n+2)}{4}+2\left(\frac{n^3+5n^2-11n+2}{2}\right)+\frac{n^4+n^3-15n^2+20n-4}{2}\\ & = & \frac{n^4+3n^3-3n^2}{2} . \end{array}$

Next, we consider the sum of the label of each vertex and the labels of all edges incident to this vertex. We have the sum of the label of each hub and the labels of all edges incident to this hub as follows.

For $1 \leq j\leq n-2$ ,

$f(v_j)+\sum\limits_{i = 1}^{n}f(u_iv_j) = \frac{S_{h}+S_{s}}{n-2} = \frac{n^3+3n^2-3n}{2}.$

We obtain the sum of the label of each rim vertex and the labels of all edges incident to this rim vertex as follows.

For $2 \leq i\leq n-1,$

$f(u_i)+f(u_iu_{i+1})+f(u_{i-1}u_i)+\sum\limits_{j = 1}^{n-2}f(u_iv_j) = \frac{ S_{rv}+2S_{re}+S_{s}}{n} = \frac{n^3+3n^2-3n}{2}.$

Similarly,

$f(u_n)+f(u_nu_1)+f(u_{n-1}u_n)+\sum\limits_{j = 1}^{n-2}f(u_nv_j) = \frac{n^3+3n^2-3n}{2}$

and

$f(u_1)+f(u_1u_2)+f(u_nu_1)+\sum\limits_{j = 1}^{n-2}f(u_1v_j) = \frac{n^3+3n^2-3n}{2}.$

Therefore, $f$ can be an even vertex magic total labeling of $W_{n, n-2}$ with the following magic constant:

$k = \frac{n^3+3n^2-3n}{2}.$

□

3. An even vertex magic $W_{n, t}$ where $3\leq n\leq 9$

In this section, we establish a characterization of an even vertex magic $W_{n, t}$ for an integer $3\leq n \leq 9$ . First, we present an $n$ -fold wheel $W_{n, n}$ which has an even vertex magic total labeling for every odd integer $3\leq n \leq 9$ as follows.

Theorem 3.1. For every odd integer $3\leq n \leq 9$ , the $n$ -fold wheel $W_{n, n}$ is an even vertex magic.

Proof. Let $n$ be an odd integer where $3\leq n \leq 9$ . We define

$f: V(W_{n, n}) \cup E(W_{n, n}) \rightarrow \{1, 2, \cdots, n^2+3n\},$

as the sufficient condition of Theorem 2.7, by

$\begin{array}{rcll} f(u_i) & = & 2i, &\mbox{if }~~ 1 \leq i \leq n , \\ f(v_j) & = & 2n+2j, &\mbox{if}~~ 1 \leq j \leq n , \\ f(u_iu_{i+1}) & = & 2n+1-2i, &\mbox{if}~~ 1 \leq i \leq n-1 , \\ f(u_nu_1) & = & 1, & \end{array}$

and for $1 \leq i, j \leq n$ , $f(u_iv_j)$ are shown in Tables 1–4,

Table 1. Labels of edges

$u_iv_j$ of

$W_{3, 3}$ by

$f$ , for

$1 \leq i, j \leq 3$ .

$f(u_iv_j)$	$v_1$	$v_2$	$v_3$
$u_1$	11	18	13
$u_2$	14	15	9
$u_3$	17	7	16

| Show Table

DownLoad: CSV

Table 2. Labels of edges

$u_iv_j$ of

$W_{5, 5}$ by

$f$ , for

$1 \leq i, j \leq 5$ .

$f(u_iv_j)$	$v_1$	$v_2$	$v_3$	$v_4$	$v_5$
$u_1$	15	21	36	38	31
$u_2$	13	30	29	33	28
$u_3$	40	24	35	19	17
$u_4$	34	37	11	23	32
$u_5$	39	27	26	22	25

| Show Table

DownLoad: CSV

Table 3. Labels of edges

$u_iv_j$ of

$W_{7, 7}$ by

$f$ , for

$1 \leq i, j \leq 7$ .

$f(u_iv_j)$	$v_1$	$v_2$	$v_3$	$v_4$	$v_5$	$v_6$	$v_7$
$u_1$	15	51	64	40	29	68	57
$u_2$	66	43	17	38	19	59	70
$u_3$	25	45	56	48	61	32	47
$u_4$	52	31	33	63	54	53	30
$u_5$	42	35	65	34	60	36	46
$u_6$	55	67	23	58	49	27	41
$u_7$	69	50	62	37	44	39	21

| Show Table

DownLoad: CSV

Table 4. Labels of edges

$u_iv_j$ of

$W_{9, 9}$ by

$f$ , for

$1 \leq i, j \leq 9$ .

$f(u_iv_j)$	$v_1$	$v_2$	$v_3$	$v_4$	$v_5$	$v_6$	$v_7$	$v_8$	$v_9$
$u_1$	73	33	102	69	19	79	63	86	91
$u_2$	43	56	35	62	49	89	85	93	87
$u_3$	75	61	48	53	104	92	95	23	50
$u_4$	80	58	72	25	108	97	77	47	39
$u_5$	51	96	59	81	99	37	55	57	70
$u_6$	60	100	44	101	67	40	41	90	64
$u_7$	38	83	103	82	74	78	29	76	46
$u_8$	88	105	94	65	42	66	52	31	68
$u_9$	107	21	54	71	45	27	106	98	84

| Show Table

DownLoad: CSV

For every odd integer $3 \leq n\leq 9$ , the labeling $f$ , as defined above, is an even vertex magic total labeling of the $n$ -fold wheel $W_{n, n}$ with magic constants $k = 50, 153, 340$ and $635$ , respectively. Therefore, $W_{n, n}$ is an even vertex magic. □

As a consequence of an even vertex magic $W_{3, 2}$ , Proposition 2.2 and Theorem 3.1, in any $t$ -fold wheel $W_{3, t}$ , we are able to show that both $W_{3, t}$ and $W_{3, t}$ are only even vertex magics.

Theorem 3.2. For every integer $t \geq 2$ , the $t$ -fold wheel $W_{3, t}$ is an even vertex magic if and only if $t = 2, 3$ .

The following result gives the necessary and sufficient condition for the $t$ -fold wheel $W_{n, t}$ to be an even vertex magic for every odd integer $5\leq n \leq 9$ .

Theorem 3.3. For every odd integer $5\leq n \leq 9$ and an integer $t \geq 2$ , the $t$ -fold wheel $W_{n, t}$ is an even vertex magic if and only if $t = n$ .

Proof. Let $n$ be an odd integer where $5\leq n \leq 9$ and $t$ is an integer where $t \geq 2$ . Assume that the $t$ -fold wheel $W_{n, t}$ is an even vertex magic. By Proposition 2.2, $2\leq t \leq n$ .

Case 1. $n = 5, 7.$ If $2\leq t \leq n-1$ , then $n^2t^2+2n^2t+n$ is not divisible by $n+t$ , and hence $k$ is not an integer, which is a contradiction. Therefore, $t = n$ .

Case 2. $n = 9.$ If either $t = 2$ or $4\leq t \leq n-1$ , then $n^2t^2+2n^2t+n$ is not divisible by $n+t$ , and hence $k$ is not an integer, which is a contradiction. If $t = 3$ , then,

$2nt^3-n^2t^2-2n^2t+6nt^2+7nt+n^2-t^2+n-t-\frac{2nt^4+4nt^3-2nt}{n+t} = -174 < 0,$

which is a contradiction with Proposition 2.6. Therefore, $t = n$ .

Conversely, assume $t = n$ . By Theorem 3.1, $W_{n, t}$ is an even vertex magic. □

We show an even vertex magic total labeling of $W_{n, n-2}$ for every even integer $4\leq n \leq 8$ as follows.

Theorem 3.4. For every even integer $4 \leq n \leq 8$ , the $(n-2)$ -fold wheel $W_{n, n-2}$ is an even vertex magic.

Proof. Let $n$ be an even integer with $4 \leq n \leq 8$ . We define

$f: V(W_{n, n-2}) \cup E(W_{n, n-2}) \rightarrow \{1, 2, \ldots, n^2+n-2\}$

as the sufficient condition of Theorem 2.8, by

$\begin{array}{rcll} f(u_i) & = & 2i, &\mbox{if}~~ 1 \leq i \leq n , \\ f(v_j) & = & 2n+2j, &\mbox{if}~~ 1 \leq j \leq n-2 , \end{array}$

for $1 \leq i\leq n-1$ , $f(u_iu_{i+1})$ and $f(u_nu_1)$ are shown in Tables 5–7.

Table 5. Labels of edges

$u_iu_{i+1}$ and

$u_4u_1$ of

$W_{4, 2}$ by

$f$ , for

$1 \leq i\leq 3$ .

$f(u_1u_2)$	$f(u_2u_3)$	$f(u_3u_4)$	$f(u_4u_1)$
18	9	11	13

| Show Table

DownLoad: CSV

Table 6. Labels of edges

$u_iu_{i+1}$ and

$u_6u_1$ of

$W_{6, 4}$ by

$f$ , for

$1 \leq i\leq 5$ .

$f(u_1u_2)$	$f(u_2u_3)$	$f(u_3u_4)$	$f(u_4u_5)$	$f(u_5u_6)$	$f(u_6u_1)$
40	39	38	13	17	19

| Show Table

DownLoad: CSV

Table 7. Labels of edges

$u_iu_{i+1}$ and

$u_8u_1$ of

$W_{8, 6}$ by

$f$ , for

$1 \leq i\leq 7$ .

$f(u_1u_2)$	$f(u_2u_3)$	$f(u_3u_4)$	$f(u_4u_5)$	$f(u_5u_6)$	$f(u_6u_7)$	$f(u_7u_8)$	$f(u_8u_1)$
70	68	66	64	62	19	13	11

| Show Table

DownLoad: CSV

And for $1 \leq i\leq n$ and $1 \leq j\leq n-2$ , $f(u_iv_j)$ are shown in Tables 8–10.

Table 8. Labels of edges

$u_iv_j$ of

$W_{4, 2}$ by

$f$ , for

$1 \leq i\leq 4$ and

$1 \leq j\leq 2$ .

$f(u_iv_j)$	$v_1$	$v_2$
$u_1$	16	1
$u_2$	14	5
$u_3$	7	17
$u_4$	3	15

| Show Table

DownLoad: CSV

Table 9. Labels of edges

$u_iv_j$ of

$W_{6, 4}$ by

$f$ , for

$1 \leq i\leq 6$ and

$1 \leq j\leq 4$ .

$f(u_iv_j)$	$v_1$	$v_2$	$v_3$	$v_4$
$u_1$	36	24	3	29
$u_2$	11	25	33	1
$u_3$	26	9	5	30
$u_4$	22	23	34	15
$u_5$	37	21	28	27
$u_6$	7	35	32	31

| Show Table

DownLoad: CSV

Table 10. Labels of edges

$u_iv_j$ of

$W_{8, 6}$ by

$f$ , for

$1 \leq i\leq 8$ and

$1 \leq j\leq 6$ .

$f(u_iv_j)$	$v_1$	$v_2$	$v_3$	$v_4$	$v_5$	$v_6$
$u_1$	69	3	40	56	39	50
$u_2$	1	67	5	59	31	35
$u_3$	21	17	65	7	44	46
$u_4$	42	38	23	33	9	57
$u_5$	61	27	37	34	30	15
$u_6$	41	53	49	32	47	25
$u_7$	29	60	54	52	63	36
$u_8$	58	55	45	43	51	48

| Show Table

DownLoad: CSV

For every even integer $4 \leq n \leq 8$ , the labeling $f$ , as defined above, is an even vertex magic total labeling of the $(n-2)$ -fold wheel $W_{n, n-2}$ with magic constants $k = 50, 153$ and $340$ , respectively. Therefore, $W_{n, n-2}$ is an even vertex magic. □

There is a similar methodology of the proof of Theorem 3.4, which is also used in the study of graph operations (see ^[2]). Next, we determine a characterization of the $t$ -fold wheel $W_{n, t}$ to be an even vertex magic for every even integer $4 \leq n \leq 8$ . In order to we need to present the following lemma involving a $3$ -fold wheel $W_{8, 3}$ .

Lemma 3.5. The $3$ -fold wheel $W_{8, 3}$ is not an even vertex magic.

Proof. On the contrary, assume that the $3$ -fold wheel $W_{8, 3}$ is an even vertex magic with a magic constant $k$ . Since $W_{8, 3}$ has an order $11$ and a size $32$ and by Proposition 2.1, $k = 160$ . We have that

$S_{rv} = 132, \ \ 2S_{re} = 1, 628-2S_{s}$

and

$S_{rv}+2S_{re}+S_{s} = 8k = 1, 280.$

Thus, $S_{s} = 480$ . However, $S_{h}+S_{s} = 3k = 480$ . This is a contradiction because $S_{h} > 0$ . Therefore, $W_{8, 3}$ is not an even vertex magic. □

We are able to show that the necessary and sufficient condition for the $t$ -fold wheel $W_{n, t}$ is an even vertex magic for every even integer $4\leq n \leq 8$ .

Theorem 3.6. For every even integer $4\leq n \leq 8$ and integer $t \geq 2$ , the $t$ -fold wheel $W_{n, t}$ is an even vertex magic if and only if $t = n-2$ .

Proof. Let $n$ be an even integer where $4\leq n \leq 8$ and $t$ is an integer where $t \geq 2$ . Assume that the $t$ -fold wheel $W_{n, t}$ is an even vertex magic. By Proposition 2.2, $2\leq t \leq n$ .

Case 1. $n = 4, 6.$ If either $2\leq t \leq n-3$ or $n-1\leq t \leq n$ , then $n^2t^2+2n^2t+n$ is not divisible by $n+t$ , and hence $k$ is not an integer, which is a contradiction. Therefore, $t = n-2$ .

Case 2. $n = 8.$ If either $4\leq t \leq n-3$ or $n-1\leq t \leq n$ , then $n^2t^2+2n^2t+n$ is not divisible by $n+t$ , and hence $k$ is not an integer, which is a contradiction. If $t = 3$ , then, by Lemma 3.5, $W_{n, t}$ is not an even vertex magic, which is a contradiction. If $t = 2$ , then,

$2nt^3-n^2t^2-2n^2t+6nt^2+7nt+n^2-t^2+n-t-\frac{2nt^4+4nt^3-2nt}{n+t} = -62 < 0,$

which is a contradiction with Proposition 2.6. Therefore, $t = n-2$ .

Conversely, assume $t = n-2$ . By Theorem 3.4, $W_{n, t}$ is an even vertex magic. □

4. Conclusions

In this paper, we have not only established the bound of an integer $t$ for the even vertex magic total labeling of the $t$ -fold wheel, but have also presented the necessary condition for such labeling in terms of $n$ and $t$ . Furthermore, we have conducted an investigation into the sufficient conditions for labelings that can serve as even vertex magic total labelings for $W_{n, n}$ when $n$ is odd, and $W_{n, n-2}$ when $n$ is even.

Our research has led us to the following significant conclusions:

● For every integer $t \geq 2$ , the $t$ -fold wheel $W_{3, t}$ is an even vertex magic total labeling if and only if $t = 2, 3$ .

● For every odd integer $5 \leq n \leq 9$ and an integer $t \geq 2$ , the $t$ -fold wheel $W_{n, t}$ is an even vertex magic total labeling if and only if $t = n$ .

● For every even integer $4 \leq n \leq 8$ and an integer $t \geq 2$ , the $t$ -fold wheel $W_{n, t}$ is an even vertex magic total labeling if and only if $t = n-2$ .

In essence, our work has discussed the characterizations of $t$ -fold wheel $W_{n, t}$ to possess an even vertex magic total labeling for an integer $3 \leq n \leq 9$ . It would be interesting to apply the results of this paper to further study under what conditions for $W_{n, t}$ will be an even vertex magic, especially for a larger $n$ .

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Conflict of interest

The authors declare that they have no conflicts of interest.

References

[1]	T. M. Whitin, Inventory control and price theory, Manag. Sci., 2 (1955), 61–80. https://doi.org/10.1287/mnsc.2.1.61 doi: 10.1287/mnsc.2.1.61
[2]	J. Eliashberg, R. Steinberg, Marketing-production joint decision-making, In: Handbooks in operations research and management science, 5 (1993), 828–880. https://doi.org/10.1016/S0927-0507(05)80041-6
[3]	W. Elmaghraby, P. Keskinocak, Dynamic pricing in the presence of inventory considerations: research overview, current practices, and future directions, Manag. Sci., 49 (2003), 1287–1309. https://doi.org/10.1287/mnsc.49.10.1287.17315 doi: 10.1287/mnsc.49.10.1287.17315
[4]	L. M. A. Chan, Z. J. M. Shen, D. Simchi-Levi, J. L. Swann, Coordination of pricing and inventory decisions: a survey and classification, In: D. Simchi-Levi, S. D. Wu, Z. J. M. Shen, Handbook of quantitative supply chain analysis, International Series in Operations Research & Management Science, Boston: Springer, 74 (2004), 335–392. https://doi.org/10.1007/978-1-4020-7953-5_9
[5]	D. Simchi-Levi, X. Chen, J. Bramel, The logic of logistics: theory, algorithms, and applications for logistics and supply chain management, 2 Eds, New York: Springer, 2005 https://doi.org/10.1007/b97669
[6]	C. A. Yano, S. M. Gilbert, Coordinated pricing and production/procurement decision: a review, In: A. K. Chakravarty, J. Eliashberg, Managing business interfaces, International Series in Quantitative Marketing, Boston: Springer, 16 (2005), 65–103. https://doi.org/10.1007/0-387-25002-6_3
[7]	R. H. Niu, I. Castillo, T. Joro, Joint pricing and inventory/production decisions in a supply chain: a state-of-the-art survey, Proceedings of the Western Decision Sciences Institute, WDSI 2001 Fortieth Annual Meeting, Portland Oregon, 2011.
[8]	X. Chen, D. Simchi-Levi, Pricing and inventory management, In: R. Philips, O. Özalp, The Oxford handbook of pricing management, Oxford University Press, 2012,784–824. https://doi.org/10.1093/oxfordhb/9780199543175.013.0030
[9]	R. Zhang, An introduction to joint pricing and inventory management under stochastic demand, 2013.
[10]	A. V. Den Boer, Dynamic pricing and learning: Historical origins, current research, and new directions, Surv. Oper. Res. Manage. Sci., 20 (2015), 1–18. https://doi.org/10.1016/j.sorms.2015.03.001 doi: 10.1016/j.sorms.2015.03.001
[11]	G. Feichtinger, R. Hartl, Optimal pricing and production in an inventory model, Eur. J. Oper. Res., 19 (1985), 45–56. https://doi.org/10.1016/0377-2217(85)90307-8 doi: 10.1016/0377-2217(85)90307-8
[12]	P. C. Lin, L. Y. Shue, Application of optimal control theory to product pricing and warranty with free replacement under the influence of basic lifetime distributions, Comput. Ind. Eng., 48 (2005), 69–82. https://doi.org/10.1016/j.cie.2004.07.009 doi: 10.1016/j.cie.2004.07.009
[13]	P. C. Lin, Optimal pricing, production rate, and quality under learning effects, J. Bus. Res., 61 (2008), 1152–1159. https://doi.org/10.1016/j.jbusres.2007.11.008 doi: 10.1016/j.jbusres.2007.11.008
[14]	Y. Feng, J. Ou, Z. Pang, Optimal control of price and production in an assemble-to-order system, Oper. Res. Lett., 36 (2008), 506–512. https://doi.org/10.1016/j.orl.2008.01.012 doi: 10.1016/j.orl.2008.01.012
[15]	M. F. Keblis, Y. Feng, Optimal pricing and production control in an assembly system with a general stockout cost, IEEE Trans. Automat. Contr., 57 (2012), 1821–1826. https://doi.org/10.1109/TAC.2011.2178335 doi: 10.1109/TAC.2011.2178335
[16]	X. Cai, Y. Feng, Y. Li, D. Shi, Optimal pricing policy for a deteriorating product by dynamic tracking control, Int. J. Prod. Res., 51 (2013), 2491–2504. https://doi.org/10.1080/00207543.2012.743688 doi: 10.1080/00207543.2012.743688
[17]	R. Chenavaz, Better product quality may lead to lower product price, B.E. J. Theor. Econ., 17 (2016), 1–22. https://doi.org/10.1515/bejte-2015-0062 doi: 10.1515/bejte-2015-0062
[18]	R. Chenavaz, Dynamic pricing rule and R & D, Econ. Bull., 31 (2011), 2229–2236.
[19]	J. Vörös, Multi-period models for analyzing the dynamics of process improvement activities, Eur. J. Oper. Res., 230 (2013), 615–623. https://doi.org/10.1016/j.ejor.2013.04.036 doi: 10.1016/j.ejor.2013.04.036
[20]	E. Adida, G. Perakisy, A nonlinear continuous time optimal control model of dynamic pricing and inventory control with no backorders, Nav. Res. Log., 54 (2007), 767–795. https://doi.org/10.1002/nav.20250 doi: 10.1002/nav.20250
[21]	M. Weber, Optimal inventory control in the presence of dynamic pricing and dynamic advertising, Ph.D. Dissertation, Humboldt-Universität of Berlin, Germany, 2015. https://doi.org/10.18452/17339
[22]	A. Herbon, E. Khmelnitsky, Optimal dynamic pricing and ordering of a perishable product under additive effects of price and time on demand, Eur. J. Oper. Res., 260 (2017), 546–556. https://doi.org/10.1016/j.ejor.2016.12.033 doi: 10.1016/j.ejor.2016.12.033
[23]	L. Yang, Y. Cai, Optimal dynamic pricing for fresh products under the cap-and-trade scheme, 2017 International Conference on Service Systems and Service Management, Dalian, IEEE, 2017. https://doi.org/10.1109/ICSSSM.2017.7996183

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)