A Formulated Approximated Solution for a Bus Transit System with Peak and Normal Periods

Abstract

For a bus transit system with peak and normal periods, we derived an approximated formulated solution, to provide a patchwork for a published paper. Based on the same numerical example of the original model, our approximated solution attains the minimum cost within 0.3% to indicate that our approximated formulated solution is a good supplementary tool to estimate the optimal solution.

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1. Introduction

Researchers developed models to mimic real-world phenomena, and then tried to apply their results from proposed models to provide useful information to deal with real-world problems. We can classify models into two categories. The first one tried to use many parameters, decision variables, and constraints to illustrate a closed description of real-world problems. Due to the complexity of their model, no partitioners knew the exact optimal solution, and then they applied many algorithms and programs to search for the acceptable near-optimal solution. On the other hand, some researchers only concentrated on a few decision variables and then they tried to find the optimal solution. This approach may be a little far from real-world phenomena such that challengers claimed that the exact optimal solution cannot be applied to a real-world problem. However, based on the knowledge of the optimal solution, researchers can decide which parameters are most influential in the optimal solution and then spend more research funds to decide the values for those important parameters. In this paper, we will adopt the second category, because the real world is too complicated, fuzzy, and chaotic such that those complicated models only considered a small portion of possible decision variables, and tedious constraints reflected a glance of the interaction among parameters and decision variables. In the following, we provide literature reviews that are closely related to our study of transit models of bus service. Chang and Schonfeld [1] extended Chang and Schonfeld [2] to treat the route length as a new variable and then derived the optimal solution for a cubic polynomial. Yang et al. [3] first pointed out the questionable results of Chang and Schonfeld [1] and then provided their revised optimal solution for the improved cubic polynomial. Tung et al. [4] showed that Chang and Schonfeld [1] and Yang et al. [3] tried to prove the uniqueness of the optimal solution for the entire real number that implies Chang and Schonfeld [1] and Yang et al. [3] only provided a partial solution for the transit model. Tung et al. [4] presented a complete solution method for Chang and Schonfeld [1] and Yang et al. [3]. Lin and Julian [5] showed that the alternative sequence constructed by Yang et al. [3] is complicated for ordinary partitioners, and then Lin and Julian [5] developed a new iterative method to generate a monotonic sequence that will converge to the optimal solution. Hung and Julianne [6] applied the bisection method to find the optimal route length for the transit model examined by Yang et al. [3]. Luo [7] pointed out that the restriction proposed by Yang et al. [3] with a route length greater than one mile may be impractical for some seashore cities. He constructed a new starting point to execute Newton’s method to locate the optimal route length. By the second-order approximation of the exponential function, Chuang and Chu [8] studied the public transit system developed by Hendrickson [9] to find an approximated formulated solution. By the third-order approximation of the exponential function, Hung et al. [10] derived a new approximated formulated solution for Hendrickson [9]. Imam [11] is a variation of Chang and Schonfeld [1]. In Chang and Schonfeld [1], they used a weighted sum to combine several costs. On the other hand, in Imam [11], he applied a weighted multiplication to combine several costs. Yang et al. [12] showed that the first model constructed by Imam [11] without the capacity constraint is unreasonable. Lin and Hopscotch [13] examined the second model of Imam [11] with the capacity constraint to show the iterative method proposed by Imam [11] is questionable and then provided their revised iterative approach. Chen and Julian [14] proved the optimal solution for the transit model proposed by Imam [11] is existing and unique. Moreover, Chen and Julian [14] derived a formulated solution for optimal headway. Chao [15] considered the transit network design for Rivera of Uruguay which was discussed by Mauttone and Urquhart [16]. Chu and Hopscotch [17] investigated the transit network in Chicago proposed by Asadi Bagloee and Ceder [18]. Chu and Lin [19] studied the bus model in the city centers of Barcelona developed by Roca-Riu et al. [20]. Chuang [21] examined the transportation system of the urban area of Rome that was constructed by Cipriani et al. [22]. We can claim that the papers of Mauttone and Urquhart [16], Asadi Bagloee and Ceder [18], Roca-Riu et al. [20], and Cipriani et al. [22] are condensed from several hundred pages of project plans. Hence, there are inconsistent expressions, unclear even chaos descriptions, and contradicted explanations in Mauttone and Urquhart [16], Asadi Bagloee and Ceder [18], Roca-Riu et al. [20], and Cipriani et al. [22] that are pointed out by Chao [15], Chu and Hopscotch [17], Chu and Lin [19], and Chuang [21], respectively. After some literature reviews, we focus on our studies. Based on the bus transit model of Jara-Diaz and Gschwender [23], Jara-Diaz et al. [24] developed a bus transit model including peak period and normal period to find the optimal solution for peak frequency and normal frequency to improve Jansson [25] and Chang and Schonfeld [2]. Jansson [25] extended the bus model of Mohring [26] with passengers boarding and alighting and passengers’ in-vehicle times to derive a closed analytical solution. Chang and Schonfeld [2] generalized Kocur and Hendrickson [27] to show that the optimal number of routes and the optimal frequency depend on the cubic root of the demand. Jara-Diaz et al. [24] dealt with their bus transit model by jointly optimizing for peak and normal periods. Their solution procedure contained a quintic polynomial such that they cannot derive the formulated optimal solution for peak and normal frequencies. Up to now, Jara-Diaz et al. [24] had been cited in seven papers. We provide a brief review of those seven papers. Daganzo and Ouyang [28] examined the transit system with door-to-door service concerning nonshared taxis, dial-a-ride, and shared taxis. Xiong et al. [29] applied a genetic algorithm to optimize the community shuttle network. Börjesson et al. [30] studied the distribution effects of transit subsidies. Connors et al. [31] estimated the travel time between on-peak and off-peak periods. Hörcher et al. [32] developed models with private cars and public transport for congestion and crowding. Proboste et al. [33] developed two models for open and closed bus rapid transit to study their social costs in medium-sized cities. Zhang et al. [34] constructed transit models for a feeder bus line to minimize the waiting time of passengers and supplier operating costs. From the above discussion, we can claim that those seven papers only listed Jara-Diaz et al. [24] in their introduction without providing a further examination of Jara-Diaz et al. [24]. The purpose of our paper is to provide a formulated approximated solution of two frequencies for peak and normal periods. We observe that the joint solution for the bus model proposed by Jara-Diaz et al. [24] can be treated as a decreasing objective function of the normal period optimal solution and an increasing function of the peak period optimal solution. Jara-Diaz et al. [24] mentioned that they would face a quintic polynomial for the frequency of the normal period such that they cannot derive a formulated optimal solution for the normal period that motivates this research. We add an extra condition to assume that the increasing rate of frequency for the peak period is the same as the decreasing rate of frequency for the normal period. Our procedure contains a quartic polynomial for the changing rate such that we can apply the formulated solution for a quartic polynomial. Based on the same numerical example of Jara-Diaz et al. [24], our formulated approximated solution for two frequencies attains the total cost within 0.3%. This paper is organized as follows: In Section 2, we provide a list of notations. In Section 3, we present a review of the transit system proposed by Jara-Diaz et al. [24]. In Section 4, we derive a formulated approximated solution for frequencies of a normal period and a peak period. In Section 5, we examine the same numerical as that of Jara-Diaz et al. [24] to show our approximated solution attains the optimal cost within 0.3%. In Section 6, we point out several questionable results of Jara-Diaz et al. [24]. In Section 7, we provide managerial insights and directions for future research. In Section 8, we conclude our discussion of the transit model proposed by Jara-Diaz et al. [24].

Recently, there is a paper, by Wang et al. [35], that discussed a bus transit system to derive a pair of approximated optimal solutions to help researchers realize the interrelationship between peak and normal periods under the considerations of two frequencies.

2. Notation

To be compatible with Jara-Diaz et al. [24], we use the same expressions as them. We recall the notation of Jara-Diaz et al. [24] as follows:

  $B$: Total fleet size, $B=t_{c}f=fT+tY$

   $c_B$: Exogenous unit cost per bus, operating cost (usage), $c_B=c_{BO}+c_{KO}K$

   $c_{BC}$: Fixed operating cost

   $c_{BO}$: Fixed capital cost

   $c_K$: Exogenous unit cost per seat, capital cost (acquisition), $c_K=c_{BC}+c_{KC}K$

   $c_{KC}$: Operating cost related to bus capacity

   $c_{KO}$: Capital cost related to bus capacity

   $E_N$: Durations for a normal period

   $E_P$: Durations for a peak period

   $f$: The frequency of the line

   $f_N$: The frequency of the line, during a normal period

   $f_P$: The frequency of the line, during a peak period

   $K$: Bus capacity, there is neither overcrowding nor denied boarding, $K=lY/fL$ for the separate transit model; for two periods model, $K=l_P{Y}_P/f_{P}L$

  $L$: The total route length

  $l$: Average trip length for a passenger

  $l_N$: Average trip length for a passenger, during a normal period

  $l_P$: Average trip length for a passenger, during a peak period

  $P_V$: Value of in-vehicle time

  $P_W$: Value of waiting time

  $Q\epsilon$: Our proposed objective function

  $t$: Boarding-alighting time per passenger

  $t_c$: The cycle time, $t_c=T+tY/f$ time in motion plus time at stops

  $T$: Time in motion, how long for a complete route

  $T_N$: Time in motion, during a normal period

  $T_P$: Time in motion, during a peak period

  $VR{C}_2{f}_{P,}{f}_N$: The objective function proposed by Jara-Diaz et al. [24]

  $Y$: The average hourly demand, the number of users that enters the transit system per hour

  $Y_N$: The number of users that enters the transit system per hour, during a normal period

  $Y_P$: The number of users that enters the transit system per hour, during a peak period

To simplify the expression, Jara-Diaz et al. [24] assumed the following abbreviations: $A_P$, $G_P$, $A_N$, $G_N$, and $\delta$, for coefficients of the objective function, $VR{C}_2{f}_{P,}{f}_N$, concerning $f_P$, $1/f_P$, $f_N$, $1/f_N$, and $f_N/f_P$ such that$$\begin{array}{}\text{(1)}& A_P=c_{BC}{T}_P+c_{BO}{E}_P{T}_P,\text{(2)}& G_P=l_{P}t{Y}_P\frac{c_{KC}{Y}_P+c_{KO}{E}_P{Y}_P+c_{KO}{E}_N{Y}_N+P_V{E}_P{Y}_P}{L}+\frac{E_P{P}_W{Y}_P}{2},\text{(3)}& A_N=c_{BO}{E}_N{T}_N,\text{(4)}& G_N=\frac{E_N{P}_W{Y}_N}{2}+\frac{E_N{l}_N{P}_{V}t{Y}_N^2}{L},\text{(5)}& \delta =\frac{c_{KO}{E}_N{l}_P{T}_N{Y}_P}{L}.\end{array}$$

We further simplify the expression so we set two variables: $x$ and $y$, for $f_P$ and $f_N$, and four constants: $a$, $b$, $c$, and $d$, for $A_P$, $G_P$, $A_N$, and $G_N$, respectively.

3. Review of the Transit System Proposed by Jara-Diaz et al. [24]

Mohring [26] developed a bus transit model where the operating cost increases with the frequency and the waiting time decreases with the frequency. Jansson [25, 36] extended Mohring [26] by considering boarding and alighting time and in-vehicle time for passengers to derive a closed-form optimal solution. Jara-Díaz and Gschwender [37] expanded Mohring [26] with costs of access time and in-vehicle time. Jara-Díaz and Gschwender [23] generalized Jansson [36] to include vehicle size and financial constraints for a single-line bus transit model. Jara-Diaz et al. [24] constructed a bus transit model with two periods: peak and normal. For their detailed derivation, please refer to Jara-Diaz et al. [24]. We directly quote their objective function (the total cost per day) for the two-period model as follows:$$\begin{array}{}\text{(6)}& VC{R}_2{f}_P,f_N=A_p{f}_P+\frac{G_P}{f_P}+A_N{f}_N+\frac{G_N}{f_N}+\delta \frac{f_N}{f_P},\end{array}$$where the abbreviations are listed in Section 2 of equations (1)–(5).

Jara-Diaz et al. [24] took partial derivatives with respect to $f_P$ and $f_N$ to derive the following:$$\begin{array}{c}\text{(7)}& \frac{\partial }{\partial f_P}VC{R}_2{f}_P,f_N=A_p-\frac{G_P}{{f_P}^2}-\delta \frac{f_N}{{f_P}^2},\frac{\partial }{\partial f_N}VC{R}_2{f}_P,f_N=A_N-\frac{G_N}{{f_N}^2}+\frac{\delta }{f_P}.\end{array}$$

If we solve the system of $\partial VC{R}_2/\partial f_P=0$ and $\partial VC{R}_2/\partial f_N=0$, to cancel out the variable $f_P$, such that we reduce the system to one variable of $f_N$, then a quintic equation appeared:$$\begin{array}{}\text{(8)}& {A_N}^2{\delta }^2{f_N}^4=A_P{G}_P+\delta f_N{A_N{G}_N-{f_N}^2}^2,\end{array}$$as mentioned in Jara-Diaz et al. [24]. Owing to there is no formulated solution for a quintic polynomial, Jara-Diaz et al. [24] claimed that “no analytical solution can be found as established by the Abel-Ruffini theorem (see, for example, Alekseev [38]).”

The goal of our paper is to derive a formulated approximated solution for $f_P$ and $f_N$, and then, we will apply our approximated solution to show that our formulated approximated solution attains the minimum cost within 0.3% estimation errors.

4. Our Derivations

We recall the minimum problem proposed by Jara-Diaz et al. [24] of equation (6). Under our further simplifications, we assume the minimum problem of equation (6) in an abstract setting is as follows:$$\begin{array}{}\text{(9)}& \text{minimize\hspace{0.17em}}gx,y=ax+\frac{b}{x}+cy+\frac{d}{y}+\delta \frac{y}{x},\end{array}$$where $x$, $y$, $a$, $b$, $c$, and $d$ are further simplifications for $f_P$, $f_N$, $A_P$, $G_P$, $A_N$, and $G_N$, respectively. Based on equation (9), we can split the objective function into three parts: $ax+b/x$, $cy+d/y$, and $\delta y/x$ such that we will derive a formulated approximated solution in the following derivations.

We break down the minimum problem of equation (9) into three parts: (a) the first and the second terms, (b) the third and the fourth terms, and (c) the fifth term.

Motivated by the first and second terms of equation (9), we consider the following minimum problem:$$\begin{array}{}\text{(10)}& fx=ax+\frac{b}{x},\end{array}$$where $x>0$, with $a>0$ and $b>0$.

$fx$ has a formulated minimum point, denoted as $x^{\#}=\sqrt{b/a}$ and the minimum value $f\sqrt{b/a}=2\sqrt{ab}$. We will take $x^{\#}=\sqrt{b/a}$ as an initial point to find an approximated optimal solution.

Similarly, motivated by the third and fourth terms of equation (9), we consider the following minimum problem:$$\begin{array}{}\text{(11)}& hy=cy+\frac{d}{y},\end{array}$$where $y>0$, with $c>0$ and $d>0$. $hy$ has a formulated minimum point, denoted as $y^{\#}=\sqrt{d/c}$ and the minimum value $h\sqrt{d/c}=2\sqrt{cd}$. We will take $y^{\#}=\sqrt{d/c}$ as an initial point to find an approximated optimal solution.

Our solution procedure is starting at $x^{\#},y^{\#}$ searching for a formulated approximated solution.

Next, there are two directions to vary $x^{\#}$, to increase to $1+\epsilon x^{\#}$ or to decrease to $1-\epsilon x^{\#}$, and then $f1+\epsilon x^{\#}$ and $f1-\epsilon x^{\#}$ are both increasing from the minimum value $f{x}^{\#}$. Moreover, we observe the fifth term as $\delta y/x$ to reduce the compact of the increasing value, we should only consider changes from $x^{\#}$ to $1+\epsilon x^{\#}$.

Hence, we vary the value of $x$, from the minimum solution of $fx$ at $x^{\#}$ to $1+\epsilon x^{\#}$, and then we estimate the change between $f{x}^{\#}$ and $f1+\epsilon x^{\#}$ to derive that$$\begin{array}{}\text{(12)}& f1+\epsilon x^{\ast }-f{x}^{\ast }=\frac{{\epsilon }^2}{1+\epsilon }\sqrt{ab}.\end{array}$$

From equation (12), if the deviation is executed from the minimum solution $x^{\#}$, to $1+\epsilon x^{\#}$, then the minimum value will increase by ${\epsilon }^{2}f{x}^{\#}/21+\epsilon$, such that the relative increasing with respect to $f{x}^{\#}$ is obtained as follows:$$\begin{array}{}\text{(13)}& \frac{f1+\epsilon x^{\#}-f{x}^{\#}}{f{x}^{\#}}=\frac{{\epsilon }^2}{21+\epsilon },\end{array}$$where $f{x}^{\ast }=2\sqrt{ab}$.

Similarly, there are two directions to vary $y^{\#}$, to increase to $1+\epsilon y^{\#}$ or to decrease to $1-\epsilon y^{\#}$, and then $h1+\epsilon y^{\#}$ and $h1-\epsilon y^{\#}$ are both increasing from the minimum value $h{y}^{\#}$. Moreover, we observe the fifth term as $\delta y/x$ to reduce the compact of the increasing value, we should only consider changing from $y^{\#}$ to $1-\epsilon y^{\#}$.

We add a restriction of symmetry to increase our solution from $x^{\#}$ to $1+\epsilon x^{\#}$, and then we decrease our solution from $y^{\#}$ to $1-\epsilon y^{\#}$. Our variation is symmetric, under the following consideration:$$\begin{array}{}\text{(14)}& \frac{1+\epsilon x^{\#}-x^{\#}}{x^{\#}}=\frac{y^{\#}-1-\epsilon y^{\#}}{y^{\#}}.\end{array}$$

Hence, we vary the value of $y$, from the minimum solution of $hy$ at $y^{\#}$ to $1-\epsilon y^{\#}$, to obtain that$$\begin{array}{}\text{(15)}& h1-\epsilon y^{\ast }-h{y}^{\ast }=\frac{{\epsilon }^2}{1-\epsilon }\sqrt{cd}.\end{array}$$

From equation (15), if the deviation is executed from the minimum solution $y^{\#}$, to $1-\epsilon y^{\#}$, then the minimum value will increase by ${\epsilon }^{2}h{y}^{\#}/21-\epsilon$, such that the relative increase with respect to $h{y}^{\#}$ is found as follows:$$\begin{array}{}\text{(16)}& \frac{h1-\epsilon y^{\#}-h{y}^{\#}}{h{y}^{\#}}=\frac{{\epsilon }^2}{21-\epsilon }.\end{array}$$

Motivated by the fifth term of equation (9), to achieve a smaller value of $gx,y$, starting from $x^{\#},y^{\#}$, we should decrease the value of $y$, from $y^{\#}$ to $1-\epsilon y^{\#}$, with $\epsilon >0$ and increase the value of $x$, from $x^{\#}$ to $1+\epsilon x^{\#}$, with $\epsilon >0$, and then we compute the relative decreasing to $p{y}^{\#}/x^{\#}$; then,$$\begin{array}{}\text{(17)}& \frac{p1-\epsilon y^{\#}/1+\epsilon x^{\#}-p{y}^{\#}/x^{\#}}{p{y}^{\#}/x^{\#}}=\frac{-2\epsilon }{1+\epsilon },\end{array}$$where $px,y=\delta y/x$.

We compare absolute values of magnitudes for three terms appeared in equations (13), (16), and (17), with $\epsilon >0$, to derive that$$\begin{array}{}\text{(18)}& \frac{{\epsilon }^2}{21+\epsilon }\sim \frac{{\epsilon }^2}{21-\epsilon }<\frac{2\epsilon }{1+\epsilon },\end{array}$$where $\epsilon$ is a small positive number.

Therefore, we can claim that when $\epsilon$ is a small positive number, then two increasing values of equations (13) and (16) by the first four terms of equation (9) will be smaller than the decreasing of equation (17) for the fifth term of equation (9). This observation motivates us to develop our approximated model to derive a formulated approximated solution for the bus transit model of Jara-Diaz et al. [24].

Our objective function, denoted as $Q\epsilon$, is to maximize the difference between $g1+\epsilon x^{\#},1-\epsilon y^{\#}$ and $g{x}^{\#},y^{\#}$, that is, for $\epsilon >0$,$$\begin{array}{}\text{(19)}& \max Q\epsilon =g{x}^{\#},y^{\#}-g1+\epsilon x^{\#},1-\epsilon y^{\#},\end{array}$$where $x^{\#}=\sqrt{b/a}$ and $y^{\#}=\sqrt{d/c}$.

We derive that$$\begin{array}{}\text{(20)}& Q\epsilon =\frac{2\epsilon }{1+\epsilon }\delta \sqrt{\frac{ad}{bc}}-\frac{{\epsilon }^2}{1-\epsilon }\sqrt{cd}-\frac{{\epsilon }^2}{1+\epsilon }\sqrt{ab}.\end{array}$$

In the following, we begin to maximize $Q\epsilon$ to find that$$\begin{array}{}\text{(21)}& Q^{\prime }\epsilon =\frac{2}{{1+\epsilon }^2}\delta \sqrt{\frac{ad}{bc}}-\frac{2\epsilon -{\epsilon }^2}{{1-\epsilon }^2}\sqrt{cd}-\frac{2\epsilon +{\epsilon }^2}{{1+\epsilon }^2}\sqrt{ab},\text{(22)}& Q^{″}\epsilon =\frac{-4}{{1+\epsilon }^3}\delta \sqrt{\frac{ad}{bc}}-\frac{2}{{1-\epsilon }^3}\sqrt{cd}-\frac{2}{{1+\epsilon }^3}\sqrt{ab}.\end{array}$$

Since $Q^{″}\epsilon <0$, for $\epsilon >0$, $Q\epsilon$ is a concave down function for $\epsilon >0$ such that the solution of $Q^{\prime }\epsilon =0$ for $\epsilon >0$, is the maximum point for $Q\epsilon$. To solve $Q^{\prime }\epsilon =0$, for $\epsilon >0$, that is, to solve the following quartic equation:$$\begin{array}{}\text{(23)}& 2{1-\epsilon }^2\delta \sqrt{\frac{ad}{bc}}-2\epsilon -{\epsilon }^2{1+\epsilon }^2\sqrt{cd}-2\epsilon +{\epsilon }^2{1-\epsilon }^2\sqrt{ab}=0.\end{array}$$

We rewrite equation (21) according to decreasing order of $\epsilon$ as follows:$$\begin{array}{}\text{(24)}& a_4{\epsilon }^4+a_2{\epsilon }^2+a_1\epsilon +a_0=0,\end{array}$$with our following abbreviations:$$\begin{array}{c}\text{(25)}& a_4=\sqrt{cd}-\sqrt{ab},a_2=-3a_4+a_0,\\ a_1=-2\sqrt{ab}+\sqrt{cd}+a_0,\\ a_0=2\delta \sqrt{\frac{ad}{bc}}.\end{array}$$

There are formulated solutions for a quartic polynomial. Please refer to appendix for the formulated solutions from Planet [39].

We recall equation (21) to imply that$$\begin{array}{}\text{(26)}& \underset{\epsilon ⟶0^{+}}{\lim }Q\prime \epsilon =2\delta \sqrt{\frac{ad}{bc}}>0,\text{(27)}& \underset{\epsilon ⟶1^{-}}{\lim }Q\prime \epsilon =\frac{\delta }{2}\sqrt{\frac{ad}{bc}}-\frac{\sqrt{cd}}{\underset{\epsilon ⟶1^{-}}{\lim }{1-\epsilon }^2}-\frac{3}{4}\sqrt{ab}=-\infty <0.\end{array}$$

From $Q″\epsilon <0$, for $\epsilon >0$, $Q\prime \epsilon$ is a decreasing function from ${\lim }_{\epsilon ⟶0^{+}}Q\prime \epsilon >0$ to ${\lim }_{\epsilon ⟶1^{-}}{Q}^{\prime }\epsilon <0$. Hence, there is a point, denoted by ${\epsilon }^{\ast }$ that satisfies $Q\prime {\epsilon }^{\ast }=0$. We recall the results of equation (22) to imply that $Q\epsilon$ is a concave function for $\epsilon >0$ such that ${\epsilon }^{\ast }$ is the maximum point for $Q\epsilon$ with $0<\epsilon <1$.

5. Numerical Example

We refer to the numerical example proposed by Jara-Diaz et al. [24] with the following data: $c_{BC}=\text{\$}4.14$, $c_{BO}=\text{\$}1.32/\text{h}$, $c_{KC}=\text{\$}0.45/\text{seat}$, $c_{KO}=\text{\$}0.10/\text{h}-\text{seat}$, $E_N=13\text{h}$, $E_p=5\text{h}$, $L=40\text{km}$, $l_N=5\text{km}$, $l_p=10\text{km}$, $p_V=\text{\$}1.48/\text{h}$, $p_W=\text{\$}4.44/\text{h}$, $t=2.5/3600\text{h}$ (in Jara-Diaz et al. [24], they used $t=2.5\text{s}$, we convert to $t=2.5/3600\text{h}$, to be consistent with other parameters), $T_N=1.5\text{h}$, $T_p=2\text{h}$, $Y_N=\mathrm{30,000}\text{pax}/\text{h}$, and $Y_p=\mathrm{100,000}\text{pax}/\text{h}$.

Hence, we find that $a=A_p=21.48$, $b=G_p=16283611$, $c=A_N=25.74$, $d=G_N=2368925$, and $\delta =4.875\times {10}^4$ to derive that$$\begin{array}{c}\text{(28)}& x^{\#}=f_P^{\#}=\sqrt{\frac{b}{a}}=870.6793,y^{\#}=f_N^{\#}=\sqrt{\frac{d}{c}}=\mathrm{303.3691.}\end{array}$$

Based on formulated solutions in appendix, we find four roots of $\epsilon$ for equation (24), ${\epsilon }_1=1.4047+0.9435i$, ${\epsilon }_2=1.4047-0.9435i$, ${\epsilon }_3=0.3452$, and ${\epsilon }_4=-3.1547$, which is consistent with our derivation that there is only one solution for $0<\epsilon <1$.

Hence, our formulated approximated solution for $y^{\ast }$ and $x^{\ast }$ are obtained as follows:$$\begin{array}{c}\text{(29)}& y^{\#}1-{\epsilon }_3=\sqrt{\frac{d}{c}}1-{\epsilon }_3=198.1647,x^{\#}1+{\epsilon }_3=\sqrt{\frac{b}{a}}1+{\epsilon }_3=1172.6195,\end{array}$$and then our approximated cost is as follows:$$\begin{array}{}\text{(30)}& {\text{VCR}}_{2}1+{\epsilon }_3{f}_P^{\#},1-{\epsilon }_3{f}_N^{\#}=\mathrm{64367.89.}\end{array}$$

In Jara-Diaz et al. [24], they did not explicitly inform us what their optimal solution is such that we must try to derive the exact optimal solution by ourselves.

In the following, we reconsider equation (8) by a sequenced approach. Under our assumptions, where $A_P$, $G_P$, $A_N$, $G_N$, and $f_N$ are simplified by $a$, $b$, $c$, $d$, and $y$, respectively, then we rewrite equation (8) as follows:$$\begin{array}{}\text{(31)}& c^2{\delta }^2{y}^4=ab+\delta y{cd-y^2}^2.\end{array}$$

Based on the expression of equation (31), we will apply a recursive approach to derive that$$\begin{array}{}\text{(32)}& y_{n+1}={\frac{ab+\delta y_n{cd-y_n^2}^2}{c^2{\delta }^2}}^{1/4},\end{array}$$with $y_1=200$, then we list our findings of the derived sequence $y_n$ in Table 1.

Table 1

Sequence solution for the optimal normal frequency.

$n$	$y_{2n+1}$	$y_{2n}$
1	187.156	173.912
2	183.979	180.923
3	183.223	182.511
4	183.045	182.879
5	183.004	182.965
6	182.994	182.985
7	182.992	182.990
8	182.991	182.991

We derive an alternative sequence with those odd terms $y_{2n+1}$ decrease and other even terms $y_{2n}$ increase and then $y_{16}=y_{17}=182.991$ to imply that$$\begin{array}{}\text{(33)}& f_N^{\ast }=182.99,\end{array}$$that has coincided with the root denoted as $182.9911739$, which is obtained by the built-in program in Excel.

Based on equation (7), we know that$$\begin{array}{}\text{(34)}& f_P^{\ast }=x^{\ast }=\frac{\delta }{d/{y^{\ast }}^2-c}=1083.23,\end{array}$$and then the optimal minimum$$\begin{array}{}\text{(35)}& VC{R}_2{f}_P^{\ast },f_N^{\ast }=\mathrm{64191.39.}\end{array}$$

If we compare the results of equation (30) and (35) to estimate the relative error of our approximation, then we find that$$\begin{array}{}\text{(36)}& \frac{64367.89}{64191.39}=\mathrm{1.00275.}\end{array}$$

Hence, we can claim that our approximated solution attains the optimal minimum within $0.3\%$.

6. Review of Findings with Jara-Diaz et al. [24]

We recall that Luo [7] mentioned that “The Transit network design problem is an important issue for a metropolitan to operate his public bus system. Traffic models with a rectangular service zone can be useful for the first step, design of routes, the second step, and setting of frequencies for a public transit planning process.” Hence, the model proposed by Jara-Diaz et al. [24] covered the first and the second steps for the traffic design.

We must point out that Jara-Diaz et al. [24] did not explicitly inform us what is their optimal solution. On the other hand, they provided several graphs to record their sensitivity analysis findings. We recall Graph 1 of Jara-Diaz et al. [24], the horizontal axis is total patronage, the left vertical axis is bus capacity and the right vertical axis is fleet size. The brown line results for fleet size and the blue line results for capacity (seats for a bus). We recall that $Y_N=\mathrm{30,000}\text{pax}/\text{h}$ and $Y_p=\mathrm{100,000}\text{pax}/\text{h}$, so the total patronage is $1.3\times {10}^5$. We use interpolation to find that $f_p=1180.556$, by Graph 1 of Jara-Diaz et al. [24], and $f_N=195.238$, by Graph 3 of Jara-Diaz et al. [24], with a ratio of $0.853$ such that we find the following equation:$$\begin{array}{}\text{(37)}& g{f}_p=1180.556,f_N=195.238=\mathrm{64372.641.}\end{array}$$

For completeness, we will list some questionable results in Jara-Diaz et al. [24] to help researchers absorb this important paper. We observe that on page 67, line 8 from the bottom of Jara-Diaz et al. [24], they mentioned that a constant term $K=Y_p{l}_p/f_{p}L$. However, this constant term $K$ contained a variable $f_p$.

We assume that “ratio = capital costs/operational costs.” For several different values of the frequency of the line, during a peak period in the following Table 2.

Table 2

Our evaluations for ratio.

$f_p$	Ratio
870	4.071
1083	4.004
1200	3.971
2000	3.8

However, in the legend of Graph 2, Jara-Diaz et al. [24] claimed that the range of ratios satisfies $0.5\le \text{ratio}\le 2$. Hence, we raise a questionable problem for their Graph 2.

Jara-Diaz et al. [24] mentioned that on page 68, line 22, $A_p=A$ and page 69, line 2, $A_N<A$. We recall the numerical example provided by Jara-Diaz et al. [24] with $A_p=21.48$ and $A_N=25.74$ such that the above assertion of Jara-Diaz et al. [24] contained questionable results.

We recall that on page 66, lines 4-5 from the bottom, capital cost $=c_{BC}+c_{KC}K=4.14+0.45K$, and operating cost $=c_{BO}+c_{KO}K=1.32+0.1K$

We need to compute the following equation:$$\begin{array}{}\text{(38)}& \frac{4.14+0.45K}{1.32+0.1K}>\frac{4.14}{1.32}=3.976,\end{array}$$that is, beyond the range proposed by Jara-Diaz et al. [24], with $0.5\le \text{ratio}\le 2$.

7. Direction for Future Research

From a formulated approximated solution, we can observe the influence of each parameter for the decision variable to find a functional relation to estimate the influence of each parameter. With formulated solutions, researchers took the partial derivative with respect to each parameter, and then they can find the elastic relation for each parameter. Hence, the benefit of a formulated approximated solution can help us directly observe the influence of each parameter. We recall that Hung and Julianne [6] claimed that “Traffic models with a formulated optimal solution can help researchers determine relations among variables and deterministic parameters that will influence the spending on how much budget to collect and synthesize real data.” Hence, our newly formulated approximated solutions will provide some managerial meaning for researchers to realize the transit system studied by Jara-Diaz et al. [24].

On the other hand, researchers used sensitivity analysis to estimate the influence of each parameter. We recall Chu and Chung [40] and Yang [41] to study the sensitivity analysis of the inventory model proposed by Park [42]. Chu and Chung [40] and Yang [41] pointed out that depending on a finite observation to predict the tendency of a variable is questionable. Hence, our formulated approximated solution provides an alternative approach to estimate the influence of parameters and variables.

To prove $r_3$ of equation (A.4) satisfying $0<{\text{r}}_3<1$ will be an interesting research topic for future researchers.

8. Conclusion

Jara-Diaz et al. [24] published a bus transit model with peak and normal (off-peak) periods to examine the synthesized effect of two periods. They mentioned that it is impossible to derive the formulated solution for the optimal peak and normal frequencies. In this paper, we provide an approximated formulated solution for the peak and normal frequencies. From the same numerical example provided by Jara-Diaz et al. [24], we demonstrate that our approximated formulated solution can reach the optimal cost with a relatively small error. Hence, our approach provides new insight into the bus transit model proposed by Jara-Diaz et al. [24].

Disclosure

This manuscript is a preprint.

Acknowledgments

This work was supported in part by the Weifang University of Science and Technology, with the unified social credit code: 52370700MJE3971020.

Appendix

The four roots $r_1$, $r_2$, $r_3$, and $r_4$ of a quartic polynomial,$$\begin{array}{}\text{(A.1)}& x^4+\alpha x^3+\beta x^2+\gamma x+\omega =0,\end{array}$$are given by the following formulas. To simplify the expressions, we assume ${\mathrm{\Delta }}_j$, for $j=\mathrm{1,2},\mathrm{...},9$, with$$\begin{array}{c}\text{(A.2)}& {\Delta }_1=\frac{{\alpha }^2}{4}-\frac{2\beta }{3},{\Delta }_2=\frac{{\alpha }^2}{2}-\frac{4\beta }{3},\\ {\Delta }_3={\beta }^2-3\alpha \gamma +12\omega ,\\ {\Delta }_4=-{\alpha }^3+4\alpha \beta -8\gamma ,\\ {\Delta }_5=2{\beta }^3-9\alpha \beta \gamma +27{\gamma }^2+27{\alpha }^2\omega -72\beta \omega ,\\ {\Delta }_6=\sqrt[3]{2}{\Delta }_7,\\ {\Delta }_7=-4{{\Delta }_7}^3+{{\Delta }_3}^2,\\ {\Delta }_8=\frac{{\Delta }_6}{3{{\Delta }_5+\sqrt{{\Delta }_7}}^{1/3}},\\ r_1=\frac{-\alpha }{4}-\frac{1}{2}\sqrt{{\Delta }_1+\frac{{\Delta }_6}{3{{\Delta }_5+\sqrt{{\Delta }_7}}^{1/3}}+{\frac{{\Delta }_5+\sqrt{{\Delta }_7}}{54}}^{1/3}}.\end{array}$$

The formulated roots $r_1,r_2,r_3,$ and $r_4$ for a quartic polynomial are listed as follows:$$\begin{array}{c}\text{(A.3)}& r_1=\frac{-\alpha }{4}-\frac{1}{2}\sqrt{{\Delta }_1+{\Delta }_8+{\Delta }_9}-\frac{1}{2}\sqrt{{\Delta }_2-{\Delta }_8-{\Delta }_9-\frac{{\Delta }_4}{\sqrt[4]{{\Delta }_1+{\Delta }_8+{\Delta }_9}}},r_2=\frac{-\alpha }{4}-\frac{1}{2}\sqrt{{\Delta }_1+{\Delta }_8+{\Delta }_9}+\frac{1}{2}\sqrt{{\Delta }_2-{\Delta }_8-{\Delta }_9-\frac{{\Delta }_4}{\sqrt[4]{{\Delta }_1+{\Delta }_8+{\Delta }_9}}},\\ \text{(A.4)}& r_3=\frac{-\alpha }{4}+\frac{1}{2}\sqrt{{\Delta }_1+{\Delta }_8+{\Delta }_9}-\frac{1}{2}\sqrt{{\Delta }_2-{\Delta }_8-{\Delta }_9-\frac{{\Delta }_4}{\sqrt[4]{{\Delta }_1+{\Delta }_8+{\Delta }_9}}},\text{(A.5)}& r_4=\frac{-\alpha }{4}+\frac{1}{2}\sqrt{{\Delta }_1+{\Delta }_8+{\Delta }_9}+\frac{1}{2}\sqrt{{\Delta }_2-{\Delta }_8-{\Delta }_9-\frac{{\Delta }_4}{\sqrt[4]{{\Delta }_1+{\Delta }_8+{\Delta }_9}}}.\end{array}$$

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