# Cost Reduction of Taxi Enterprises at the Expense of Automobile Fleet Optimization

.V Novytskyi, Yu. Melnikova

► To cite this version:

V Novytskyi, Yu. Melnikova. Cost Reduction of Taxi Enterprises at the Expense of Automobile Fleet Optimization. Mechanics, Materials Science & Engineering Journal, 2017, 7, 10.13140/RG.2.2.24945.89447. hal-01508561

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# Cost Reduction of Taxi Enterprises at the Expense of Automobile Fleet Optimization

Novytskyi A.V.¹, Melnikova Yu. I.¹

1 - Department of Transport Management, National Mining University, Dnipropetrovsk, Ukraine

<figure><img src="image_2.png" /></figure>

DOI 10.13140/RG.2.2.24945.89447

**Keywords:** taxi service, queuing system, probability of service denial, cost

**ABSTRACT.** Results of taxi service operation using techniques of queuing system theory have been demonstrated. It has been shown that probability of service denial is the key quality criterion of transport services for taxi services. It is expedient to use total expenditures of queuing system as target function to estimate the efficiency of taxi service. It has been determined that application of queuing theory techniques makes it possible to identify optimum value of the number of operating motor vehicles for specific environment. The value is optimum according to minimum-cost criterion.

**Introduction.** Cost saving to provide services under the conditions of competitive indicators of quality is one of the most important problems for any transport enterprise. The problem becomes topical in the context of excessive supply. On the one hand, customer acquisition involves improvement of quality indicators which results in extra costs; on the other hand, economic situation requires cost cutting. Taxi enterprises should operate under those conditions.

Currently more than 200,000 motor vehicles of various ownership forms operate in the market (data by the Trade Union of taxi drivers of Ukraine). That is an obvious excess of supply. Except that the figure experiences constant expansion due to private car owners engaged in private cabbing to repay loans. In this context, increase in the number of taxi supply is followed by quality degradation. That depends chiefly on poor skills of staff of taxi enterprises resulting in protraction of waiting period and travel time, nonoptimal delivery routes, and high-cost transportation. Analysis of operation of taxi enterprises in Ukrainian cities shows that the majority of organizational decisions are made relying upon the experience of prior periods. Even if economic and mathematical substantiation is performed, it is based upon simplified techniques using averaged values of influencing parameters. The authors have analysed six enterprises in Dnipropetrovsk region. Four of the six enterprises keep records of the number of orders according to oral information by drivers. No enterprise accumulates and analyses information concerning the period of bringing the order to effect, the number of unexecuted orders etc. Moreover, in many cases the number of motor vehicles operating during a shift depends on the availability of serviceable motor vehicles. As a result, there is no necessary information to develop transportation scheme of transport services.

The number of service denials is one of the most important qualitative indicators in the process of taxi service management. To attract clients, transport operators put up considerable capital. According to experts' research, acquisition of a new client costs a company six times more than retention of available one. If however a client leaves unsatisfied, her/his return will cost twenty-five times more [1]. Practically the number of service denials or their possibility is controlled by the number of operating motor vehicles: the more motor vehicles operate during certain period, the higher is the probability to execute order and the less is probability of denial. In the context of favourable economic situation, cost escalation is covered with extra income from executed orders. However, in the context of purchasing power erosion and increase in expenditures connected with maintenance of road transport vehicles such a solution not always results in expected outcomes.
Currently techniques of queuing theory are often used to solve a problem concerning substantiation of transport service parameters. The techniques are more advantageous to compare with traditional modelling methods as they consider random nature of inflow of orders and time to service them [2]. This very fact transforms queuing theory into powerful tool to model various processes including a process of transport service. Use of queuing theory techniques makes it possible to determine probability parameters of inflows of orders, operating parameters, and qualitative indicators concerning service of orders by taxi enterprise.

**Objective.** Identification of rules to change expenditures of taxi enterprise in the context of varying parameters of inflows of orders and services to substantiate optimum number of car park.

**Data for the analysis.**

Taxi service «Elit taxi» operating in the town of Novomoskovsk (Dnipropetrovsk region) and neighbouring districts has been analysed. The enterprise renders services within 24 hours operating by means of three 8-hour shifts. The accepted practice covers service denial if vacant motor vehicles are not available. Hence, it is possible to consider the enterprise as multichannel queuing system with denials.

Automobile park consists of 27 units. According to data by finance department of the enterprise, specific expenditures connected with motor vehicle movement are 136 UAH/(motor vehicles·hour), expenditures connected with unproductive time of motor vehicle are 41 136 UAH/(motor vehicles·hour), and expenditures connected with service denials are 176 UAH/(motor vehicles·hour).

Average number of inflowing orders is taken to be equal to: shift 1 – 32.33 orders per hour, shift 2 – 20.11 orders per hour, shift 3 – 11.15 orders per hour. The enterprise normalizes average time to execute order as follows: shift 1 – 0.7 of hour (intensity of service flow is $μ_{16} = 1.42$ orders per hour), shift 2 – 0.47 of hour ($μ_{26} = 2.12$ orders per hour), and shift 3 – $μ_{36} = 0.34$ of hour (2.96 orders per hour).

Stage one of the research involved accumulation of information and its analysis concerning the number of orders and average service time (Fig.1). Results of data processing according to technique [2] have helped determine that values of intensity of flow of orders taken at the enterprise are valid; values of service flow intensity differ greatly. Thus, for shift 1 actual average time to execute order is 0.64 of hour, for shift 2 it is 0.54 of hour, and for shift 3 it is 0.41 of hour. Thus, relying upon comparison of experimental values and critical values of Pearson criterion with $α = 0.05$ significance level it has been determined that the both flows are described by means of Poisson distribution law with following intensities:

- For flow of orders: $λ_{1φ} = 32.33$ orders per hour, $λ_{2φ} = 20.11$ orders per hour, $λ_{3φ} = 11.15$ orders per hour;

- For service flow: $μ_{1φ} = 1.56$ orders per hour, $μ_{2φ} = 1.85$ orders per hour, $μ_{3φ} = 2.44$ orders per hour.

Comparison of information obtained at the enterprise with experimental data has shown that gaps are as follows: 9% for shift 1; 15% for shift 2; and 19% for shift 3 (Fig.1).
<figure><img src="image_3.png" /><figcaption>Fig. 1. Results of data processing.</figcaption></figure>

It is evident, that use of averaged data introduces significant errors into determination of flow parameters preventing from adequate evaluation of queuing system.

Stage two of the research involved determination of basic parameters of queuing system for basic scenario and design scenario to organize service of orders. Key indices of multichannel queuing systems with denials are [3]:

- The number of service channels, i.e. total number of motor vehicles operating during a shift;

- Probability of service denial, i.e. probability that order will not be completed and will leave queuing system.

Besides substation of optimum number of motor vehicle operating during every shift using criterion of minimal total expenditures of queuing system is one of the research tasks.

General costs of queuing system with denials are determined by formula [4]:

$$C_{CMO} = C_{np} \bar{n}_{CB} + C_{\delta_{B}} \bar{n}_{3} + C_{omk} p_{omk} \lambda,$$

where $C_{np}$ – is specific cost connected with unproductive time of motor vehicle, UAH/(motor vehicles·hour);

$C_{\delta_{B}}$ – is specific cost connected with motor vehicle movement, UAH/(motor vehicles·hour);

$C_{omk}$ – is specific cost connected with service denial, UAH/(motor vehicles·hour);

$\bar{n}_{CB}, \bar{n}_{3}$ – is average number of vacant and motor vehicles under service respectively;

$p_{omk}$ – is probability of service denial. Thus, target function is expressed as

$$[C_{np} \bar{n}_{CB} + C_{\delta_{B}} \bar{n}_{3} + C_{omk} p_{omk} \lambda] \rightarrow \min .$$

Basic scenario used data obtained at the enterprise under study. On the ground of cost saving every shift involves minimum quantity of operating motor vehicles to achieve predetermined load intensity. According to information by the enterprise, shift 1 involves 23 motor vehicles; shift 2 involves 10 motor
vehicles, and shift 3 involves 4 motor vehicles. Table 1 demonstrates calculation results of key indices of queuing system operation.

<table><caption>Table 1. Calculation results concerning queuing system operation (basic scenario).</caption><thead><tr><th>Index</th><th>Shift 1</th><th>Shift 2</th><th>Shift 3</th></tr></thead><tbody><tr><td>The number of operating motor vehicles, units</td><td>23</td><td>10</td><td>4</td></tr><tr><td>Probability of service denial</td><td>0.143</td><td>0.235</td><td>0.212</td></tr><tr><td>“Currency lost /income”</td><td>0.167</td><td>0.307</td><td>0.269</td></tr><tr><td>Expenditures, connected with motor vehicle movement, UAH per hour</td><td>2745</td><td>986</td><td>402</td></tr><tr><td>Expenditures, connected with unproductive time of motor vehicle, UAH per hour</td><td>145</td><td>113</td><td>43</td></tr><tr><td>Expenditures, connected with service denial, UAH per hour</td><td>902</td><td>832</td><td>414</td></tr><tr><td>Queuing system expenditures, UAH per hour</td><td>3792</td><td>1931</td><td>859</td></tr></tbody></table>

Analysis of the results demonstrates poor efficiency of queuing system operation in terms of basic scenario of transport service management. During every shift the enterprise uses minimum possible number of motor vehicles being geared to load intensity and trying to cut expenditures connected with movement of motor vehicles. That very time, possibility of service denial is 23.5%, and “currency lost /income” index is 0.307 to be invalid under the conditions of competitive market.

Following calculations were performed with the help of identical technique. However, the calculations were required to determine optimum number of operating motor vehicles providing a condition for minimum aggregate expenditure. To do that, basic parameters of multichannel queuing system were calculated. The calculations involved denials in the context of various numbers of operating motor vehicles. Taking into account the fact that actual values of order service flow intensity differ greatly from those taken before, in terms of design scenario, load intensity of the system will be as follows: $\rho_{\phi I} = 20.69$ orders per hour for shift 1; $\rho_{\phi II} = 10.86$ orders per hour for shift 2; and $\rho_{\phi III} = 4.55$ orders per hour for shift 3.

Table 2 demonstrates an example of calculation results.

<table><caption>Table 2. Calculation results of queuing system operation indices (design scenario, shift 2).</caption><thead><tr><th rowspan="2">Index</th><th colspan="7">The number of motor vehicles</th></tr><tr><th>11</th><th>12</th><th>13</th><th>14</th><th>15</th><th>16</th><th>17</th></tr></thead><tbody><tr><td>Probability of service denial</td><td>0.2002</td><td>0.1534</td><td>0.1136</td><td>0.0810</td><td>0.0554</td><td>0.0362</td><td>0.023</td></tr><tr><td>Average number of busy channels</td><td>8.69</td><td>9.20</td><td>9.63</td><td>9.98</td><td>10.26</td><td>10.47</td><td>10.62</td></tr><tr><td>Expenditures, connected with movement, UAH per hour</td><td>1181</td><td>1251</td><td>1309</td><td>1357</td><td>1395</td><td>1424</td><td>1444</td></tr><tr><td>Expenditures, connected with unproductive time, UAH per hour</td><td>95</td><td>115</td><td>138</td><td>165</td><td>194</td><td>227</td><td>262</td></tr><tr><td>Expenditures, connected with denial, UAH per hour</td><td>709</td><td>543</td><td>402</td><td>287</td><td>196</td><td>128</td><td>80</td></tr><tr><td>Total expenditures of queuing system, UAH per hour</td><td>1985</td><td>1908</td><td>1850</td><td>1809</td><td>1786</td><td>1779</td><td>1786</td></tr></tbody></table>
As analysis of the results has shown in terms of increase of the number of operating motor vehicles, possibility of service denials decrease according to exponential law; average number of motor vehicles engaged in order servicing also increases nonlinearly resulting in proportional growth of expenses connected with movement of motor vehicles and their unproductive time. Dependence graph $C_{CMO}(n)$ has its minimum when $n = 16$ ($C_{CMO} = 1779$ UAH per hour). It has been determined analogously that in the context of concerned conditions, optimum number of motor vehicles operating during shift 3 is 9 automobiles and for shift 1 the number is 27 automobiles. That is, minimum expenses will involve increase in 4 motor vehicles (shift 1) and 5 motor vehicles (shifts 2 and 3) (Table 3).

<table><caption>Table 3. Comparison of efficiency indices of queuing system for basic scenario and design one.</caption><thead><tr><th rowspan="2">Index</th><th colspan="2">Shift 1</th><th colspan="2">Shift 2</th><th colspan="2">Shift 3</th></tr><tr><th>Basic scenario</th><th>Design scenario</th><th>Basic scenario</th><th>Design scenario</th><th>Basic scenario</th><th>Design scenario</th></tr></thead><tbody><tr><td>Intensity of influent flow of orders</td><td>32.33</td><td>32.33</td><td>20.11</td><td>20.11</td><td>11.10</td><td>11.10</td></tr><tr><td>Average service time</td><td>0.70</td><td>0.64</td><td>0.47</td><td>0.54</td><td>0.34</td><td>0.41</td></tr><tr><td>Intensity of order servicing</td><td>1.42</td><td>1.56</td><td>2.12</td><td>1.85</td><td>2.96</td><td>2.44</td></tr><tr><td>Intensity of queuing system load</td><td>22.72</td><td>20.69</td><td>9.48</td><td>10.86</td><td>3.75</td><td>4.55</td></tr><tr><td>The number of motor vehicles</td><td>23</td><td>27</td><td>10</td><td>16</td><td>4</td><td>9</td></tr><tr><td>Probability of service denial</td><td>0.143</td><td>0.034</td><td>0.235</td><td>0.036</td><td>0.212</td><td>0.025</td></tr><tr><td>Expenditures, connected with movement</td><td>2745</td><td>2817</td><td>986</td><td>1424</td><td>402</td><td>603</td></tr><tr><td>Expenditures, connected with unproductive time</td><td>145</td><td>288</td><td>113</td><td>227</td><td>43</td><td>187</td></tr><tr><td>Expenditures, connected with probability of service denial</td><td>902</td><td>216</td><td>832</td><td>128</td><td>414</td><td>50</td></tr><tr><td>Total expenditures of queuing system, UAH per hour</td><td>3793</td><td>3321</td><td>1931</td><td>1779</td><td>859</td><td>840</td></tr><tr><td>Changes in queuing system expenditures, UAH per hour</td><td colspan="2">-472</td><td colspan="2">-152</td><td colspan="2">-19</td></tr></tbody></table>

Increase in the number of operating motor vehicles is favourable for service quality as probability in service denial decreases (Fig. 2) to be particularly relevant for taxi enterprises. Meanwhile it should be noted that in terms of the number of motor vehicles increase, deviation of shiftable value of denial probability from daily average one is 14.3 %, while it is 21.2% for basic scenario. Minor spread of denial probability makes it possible to control quality of passenger service.
<figure><img src="image_4.png" /><figcaption>Fig. 2. Probability of service denial in the context of different number of operating motor vehicles.</figcaption></figure>

Analysis of changes in queuing system has shown that increase in the number of operating motor vehicles results in escalation of costs connected with movement (up to 50%) and unproductive time (up to 100%). That very time expenses connected with service denial decrease proportionally to increase in service denial probability (by 80%); as a result it compensates cost escalation for movement and unproductive time. It should also be noted that shift 1 demonstrates the greatest reduction of general costs when demand is the most intensive (Fig. 3).

<figure><img src="image_5.png" /><figcaption>Fig. 3. Changes in queuing system expenses.</figcaption></figure>

**Summary.** Results of order servicing modelling process by taxi enterprise with the help of queuing system have helped determine the following:

1. To provide high-level passenger service and effective use of motor vehicles it is required to perform constant (automated if possible) on-line collecting and processing of parameters of inflow of orders and service. Along with collecting and analysing information concerning the number of transportation orders, constant control over the order execution time is required. Otherwise, considerable deviations of the obtained parameters from optimum values are possible;
2) Substantiated choice of rational parameters of technological scheme of taxi transportation involves queuing system theory as it fives ability to consider random nature of inflowing orders and service time;

3) Reduction of total expenses of taxi enterprise is possible owing to attraction of extra motor vehicles to serve orders. It allows reducing costs connected with transportation denials at the expense of increase in service possibility;

4) Effect resulting from the use of queuing system theory is the most evident in terms of sharp shiftable variations in inflow order intensity and service as well as in terms of loads on a system close to maximum ones.

References

[1] Kleinrock L. (1979). Queueing theory: translation from English [Teoriia massovogo obsluzhivaniia]. Moscow, "Mashinostroeniie", 432 pp.

[2] Wentzel E. S. (1991). Theory of random processes and its engineering applications [Teoriia sluchaynykh protsessov i eio inzhenernyie prilozheniia]. Moscow, "Vysshaia shkola". 384 pp.

[3] Koroliuk V. (1985). Reference book on the theory of probability and mathematical statistics [Spravochnik po teorii veroiatnostei i matematicheskoi statistike]. Moscow, "Nauka". 640 pp.

[4] Khinchin A. Mathematical methods of queuing theory [Matematicheskie metody teorii massovogo obsluzhivaniia]. Moscow, "Nauka". 248 pp.

[5] Ruibin Bai, Jiawei Li, Jason A. D. Atkin, Graham Kendall. A novel approach to independent taxi scheduling problem based on stable matching, Journal of the Operational Research Society, (2014) 65: 1501. doi:10.1057/jors.2013.96