Research on Optimizing Human Resource Expenditure in the Allocation of Materials in Universities

Abstract

This paper establishes a multivariate function model for natural human load-carrying walking in some typical scenarios such as college equipment and material relocation by students and a large amount of identical freight relocation in commercial activities. For classified material relocation needs and constraints, we obtain the relationship between walking speed and load weight for a single person, as well as the time cost for different round trips. By establishing an integer programming model with the minimum total transportation time cost and shelf life as the objective function and the requirements of negative weight and speed as the constraint conditions, we reach the optimal item allocation methods considering time cost and shelf life. We discover that there is an approximate linear relationship between the change in natural walking speed and travel time when the load is small, thus obtaining the time cost of student transportation under different round-trip situations. The Monte Carlo simulation algorithm, which is more efficient compared with other methods such as the integer programming method, is used to obtain the optimal allocation scheme that meets the efficiency and quality requirements. The analysis methods and results can be used as guidance for task scheduling optimization for material relocation in educational organizations as well as commercial agencies.

1. Introduction

1.1. Background of the Problem

Over the past several decades, with constantly escalating logistic demand, a scheduling problem is gradually emerging and developing rapidly. For example, logistic scheduling problems are typical in universities. With the increase in online shopping, increasing quantities of parcels pour into campuses daily, and one or more large-scale parcel distribution centers are established in a single campus, while many city blocks share only one small distribution center in a city area. Moreover, with fast university development in China, constant facility upgrades or new campus plans call for many freight relocation needs. For example, a new university’s physics experiment classroom relocation calls for large-scale teaching equipment relocation. The university hosted a national academic conference that called for large-scale logistic material transportation. In most of such scenarios for freight transportation, at least part of the relocation trip requires manual labor for transport [1,2,3], for example, when complicated geographical locations or unmovable obstacles are involved. Therefore, there are general needs regarding the specified freight and the type of relocation, so an efficient relocation scheme is needed to optimize the relocation task [4,5], partially to meet stringent timing restrictions and minimum human and financial resource expenditures.

The general problem is that, depending on the information of different items to be relocated and the details of resource dispatch requirements, the university logistic unit, for example, tries to develop corresponding solutions. For items such as vegetables or fresh items, special attention should be paid to the issue of squeezing and maintaining quality and efficiency during transportation. In some special road sections or situations, manual load consumption may be high, and students (in universities where transportation tools have objective restrictions) need to be transported one or more times with heavy loads [6,7,8]. Logistics units need to further optimize personnel configuration and focus on sorting and distribution. How to build a logistics distribution and allocation model that meets these requirements, improves the level of logistics services, and proposes solutions for the logistics distribution model [9,10,11,12,13] to achieve the highest resource utilization rate is of great significance for the development of the logistics problem today, not only in universities but also in commercial locations with special needs. Many prior works [14] have partially analyzed this problem by considering the fixed weight or only in a biomechanical way. This study combined variable weight (load) and factors from a biomechanical origin to reach an optimized solution of the model.

1.2. Problem Formulation

We adapted one typical problem scenario. The methodology introduced can be easily extended to different combinations of problematic parametric value settings [15,16,17]. The following problem, which is representative of a vast category of similar problems, is stated as follows (considering the sample data in Appendix A):

Scheduling task one: Establish a dual leg weight-bearing walking model, study the relationship between the natural walking speed and the weight carried by nine students carrying items of different mass in a short period of time, and predict the changes in the speed of the second and third equal weight-bearing round trips, namely the change in time cost.

Scheduling task two: Based on the results obtained from task one, distribute all 16 freight packages to nine or fewer transport workers once or multiple times, and establish a mathematical model to determine the method of assembling and matching transportation packages for the student with the lowest transportation cost (each time).

Scheduling task three: Distribute all 16 freight packages to nine or fewer transport workers one or multiple times, establish a mathematical model to determine the mixed solution with the highest number of transportation packages with an average shelf life rate (which equals one subtracting damage rate) exceeding 80%, and provide the shelf life rate for each transportation package. The above three tasks choose total time cost, human resource expenditure and shelf life rate as objective functions for optimization. The notations utilized in this study are shown in

Table 1. Symbols description.

Symbol	Symbol Description
$E_W$	Energy rate per unit weight (J/Min)
s	Step
n	Step rate (number/Min)
$s_u^2$	Extreme step size (meter)
$n_u^2$	Extreme step rate (number/Min)
$v_u$	Natural ultimate speed (M/Min)
T	Time cost (Mins)
$m_{ij}$	The i-th student’s j-th load (Kg)
$m_{imin}$	lower limit of load for i-th student (Kg)
$m_{imax}$	Upper limit of load for i-th student (Kg)
$X_i$	The weight of student i’s load (Kg)
$Y_{ij}$	Item allocation matrix
$P_i$	i-th student’s item shelf rate (%)

.

2. Models and Methods

2.1. Problem Analysis

2.1.1. Analysis of Task One

Regarding the analysis of Problem 1, natural walking is a special way of walking. Through a literature review, it is known that at any given speed, the oxygen consumption required for a freely chosen step rate is the smallest, and there is a direct correspondence between the load and the walking rate [18,19,20]. A multivariate process group can be established. According to Question 1, it can be seen that each student knows two sets of natural walking data. Based on the known data, the unknown variables of the relevant equation system are eliminated, and the data of students with the same characteristics are then connected for solving. Finally, the relationship between the natural walking speed and the weight carried [21,22,23,24] by each group of students in a short period of time is obtained. At the same time, through the literature review, it is known that there is a correlation between the speed of natural walking and the travel time during long-term travel, which can be used to obtain the changes in the speed of the second and third round trips with equal load.

2.1.2. Analysis of Task Two

For the analysis of question one, it is required to allocate all 16 freight packages to 9 or fewer transport workers based on the results obtained from question one. It is known that the cost of human transportation is directly related to the cost of transportation time. In the process of solving problem one, the relationship between student transportation time cost and load and transportation time is known. Based on this, the objective function of a single-objective programming model can be established, which is to minimize the total transportation time. Meanwhile, as the impact of item type on transportation time is not considered, the optimal allocation method for items can be directly calculated multiple times using the Monte Carlo simulation algorithm with the student’s load-bearing capacity as the constraint condition. The reason that we utilize the Monte Carlo simulation method is that it is basically linearly dependent on the data size (9 in this test data) and is applicable to size values higher than 9. Other methods such as integer programming may exponentially depend on the size value. It is estimated that classic integer programming method calculation consumes about 24 h for data size value 15. So the Monte Carlo method is more applicable.

2.1.3. Analysis of Task Three

Regarding the analysis of question three, as-the weight and shelf life of each item are fixed, different types of items are assigned to 9 or fewer transporters. During-the mixed allocation process, the shelf life of the transport package changes, so the items can be combined and allocated based on weight. Due to the neglect of transportation time cost and item quality, a single-objective programming model can be established with the objective function of maximizing the number of transportation packages with a shelf life rate exceeding 80% using the student’s load-bearing capacity as a constraint. The Monte Carlo simulation algorithm can be used for multiple simulation calculations to obtain the optimal allocation method and the shelf life rate of each transportation package.

2.2. Model Assumptions and Symbols Description

• Assuming that the physical attributes of students are only related to the data table and meet the requirements of the experimental model;
• Assuming that there is no strict adherence to natural speed during student transportation;
• Neglecting objective factors beyond the set conditions during transportation;
• Assuming that the weight of an item can be divided in sufficiently small mass units.

2.3. Establishment and Solution of the Model for Task One

2.3.1. Analysis and Solution of Task One

For question one, it is observed that 9 students utilized natural walking mode during transportation. By referring to the relevant literature and experimental reports on this mode, we can establish the relationship between natural walking time and load weight, and derive appropriate model equations. Additionally, considering the correlation between students’ idle transportation time, the model equation can be further simplified and solved accordingly.

2.3.2. Model Preparation-Energy Consumption Model for Natural Walking

The energy consumption during walking is typically expressed as a function of walking speed. However, this relationship only applies to the stride frequency pattern during natural walking, which is used to predict the energy consumption of any combination of stride length and stride frequency. A literature review indicates that the energy demand for this combination can be determined through two experiments. In the first scenario, subjects are free to choose their own walking mode to achieve the specified speed. In the second scenario, the speed remains constant, but the subject is forced to use the prescribed range of step rates. The combination of the results of these two experiments has generated enough data to determine the energy equation for patterns, including “free” and “forced” gaits. The results indicate that at any given speed, the oxygen consumption required for a freely chosen step rate is the lowest. Any other forced step rate at the same speed will increase the oxygen cost, exceeding the oxygen cost required for natural step rates. Research has shown [25] that in natural conditions, the total energy consumption during horizontal walking at all stride and speed is

$$E_w=E_0/(1-s^2/s_u^2)(1-n^2/n_u^2)$$

(1)

where $E_w$ is energy rate per unit weight, in cal/kg/min, s is the step length in meters, N is the step rate in steps per minute, $s_u$ is the limit step size in meters, $n_u$ is the step rate measured in steps per minute, $v_u$ is moving speed, the energy consumption rate at $t=0$ is $E_0$. The condition for the minimum E derived from Equation (1) is applicable to a fixed velocity v, which has been proven to be

$$s=(s_u/n_u)n$$

(2)

Research has shown that the minimum $E_w$ occurs at $v=v_u/3$, where $v=s{n}_u$. If the speed remains constant, the minimum $E_w$ step size for that speed needs to be proportional to the step length, and the proportional constant $s_u/n_u$ remains the same regardless of the speed.

When the subject walks freely, they usually adopt a pattern that is almost identical to the one described in Equation (2). Figure 1 shows the s/n walking mode of the example. A bold continuous line was drawn in the form of Equation (2). The sin relationship shown in Figure 1 is basically linear for all subjects, but there are significant differences between subjects (such as B, C, and G). However, due to a 10% increase in the s/n ratio of typical subjects (walking speed = $V_u=80$ m/min), energy expenditure only increased by 0.7% compared to the optimal level. Therefore, subjects can change their walking mode near the optimal ratio without significantly increasing their energy expenditure. At the same speed, even if sin increases by 20%, it will only increase by 2.6%. In natural walking, step length is uniquely correlated with step frequency through Equation (2). If speed is chosen as the independent variable and Equation (2) is used, Equation (1) will be simplified to the following natural walking equation:

$$E_w=E_0/(1-v/v_u)$$

(3)

where v is walking speed in unit m/min.

2.4. Establishment of Energy Consumption Model for Load Travel of Task One

The relationship between oxygen consumption $V_{O_2}$ [mL/kg. min] and load $L/W_b$ (%) at various speed levels is shown in Figure 2, using the mean measured values of 12 subjects as the experimental results.

The results in Figure 2 indicate that the relationship between oxygen consumption (VO) and load (LW) during flat ground load travel should be strictly nonlinear. However, when the speed is low, for example, $V<5$ km/h, the relationship is basically linear, when the speed is high, the nonlinear relationship will gradually become significant. There have been many studies on weight-bearing travel and some empirical equations have been proposed to represent this relationship, with international legal units being used uniformly. Some of the main equations are as follows [14]:

$$M/W_b=(1+{\displaystyle \frac{L}{W_b}}){[2.7+3.2{(V-0.7)}^{1.65}+G(0.23+0.29(V-0.7))]}^{1.3}$$

(4)

When $G=0$, there is

$$M/W_b=1.5+2(1+{\displaystyle \frac{L}{W_b}}){(L/W_b)}^2+1.5V^2(1+L/W_b)$$

(5)

The empirical equation indicates that there is a basic linear relationship between $V_{O_2}$ and $L/W_b$. However, the research results show that for flat terrain, when V is very high or $L/W_b$ very large, the nonlinear relationship between $V_{O_2}$ and $L/W_b$ will gradually become significant (Figure 1). Some studies have also found that the energy consumption during flat ground moving is not always linearly related to the load. It has been pointed out that when the load is greater than 40% to 45% of the body weight, the energy consumption during driving will increase nonlinearly, and when the load is larger, the energy consumption at a faster rate.

Considering that all 9 students are moving objects at a natural walking speed, the equation system can be obtained by combining Equations (3) and (6) as,

$$\{\begin{array}{}\mathrm{(6a)}& E_w=E_0/(1-v/v_u\mathrm{(6b)}& E_w/W_b=1.5+2(1+{\displaystyle \frac{L}{W_b}}){(L/W_b)}^2+1.5V^2(1+{\displaystyle \frac{L}{W_b}})\end{array}$$

Meanwhile, due to the same natural time cost for the first empty load of every three passengers, it can be inferred that the maximum speed $v_u$ is the same. The function relationship between the natural walking speed and the load of the i-th student carrying items of different masses in a short period of time, without considering the influence of load time on the natural walking speed, can be obtained as follows

$$E_{0i}/(1-v/v_{uj})/W_{bi}=1.5+2(1+{\displaystyle \frac{L}{W_{bi}}}){(L/W_{bi})}^2+1.5V^2(1+{\displaystyle \frac{L}{W_{bi}}})$$

(7)

where v is the speed of item handling, L is the mass of item. If we consider the cost of transportation time under the influence of time, as the quality of the goods is much lower than 40% of the weight of the transporter themselves, the transportation time can be regarded as positively correlated with the rate of physical exertion, which can be calculated linearly by 2.6%.

2.5. Solving the Model for Task One

Due to the fixed transportation distance, if the distance is considered as a unit length, then $v=1/$one-way time cost. Using programming software to numerically solve the model, the relationship between the natural walking speed of students and their weight is obtained as shown in

Table 2. Student speed–load relationship ($a/(1-V)/b=1.5+2(1+L/b)(L/b)+1.5\ast V2$).

Student Code	a	b
1	3.9638	2.0736
2	3.6729	1.9214
3	3.4442	1.8018
4	8.2578	3.5795
5	8.0229	3.4777
6	7.5248	3.2617
7	13.7822	5.1146
8	12.9585	4.8089
9	12.1348	4.5032

.

Considering that the physical exertion during unloaded natural walking can be ignored, the time cost under load is linearly correlated with the speed of physical exertion. After calculation, the time cost for each load stage is shown in

Table 3. Time cost for each round trip stage.

Student Code	First Handling Time Cost/min	Second Handling Time Cost/min	Third Handling Time Cost/min
1	7.5	7.77	8.05
2	8.00	8.29	8.59
3	8.50	8.81	9.12
4	8.80	9.12	9.45
5	9.00	9.32	9.66
6	9.50	9.84	10.20
7	9.60	9.95	10.30
8	10.00	10.36	10.73
9	10.50	10.88	11.27

. The Matlab codes are shown in the Supplementary Materials of this work.

2.6. Conclusion of Model for Task One

According to the above analysis, it can be seen that under natural walking conditions, there is a corresponding relationship between time cost and speed. In the case of small loads, the increase in transportation time cost is linearly correlated with transportation time.

2.7. Analysis for Task Two

For question two, it can be seen that the transportation process still adopts a natural walking mode. From question one, the relationship between transportation speed and load weight in this mode is known. Considering that the impact of item types on transportation time and cost can be ignored, the establishment of a single-objective programming model mainly considers the relationship between transportation time, load weight, and transportation frequency for 9 students.

2.8. Model Establishment for Task Two-Data Preprocessing

The total mass of the i-th item is $a_i$, and its shelf life rate is $b_i$, both of which are constant real numbers. The values are shown in

Table 4. Total weight and shelf life of items.

Item Type	Total Weight ${\mathit{a}}_{\mathit{i}}$ (Kg)	Shelf Life Rate ${\mathit{b}}_{\mathit{i}}$
1	10	0.88
2	15	0.6
3	7	0.93
4	17	0.9
5	10	0.9
6	14	0.78
7	10	0.7
8	10	0.83
9	13	0.95
10	20	0.87
11	3	0.65
12	20	0.75
13	16	0.8
14	12	0.68
15	10	0.87
16	10	0.83

.

Due to the lack of consideration for transportation time costs and the fact that the student load limit is greater than the total mass of the item, it is assumed that students will only participate in a single transportation activity. If the transportation weight of the i-th student is $X_i$, then

$$X_i={\mathsf{\Sigma }}_{i=1}^{16}{Y}_{ij}{a}_j$$

(8)

where $Y_{ij}$ is coefficient matrix to be determined.

2.9. Model for Task Two

Due to limitations in the number and handling capacity of students, with their load-bearing capacity as the limiting condition and the maximum number of transportation packages with a shelf life rate exceeding 80% as the objective function, a single-objective programming model is established as follows:

(1)

Establishment of objective function

As can be seen from the question, the maximum number of transportation packages that require a shelf life rate exceeding 80% is the variable z, which can be recorded as transportation packages with a shelf life rate exceeding 80% in a certain transportation allocation situation, Objective function: T = max (z)

(2)

Establishment of restrictive conditions

Due to not considering the impact of item types and time costs, the shelf life rate of the i-th student’s transport bag is $P_i$, then $P_i={\mathsf{\Sigma }}_{j=1}^{16}{Y}_{ij}{a}_j{b}_j/Sigm{a}_{i=1}^{16}{Y}_{ij}{a}_j$. The single-objective programming model is as follows:

$$\{\begin{array}{}\mathrm{(9a)}& X_i={\mathsf{\Sigma }}_{i=1}^{16}{Y}_{ij}{a}_j\mathrm{(9b)}& P_i={\mathsf{\Sigma }}_{j=1}^{16}{Y}_{ij}{a}_j{b}_j/{\mathsf{\Sigma }}_{i=1}^{16}{Y}_{ij}{a}_j\mathrm{(9c)}& m_{imin}<X_{ij}<m_{imax}\mathrm{(9d)}& T=\underset{i=1}{\overset{16}{\mathsf{\Sigma }}}H(P_i-0.8)\end{array}$$

2.10. Solving the Model for Task Two

We use the Monte Carlo simulation algorithm to obtain the item allocation matrix T for different situations. Through programming software, we perform 10,000 cycles under restricted conditions. During the cycle, we store the maximum value of z in MAX for a single cycle and use matrix DMAX to store the optimal allocation matrix for the previous several simulation conditions. When there is a value greater than the original DMAX value, we replace z* with z and replace DMAX with MAX. After the loop ends, the obtained optimal simulation matrix DMAX is the optimal allocation scheme Y* After calculation, the optimal allocation plan and the parcel shelf life rate are shown in

Table 5. Optimal package calculated shelf life rate.

Package Code	Parcel Retention Rate
1	0.821629
2	0.804405
3	0.801495
4	0.833797
5	0.804377
6	0.809537
7	0.802351
8	0.808308
9	0.812711

.

2.11. Conclusion for Model of Task Two

Based on the analysis above, it can be concluded that the optimal allocation plan that meets the shelf life rate requirements can be obtained by using the method of single transportation of items by students, without considering the time cost of item transportation.

3. Evaluation of the Models

3.1. Evaluation of the Model

3.1.1. Advantages of the Models

(1)

The model adopts the Monte Carlo simulation algorithm to randomly generate multiple feasible operation plans and compare them pairwise to obtain the optimal plan. This method has a high degree of randomness, and the generated multiple excellent transportation plans can avoid the impact of some students being injured during transportation and unable to transport heavier packages.

(2)

The model refers to many widely recognized references within the industry and has great scientific and practical value.

3.1.2. Disadvantages of the Models

The model adopts the Monte Carlo simulation algorithm, which has randomness and may not be able to obtain the best solution within the limited number of times specified by the program. Therefore, it is necessary to run the program repeatedly and verify the results.

3.2. Improvement of the Models

The establishment of the model initially involved gathering several widely recognized literature materials in the industry, comparing and analyzing them, and then selecting the literature that is most suitable for this type of simulation situation for further reference. The relationship formula established has a certain degree of scientific validity and feasibility. Additionally, we employed the Monte Carlo simulation algorithm to randomly simulate various situations of transporting student packages and compared them pairwise in order to determine the most suitable transportation plan. The random simulation algorithm not only accommodates the randomness of package allocation to different carriers in real situations but also avoids the impact of potential accidents on certain carriers. Moreover, each model run can provide an equally efficient and different transportation plan for selection. The establishment of this model can not only be applied to the logistics transportation industry but can also be extended to all industries involved in the distribution process.

4. Discussion

In this study, factors such as walking speed, step length, and load capacity were taken into account, to optimize the relocation cost of a large quantity of goods. Objective functions such as transportation time, labor resource expenditure (referring to the number of workers involved), and shelf life of the goods were also considered. We not only developed an optimization method based on the Monte Carlo algorithm but also refined previous work that focused on optimizing the costs of loading and walking. Prior work, however, failed to take into account variable load weights or group relocation that involve multiple individuals. The algorithm we developed is highly suitable for large data sizes in real-world applications that involve manual relocation of freight because it only relies linearly on the size of the data(number of people or number of packages). To illustrate the effectiveness of the Monte Carlo algorithm, we conducted a comparative analysis of its performance against pure integer programming using different data sizes (the number of students) being 7, 8, 9, and 11, as shown in

Table 6. Computation error for Monte Carlo simulation compared with integer programming.

Student Size	Running Time of Monte Carlo	Running Time of Integer Prog	Error of Monte Carlo Simulation	Diff. btw. Monte Carlo and Interpolation of Table 3
7	1 min	30 min	1.5%	-
8	1.5 min	70 min	0.9%	-
9	2.4 min	160 min	1.2%	-
11	5 min	900 min	0.8%	-
20	15 min	-	-	5.1%
30	26 min	-	-	6.3%

. The results indicated that the Monte Carlo algorithm exhibited a commendably low level of uncertainty. Due to the inherent systematic errors of the model, the integer programming results might be overdetermined. However, the Monte Carlo algorithm has the potential to introduce random errors that, in some cases, could counterbalance and reduce the overall error margin. Furthermore, for large data volumes N = 20, 30, we compared the results of the Monte Carlo algorithm with the interpolation results presented in Table 3 and evaluated the differences, which allowed us to assess the discrepancies between the two approaches.

5. Conclusions

In this study, we develop an integer programming model for a typical manual freight relocation problem in scenarios like colleges with student laborers. The model aims to minimize transportation time costs and incorporates a personal load-time–cost relationship while also considering possible shelf life quality constraints. The Monte Carlo simulation algorithm is used to identify the optimized freight allocation scheme under different round trip conditions. The optimized schemes are determined based on the objectives of minimum time cost, lowest labor cost and maximum shelf life rates. The analysis results can be applied to freight relocation scheduling problems to optimize resource dispatch in educational institutions or commercial organizations. Furthermore, the algorithm developed in this study can be used to solve a large amount of parcel machine dispatch problems using artificial intelligence technology, with partial updates of the currently developed algorithm.