## 1. INTRODUCTION

Due to numerous applications, vehicle routing has become one of the widest and most important issues in the field of decision-making and optimization. The present study aimed to design a new robust mathematical model for vehicle routing, taking into account the simultaneous pickup and delivery, the time window, and the worker allocation for transportation. Simultaneous pickup and delivery method is a highly applied technique in goods transportation systems due to its ability to satisfy the needs of all customers with the products of others through proper routing. In addition, most products have a shelf life or tardiness in delivery leads to a financial penalty. To this end, time windows for product pickup and delivery are considered more applicable and closer to real-world situations. In case of early or late delivery of goods to customers, the distribution center must pay a price as the pre-determined penalty. Most studies performed in the field have only focused on finding an optimal route for vehicles (Amouzad Mahdiraji and Yousefi Talouki, 2021;Nasiri and Ahmadi Daryakenari, 2021;Sharma and Sharma, 2021). In the real world, however, there are important factors in addition to finding the optimal route. In this regard, an issue is allocating workers based on the ability to do relevant work to serve. An example of these allocations is security centers such as banks in order to deliver valuables or moving important goods, which requires a specialized workforce for transportation. For instance, a large volume of financial documents and reports of a bank can be transmitted by appropriate means and with the help of security forces due to the low financial value of these bonds and the low probability of their theft. At the same time, special equipment and highly specialized security forces must be used when an object of great financial value is planned to be transported. Optimal use of vehicles and specialized personnel for maintenance is crucial in centers where such transfers take place. For instance, sometimes a large number of criminals that have done insignificant crimes and the chance of their escape is very low are transferred by a normal vehicle and one security force. However, things are more complicated in the case of certain criminals, for which special vehicles and specialized security forces are required.

The goal of this model is to optimize these problems and propose proper solutions. In many cases in the real world, the parameters involved in the problem cannot be accurately identified and these parameters must be considered under uncertainty. The robust approach largely satisfies the justification and optimality of responses by identifying control variables. In this research, a robust planning approach is used to deal with the uncertainty of the parameters. In fact, the parameters are entered into the model at several levels under the heading of a scenario with a certain probability of occurrence, and finally, an appropriate answer to the problem is obtained. Some examples are presented by GAMS to evaluate the accuracy of the model. Since the routing problem is an NP-hard problem, with increasing variables, the solution time increases exponentially. Naturally, solving large-scale problems is not possible with GAMS software. GAMS optimization software is used to analyze the answers to the problem.

Vehicle routing has been studied in the field of operations research, and has been considered in the form of distribution by truck issues since more than 50 years ago (Dantzig and Ramser, 1959). The issue of vehicle routing has been studied in the field of operations research, and has been taken into account in the form of distribution by truck issues since more than 50 years ago (Dantzig and Ramser, 1959). The issue is defined as a series of optimal routes for a set of vehicles to satisfy customers’ needs. Significant attention has been paid to routing due to the applicability of this matter in the real world and its deep impact on the economic area for suppliers. Therefore, it has been assessed in the form of rich vehicle routing. The main goal is to determine the best tour for product delivery while considering customers’ demands. Numerous books and articles have been published in this area, including a research by Laporte and Osman (1995). The traveling salesman problem was first presented and solved by Laporte and Osman (1959) in the literature of vehicle routing articles. Because this problem is a special form of the vehicle routing issue, it is recognized as the basis for articles related to the vehicle routing area. Ultimately, Clarke and Wright (1964) solved the aforementioned problem with more than one vehicle. Golden *et al*. (1984) published the first article entitled “vehicle routing”. Other types of problems in this field were presented in the 1970s, including the research by Lieberman and Simpson (1960), which was carried out to solve the problem of solid waste collection. Solomon (1987) added a time window to the problem, and extensive research was performed in the 1990s due to the accessibility and rapid growth of microcomputers. In addition, metaheuristic methods were applied to solve the problem. In this regard, the first research on stochastic vehicle routing was carried out by Cook and Russell (1978). Laporte *et al*. (1992) took another step toward solving the routing problem correctively through the development of the branch and bound technique. In addition, Toth and Vigo (1999) solved symmetric and asymmetric routing problems using the extended energy-saving algorithm. In another research, Osman *et al*. (2005) used the genetic algorithm to solve the multi-objective routing problem. Moreover, Calvete *et al*. (2007) applied an ideal programming method to solve routing problems with a time window while considering several objective functions and constraints. In this problem, the vehicle capacity and time to refer to the customer were considered as objective. In a study, Erdoğan and Miller-Hooks (2012) proposed the green vehicle routing problem, which received special attention due to the importance of issues related to the environment.

A blended number programming model was proposed, and a branch-and-value calculation was created (Poikonen *et al*., 2017). Vehicle steering issue is one of the broadly explored regions in transportation science, chiefly because of the possible expense investment funds and administration improvement openings that brings to associations engaged with actual conveyance of products. A multi-terminal green vehicle directing issue was created by expanding income and limiting expenses, time and emanation, and afterward, an improved subterranean insect settlement streamlining (IACO) calculation was applied that means to productively take care of the issue (Li *et al*., 2019).

Moreover, Azi *et al*. (2010) presented an accurate model based on the branch and price approach, where, given that not all customers could receive service, benefits are considered for any client that receives service, which leads to a decrease in the service cost. Recently higher attention is paid to the metaheuristic approaches, and the more problems are changing from simple issues to those that fit the daily needs of individuals. Setak *et al*. (2015) used the time-dependent and multi-graph vehicle routing and the FIFO method (first in, first out) while considering more than one node and one bound in the vehicle routing problem simultaneously. In addition, they presented the tabu search algorithm as a problem-solving method and compared it with other accurate techniques. One of the advantages of their proposed model was considering the vehicle speed factor in addition to time window and FIFO policy. Küçükoğlu *et al*. (2015) presented an advanced hybrid metaheuristic method for vehicle routing problem while considering a time window and delivery from the distribution center. They also used the time window and the k-nearest neighbors’ algorithm to achieve an initial solution. Moreover, their work is distinguished from the work of others due to the use of two metaheuristic methods and a heuristic approach to obtain the initial solution, which led to obtaining suitable solutions. Letchford and Salazar-González (2015) carried out the modeling of the multi-commodity vehicle routing problem with capacity constraints using the knapsack method. In addition, the method yielded a more favorable solution, compared to other metaheuristic approaches. Kramer *et al*. (2015) presented the speed and travel time optimization algorithm for the pollution-routing problem. They also added the use of definitive equations to make the released algorithm more robust while considering additional variables and an alternative model. In a research, Li *et al*. (2015) applied an iterative deepening algorithm based on neighbor selection for a multi-depot vehicle routing problem with simultaneous pickup and delivery and a neighbor selection method to optimize the initial solution of the algorithm.

In a study, Belgin *et al*. (2018) optimized the twolevel vehicle routing problem with simultaneous pickup and delivery (VRPSDP). In the designed two-level routing problem, the goods are sent from the factory to the middle warehouses in the first level and then sent to the customers from the middle warehouses in the second level. These scholars used the variable neighborhood descent (VND) metaheuristic algorithm to optimize this issue. In another study, Qin *et al*. (2019) optimized VRPSDP while considering environmental pollutions. These researchers took the amount of carbon produced by the travel into account as taxation, and their main goal was to reduce the routing costs. In addition, an adaptive genetic algorithm was used to optimize the problem, and the efficiency of the technique was assessed by solving various numerical examples. Azizi and Hu (2020) optimized simultaneous pickup and delivery in a multi-level and multi- commodity supply chain. In this problem, depot location and routing were optimized in a three-level supply chain. To this end, the CPLEX software was employed and the problem was validated in three tests.

The term robustness is discussed versus words such as uncertainty or lack of confidence, inaccuracy, and continuous variability. In other words, stability and related models are used to deal with uncertainty and similar words. However, there are other techniques such as stochastic programming and sensitivity analysis to deal with uncertainty. Historically speaking, optimization in an unreal situation initiated in the late 1950s, which rapidly advanced both in theory and algorithm fields. Numerous approaches have been used for optimization under uncertainty, including mathematical expectation minimization, minimized deviations from ideals, and minimizing maximum costs. Meanwhile, three main approaches of stochastic programming, fuzzy programming, and stochastic dynamic programming can be distinguished from others. According to Mulvey *et al*. (1995), management scientists have used sensitivity analysis techniques to adapt realworld data to the realm of mathematical programming. The main objective of these post-optimality studies is to discover the effect of worry on the model’s outputs. In addition, such assessments are of a reactive type or have reactivity trait. This type of study only evaluates the effect of data uncertainty on the model’s suggestions (outputs of the proposed model. Therefore, Mulvey *et al*. (1995) mentioned the need for a proactive method. As such, it is needed to design models that have less sensitivity to the data, compared to classic mathematical programming models. One of these models is probabilistic linear programming. However, dealing with this type of data through sensitivity analysis or stochastic programming formulation is associated with some issues. Robust optimization is another approach developed in the past few years to deal with data uncertainty. In this method, optimization is carried out when the worst happens, which might lead to a min-max objective function. In this method, near-optimum solutions was found that face high probability. In other words, the solution obtained is justified by slightly for bearing the (optimization of) objective function. However, in objective function coefficients under uncertainty, a solution can be find, which most likely is worse than real answers, by slightly skipping the value of the optimal objective function (Deb *et al*., 2002: Masoumnezhad *et al*., 2020).

Overall, input data is assumed equal to nominal values in definitive mathematical programming. This attitude does not take the effect of uncertainty on the quality and justification of the model into account. In fact, data that take values that are different from their nominal values may lead to the violation of a number of constraints, and the optimal solution may not be optimized for a long time or even be justified. This discussion raises the natural urge to design and provide solution methods that provide security against data uncertainty. In general, these methods are called “robust solutions” (Bertsimas and Sim, 2004a). Soyster performed the first research in this area, where a linear programming model was presented to produce a solution that fit all data related to the convex set. The mentioned model presented solutions that acted extremely conservatively in relation to the optimality of the nominal problem in order to ensure robustness. In this approach, the nominal problem is removed from the optimality largely in order to ensure a robust solution. In this model, any input data can receive any value of an interval (Ben-Tal and Nemirovski, 2000;Bertsimas and Sim, 2004b). Along with this research, Bertsimas and Sim (2004c) presented a different approach to control the level of conservatism. One of the advantages of the approach is developing a linear optimization model and its utility in discrete optimization models. In addition, it adjusts the conservatism level. In addition to the mentioned studies, which were based on the fluctuation of parameters in an interval, other studies have been carried out in the field of mathematical modeling. For instance, Mulvey *et al*. (1995) conducted a research based on the scenario concept, which will be explained below. Other research in the field of robustness includes fuzzy robust programming, in which it is assumed that all or some of the constraints or input data are fuzzy numbers. Golsefidi and Jokar (2020) used robust optimization for simultaneous decisionmaking about production, inventory, and routing. The mentioned research applied the vendor-managed inventory (VMI) approach to present a mathematical model under uncertainty. In this respect, the Bertsimas and Sim approach was exploited to deal with uncertainty. In another study, Kahfi *et al*. (2021) solved the multiperiodic and multi-commodity routing under uncertainty using a robust optimization approach. To this end, two mathematical models called LRP and LARP were presented, and a comparison of their results proved the more efficient performance of LARP in terms of timing and optimal solution. Furthermore, the comparison of the results of definitive and robust LARP models showed the validity of the robust optimization approach.

## 2. METHODOLOGY

The mathematical model of the research is presented in this section. First, the mathematical model was presented in definite terms and then the rewrite model was presented using the robust optimization approach. In general, a network of points was considered for the mathematical model, which was divided into two groups of customers and distribution centers. The goal was to send a set of goods from the distribution center to customers and collect another set of commodities from customers and deliver them to the distribution center. To this end, a set of vehicles and workers had to be assigned to different routes. The components of the mathematical model are introduced below.

The objective function of the presented model included five phrases; the first phrase was related to the estimation of moving costs between demand points. In fact, if vehicle v was sent from the i-th point to the j-th point, the decision variable x would be equal to one, and a cost equal to the cost of transportation from the i-th point to the j-th point would be added. Since the objective function was of minimization type, points were selected for relocation that ultimately imposed the least cost to the system based on other constraints and function phrases. The phrase of the objective function estimated the service earliness penalty. In terms of this penalty, it should be explained that customers want to receive their goods in a certain period with a low limit due to various reasons such as lack of storage space and a special ordering system. If their order is made ahead of time, the company should pay a payment for not adhering to the contracts’ principles. The third phrase of the objective function is in the form of costs of delivery tardiness and receiving the product. Therefore, the model aimed to deliver the products at the prespecified time as much as possible. The fourth phrase of the objective function estimated the wage of workers. According to this phrase, if a worker is assigned to a service vehicle, they will be paid according to the level of specialization and the length of service. The final phrase of the objective function estimated the cost of setting up the vehicles. In fact, some arrangements must be made on a vehicle whenever it is selected to serve customers, which can include refueling costs and driver services. Moreover, this sentence causes the model not to use the desired number of vehicles and to consider the optimal state mathematically. The mentioned phrases calculated the total costs of the system and attempted optimally to me*et al*l needs.

Since the mathematical programming was of accurate programming type, the problem faced several constraints, which are explained below. Constraints (2) ensured exclusive customer service, meaning that each customer will be served only once by one vehicle. Constraints (3) ensured the proper performance of the model regarding vehicle movement. In this regard, any vehicle that enters the customer’s location must move from there to other customers or the distribution center. In fact, the constraints guaranteed the return of vehicles to the distribution center. In addition, constraints (4) were indicative of limited use of vehicles, meaning that each vehicle could be used only once. Based on the presented problem, each vehicle carries the number of goods needed to serve its customers when leaving the distribution center. This value was calculated by constraints (5). Since it is possible to deliver and load goods simultaneously in any customer location in the designed model, the load of the vehicle when leaving any point (including the distribution center) must be equal to the difference between the number of goods delivered and the loaded goods, which was calculated by constraints (6-9). Moreover, loaded goods in each vehicle should not exceed its capacity, which was mentioned in constraints (10, 11). One of the most important issues in the routing problem is focusing on the lack of formation of sub-tour. In fact, the tour formed for the vehicle should occur continuously and its breaking is not allowed. This important issue was guaranteed by constraints (12), which allocated the desired number of each point in the tour and then allocated a larger number by moving to the next point, which automatically eliminates a break in the tours. One of the features of the presented model was having a time window. Therefore, it is needed to estimate the time to reach each point, which was calculated by constraints (13) while considering the path of each device and the time of reaching each point. In addition, constraints (14) and (15) dealt with assigning or lack of assigning earliness penalty to each point and the probability of allocation of tardiness penalty to customer demand, similar to constraints (16) and (17). Up to this point, all constraints related to the proper routing of vehicles to provide services to customers based on the possibility of simultaneous pickup and delivery and time windows are considered. However, workforce allocation should be estimated simultaneously, which is explained below. As mentioned before, workers are divided into several categories based on their level of expertise and ability, and a specific number of each type is available. However, the number of workforces is estimated in the presented study to optimize the process. In this regard, it should be noted that the number of workers is specified based on their abilities in work. It is clear that more workers will be needed if a large number of goods must be transferred, which was calculated by constraints (18). However, due to the nature and context of the model applications, the number does not simply mean the numerical number of the product and is considered as the value and weight of the product. In other words, products may have low physical weight and volume, but are of great value and require a large amount of work force to provide security and transfer (e.g., transfer of precious diamonds between two banks or transfer of professional criminals). However, it is clear that the number of workers should not exceed their available number, which was realized by constraints (19). Workers are also not available indefinitely. In fact, each employer is available for a certain period according to the contracts concluded, and the total hours of work force use must be less than their available hours. As such, constraints (20) satisfied this need. Constraints (21) and (22) focused on the feasibility of allocation of workers to vehicles. Nonetheless, the capacity of vehicles to transport workers is also limited, which makes the issue more real. This point was satisfied by constraints (23). Ultimately, constraints (24) mentioned the type of variables used in the model.

In the present research, some of the system’s costs, such as transportation costs between each point, cost of setting up the device, the time required to move from one point to another, the demand of each customer, the time required for loading and delivery of products and time of availability of workforces, were considered indefinitely and under the scenario. Similar to robust optimization explained above, a VRPSDP with time windows and worker allocation is presented below:

where it is considered:

Therefore, the objective function model of robust optimization of the transportation routing problem was, as follows:

However, the objective function (63) is non-linear due to its absolute value and the problem was turned into linear programming by introducing two new variables of *q _{s}* and

*p*. In addition, constraint ${q}_{s}-\text{}{p}_{s}=T{C}_{s}-{\displaystyle {\sum}_{s\text{'}}{P}_{s\text{'}}\text{}T{C}_{s\text{'}}}$ was added to the main model.

_{s}

s.t

Constraints of the main problem:

## 3. RESULTS AND DISCUSSION

In this section, two examples in different dimensions are proposed and analyzed in order to check and ensure the correctness of the proposed model. A company is assumed to intend to serve its inclusive customers and should use specialized forces for transportation and protection according to its offered type of goods. Therefore, the company is willing to use the model. Characteristics of the customers and the number of available equipment are presented in the table below. It is worth noting that in order to evaluate more examples; the company is examined in two different levels of access to the equipment so that the input data of the presented examples are randomly generated from a certain interval according to a monotonic function.

After solving the examples in Gams11 software, the obtained solutions were analyzed, as presented in Table 2.

In example 1, the model suffered no tardiness penalty considering the time that was pre-specified by customers and the time required for moving between demand points, and the time of access to workers. However, the model was subject to a small earliness penalty. Meanwhile, the lack of paying the penalty might increase other costs, which could make the problem non-optimized. In example 2, the tardiness level was relatively high, which led to an increase in penalty amount. At the same time, there was no difference between examples 1 and 2 in terms of travel costs, which could indicate proper routing in example 1. In addition, the final routes selected for coverage are shown in Table 3:

Only two vehicles were used in example 1, and the total travel duration was Max (48, 62) =62. In example 2, however, all vehicles were employed and the total travel duration was Max (57, 69, 74, 53) =74.

At this stage, the sensitivity of the model regarding changes in the problem’s parameters was analyzed to assess the solutions presented by the model. The value of *ω* was one of the effective factors for robust optimization problem solving since it was directly related to the justified or unjustified nature of the issue. Therefore, the presented examples were implemented per various amounts of *ω* , the results of which are presented below.

As observed, the value of the objective function changed drastically with a change in the value of *ω* . In fact, when *ω* =0 , the value of unmet ${\delta}_{ijs}^{v}$ increased, which means that no demand was satisfied. The unmet value of ${\delta}_{ijs}^{v}$ decreased gradually with an increase in the *ω* value, which increased the cost of system. In example 1, the value of ${\delta}_{ijs}^{v}$ reached zero and the model was justified when *ω* = 60 . In example 2, the value of *ω* was equal to 70. The comparison chart below shows the growth rate of the system cost per different amount of *ω* .

As observed in Figure 1, the increased value of *ω* led to an uptrend in total costs. Since the model was justified at *ω* = 60 in example 1 and at *ω* = 70 in example 2, the higher *ω* values had no longer an impact and similar results were obtained.

## 4. CONCLUSION

In the present research, a mathematical model for vehicle routing by considering real-life situations in transportation was developed, which included simultaneous pickup and delivery at customer points, time windows, and worker allocation. It is notable that simultaneous considering of worker allocation with routing problem can be extremely beneficial for organizations such as banks, prisons, and all organizations that deal with goods transportation by specialized workforces, thereby extremely decreasing the system’s costs. In order to increase the flexibility of the model and have real-world situations, some of the problem’s parameters such as time required to travel between points, time of access to human resources, customer demands, and vehicle setting-up costs were considered under uncertainty. The methods of dealing with problems under conditions of uncertainty are extremely wide and can be changed according to the type of problem and the type of uncertainties. However, one of the most effective parts of these methods is to define the desired parameters under different scenarios with their unique probability of occurrence. This type of parameter uncertainty makes the problem unjustified and nonoptimal. Therefore, an efficient approach must be taken to deal with it. In this regard, the most appropriate method was introduced by Mulvey *et al*. (1995) named the robust optimization approach. The mentioned technique was used in the present study, and two examples with different levels were solved and assessed and the results of problem sensitivity analysis were presented to ensure the accuracy of the model.