Reagents and swab tests during the COVID-19 Pandemic: An optimized supply chain management with UAVs

Colajanni, Gabriella; Daniele, Patrizia; Sciacca, Daniele Article Reagents and swab tests during the COVID-19 Pandemic: An optimized supply chain management with UAVs Operations Research Perspectives Provided in Cooperation with: Elsevier Suggested Citation: Colajanni, Gabriella; Daniele, Patrizia; Sciacca, Daniele (2022) : Reagents and swab tests during the COVID-19 Pandemic: An optimized supply chain management with UAVs, Operations Research Perspectives, ISSN 2214-7160, Elsevier, Amsterdam, Vol. 9, pp. 1-15, https://doi.org/10.1016/j.orp.2022.100257 This Version is available at: https://hdl.handle.net/10419/325744 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/bync-nd/4.0/). Contents lists available at ScienceDirect Operations Research Perspectives journal homepage: www.elsevier.com/locate/orp Reagents and swab tests during the COVID-19 Pandemic: An optimized supply chain management with UAVs Gabriella Colajanni∗, Patrizia Daniele, Daniele Sciacca Department of Mathematics and Computer Science of Catania, Viale Andrea Doria, 6 - 95125 Catania, Italy ARTICLE INFO Keywords: COVID-19 pandemic Swab tests supply chain optimization Variational inequality theory UAVs and 5G technology ABSTRACT In this paper, we develop a supply chain optimization model for the preparation, provision, transportation, and execution of swab tests during COVID-19 pandemic. The proposed approach is based on a multi-tiered network consisting of manufacturing companies of reagents, processing laboratories (where the swab kits are prepared and some swab tests are analyzed), landing stations for UAVs and test centers. As innovations in the supply chain, the sharing of reagents between processing laboratories and the use of UAVs, using 5G technology, are contemplated in the management of the COVID-19 Pandemic. To obtain the optimal solutions of the underlying optimization problem, we provide a variational formulation problem for which results of existence and uniqueness will be provided. Finally, some numerical simulations are examined to validate the effectiveness of our approach. 1. Introduction The COVID-19 pandemic, which broke out in China at the end of 2019, has certainly revolutionized our habits. The sudden and uncontrollable spread of infections has forced the entire population to periods of quarantines and restrictions and normal daily activities were closely monitored in order to ensure a slowdown in the spread of infections. Face-to-face education was suspended, giving rise to distance learning; work activities, when possible, was converted into work from home activities. On March 11, 2020, after assessing the levels of spread and severity of the SARS-CoV-2 infection, the World Health Organization (WHO) declared that the COVID-19 outbreak recorded over the past months can be characterized as a pandemic (see [1]), but, despite that, it can still be controlled. On 30 January 2020, following a second meeting of the Emergency Committee, the WHO Director-General had already declared the international SARS-CoV-2 outbreak a Public Health Emergency of International Concern (PHEIC), as defined in the International Health Regulations (IHR, 2005), [2]. Authors in [3] determined the criteria to be used in the decision of admission of COVID-19 patients to the intensive care units, while many researchers studied the future trend of the pandemic (see, for example, [4], where a hybrid reinforcement learning based algorithm is designed or [5], where a regression-based Robust Optimization (RO) approach to efficiently predict the number of patients with confirmed infection caused by the COVID-19 is developed). Despite the initial uncertainties, the use of masks and protective devices has become one of the most important means in countering ∗Corresponding author. E-mail addresses: [email protected] (G. Colajanni), [email protected] (P. Daniele), [email protected] (D. Sciacca). the spread of the virus [6]. The mandatory use of Personal Protective Equipment (PPE) was associated with an integrated surveillance system for the spread of COVID-19 [7], which continuously and systematically collected, compared and analyzed information on all cases of SARS- CoV-2 infection, confirmed by molecular diagnosis performed in public and private health facilities and in diagnostic and analysis laboratories (also the Computer Tomography scan images can be suitable for COVID-19 detection, see [8]). It was, and still is, a necessary and useful observation tool both for informing citizens about the impact and evolution of the pandemic and for offering decision support for public health responses from health authorities. This surveillance system is based on the election of a sample of the respiratory tract. Generally, this sample is collected from the upper respiratory tract through a rhino-pharyngeal and oropharyngeal swab. According to the American Centers of Disease Control and Prevention (CDC), anterior nasal swab and turbinate nasal swab med can also be used. From the sample of biological material, transported to the laboratory at a controlled temperature, the extraction and purification of RNA is performed for the subsequent search for viral RNA using a rapid molecular method called Reverse Real-Time PCR (rRT-PCR) [9]. To perform the diagnostic tests, it is necessary to use specific reagents, which, in some phases, especially in the phases of peaks of infections, are difficult to produce, to find and store (see [10,11]). Moreover, also the production and provision of swab kits and swab tests encountered delays due to the huge demand. The difficulties https://doi.org/10.1016/j.orp.2022.100257 Received 19 August 2022; Received in revised form 17 October 2022; Accepted 18 October 2022 Operations Research Perspectives 9 (2022) 100257 2 G. Colajanni et al. in finding PPE, swab kits and tests and reagents, due to the large and not expected number of requests, dictate the need to provide suppliers and companies involved in the supply chain with optimization models, based on mathematical models, in order to avoid delays and/or shortages in supply (see [12] for a mathematical model devoted to design a sustainable mask Closed-Loop Supply Chain Network (CLSCN) during the COVID-19 outbreak in which the locational, supply, production, distribution, collection, quarantine, recycling, reuse, and disposal decisions are taken into account). For these reasons, in this paper we provide a network-based supply chain optimization model for the execution of swab tests and the provision of reagent needed to execute a swab test (see [13] for a supply chain network with inclusion of labor, [14] for a reverse supply chain which consists of a pharmaceutical manufacturer and a retailer, [15] for a stochastic optimization of supply chain operations under ripple effect caused by pandemic and [16] for an integer programming formulation to maximize the number of swab tests during a pandemic). We assume that the manufacturing companies produce reagents and sell them to processing laboratories, where the swab kits are prepared, and some swab tests are performed on people and analyzed. As innovation, we suppose that some processing laboratories can self-produce some kinds of reagents and send them to other laboratories, to guarantee a fluidity in the management of a large number of requests for the execution of swabs. An absolute novelty that for the first time is taken into consideration in an optimization model that concerns the provision and supply chain of reagents, and the execution of swabs is to consider, as a means of transport, in addition to the usual ones (cars, trucks, ...), the Unmanned Aerial Vehicles (UAVs), managed and orchestrated through innovative communications infrastructures, such as 5G Networks. UAVs, recently, have been proposed as an alternative in order to overcome the limitations and shortcomings of current practices (see [17] for an extensive and comprehensive review of papers dealing with the use of UAVs in engineering transportation and [18] for the use of fleets of Multi-access Edge Computing UAVs in 5G environments). The great potential of their use in transportation and delivering has been observed by the engineering community, although today some limitations are present. In [19], the author emphasizes the multiple possibilities of using UAVs in improving and facilitating transport problems, in addition to the possibility to use them in the planning, designing and monitoring of transport and other infrastructure. Particularly, their use could be of fundamental importance in transportation of goods, in medical supply, in civil and transportation engineering and in traffic engineering problems. There are many examples of UAVs used in the delivering of medical items. For instance, in 2019, the government of Malawi set up drone corridor networks to serve the East African country of nearly 20 million people that has an inadequate health infrastructure. Starting with childhood vaccines for malaria, TB and rotavirus, the drone network quickly expanded to warehouse or ‘beehive’ pharmaceuticals for delivery on-demand. Similarly, thanks to their ability to fly over the country’s dense rainforest, the Democratic Republic of the Congo decided in 2021 to active an Ebola vaccine project with the aim of vaccinating people in remote locations using bi-directional medical delivery drones, which drop-off the Ebola vaccine and then collect biometric data from vaccines, together with other health reports. Finally, since 2011, Zipline company delivers whole blood, platelets, frozen plasma, and cryoprecipitate along with medical products, including vaccines, infusions, and common medical commodities in Nigeria, Cote d’Ivoire and Kenya. Despite the obvious benefits and advantages of using UAVs in medical items delivering, however, it is clear that we must take into account the engineering limitations of UAVs, first of all the need to recharge their batteries. Therefore, in this framework, we consider a network with five levels in which, at the highest level of the network, manufacturing companies produce reagents and sell them to processing laboratories, placed in the second level of the network. Processing laboratories purchase reagents from manufacturing companies, prepare the swab kits, analyze/process the swab tests and can execute swab tests directly on people. We also take into account the possibility for processing laboratories to selfproduce some kinds of reagent and to exchange reagents with each other. The third level of the network consists of the landing stations that can be used (or not) by drones for the delivery of the swab kits to the test centers, that constitutes the fourth level of the network. Here, the test centers receive the swab kits from the processing laboratories (through the traditional transportation or via UAVs), execute the swab tests directly on people, representing the last level of the network, and analyze them or send them to the processing laboratories for the tests. We emphasize that the use of UAVs as means of transport is supported by a 5G infrastructure through which all requests (for both kits to be sent and swabs to be collected), the geographical position of the people, the type of swab requested, the type of means of transport to be sent, the results of the swabs analyzed etc, are transmitted. It is known, in fact, that the recent development of such an innovative technology allows a secure and fast transmission of information which is therefore shared in real-time. On the other hand, assuming that geographically difficult to reach areas can be also considered, the combination of UAVs-5G infrastructure (which do not require physical connections with the ground) is considerably advantageous (see [20]). In a system optimization point of view, this mathematical model aims to determine the optimal flow of reagents, swab kits and swab tests in order to maximize the quantity of executed swab tests and, simultaneously, to maximize the profits of laboratories. Moreover, a multitude of linear and non-linear constraints, in terms of capacity, execution times and conservation of flows, must be satisfied. To the obtained constrained optimization problem, we associate a variational formulation, for which results in terms of existence and uniqueness of the solution are provided. In addition, the duality theory is examined to guarantee an alternative variational formulation more tractable in computational terms. The rest of the paper is organized as follows. In Section 2, we describe in a more detailed way the multi-tiered network on which our optimization model is based, that we describe in Section 3. Section 4 is devoted to the variational formulation of the model and, finally, in Section 7we provide three numerical simulations to validate the effectiveness and the potentiality of our model. Particularly, we show how the sharing of reagents among laboratories and the use of UAVs through 5G technology as means of transport is of fundamental importance (and suitable in terms of objective function) in the supply chain of medical items, and, specifically, of reagents, kits and swab tests. We also show how our model is able to capture the most important objective, that is the maximization of the swabs analyzed, while maximizing the profit at the same time. 2. Network description In this section we describe the topology of the network on which the model proposed in the following section is based (see [21] for another optimization model on the management of reagents and swabs, based on a multi-level network). As already mentioned above, it is well known that a large number of people require to undergo COVID-19 swab tests for various reasons (such as to know if they are positive following contact with a positive person or before a particular event, for work reasons, to be able to travel, and so on). There exist different types of tests that can be used for COVID-19 detection, including diagnostic tests that look for active coronavirus infection in mucus or saliva using a swab (which is inserted into the throat and the nose). Some swab-tests, the antigen tests, are generally quick and cheap and for this reason are called rapid tests; they look for a piece of the coating of the virus. Other types of swab-tests, the molecular tests, are often called Real Time-Polymerase Chain Reaction Operations Research Perspectives 9 (2022) 100257 3 G. Colajanni et al. (RT-PCR) tests for the lab technique used to detect the nucleic acid (such as RNA) belonging to the coronavirus, and they are considered the most sensitive and highly accurate swab-tests. Since there exist different types of swab-tests, in this paper we consider 𝑆types of swab-tests and denote by 𝑠the generic one. Furthermore, preparing a swab-kit, running a swab-test and reading its results requires particular equipment but especially one or more reagents, that are specific chemicals without which it is not possible to obtain the necessary results and are often in short supply because of the high number of requests. Moreover, each type of swab-test and kit requires different quantities of each type of reagent. Therefore, we denote by 1,…, 𝑟, …, 𝑅 the different types of reagents. People can be grouped into 𝐺groups (1,…, 𝑔, …, 𝐺), distinguished both by geographical area and by category: minors or adults (for which there are different prices established by the Government or sales policies), people who can swab for a fee or for free of charge (for the latter it is the National Health Service that pays, so the laboratory or the test center that execute the swab test get a different contribution than that received when a person pays directly), people who move or stay ‘‘at home’’ (for the latter the price is increased by the travel costs, also based on the geographical area of residence). Note that the model we present in the next section is generic, hence, different and/or other categories of people can be assumed. People can get any type of swab-test by going (or contacting, for home execution) to one of the 𝐻test centers, such as hospitals, pharmacies, analysis laboratories, local health centers, dedicated hubs, etc. In the generic test center ℎ, after swabbing people, the swab tests can be analyzed there or sent to a processing laboratory, depending on the type of swab test to be analyzed and the available equipment. The processing laboratories are specific laboratories, accredited by the National Health System, in which all types of swab tests can be analyzed. We denote such processing laboratories with 1,…, 𝑝, …, 𝑃 . Furthermore, we assume that these laboratories deal not only with analyzing the swab tests received from the test centers (or executed on people who decide to go directly to these processing laboratories), but also with the preparation of the swab kits to be sent to the test centers. Moreover, each processing laboratory buys the reagents from the manufacturing companies (1,…, 𝑎, …, 𝐴) or could auto-produce one or more types of reagents. In this paper we also assume that each processing laboratory could buy or sell reagents from or to other laboratories. One of the innovative aspects of this paper, as mentioned in the previous Section, concerns the possibility to use some UAVs as a means of transporting swab tests to be performed or analyzed. Therefore, in Fig. 1 the network topology is depicted, where continuous links and dashed links indicate traditional means of transport and through UAVs, respectively. Moreover, we suppose that the test center could be reached using the traditional means of transport or through UAVs. The UAVs can directly reach the test centers or stop at the landing stations 1,…, 𝑙, …, 𝐿. We now explain the network in detail. It consists of five levels. The highest level of the network is composed by the manufacturing companies (1,…, 𝑎, …, 𝐴) which produce the reagents (1,…, 𝑟, …, 𝑅). The reagents are purchased by the processing laboratories (1,…, 𝑝, …, 𝑃 ), which could also self-produce some kinds of reagent and which are mainly concerned with preparing the swab kits and analyzing/ processing the swab tests (if the test centers do not do it). In addition, processing laboratories can also swab the test directly on people. Therefore, the second layer of the network consists of the combination between the processing laboratory and the swab tests. Note that the links between the processing laboratories refer to the purchase and sale of reagents. The intermediate level of the network consists of the landing stations (1,…, 𝑙, …, 𝐿) that can be used (or not) by UAVs to transport the swab kits to the test centers. The combination of test centers and swab tests constitutes the fourth level of the network. Each test center receives the swab kits from the processing labs (through the traditional transportation or via UAVs), gets the swab tests (on people) and then analyzes them or sends them to the processing laboratories for the test analysis. The lowest level of the network consists of the groups of people (as previously described) requiring the swab tests. We specify that: •the transaction of reagents between companies and processing laboratories takes place only through the traditional method (black continuous links), while between laboratories the transaction can take place both through the traditional method and through UAVs (black continuous links and blue dashed links, respectively); •the transport of swab kits and swab tests between the second (the processing laboratories) and the third layer (landing stations) takes place only via UAVs (blue dashed links); •the transport of swab kits and swab tests between the third (landing stations) and the fourth layer (test centers) takes place only via the traditional transportation means (black continuous links); •the transport of swab kits and swab tests between the second (the processing laboratories) and the fourth layer (test centers) can take place both through the traditional method and through UAVs (for simplicity we denoted them with gray dashed links, but each of them represents a pair of parallel links to indicate both modes of transport); moreover, note that these links connect nodes of the same swab-type; •people could reach (or be reached by specialized personnel from) the test centers or directly the processing laboratories (black continuous links). We underline that the main extension, respect to the paper in [21] (where a multi-period resource allocation model is analyzed, with the objective of simultaneously maximize the quantity of all performed swabs and minimize the time required to obtain the swabs result), consists in using some UAVs as a means of transporting. Therefore, also the intermediate level of the network, which consists of the landing stations (that can be used, or not, by UAVs to transport the swab kits to the test centers) represents a crucial innovation of this paper. Furthermore, in this paper we assume that the processing laboratories deal not only with analyzing the swab tests received from the test centers (or directly executed on people), but also with the preparation of the swab kits to be sent to the test centers. On the contrary, in [21] the swab kits are not taken into account. Moreover, here we are supposing that each laboratory could sell or buy reagents (not swab tests) to or from other laboratories and that the test centers are also able to analyze some types of swab test. Finally, the models deal with different aspects: both of them aim to determine the optimal flow of reagents and swab tests in order to maximize the quantity of executed swab tests, but in this paper the optimal flow of swab kits are also established and the laboratories’ profits are maximized. 3. Optimization model In this section we describe the optimization model which allows us to determine the optimal flow of reagents, swab kits and swab tests in order to maximize the quantity of executed swab tests while maximizing the labs’ profits and ensuring that capacity, time and processing capacities constraints are met. We firstly introduce the used notation (variables, parameters, cost and time functions) and then we propose a mathematical formulation. Operations Research Perspectives 9 (2022) 100257 4 G. Colajanni et al. Fig. 1. Network Topology. Table 1 Variables of the model. Variable Description 𝑦𝑎𝑟𝑝 Quantity of reagent 𝑟purchased by the processing laboratory 𝑝 from the company 𝑎 𝑦(𝐴𝑅) 𝑝𝑟 Quantity of reagent 𝑟self-produced by the processing laboratory 𝑝 𝑦(𝑆𝑅) 𝑝 𝑝𝑟𝑚 Quantity of reagent 𝑟sent by the processing laboratory 𝑝to 𝑝 in mode 𝑚, where 𝑚= 1 means that the traditional transportation is chosen, while 𝑚= 2 means that the transport by UAV is used 𝑥(𝐿𝑆) 𝑝𝑠𝑙 Quantity of 𝑠-type swab tests sent from the processing laboratory 𝑝to the landing station 𝑙(via UAV) 𝑥(𝑇 𝐶) 𝑝𝑠ℎ𝑚 Quantity of 𝑠-type swab tests sent from the processing laboratory 𝑝to the test center ℎthrough mode 𝑚 𝑥(𝑃) 𝑝𝑠𝑔 Quantity of 𝑠-type swab tests requested by group 𝑔of people and processed directly by the processing laboratory 𝑝 𝑧𝑝𝑠𝑙ℎ Quantity of 𝑠-type swab tests (coming from the processing laboratory 𝑝) sent from the landing station 𝑙to the test center ℎ (through the traditional transportation mode) 𝑤ℎ𝑠𝑔 Quantity of 𝑠-type swab tests requested by group 𝑔of people at the test center ℎ 3.1. Notation The main purpose of the model is to maximize the number of swab tests obtained and analyzed, taking into account that particular reagents are required for each swab. The variables of the model are reported in Table 1, in which the used notation and the description of each variable are shown. The mathematical model includes two types of functions: the cost functions, denoted by 𝑐(⋅), and the time functions, denoted by 𝑡(⋅). All the functions are defined as follows. Let: •𝑐(𝐵𝑅) 𝑎𝑟𝑝 be the cost for the processing laboratory 𝑝to buy reagent 𝑟 from company 𝑎and we assume that such a cost is a function of the amount of purchased reagent: 𝑐(𝐵𝑅) 𝑎𝑟𝑝 =𝑐(𝐵𝑅) 𝑎𝑟𝑝 (𝑦𝑎𝑟𝑝),∀𝑎= 1,…, 𝐴, ∀𝑟= 1,…, 𝑅, ∀𝑝= 1,…, 𝑃 ; •𝑐(𝐴𝑅) 𝑝𝑟 be the cost that the processing laboratory 𝑝has to pay to auto-produce reagent 𝑟and we assume that such a cost is a function of the amount of self-produced reagent: 𝑐(𝐴𝑅) 𝑝𝑟 = 𝑐(𝐴𝑅) 𝑝𝑟 (𝑦(𝐴𝑅) 𝑝𝑟 ),∀𝑝= 1,…, 𝑃 , ∀𝑟= 1,…, 𝑅; •𝑐(𝑆𝑅) 𝑝𝑝𝑟𝑚 be the cost that the processing laboratory 𝑝has to pay to receive reagent 𝑟by the processing laboratory 𝑝 in mode 𝑚and we assume that such a cost is a function of the amount of transported reagent: 𝑐(𝑆𝑅) 𝑝𝑝𝑟𝑚 =𝑐(𝑆𝑅) 𝑝𝑝𝑟𝑚 (𝑦(𝑆𝑅) 𝑝𝑝𝑟𝑚),∀𝑝, 𝑝 = 1,…, 𝑃 , ∀𝑟= 1,…, 𝑅, ∀𝑚= 1,2; •𝑐(𝐾) 𝑝𝑠 be the cost to prepare an 𝑠-type swab kit in laboratory 𝑝(to send at the test centers, directly or through the landing stations, or to get it on a person) and we assume that such a cost is a function of the amount of prepared kits: 𝑐(𝐾) 𝑝𝑠 = 𝑐(𝐾) 𝑝𝑠 (∑𝐿 𝑙=1 𝑥(𝐿𝑆) 𝑝𝑠𝑙 +∑2 𝑚=1 ∑𝐻 ℎ=1 𝑥(𝑇 𝐶) 𝑝𝑠ℎ𝑚 +∑𝐺 𝑔=1 𝑥(𝑃) 𝑝𝑠𝑔),∀𝑝= 1,…, 𝑃 , ∀𝑠= 1,…, 𝑆; •𝑐(𝑃 𝑟) 𝑝𝑠 be the cost to process the swab test 𝑠in laboratory 𝑝and we assume that such a cost is a function of the amount of swab tests processed by 𝑝(we will define this quantity later); •𝑐(𝐿𝑆) 𝑝𝑠𝑙 be the cost to send a UAV from 𝑝to the land station 𝑙, transporting the 𝑠-type swab kits and we assume that such a cost is a function of the amount of transported swab tests: 𝑐(𝐿𝑆) 𝑝𝑠𝑙 = 𝑐(𝐿𝑆) 𝑝𝑠𝑙 (𝑥(𝐿𝑆) 𝑝𝑠𝑙 ),∀𝑝= 1,…, 𝑃 , ∀𝑠= 1,…, 𝑆, ∀𝑙= 1,…, 𝐿; •𝑐(𝑇 𝐶) 𝑝𝑠ℎ𝑚 be the cost to transport the 𝑠-type swab kits from 𝑝to the test center ℎin mode 𝑚and we assume that such a cost is a function of the amount of transported swab tests: 𝑐(𝑇 𝐶) 𝑝𝑠ℎ𝑚 =𝑐(𝑇 𝐶) 𝑝𝑠ℎ𝑚(𝑥(𝑇 𝐶) 𝑝𝑠ℎ𝑚),∀𝑝= 1,…, 𝑃 , ∀𝑠= 1,…, 𝑆, ∀ℎ= 1,…, 𝐻, ∀𝑚= 1,2; •𝑐(𝐿𝐻) 𝑙ℎ𝑠 be the cost to transport the 𝑠-type swab kits from 𝑙to ℎ and we assume that such a cost is a function of the amount of transported swab tests: 𝑐(𝐿𝐻) 𝑙ℎ𝑠 =𝑐(𝐿𝐻) 𝑙ℎ𝑠 (𝑧𝑝𝑠𝑙ℎ),∀𝑙= 1,…, 𝐿, ∀ℎ= 1,…, 𝐻, ∀𝑠= 1,…, 𝑆; •𝑡(𝑆𝑅) 𝑝 𝑝𝑚 be the time to transport reagents from the processing laboratory 𝑝to 𝑝 and we suppose that such a time is a function of the total amount of reagents transported between 𝑝and 𝑝: 𝑡(𝑆𝑅) 𝑝 𝑝𝑚 =𝑡(𝑆𝑅) 𝑝 𝑝𝑚 (∑𝑅 𝑟=1 𝑦(𝑆𝑅) 𝑝 𝑝𝑟𝑚),∀𝑝, 𝑝 = 1,…, 𝑃 , ∀𝑚= 1,2; •𝑡(𝑇 𝐶) 𝑝ℎ𝑚 be the time to transport swab kits from the processing laboratory 𝑝to the test center ℎin mode 𝑚and we suppose that such a time is a function of the total amount of swab kits transported between 𝑝and ℎ:𝑡(𝑇 𝐶) 𝑝ℎ𝑚 =𝑡(𝑇 𝐶) 𝑝ℎ𝑚 (∑𝑆 𝑠=1 𝑥(𝑇 𝐶) 𝑝𝑠ℎ𝑚),∀𝑝= 1,…, 𝑃 , ∀ℎ= 1,…, 𝐻, ∀𝑚= 1,2; •𝑡(𝐿𝑆) 𝑝𝑙 be the time to transport swab kits from the processing laboratory 𝑝to the landing station 𝑙(via UAV) and we suppose that such a time is a function of the total amount of swab kits transported between 𝑝and 𝑙:𝑡(𝐿𝑆) 𝑝𝑙 =𝑡(𝐿𝑆) 𝑝𝑙 (∑𝑆 𝑠=1 𝑥(𝐿𝑆) 𝑝𝑠𝑙 ),∀𝑝= 1,…, 𝑃 , ∀𝑙= 1,…, 𝐿; •𝑡𝑙ℎ be the time to transport swab kits from the landing station 𝑙to the test center ℎ(through the traditional mode) and we suppose that such a time is a function of the total amount of swab kits transported between 𝑙and ℎ:𝑡𝑙ℎ =𝑡𝑙ℎ (∑𝑃 𝑝=1 ∑𝑆 𝑠=1 𝑧𝑝𝑠𝑙ℎ),∀𝑙= 1,…, 𝐿, ∀ℎ= 1,…, 𝐻; Operations Research Perspectives 9 (2022) 100257 5 G. Colajanni et al. •𝑡(𝑃) 𝑝𝑔 be the time to transport swab kits from the processing laboratory 𝑝to the group of people 𝑔and we suppose that such a time is a function of the total amount of swab kits transported between 𝑝and 𝑔:𝑡(𝑃) 𝑝𝑔 =𝑡(𝑃) 𝑝𝑔 (∑𝑆 𝑠=1 𝑥(𝑃) 𝑝𝑠𝑔),∀𝑝= 1,…, 𝑃 , ∀𝑔= 1,…, 𝐺. We now describe all the parameters used for the formulation. Let: •𝜌(𝑃) 𝑝𝑠𝑔 be the revenue obtained by the processing laboratory 𝑝from the group 𝑔of people for the 𝑠-type swab test; it includes the kit preparation, swabbing the patient, the possible transport (which depends on the category of 𝑔) and the analysis of the swab test; •𝜌(𝑇 𝐶) 𝑝𝑠ℎ be the revenue obtained by the processing laboratory 𝑝from the test center ℎfor the 𝑠-type swab test and it refers only to the prepared kit; •𝜌(𝑃 𝑟𝑇 𝐶) 𝑝𝑠ℎ be the revenue obtained by the processing laboratory 𝑝 from the test center ℎfor the analysis of an 𝑠-type swab test (we underline that the processing laboratory 𝑝receives such a revenue only if the test center ℎis not able to analyze the 𝑠-type swab test and sends it to 𝑝); •𝜌(𝐿𝐻) 𝑝𝑠𝑙ℎ be the revenue obtained by the processing laboratory 𝑝for a prepared 𝑠-type swab kit sent to the landing station 𝑙for the test center ℎ(observe that 𝜌(𝑇 𝐶) 𝑝𝑠ℎ and 𝜌(𝐿𝐻) 𝑝𝑠𝑙ℎ are analogous but we are assuming that they could be different for sales strategies; clearly nothing prevents them from being equal); •𝜌(𝑃 𝑟𝐿𝐻) 𝑝𝑠𝑙ℎ be the revenue analogous of 𝜌(𝑃 𝑟𝑇 𝐶) 𝑝𝑠ℎ obtained using the landing station 𝑙; •𝐶𝑎𝑟 be the overall amount of reagent 𝑟produced by the company 𝑎; •𝐶𝑝𝑟 be the maximum quantity of reagent 𝑟that the processing laboratory 𝑝is able to self-produce; •𝑓(𝐾) 𝑠𝑟 be the quantity of reagent 𝑟needed to prepare an 𝑠-type swab test kit; •𝑓(𝑃 𝑟) 𝑠𝑟 be the quantity of reagent 𝑟needed to analyze/process an 𝑠-type swab test; •𝐷𝑠𝑔 be the quantity of 𝑠-type swab tests required by the group 𝑔 of people; •𝑏(𝐾) 𝑝𝑠 be the processing resources required by the processing laboratory 𝑝to prepare an 𝑠-type swab test; •𝑏(𝑃 𝑟) 𝑝𝑠 be the processing resources required by the processing laboratory 𝑝to analyze an 𝑠-type swab test; •𝑏(𝐸) 𝑝𝑠 be the processing resources required by the processing laboratory 𝑝to get an 𝑠-type swab test; •𝑏(𝐸) ℎ𝑠 be the processing resources required by the test center ℎto get an 𝑠-type swab test; •𝑏(𝑃 𝑟) ℎ𝑠 be the processing resources required by the test center ℎto analyze an 𝑠-type swab test; •𝐵𝑝be the maximum processing capacity available for the processing laboratory 𝑝, it represents the productivity efficiency and depends on machinery, on time, but above all on the workforce available at 𝑝; •𝐵ℎbe the maximum processing capacity available for the test center ℎ; •𝐵𝑝𝑠 be the maximum quantity of 𝑠-type swab tests that the processing laboratory 𝑝is able to analyze (note that such a quantity could be zero if 𝑝cannot analyze 𝑠-type swab tests); •𝐵ℎ𝑠 be the maximum quantity of 𝑠-type swab tests that the test center ℎis able to analyze (note that such a quantity could be zero if ℎcannot analyze 𝑠-type swab tests); •𝛩(𝑅) 𝑝 𝑝𝑚 be the maximum reagents transportation capacity in the link between the processing laboratories 𝑝and 𝑝 in mode 𝑚(this is the maximum capacity of the used transport mode); •𝛩(𝑆) 𝑝ℎ𝑚 be the maximum swab tests transportation capacity in the link between the processing laboratory 𝑝and the test center ℎin mode 𝑚; •𝛩(𝑆𝐿) 𝑝𝑙 be the maximum swab tests transportation capacity in the link between the processing laboratory 𝑝and the landing station 𝑙(this is the UAV’s maximum capacity); Table 2 Parameters of the model. Parameters Description 𝜌(𝑃) 𝑝𝑠𝑔 ,𝜌(𝑇 𝐶) 𝑝𝑠ℎ ,𝜌(𝑃 𝑟𝑇 𝐶) 𝑝𝑠ℎ , 𝜌(𝐿𝐻) 𝑝𝑠𝑙ℎ ,𝜌(𝑃 𝑟𝐿𝐻) 𝑝𝑠𝑙ℎ Revenues obtained by the processing laboratory 𝑝for the kit preparation and/or test analysis 𝐶𝑎𝑟,𝐶𝑝𝑟 Amount of reagent 𝑟produced 𝑓(𝐾) 𝑠𝑟 ,𝑓(𝑃 𝑟) 𝑠𝑟 Quantity of reagent 𝑟needed to prepare a kit or analyze a test 𝐷𝑠𝑔 Quantity of 𝑠-type swab tests required by the group 𝑔 of people 𝑏(𝐾) 𝑝𝑠 ,𝑏(𝑃 𝑟) 𝑝𝑠 ,𝑏(𝐸) 𝑝𝑠 , 𝑏(𝐸) ℎ𝑠 ,𝑏(𝑃 𝑟) ℎ𝑠 Processing resources required to prepare a kit or analyze a test 𝐵𝑝,𝐵ℎ,𝐵𝑝𝑠,𝐵ℎ𝑠 Maximum processing capacity available at 𝑝or ℎand maximum quantity of 𝑠-type swab tests that 𝑝or ℎis able to analyze 𝛩(𝑅) 𝑝 𝑝𝑚,𝛩(𝑆) 𝑝ℎ𝑚,𝛩(𝑆𝐿) 𝑝𝑙 Maximum reagents or swab tests transportation capacity 𝑇𝑟,𝑇𝑠Maximum allowed reagent or swab tests transport time 𝑑ℎ𝑠𝑚 The roundtrip parameter •𝑇𝑟be the maximum allowed reagent transport time, so that the reagent 𝑟does not deteriorate; •𝑇𝑠be the maximum allowed swab kit and test transport time, so that the 𝑠-type swab test or kit does not deteriorate; •𝑑ℎ𝑠𝑚 be the ‘‘roundtrip’’ parameter. Such a parameter is equal to 1if the test center ℎis able to analyze the 𝑠-type swab tests, and, hence, the swab kits do not come back to the processing laboratory. On the contrary, if the test center ℎcannot analyze the 𝑠-type swab tests, the ‘‘roundtrip’’ parameter is equal to 2because the swab kits have to come back to the processing laboratory to be analyzed. Note that the ‘‘roundtrip’’ parameter 𝑑ℎ𝑠𝑚 also has the index 𝑚because in some cases the transportation means have to return to the laboratory. Indeed, in this paper we will assume that the drone always returns, however, the traditional vehicle (equipped cars, vans, helicopters, etc.) returns only if some swabs have to go back (to be processed in the laboratory). For further clarity, all the parameters of the model are summarized in Table 2. 3.2. Mathematical formulation The main aim of our model is to maximize the amount of the analyzed swab tests because, as previously discussed, it is very important to know who the positive (infected) people are, to prevent the contagion from increasing more and more. At the same time, the model is able to determine the optimal distribution flows that allow us to maximize the profit of each processing laboratory, that is maximize the revenues while minimizing the sum of costs. Therefore, since the model involves more than one objective function which must be maximized, we have a multi-objective programming problem and we use the weighted sum method that combines and converts all the objective functions into a single-objective composite function. Let 𝛼,𝛽∈ [0,1] two weights, then the problem becomes: max {𝛼 𝑃 ∑ 𝑝=1 𝑆 ∑ 𝑠=1 [𝐿 ∑ 𝑙=1 𝑥(𝐿𝑆) 𝑝𝑠𝑙 + 2 ∑ 𝑚=1 𝐻 ∑ ℎ=1 𝑥(𝑇 𝐶) 𝑝𝑠ℎ𝑚 + 𝐺 ∑ 𝑔=1 𝑥(𝑃) 𝑝𝑠𝑔]+ 𝛽 𝑃 ∑ 𝑝=1 [𝑆 ∑ 𝑠=1 [𝐺 ∑ 𝑔=1 𝜌(𝑃) 𝑝𝑠𝑔𝑥(𝑃) 𝑝𝑠𝑔 + 𝐻 ∑ ℎ=1 2 ∑ 𝑚=1 (𝜌(𝑇 𝐶) 𝑝𝑠ℎ + (𝑑ℎ𝑠1− 1)𝜌(𝑃 𝑟𝑇 𝐶) 𝑝𝑠ℎ )𝑥(𝑇 𝐶) 𝑝𝑠ℎ𝑚+ 𝐿 ∑ 𝑙=1 𝐻 ∑ ℎ=1 (𝜌(𝐿𝐻) 𝑝𝑠𝑙ℎ + (𝑑ℎ𝑠1− 1)𝜌(𝑃 𝑟𝐿𝐻) 𝑝𝑠𝑙ℎ )𝑧𝑝𝑠𝑙ℎ]− ⎡⎢⎢⎢⎣ 𝐴 ∑ 𝑎=1 𝑅 ∑ 𝑟=1 𝑐(𝐵𝑅) 𝑎𝑟𝑝 (𝑦𝑎𝑟𝑝) + 𝑅 ∑ 𝑟=1 𝑐(𝐴𝑅) 𝑝𝑟 (𝑦(𝐴𝑅) 𝑝𝑟 ) + 𝑃 ∑ 𝑝=1 𝑝≠𝑝 𝑅 ∑ 𝑟=1 2 ∑ 𝑚=1 𝑐(𝑆𝑅) 𝑝𝑝𝑟𝑚 (𝑦(𝑆𝑅) 𝑝𝑝𝑟𝑚)+ Operations Research Perspectives 9 (2022) 100257 6 G. Colajanni et al. 𝑆 ∑ 𝑠=1 (𝑐(𝐾) 𝑝𝑠 (𝐿 ∑ 𝑙=1 𝑥(𝐿𝑆) 𝑝𝑠𝑙 + 2 ∑ 𝑚=1 𝐻 ∑ ℎ=1 𝑥(𝑇 𝐶) 𝑝𝑠ℎ𝑚 + 𝐺 ∑ 𝑔=1 𝑥(𝑃) 𝑝𝑠𝑔)+ 𝑐(𝑃 𝑟) 𝑝𝑠 (𝐿 ∑ 𝑙=1 𝐻 ∑ ℎ=1 (𝑑ℎ𝑠1− 1)𝑧𝑝𝑠𝑙ℎ + 2 ∑ 𝑚=1 𝐻 ∑ ℎ=1 (𝑑ℎ𝑠1− 1)𝑥(𝑇 𝐶) 𝑝𝑠ℎ𝑚 + 𝐺 ∑ 𝑔=1 𝑥(𝑃) 𝑝𝑠𝑔)+ 𝐻 ∑ ℎ=1 2 ∑ 𝑚=1 𝑑ℎ𝑠𝑚𝑐(𝑇 𝐶) 𝑝𝑠ℎ𝑚(𝑥(𝑇 𝐶) 𝑝𝑠ℎ𝑚)+2 𝐿 ∑ 𝑙=1 𝑐(𝐿𝑆) 𝑝𝑠𝑙 (𝑥(𝐿𝑆) 𝑝𝑠𝑙 ) + 𝐿 ∑ 𝑙=1 𝐻 ∑ ℎ=1 𝑑ℎ𝑠1𝑐(𝐿𝐻) 𝑙ℎ𝑠 (𝑧𝑝𝑠𝑙ℎ))]]} (1) subject to: 𝑃 ∑ 𝑝=1 𝑦𝑎𝑟𝑝 ≤𝐶𝑎𝑟,∀𝑎= 1,…, 𝐴, ∀𝑟= 1,…, 𝑅 (2) 𝑦(𝐴𝑅) 𝑝𝑟 ≤𝐶𝑝𝑟,∀𝑝= 1,…, 𝑃 , ∀𝑟= 1,…, 𝑅 (3) 𝑃 ∑ 𝑝=1 𝑝≠𝑝 2 ∑ 𝑚=1 𝑦(𝑆𝑅) 𝑝 𝑝𝑟𝑚 ≤ 𝐴 ∑ 𝑎=1 𝑦𝑎𝑟𝑝 +𝑦(𝐴𝑅) 𝑝𝑟 + 𝑃 ∑ 𝑝=1 𝑝≠𝑝 2 ∑ 𝑚=1 𝑦(𝑆𝑅) 𝑝𝑝𝑟𝑚, ∀𝑝= 1,…, 𝑃 , ∀𝑟= 1,…, 𝑅 (4) 𝑆 ∑ 𝑠=1 𝑓(𝐾) 𝑠𝑟 (𝐿 ∑ 𝑙=1 𝑥(𝐿𝑆) 𝑝𝑠𝑙 + 𝐻 ∑ ℎ=1 2 ∑ 𝑚=1 𝑥(𝑇 𝐶) 𝑝𝑠ℎ𝑚 + 𝐺 ∑ 𝑔=1 𝑥(𝑃) 𝑝𝑠𝑔)+ 𝑆 ∑ 𝑠=1 𝑓(𝑃 𝑟) 𝑠𝑟 (𝐿 ∑ 𝑙=1 𝐻 ∑ ℎ=1 (𝑑ℎ𝑠1− 1)𝑧𝑝𝑠𝑙ℎ + 2 ∑ 𝑚=1 𝐻 ∑ ℎ=1 (𝑑ℎ𝑠1−1)𝑥(𝑇 𝐶) 𝑝𝑠ℎ𝑚+ 𝐺 ∑ 𝑔=1 𝑥(𝑃) 𝑝𝑠𝑔)≤ 𝐴 ∑ 𝑎=1 𝑦𝑎𝑟𝑝 +𝑦(𝐴𝑅) 𝑝𝑟 − 𝑃 ∑ 𝑝=1 𝑝≠𝑝 2 ∑ 𝑚=1 𝑦(𝑆𝑅) 𝑝 𝑝𝑟𝑚 + 𝑃 ∑ 𝑝=1 𝑝≠𝑝 2 ∑ 𝑚=1 𝑦(𝑆𝑅) 𝑝𝑝𝑟𝑚, ∀𝑟= 1,…, 𝑅, ∀𝑝= 1,…, 𝑃 (5) 𝐻 ∑ ℎ=1 𝑧𝑝𝑠𝑙ℎ =𝑥(𝐿𝑆) 𝑝𝑠𝑙 ,∀𝑝= 1,…, 𝑃 , ∀𝑠= 1,…, 𝑆, ∀𝑙= 1,…, 𝐿 (6) 𝐺 ∑ 𝑔=1 𝑤ℎ𝑠𝑔 = 𝑃 ∑ 𝑝=1 2 ∑ 𝑚=1 𝑥(𝑇 𝐶) 𝑝𝑠ℎ𝑚 + 𝑃 ∑ 𝑝=1 𝐿 ∑ 𝑙=1 𝑧𝑝𝑠𝑙ℎ,∀ℎ= 1,…, 𝐻, ∀𝑠= 1,…, 𝑆 (7) 𝐻 ∑ ℎ=1 𝑤ℎ𝑠𝑔 + 𝑃 ∑ 𝑝=1 𝑥(𝑃) 𝑝𝑠𝑔 ≤𝐷𝑠𝑔,∀𝑠= 1,…, 𝑆, ∀𝑔= 1,…, 𝐺 (8) 𝑆 ∑ 𝑠=1 𝑏(𝐾) 𝑝𝑠 (𝐿 ∑ 𝑙=1 𝑥(𝐿𝑆) 𝑝𝑠𝑙 + 𝐻 ∑ ℎ=1 2 ∑ 𝑚=1 𝑥(𝑇 𝐶) 𝑝𝑠ℎ𝑚 + 𝐺 ∑ 𝑔=1 𝑥(𝑃) 𝑝𝑠𝑔)+ 𝑆 ∑ 𝑠=1 𝑏(𝑃 𝑟) 𝑝𝑠 (𝐿 ∑ 𝑙=1 𝐻 ∑ ℎ=1 (𝑑ℎ𝑠1− 1)𝑧𝑝𝑠𝑙ℎ + 2 ∑ 𝑚=1 𝐻 ∑ ℎ=1 (𝑑ℎ𝑠1−1)𝑥(𝑇 𝐶) 𝑝𝑠ℎ𝑚 + 𝐺 ∑ 𝑔=1 𝑥(𝑃) 𝑝𝑠𝑔)+ 𝑆 ∑ 𝑠=1 𝐺 ∑ 𝑔=1 𝑏(𝐸) 𝑝𝑠 𝑥(𝑃) 𝑝𝑠𝑔 ≤𝐵𝑝,∀𝑝= 1,…, 𝑃 (9) 𝐿 ∑ 𝑙=1 𝑥(𝐿𝑆) 𝑝𝑠𝑙 + 𝐻 ∑ ℎ=1 2 ∑ 𝑚=1 𝑥(𝑇 𝐶) 𝑝𝑠ℎ𝑚 + 𝐺 ∑ 𝑔=1 𝑥(𝑃) 𝑝𝑠𝑔 ≤𝐵𝑝𝑠, ∀𝑝= 1,…, 𝑃 , ∀𝑠= 1,…, 𝑆 (10) 𝑆 ∑ 𝑠=1 𝐺 ∑ 𝑔=1 [𝑏(𝐸) ℎ𝑠 + (2 − 𝑑ℎ𝑠1)𝑏(𝑃 𝑟) ℎ𝑠 ]𝑤ℎ𝑠𝑔 ≤𝐵ℎ,∀ℎ= 1,…, 𝐻, (11) 𝐺 ∑ 𝑔=1 𝑤ℎ𝑠𝑔 ≤𝐵ℎ𝑠,∀ℎ= 1,…, 𝐻, ∀𝑠= 1,…, 𝑆, (12) 𝑅 ∑ 𝑟=1 𝑥𝑝 𝑝𝑟𝑚 ≤𝛩(𝑅) 𝑝 𝑝𝑚,∀𝑝, 𝑝 = 1,…, 𝑃 , ∀𝑚= 1,2(13) 𝑆 ∑ 𝑠=1 𝑥(𝑇 𝐶) 𝑝𝑠ℎ𝑚 ≤𝛩(𝑆) 𝑝ℎ𝑚,∀𝑝= 1,…, 𝑃 , ∀ℎ= 1,…, 𝐻, ∀𝑚= 1,2(14) 𝑆 ∑ 𝑠=1 𝑥(𝐿𝑆) 𝑝𝑠𝑙 ≤𝛩(𝑆𝐿) 𝑝𝑙 ,∀𝑝= 1,…, 𝑃 , ∀𝑙= 1,…, 𝐿 (15) 𝑡(𝑆𝑅) 𝑝 𝑝𝑚 (𝑅 ∑ 𝑟=1 𝑦(𝑆𝑅) 𝑝 𝑝𝑟𝑚)≤𝑇𝑟,∀𝑝, 𝑝 = 1,…, 𝑃 , ∀𝑟= 1,…, 𝑅, ∀𝑚= 1,2(16) 𝑡(𝑇 𝐶) 𝑝ℎ𝑚 (𝑆 ∑ 𝑠=1 𝑥(𝑇 𝐶) 𝑝𝑠ℎ𝑚)≤𝑇𝑠, ∀𝑝= 1,…, 𝑃 , ∀𝑠= 1,…, 𝑆, ∀ℎ= 1,…, 𝐻, ∀𝑚= 1,2(17) 𝑡(𝐿𝑆) 𝑝𝑙 (𝑆 ∑ 𝑠=1 𝑥(𝐿𝑆) 𝑝𝑠𝑙 )+𝑡𝑙ℎ (𝑃 ∑ 𝑝=1 𝑆 ∑ 𝑠=1 𝑧𝑝𝑠𝑙ℎ)≤𝑇𝑠, ∀𝑝= 1,…, 𝑃 , ∀𝑠= 1,…, 𝑆, ∀𝑙= 1,…, 𝐿, ∀ℎ= 1,…, 𝐻 (18) 𝑡(𝑃) 𝑝𝑔 (𝑆 ∑ 𝑠=1 𝑥(𝑃) 𝑝𝑠𝑔)≤𝑇𝑠,∀𝑝= 1,…, 𝑃 , ∀𝑠= 1,…, 𝑆, ∀𝑔= 1,…, 𝐺, (19) 𝑦𝑎𝑟𝑝, 𝑦(𝐴𝑅) 𝑝𝑟 , 𝑦(𝑆𝑅) 𝑝 𝑝𝑟𝑚, 𝑥(𝐿𝑆) 𝑝𝑠𝑙 , 𝑥(𝑇 𝐶) 𝑝𝑠ℎ𝑚, 𝑥(𝑃) 𝑝𝑠𝑔, 𝑧𝑝𝑠𝑙ℎ, 𝑤ℎ𝑠𝑔 ≥0, ∀𝑎= 1,…, 𝐴, ∀𝑟= 1,…, 𝑅, ∀𝑝, 𝑝 = 1,…, 𝑃 , ∀𝑚= 1,2,∀𝑠= 1,…, 𝑆, ∀𝑙= 1,…, 𝐿, ∀ℎ= 1,…, 𝐻, ∀𝑔= 1,…, 𝐺 (20) The objective function (1) consists of two main terms: the total amount of analyzed swab tests, multiplied by the weight 𝛼, and the total profit, multiplied by 𝛽. The first term, that is the total amount of analyzed swab tests is given by the sum of (all types of) swab test kits sent by all processing laboratories to all the test centers (using or not the landing stations: ∑𝑃 𝑝=1 ∑𝑆 𝑠=1 ∑𝐿 𝑙=1 𝑥(𝐿𝑆) 𝑝𝑠𝑙 and ∑𝑃 𝑝=1 ∑𝑆 𝑠=1 ∑2 𝑚=1 ∑𝐻 ℎ=1 𝑥(𝑇 𝐶) 𝑝𝑠ℎ𝑚, respectively) and the tests performed directly on people (∑𝐺 𝑔=1 𝑥(𝑃) 𝑝𝑠𝑔). Observe that all the swab test kits sent by the processing labs are executed and analyzed (at the test centers or return at the labs). The latter sentence is justified from realty, since the number of requests of swab tests from people is very high and the case of swab test kits prepared, sent and ready but not used is excluded. On the contrary, it may happen that customer requests cannot be met due to a lack of swab tests (or rather, reagents needed for the analysis of the swab tests). The second term of the objective function (1), multiplied by the weight 𝛽, represents the total profit, given by the difference between the revenues and the costs. Specifically, the overall revenue is obtained summing the following terms: •∑𝑃 𝑝=1 ∑𝑆 𝑠=1 ∑𝐺 𝑔=1 𝜌(𝑃) 𝑝𝑠𝑔𝑥(𝑃) 𝑝𝑠𝑔, the total revenue obtained by the processing laboratories from the people for the executed swab tests (including the kit preparation, swabbing the patient, the possible transport and the analysis of the swab test); •∑𝑃 𝑝=1 ∑𝑆 𝑠=1 ∑𝐻 ℎ=1 ∑2 𝑚=1 (𝜌(𝑇 𝐶) 𝑝𝑠ℎ + (𝑑ℎ𝑠1− 1)𝜌(𝑃 𝑟𝑇 𝐶) 𝑝𝑠ℎ )𝑥(𝑇 𝐶) 𝑝𝑠ℎ𝑚, the total revenue obtained by the processing laboratories from the test centers for both the prepared and sent swab test kits and the possible (if done by the processing labs) analysis of the swab tests; we remind that 𝑑ℎ𝑠1= 2 (⇒(𝑑ℎ𝑠1− 1) = 1) if the swab test has to come back to the processing laboratory where it is analyzed, while 𝑑ℎ𝑠1=1(⇒(𝑑ℎ𝑠1− 1) = 0) if the swab test is analyzed at the test center (hence, in the latter case, the processing laboratory does not obtain any revenue for the swab analysis); •∑𝑃 𝑝=1 ∑𝑆 𝑠=1 ∑𝐿 𝑙=1 ∑𝐻 ℎ=1 (𝜌(𝐿𝐻) 𝑝𝑠𝑙ℎ + (𝑑ℎ𝑠1− 1)𝜌(𝑃 𝑟𝐿𝐻) 𝑝𝑠𝑙ℎ )𝑧𝑝𝑠𝑙ℎ, the total revenue obtained by the processing laboratories from the test centers, using the landing stations, for both the prepared and sent swab test kits and the possible (if done by the processing labs) analysis of the swab tests. The overall cost is obtained summing the following terms: -∑𝑃 𝑝=1 ∑𝐴 𝑎=1 ∑𝑅 𝑟=1 𝑐(𝐵𝑅) 𝑎𝑟𝑝 (𝑦𝑎𝑟𝑝), the cost for the processing laboratories to buy reagents from companies; -∑𝑃 𝑝=1 ∑𝑅 𝑟=1 𝑐(𝐴𝑅) 𝑝𝑟 (𝑦(𝐴𝑅) 𝑝𝑟 ), the cost for the processing laboratories to self-produce reagents; -∑𝑃 𝑝=1 ∑𝑃 𝑝=1 𝑝≠𝑝∑𝑅 𝑟=1 ∑2 𝑚=1 𝑐(𝑆𝑅) 𝑝𝑝𝑟𝑚 (𝑦(𝑆𝑅) 𝑝𝑝𝑟𝑚), the cost for the processing laboratories to receive reagents from other laboratories; note that we are assuming that only the laboratory that receives the reagents pays, moreover, this cost refers to the cost of transport (since the purchase cost vanishes in the objective function); Operations Research Perspectives 9 (2022) 100257 7 G. Colajanni et al. -∑𝑃 𝑝=1 ∑𝑆 𝑠=1 𝑐(𝐾) 𝑝𝑠 (∑𝐿 𝑙=1 𝑥(𝐿𝑆) 𝑝𝑠𝑙 +∑2 𝑚=1 ∑𝐻 ℎ=1 𝑥(𝑇 𝐶) 𝑝𝑠ℎ𝑚 +∑𝐺 𝑔=1 𝑥(𝑃) 𝑝𝑠𝑔), the cost for the processing laboratories to prepare all the swab test kits; -∑𝑃 𝑝=1 ∑𝑆 𝑠=1 𝑐(𝑃 𝑟) 𝑝𝑠 (∑𝐿 𝑙=1 ∑𝐻 ℎ=1(𝑑ℎ𝑠1− 1)𝑧𝑝𝑠𝑙ℎ +∑2 𝑚=1 ∑𝐻 ℎ=1(𝑑ℎ𝑠1− 1) 𝑥(𝑇 𝐶) 𝑝𝑠ℎ𝑚 +∑𝐺 𝑔=1 𝑥(𝑃) 𝑝𝑠𝑔), the cost for the processing laboratories to analyze all the swab tests; note that these costs are functions of the amount of swab tests processed by each lab (hence, the total quantity of the swab tests coming back from the test centers to the labs summed to the swab tests performed on people); -∑𝑃 𝑝=1 ∑𝑆 𝑠=1 ∑𝐻 ℎ=1 ∑2 𝑚=1 𝑑ℎ𝑠𝑚𝑐(𝑇 𝐶) 𝑝𝑠ℎ𝑚(𝑥(𝑇 𝐶) 𝑝𝑠ℎ𝑚), the cost to transport all the swab tests from the processing laboratories to the test centers; we underline that these costs are multiplied by the round trip parameter 𝑑ℎ𝑠𝑚 that equals 2 or 1 if the transportation means come back or not to the laboratory, respectively; -2∑𝑃 𝑝=1 ∑𝑆 𝑠=1 ∑𝐿 𝑙=1 𝑐(𝐿𝑆) 𝑝𝑠𝑙 (𝑥(𝐿𝑆) 𝑝𝑠𝑙 ), the cost to transport all the swab tests from the processing laboratories to the landing stations; we underline that these costs are multiplied by 2 because these routes are traveled only by UAVs, which must always return to the starting point (with or without the swab tests to be analyzed); -∑𝑃 𝑝=1 ∑𝑆 𝑠=1 ∑𝐿 𝑙=1 ∑𝐻 ℎ=1 𝑑ℎ𝑠1𝑐(𝐿𝐻) 𝑙ℎ𝑠 (𝑧𝑝𝑠𝑙ℎ), the cost to transport all the swab tests from the landing stations (and coming from the labs) to the test centers; clearly these costs are multiplied by the round trip parameter 𝑑ℎ𝑠1, where 𝑚= 1 because the used transportation means for these routes (from the landing stations to the test centers) are the traditional ones which come back only if any swab tests have to be analyzed at the processing laboratories. We now explain all the constraints. Constraints (2) and (3) represent the capacity constraints, according to which, each company cannot send more reagent than produced (see constraint (2)) and, analogously, at each processing laboratory the maximum quantity of each reagent that it is possible to self-produce must not be exceed (see constraint (3)). Constraint (4) refers to the sharing of reagents among the processing laboratories. Indeed, it establishes that the amount of each reagent that each laboratory could share with (that is, send to) other laboratories (through the traditional mean of transport and through UAVs) must not exceed the total amount of reagent possessed, that is the sum of the quantity of reagent purchased by all the manufacturing companies, the quantity of reagent self-produced and the quantity of reagent received from other laboratories. The relation between the swab tests and the amount of reagents needed to prepare and analyze them is expressed by constraint (5). It guarantees that, for each processing laboratory, the amount of each reagent needed to: - prepare the swab test kits to send at the landing stations, at the test centers and to use directly on people and - analyze the swab tests received from the test centers (using or not the landing station) and performed on people must be less than or equal to the total amount of such a reagent owned by the laboratory. Particularly, the owned reagent is determined by the sum of the amount purchased by all the manufacturing companies, the quantity of reagent self-produced and the quantity of reagent received from other laboratories, minus the quantity of reagent sent to other laboratories. Constraints (6) and (7) represent the conservation laws, according to which in each landing station and each test center, respectively, the inflows and outflows must be equal. Specifically, the quantity of each swab test type sent by a processing laboratory to a landing station must be equal to the sum of swab tests sent by the landing station to all the test centers; analogously, the quantity of each swab test type sent by all the processing laboratories and landing stations to a test center must be equal to the swab tests executed by the test center to all groups of people. Clearly, the number of requests from people cannot be exceeded, as ensured by constraint (8). Each processing laboratory has a maximum processing capacity, depending on the machinery, the time and the workforce, as determined by constraint (9). Moreover, constraint (10) establishes whether a processing laboratory is able to prepare (and analyze) a swab test type: if the parameter 𝐵𝑝𝑠 = 0, then the processing laboratory 𝑝is not able to create the 𝑠-type swab tests, otherwise 𝑝can create and send them. Constraints (11) and (12) are the analogous constraints of (9) and (10) for each test center. We underline that in constraint (11) the multiplication of 𝑏(𝑃 𝑟) ℎ𝑠 by (2 − 𝑑ℎ𝑠1)allows us to add such processing resources usage only if the (𝑠-type) swab tests are analyzed in the test center ℎ(and they are not sent back to the laboratories). Furthermore, in realty constraint (12) is fundamental since it allows us to formulate the case in which a test center is able or not to get some type of swab tests. The space capacities of each transport mean used for the links between the laboratories, between laboratories and test centers, and between laboratories and landing stations are guaranteed by constraints (13),(14) and (15). It is well known that each reagent, each swab test kit and swabbed test have a shelf-life, that is the maximum allowed transport time. Therefore, constraints (16)–(19) ensure that the shelf-life of each reagent and each swab test is not exceeded. Finally, constraint (20) represents the domain of the variables of the model. 4. Variational formulation In this section, we provide a variational formulation of the proposed model presented in Section 3. This alternative formulation allows us to use the well-known variational inequality theory, which provides, under appropriate hypotheses, results of existence and uniqueness of the solution. Moreover, the variational inequality formulation allows also the rigorous computation of the optimal solutions, as described in the next section. First, we assume that all the cost and time functions involved in the objective function (1) are continuously differentiable and convex. Theorem 4.1. The optimal solutions to the maximization problem (1);(3)– (20) are equivalent to the solutions to the variational inequality problem given by: determine (𝑥(𝐿𝑆)∗, 𝑥(𝑇 𝐶)∗, 𝑥(𝑃)∗, 𝑦∗, 𝑦(𝐴𝑅)∗, 𝑦(𝑆𝑅)∗, 𝑤∗, 𝑧∗) ∈ K, satisfying: 𝑃 ∑ 𝑝=1 𝑆 ∑ 𝑠=1 𝐿 ∑ 𝑙=1 ⎡⎢⎢⎢⎣ 𝛽 𝜕𝑐(𝐾) 𝑝𝑠 (𝑥(𝐿𝑆)∗, 𝑥(𝑇 𝐶)∗, 𝑥(𝑃)∗) 𝜕𝑥(𝐿𝑆) 𝑝𝑠𝑙 + 2𝛽 𝜕𝑐(𝐿𝑆) 𝑝𝑠𝑙 (𝑥(𝐿𝑆)∗ 𝑝𝑠𝑙 ) 𝜕𝑥(𝐿𝑆) 𝑝𝑠𝑙 −𝛼⎤⎥⎥⎥⎦ ×(𝑥(𝐿𝑆) 𝑝𝑠𝑙 −𝑥(𝐿𝑆)∗ 𝑝𝑠𝑙 ) + 𝑃 ∑ 𝑝=1 𝑆 ∑ 𝑠=1 𝐻 ∑ ℎ=1 2 ∑ 𝑚=1 [𝛽 𝜕𝑐(𝐾) 𝑝𝑠 (𝑥(𝐿𝑆)∗, 𝑥(𝑇 𝐶)∗, 𝑥(𝑃)∗) 𝜕𝑥(𝑇 𝐶) 𝑝𝑠ℎ𝑚 +𝛽 𝜕𝑐𝑃 𝑟 𝑝𝑠 (𝑥(𝑇 𝐶)∗, 𝑥(𝑃)∗, 𝑧∗) 𝜕𝑥(𝑇 𝐶) 𝑝𝑠ℎ𝑚 +𝛽𝑑ℎ𝑠𝑚 𝜕𝑐(𝑇 𝐶) 𝑝𝑠ℎ𝑚 (𝑥(𝑇 𝐶)∗ 𝑝𝑠ℎ𝑚 ) 𝜕𝑥(𝑇 𝐶) 𝑝𝑠ℎ𝑚 −𝛼−𝛽(𝜌(𝑇 𝐶) 𝑝𝑠ℎ + (𝑑ℎ𝑠1− 1)𝜌(𝑃 𝑟𝑇 𝐶) 𝑝𝑠ℎ )⎤⎥⎥⎥⎦ ×(𝑥(𝑇 𝐶) 𝑝𝑠ℎ𝑚 −𝑥(𝑇 𝐶)∗ 𝑝𝑠ℎ𝑚 ) + 𝑃 ∑ 𝑝=1 𝑆 ∑ 𝑠=1 𝐺 ∑ 𝑔=1 [𝛽 𝜕𝑐(𝐾) 𝑝𝑠 (𝑥(𝐿𝑆)∗, 𝑥(𝑇 𝐶)∗, 𝑥(𝑃)∗) 𝜕𝑥(𝑃) 𝑝𝑠𝑔 −𝛼−𝛽𝜌(𝑃) 𝑝𝑠𝑔 +𝛽 𝜕𝑐(𝑃 𝑟) 𝑝𝑠 (𝑥(𝑇 𝐶)∗, 𝑥(𝑃)∗, 𝑧∗) 𝜕𝑥(𝑃) 𝑝𝑠𝑔 ]×(𝑥(𝑃) 𝑝𝑠𝑔 −𝑥(𝑃)∗ 𝑝𝑠𝑔 ) +𝛽 𝐴 ∑ 𝑎=1 𝑅 ∑ 𝑟=1 𝑃 ∑ 𝑝=1 ⎡⎢⎢⎢⎣ 𝜕𝑐(𝐵𝑅) 𝑎𝑟𝑝 (𝑦∗ 𝑎𝑟𝑝) 𝜕𝑦𝑎𝑟𝑝 ⎤⎥⎥⎥⎦ ×(𝑦𝑎𝑟𝑝 −𝑦∗ 𝑎𝑟𝑝) +𝛽 𝑃 ∑ 𝑝=1 𝑅 ∑ 𝑟=1 ⎡⎢⎢⎢⎣ 𝜕𝑐(𝐴𝑅) 𝑝𝑟 (𝑦(𝐴𝑅)∗ 𝑝𝑟 ) 𝜕𝑦(𝐴𝑅) 𝑝𝑟 ⎤⎥⎥⎥⎦ × (𝑦(𝐴𝑅) 𝑝𝑟 −𝑦(𝐴𝑅)∗ 𝑝𝑟 ) Operations Research Perspectives 9 (2022) 100257 8 G. Colajanni et al. +𝛽 𝑃 ∑ 𝑝=1 𝑝≠𝑝 𝑃 ∑ 𝑝=1 2 ∑ 𝑚=1 ⎡⎢⎢⎢⎣ 𝜕𝑐(𝑆𝑅) 𝑝𝑝𝑟𝑚 (𝑦(𝑆𝑅)∗ 𝑝𝑝𝑟𝑚 ) 𝜕𝑦(𝑆𝑅) 𝑝𝑝𝑟𝑚 ⎤⎥⎥⎥⎦ ×(𝑦(𝑆𝑅) 𝑝𝑝𝑟𝑚 −𝑦(𝑆𝑅)∗ 𝑝𝑝𝑟𝑚 ) + 𝑃 ∑ 𝑝=1 𝑆 ∑ 𝑠=1 𝐿 ∑ 𝑙=1 𝐻 ∑ ℎ=1 [𝛽 𝜕𝑐(𝑃 𝑟) 𝑝𝑠 (𝑥(𝑇 𝐶)∗, 𝑥(𝑃)∗, 𝑧∗) 𝜕𝑧𝑝𝑠𝑙ℎ −𝛽(𝜌(𝐿𝐻) 𝑝𝑠𝑙ℎ + (𝑑ℎ𝑠1− 1)𝜌(𝑃 𝑟𝐿𝐻) 𝑝𝑠𝑙ℎ ) +𝛽𝑑ℎ𝑠1 𝜕𝑐(𝐿𝐻) 𝑙ℎ𝑠 (𝑧∗ 𝑝𝑠𝑙ℎ) 𝜕𝑧𝑝𝑠𝑙ℎ ]×(𝑧𝑝𝑠𝑙ℎ −𝑧∗ 𝑝𝑠𝑙ℎ) ∀(𝑥(𝐿𝑆), 𝑥(𝑇 𝐶), 𝑥(𝑃), 𝑦, 𝑦(𝐴𝑅), 𝑦(𝑆𝑅), 𝑤, 𝑧) ∈ K,(21) where K∶= {(𝑥(𝐿𝑆), 𝑥(𝑇 𝐶), 𝑥(𝑃), 𝑦, 𝑦(𝐴𝑅), 𝑦(𝑆𝑅), 𝑤, 𝑧) ∈ R𝑁 +∶(3)–(20) hold}, (22) and where 𝑁=𝑃 𝐿𝑆 +2𝑃 𝑆𝐻 +𝑃 𝑆𝐺+𝐴𝑅𝑃 +𝑅𝑃 +(𝑃−1)𝑃 𝑅𝑀 +𝑃 𝑆𝐿𝐻. Proof. It follows by results presented in [22]. □ For easy reference in the subsequent discussions, we put variational inequality (21) into standard form (see [22–25]), that is: determine 𝑋∗∈satisfying: ⟨𝐹(𝑋∗), 𝑋 −𝑋∗⟩≥0,∀𝑋∈.(23) We set 𝑋≡(𝑥(𝐿𝑆), 𝑥(𝑇 𝐶), 𝑥(𝑃), 𝑦(𝑆𝑅), 𝑧),𝐹(𝑋) = (𝐹𝑖(𝑋))𝑖=1,…,7, with the generic (𝑝, 𝑙, 𝑠)-th component of 𝐹1(𝑋)given by: 𝐹1 𝑝𝑙𝑠(𝑋)≡⎡⎢⎢⎢⎣ 𝛽𝜕𝑐(𝐾) 𝑝𝑠 (𝑥(𝐿𝑆), 𝑥(𝑇 𝐶), 𝑥(𝑃)) 𝜕𝑥(𝐿𝑆) 𝑝𝑠𝑙 − 2𝛽 𝜕𝑐(𝐿𝑆) 𝑝𝑙𝑠 (𝑥(𝐿𝑆) 𝑝𝑙𝑠 ) 𝜕𝑥(𝐿𝑆) 𝑝𝑙𝑠 −𝛼⎤⎥⎥⎥⎦ ,∀𝑝, 𝑙, 𝑠 (24) the generic (𝑝, 𝑠, ℎ, 𝑚)-th component of 𝐹2(𝑋)given by: 𝐹2 𝑝𝑠ℎ𝑚(𝑋)≡⎡⎢⎢⎣ 𝛽𝜕𝑐(𝐾) 𝑝𝑠 (𝑥(𝐿𝑆), 𝑥(𝑇 𝐶), 𝑥(𝑃)) 𝜕𝑥(𝑇 𝐶) 𝑝𝑠ℎ𝑚 +𝛽 𝜕𝑐𝑃 𝑟 𝑝𝑠 (𝑥(𝑇 𝐶), 𝑥(𝑃), 𝑧) 𝜕𝑥(𝑇 𝐶) 𝑝𝑠ℎ𝑚 +𝛽𝑑ℎ𝑠𝑚 𝜕𝑐(𝑇 𝐶) 𝑝𝑠ℎ𝑚 (𝑥(𝑇 𝐶) 𝑝𝑠ℎ𝑚) 𝜕𝑥(𝑇 𝐶) 𝑝𝑠ℎ𝑚 −𝛼−𝛽(𝜌(𝑇 𝐶) 𝑝𝑠ℎ + (𝑑ℎ𝑠1− 1)𝜌(𝑃 𝑟𝑇 𝐶) 𝑝𝑠ℎ )⎤⎥⎥⎥⎦ , ∀𝑝, 𝑠, ℎ, 𝑚, (25) the generic (𝑝, 𝑠, 𝑔)-th component of 𝐹3(𝑋)given by: 𝐹3 𝑝𝑠𝑔(𝑋)≡[𝛽𝜕𝑐(𝐾) 𝑝𝑠 (𝑥(𝐿𝑆), 𝑥(𝑇 𝐶), 𝑥(𝑃)) 𝜕𝑥(𝑃) 𝑝𝑠𝑔 +𝛽 𝜕𝑐𝑃 𝑟 𝑝𝑠 (𝑥(𝑇 𝐶), 𝑥(𝑃), 𝑧) 𝜕𝑥(𝑃) 𝑝𝑠𝑔 −𝛼−𝛽𝜌(𝑃) 𝑝𝑠𝑔],∀𝑝, 𝑠, 𝑔, (26) the generic (𝑎, 𝑟, 𝑝)-th component of 𝐹4(𝑋)given by: 𝐹4(𝑋) = [𝛽𝜕𝑐(𝐵𝑅) 𝑎𝑟𝑝 (𝑦𝑎𝑟𝑝) 𝜕𝑦𝑎𝑟𝑝 ],∀𝑎, 𝑟, 𝑝, (27) the generic (𝑝, 𝑟)-th component of 𝐹5(𝑋)given by: 𝐹5(𝑋) = ⎡⎢⎢⎢⎣ 𝛽 𝜕𝑐(𝐴𝑅) 𝑝𝑟 (𝑦(𝐴𝑅) 𝑝𝑟 ) 𝜕𝑦(𝐴𝑅) 𝑝𝑟 ⎤⎥⎥⎥⎦ ,∀𝑝, 𝑟, (28) the generic (𝑝, 𝑝, 𝑟, 𝑚)-th component of 𝐹6(𝑋)given by: 𝐹6 𝑝𝑝𝑟𝑚(𝑋)≡⎡⎢⎢⎢⎣ 𝛽 𝜕𝑐(𝑆𝑅) 𝑝𝑝𝑟𝑚 (𝑦(𝑆𝑅) 𝑝𝑝𝑟𝑚) 𝜕𝑦(𝑆𝑅) 𝑝𝑝𝑟𝑚 ⎤⎥⎥⎥⎦ ,∀𝑝, 𝑝, 𝑟, 𝑚, (29) and the generic (𝑝, 𝑠, 𝑙, ℎ)-th component of 𝐹7(𝑋)given by: 𝐹7 𝑝𝑠𝑙ℎ(𝑋)≡⎡⎢⎢⎣ 𝛽𝜕𝑐(𝑃 𝑟) 𝑝𝑠 (𝑥(𝑇 𝐶)∗, 𝑥(𝑃)∗, 𝑧∗) 𝜕𝑧𝑝𝑠𝑙ℎ +𝛽𝑑ℎ𝑠1 𝜕𝑐(𝐿𝐻) 𝑙ℎ𝑠 (𝑧∗ 𝑝𝑠𝑙ℎ) 𝜕𝑧𝑝𝑠𝑙ℎ −𝛽(𝜌(𝐿𝐻) 𝑝𝑠𝑙ℎ + (𝑑ℎ𝑠1− 1)𝜌(𝑃 𝑟𝐿𝐻) 𝑝𝑠𝑙ℎ )],∀𝑝, 𝑠, 𝑙, ℎ, (30) and  ≡ K. Therefore, variational inequality (21) can be rewritten into standard form (23). We observe that in the variational inequality problem in standard form (23), the feasible set is closed and convex, properties guaranteed by the nature of constraints (3)–(20) and the assumption of continuously differentiability of all the time functions, and that the function that enters the variational inequality, 𝐹, is continuous, property that follows from the continuously differentiability of the all cost functions involved in the previous formulation. Hence, the existence of a solution to variational inequality (23), or equivalently, (21), is guaranteed from the classical variational inequality theory (see [26]). Finally, following [22,26], we can state the following uniqueness result. Theorem 4.2. The uniqueness of the solution to the variational inequality (21) or, equivalently, to the variational inequality (23) is guaranteed if the function 𝐹(𝑋)is strictly monotone on , that is: ⟨𝐹(𝑋1) − 𝐹(𝑋2), 𝑋1−𝑋2⟩>0,∀𝑋1, 𝑋2∈, 𝑋1≠𝑋2.(31) The following result ensures a sufficient condition to the strictly monotonicity of the function 𝐹. Proposition 4.3. If the cost functions 𝑐(𝐵𝑅) 𝑎𝑟𝑝 (⋅),𝑐(𝐴𝑅) 𝑝𝑟 (⋅),𝑐(𝑆𝑅) 𝑝𝑝𝑟𝑚 ,𝑐(𝐾) 𝑝𝑠 (⋅), 𝑐(𝑃 𝑟) 𝑝𝑠 (⋅),𝑐(𝑇 𝐶) 𝑝𝑠ℎ𝑚(⋅),𝑐(𝐿𝑆) 𝑝𝑠𝑙 (⋅), and 𝑐(𝐿𝐻) 𝑙ℎ𝑠 (⋅)are strictly convex with respect to their own variables, 𝑐(𝐾) 𝑝𝑠 (⋅)are additive with respect to 𝑥(𝐿𝑆),𝑥(𝑇 𝐶)and 𝑥(𝑃)and 𝑐𝑃 𝑟 𝑝𝑠 (⋅)are additive with respect to 𝑥(𝑇 𝐶),𝑥(𝑃)and 𝑧, then 𝐹(𝑋)is a strictly monotone function according to (31). Proof. Let 𝑋1, 𝑋2∈be two feasible vectors such that 𝑋1≠𝑋2. We evaluate the following quantity: ⟨𝐹(𝑋1) − 𝐹(𝑋2)⟩= 𝛽 𝑃 ∑ 𝑝=1 𝑆 ∑ 𝑠=1 𝐿 ∑ 𝑙=1 [𝜕𝑐(𝐾) 𝑝𝑠 (𝑥(𝐿𝑆),1, 𝑥(𝑇 𝐶),1, 𝑥(𝑃),1) 𝜕𝑥(𝐿𝑆) 𝑝𝑠𝑙 − 𝜕𝑐(𝐾) 𝑝𝑠 (𝑥(𝐿𝑆),2, 𝑥(𝑇 𝐶),2, 𝑥(𝑃),2) 𝜕𝑥(𝐿𝑆) 𝑝𝑠𝑙 ] ×(𝑥(𝐿𝑆),1 𝑝𝑙𝑠 −𝑥(𝐿𝑆),2 𝑝𝑙𝑠 ) +2𝛽 𝑃 ∑ 𝑝=1 𝑆 ∑ 𝑠=1 𝐿 ∑ 𝑙=1 ⎡⎢⎢⎢⎣ 𝜕𝑐(𝐿𝑆) 𝑝𝑙𝑠 (𝑥(𝐿𝑆),1 𝑝𝑙𝑠 ) 𝜕𝑥(𝐿𝑆) 𝑝𝑙𝑠 − 𝜕𝑐(𝐿𝑆) 𝑝𝑙𝑠 (𝑥(𝐿𝑆),2 𝑝𝑙𝑠 ) 𝜕𝑥(𝐿𝑆) 𝑝𝑙𝑠 ⎤⎥⎥⎥⎦ ×(𝑥(𝐿𝑆),1 𝑝𝑙𝑠 −𝑥(𝐿𝑆),2 𝑝𝑙𝑠 ) +𝛽 𝑃 ∑ 𝑝=1 𝑆 ∑ 𝑠=1 𝐻 ∑ ℎ=1 2 ∑ 𝑚=1 [𝜕𝑐(𝐾) 𝑝𝑠 (𝑥(𝐿𝑆),1, 𝑥(𝑇 𝐶),1, 𝑥(𝑃),1) 𝜕𝑥(𝑇 𝐶) 𝑝𝑠ℎ𝑚 − 𝜕𝑐(𝐾) 𝑝𝑠 (𝑥(𝐿𝑆),2, 𝑥(𝑇 𝐶),2, 𝑥(𝑃),2) 𝜕𝑥(𝑇 𝐶) 𝑝𝑠ℎ𝑚 ] ×(𝑥(𝑇 𝐶),1 𝑝𝑠ℎ𝑚 −𝑥(𝑇 𝐶),2 𝑝𝑠ℎ𝑚 ) +𝛽 𝑃 ∑ 𝑝=1 𝑆 ∑ 𝑠=1 𝐻 ∑ ℎ=1 2 ∑ 𝑚=1 [𝜕𝑐𝑃 𝑟 𝑝𝑠 (𝑥(𝑇 𝐶),1, 𝑥(𝑃),1, 𝑧1) 𝜕𝑥(𝑇 𝐶) 𝑝𝑠ℎ𝑚 − 𝜕𝑐𝑃 𝑟 𝑝𝑠 (𝑥(𝑇 𝐶),2, 𝑥(𝑃),2, 𝑧2) 𝜕𝑥(𝑇 𝐶) 𝑝𝑠ℎ𝑚 ] ×(𝑥(𝑇 𝐶),1 𝑝𝑠ℎ𝑚 −𝑥(𝑇 𝐶),2 𝑝𝑠ℎ𝑚 ) +𝛽 𝑃 ∑ 𝑝=1 𝑆 ∑ 𝑠=1 𝐻 ∑ ℎ=1 2 ∑ 𝑚=1 𝑑ℎ𝑠𝑚 ⎡⎢⎢⎢⎣ 𝜕𝑐(𝑇 𝐶) 𝑝𝑠ℎ𝑚 (𝑥(𝑇 𝐶),1 𝑝𝑠ℎ𝑚 ) 𝜕𝑥(𝑇 𝐶) 𝑝𝑠ℎ𝑚 − 𝜕𝑐(𝑇 𝐶) 𝑝𝑠ℎ𝑚 (𝑥(𝑇 𝐶),2 𝑝𝑠ℎ𝑚 ) 𝜕𝑥(𝑇 𝐶) 𝑝𝑠ℎ𝑚 ⎤⎥⎥⎥⎦ ×(𝑥(𝑇 𝐶),1 𝑝𝑠ℎ𝑚 −𝑥(𝑇 𝐶),2 𝑝𝑠ℎ𝑚 ) +𝛽 𝑃 ∑ 𝑝=1 𝑆 ∑ 𝑠=1 𝐺 ∑ 𝑔=1 [𝜕𝑐(𝐾) 𝑝𝑠 (𝑥(𝐿𝑆),1, 𝑥(𝑇 𝐶),1, 𝑥(𝑃),1) 𝜕𝑥(𝑃) 𝑝𝑠𝑔 − 𝜕𝑐(𝐾) 𝑝𝑠 (𝑥(𝐿𝑆),2, 𝑥(𝑇 𝐶),2, 𝑥(𝑃),2) 𝜕𝑥(𝑃) 𝑝𝑠𝑔 ] ×(𝑥(𝑃),1 𝑝𝑠𝑔 −𝑥(𝑃),2 𝑝𝑠𝑔 ) Operations Research Perspectives 9 (2022) 100257 15 G. 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