The optimum delivery of safeguarding substances is a major part of supply chain management and a crucial issue in the mitigation against the outbreak of pandemics. A problem arises for a decision maker who wants to optimally choose a subset of candidate consumers to maximize the distributed quantities of the needed safeguarding substances within a specific time period. A nonlinear binary mathematical programming model for the problem is formulated. The decision variables are binary ones that represent whether to choose a specific consumer, and design constraints are formulated to keep track of the chosen route. To better illustrate the problem, objective, and problem constraints, a real application case study is presented. The case study involves the optimum delivery of safeguarding substances to several hospitals in the Al-Gharbia Governorate in Egypt. The hospitals are selected to represent the consumers of safeguarding substances, as they are the first crucial frontline for mitigation against a pandemic outbreak. A distribution truck is used to distribute the substances from the main store to the hospitals in specified required quantities during a given working shift. The objective function is formulated in order to maximize the total amount of delivered quantities during the specified time period. The case study is solved using a novel Discrete Binary Gaining Sharing Knowledge-based Optimization algorithm (DBGSK), which involves two main stages: discrete binary junior and senior gaining and sharing stages. DBGSK has the ability of finding the solutions of the introduced problem, and the obtained results demonstrate robustness and convergence toward the optimal solutions.

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It is worthwhile to determine the importance of safeguarding substances such as cleaning materials, alcohol, respirator masks, medical gloves, and disinfection fluids both to hospital crews and to patients, especially those suspected to have caught the virus. Countries that have succeeded in slowing the prevalence of the virus have forced their citizens to wear masks in public places. Most of the infected people have contracted the infection in closed environments and in poorly ventilated spaces, such as public places, transportation means, restaurants, cinemas, stores, hospitals, and homes. Therefore, it is essential to provide such places with necessary safeguarding substances.

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The optimum delivery of the safeguarding substances for mitigating an outbreak in network optimization is defined for scheduling the distribution system with a limited load capacity to a list of consumers with known demanded quantities. The objective is to identify the most effective route for the distribution system as measured by maximizing the total delivered quantities in a certain limited time period. The route starts from a predetermined main store location and proceeds to each chosen consumer exactly once and then returns to the main store.

The rest of this article is organized as follows. Section 2 is devoted to demonstrating the importance of providing hospitals with safeguarding substances to protect medical personnel, patients, and visitors. This section describes the problem under consideration to distribute safeguarding substances to a group of consumers in an optimal way in order to meet the needs of each consumer while increasing the total distributed safeguarding substances during a specified time period.

The mathematical model of the problem is developed in Section 3. The proposed formulation is a nonlinear binary mathematical model with a dimension depending on the number of candidate consumers to be visited. The steps of the solution procedure are also explained.

A real application case study for the optimum delivery of safeguarding substances among candidate hospitals representing an essential type of consumers for such preventive materials is presented in Section 4. In Section 5, a novel discrete binary version of a recently developed Gaining Sharing Knowledge-based Optimization Technique (GSK) is introduced for solving the problem. GSK cannot solve the problem in binary space; therefore, a Discrete Binary GSK Optimization Algorithm (DBGSK) is proposed with two new junior and senior stages. These stages allow DBGSK to inspect the problem search space efficiently.

Section 6 presents the experimental results of the problem obtained by DBGSK, and Section 7 summarizes the conclusions and the suggested points for future research.

Safeguarding substances can disinfect contaminated places and limit the quantities of viruses that travel through respiratory droplets. They also contribute to limiting transmission of infection between people in gatherings, especially in public transport and crowded places. Frontline workers in medical institutions who wore the “N95” respirator mask did not catch the virus despite closely taking care of the infected patients. That is why it is very important to provide public places and hospitals with adequate amounts of these safeguarding substances regularly.

The importance of safeguarding substances and wearing masks in public places becomes clear considering that between 6% and 18% of infected people may not show any symptoms of the disease despite being able to spread the infection. The incubation period of the virus may be up to 14 days before symptoms appear. If everyone, especially the asymptomatic, wears masks, the number of viruses circulating in the air will decrease, and the risk of transmission will be reduced.

About 3,000 drops of spray may come out of the mouth of a person during one sneeze, and some fear that the virus can spread through the spray that comes out of the mouth while speaking. Once the spray comes out of the mouth, the larger droplets settle on surfaces, while the smaller droplets remain suspended in the air for hours, which may be inhaled by a healthy person. After the virus enters the body and multiplies, viral particles exit from the cells and enter the body fluids in the lungs, mouth, and nose. When the person coughs, the tiny droplets that are filled with viruses disperse in the air.

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The conclusion is that if a virus is given favorable opportunities, it may remain suspended in the air for several hours and still be able to transmit to people who inhale the droplets. The virus appears to be able to circulate through the air particularly in closed environments.

Hospitals and health centers are at the top of the list of public places that need safeguarding substances, and attention should be paid to continuous and timely supply to them of the required amounts. Medical employees are the first line of defense in the fight against the pandemic because they are exposed to patients and visitors who may be infected with the virus.

The problem under consideration is how to distribute safeguarding substances to a group of hospitals—as a good example of public places—in an optimal way. A transport vehicle carrying an amount of these safeguarding substances moves from one store and distributes them to the selected hospitals to meet the needs of each one while maximizing the total distributed quantities. The needed amount for a hospital is determined according to the number of medical crews, attending patients, and number of visitors. The distribution process takes place within a limited time window, namely the shift of the driver and their companions. The problem begins by considering several hospitals to be supplied. Then, the problem is to determine the optimal choice between these hospitals during the limited time shift so that the largest amount of these safeguarding substances is distributed. With the addition of a new list of hospitals including those that were not provided with safeguarding substances in the first round, the process is repeated in the same way until the provision to all hospitals.

The distribution of safeguarding substances problem is mapped to a graph

The problem is then defined as:

Each consumer is visited only once to deliver the required safeguarding substances.

The route starts and ends at the main store location from where the safeguarding substances are supplied.

The overall goal of the problem is the best utilization of the available time, calculated as maximizing the total amount of distributed safeguarding substances during the predetermined time interval.

Let:

(i) Avoid the Trivial Solution:

In order to avoid the trivial solution that the distribution system will be saturated by supplying one consumer only, the following condition should hold: The quantity of safeguarding substances to be supplied to any candidate consumer should fill up completely the maximum carrying capacity of the used distribution system. In case of violation, the used system will travel to that consumer, deliver its total capacity, and obviate the need to perform the scheduling process.

In order to avoid the solution that the distribution system can deliver the needed safeguarding substances to all consumers, the problem in this case will be to minimize the total transportation distances or times (the well-known Travelling Salesman Problem). The following condition should hold: The total quantities of safeguarding substances to be delivered to all the candidate consumers should be greater than the maximum carrying capacity of the used distribution system.

These two conditions should be checked before designing the appropriate mathematical model.

(ii) Positions Constraints:

Each position

(iii) Consumer Constraints:

Each consumer

(iv) Consecutive Positions Constraints:

A position (

If

If

(v) Shift Hours Constraint:

The total time

These four parts are calculated as follows:

The last visited position in the optimum route is characterized by a unique particularity that it does not have any adjacent subsequent positions. The expression (_{3}:

_{3} is added to cover the case when the optimum route will contain all the candidate

The following expression is used to calculate _{4}:

The total time of the whole optimum route _{1} +_{2} +_{3} +_{4}.

Substituting for

where

(vi) Maximum Load Capacity of the Distribution System Constraint:

The maximum quantity distributed to all the visited consumers in the considered time shift should not exceed the maximum load capacity of the transportation system, see

where

(vii) Binary Constraints:

All the decision variables are 0–1; see

The objective function is formulated in order to maximize the total amount of delivered safeguarding substances; see

where

Finally, we have a suggested model that contains (^{2}) binary variables and (4

An optimum solution to the problem will produce two distinct situations:

If

If

Structured English is used to present the solution procedure. The use of Structured English to describe the steps of the algorithm is clear and unambiguous and can be read from start to finish. The use of Structured English keywords provides a syntax similar to that of a programming language to assist with identifying logical steps necessary to properly describe the algorithm. Structured English aims to provide the benefits of both programming logic and natural language; program logic helps to attain precision, and natural language helps with the familiarity of the spoken word. See

A real example is presented to apply the given mathematical model for very important consumers during an epidemic outbreak. An example of an important consumer of the safeguarding substances for an outbreak are hospitals that exist in different locations in Al-Gharbia Governorate, Egypt. A small truck with 1.7-ton capacity starts its route from the main store of safeguarding substances located in Tanta, the capital of the Governorate denoted by (STORE), as shown in

In one special emergency night shift that lasts five hours, five hospitals are identified as candidate consumers, denoted by serial numbers (_{ij}

The mathematical formulation for the given case study is worked out by substituting in the previously described model, that is, formulas 1–7.

Metaheuristic approaches have been developed for complex optimization problems with continuous variables. References [

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The proposed methodology is described below.

An optimization problem with constraints can be formulated as:

where

The human-based GSK algorithm has two stages: junior and senior gaining and sharing stage. All persons acquire knowledge and share their views with others. The people in early stage gain knowledge from their small networks, such as family members, relatives, and neighbours, and want to share their opinions with others who might not be from their networks, because the curiosity about other people. These people may not have the experience to categorize the people. In the same way, people in the middle or later stages enhance their knowledge by interacting with friends, colleagues, and social media friends, among others, and share their views with the most suitable person so that they can improve their knowledge. These people have the experience to judge other people and can categorize them (good or bad). The process mentioned above can be formulated mathematically in the following steps.

Step 1: To get a starting point of the optimization problem, the initial population must be obtained. The initial population is created randomly within the boundary constraints. See

where t is for the number of populations; _{p}

Step 2: At this step, the dimensions of junior and senior stages should be computed through the following formula. See

where _{J}_{S}^{max} is the maximum count of generations, and G is the count of generation.

Step 3: Junior gaining sharing knowledge stage: In this stage, the early aged people gain knowledge from their small networks and share their views with other people who may or may not belong to their groups. Thus, individuals are updated as:

According to the objective function values, the individuals are arranged in ascending order. For every _{t}_{t −1}) and worst (_{t+1}) to gain knowledge and also randomly choose (_{r}_{f}

Step 4: Senior gaining sharing knowledge stage: This stage comprises the impact and effect of other people (good or bad) on the individual. The updated individuals can be determined as follows:

The individuals are classified into three categories (best, middle, and worst) after sorting individuals into ascending order based on the objective function values.

Best _{best}

For every individual _{t}

For solving problems in discrete binary space, a novel Discrete Binary Gaining Sharing Knowledge-based Optimization algorithm is proposed. In DBGSK, the new initialization and the working mechanism of both stages (junior and senior gaining sharing stages) are introduced over discrete binary space, and the remaining algorithms remain the same as the previous ones. The working mechanism of DBGSK is presented in the following subsections.

Discrete Binary Initialization:

The initial population is obtained in GSK using

where the round operator is used to convert the decimal number into the nearest binary number.

Discrete Binary Junior Saining and Sharing Stage:

The discrete binary junior gaining and sharing stage is based on the original GSK with _{f}

Case 1. When ^{3} combinations are possible, which are listed in

_{t −1} |
_{t+1} |
_{r} |
Results | Modified results | |
---|---|---|---|---|---|

Subcase (a) | 0 | 0 | 0 | 0 | 0 |

0 | 0 | 1 | 1 | 1 | |

1 | 1 | 0 | 0 | 0 | |

1 | 1 | 1 | 1 | 1 | |

Subcase (b) | 1 | 0 | 0 | 1 | 1 |

1 | 0 | 1 | 2 | 1 | |

0 | 1 | 0 | −1 | 0 | |

0 | 1 | 1 | 0 | 0 |

Subcase (a): If _{t −1} is equal to _{t+1}, the result is equal to _{r}

Subcase (b): When _{t −1} is not equal _{t+1}, the result is the same as _{t −1} by taking −1 as 0 and 2 as 1.

The mathematical formulation of Case 1 is as follows:

Case 2. When ^{4} possible combinations that are presented in

Subcase (c): If _{t −1} is not equal to _{t+1} and _{t+1} is equal to _{r}_{t −1}.

Subcase (d): If any of the conditions arise _{t}

_{t −1} |
_{t} |
_{t+1} |
_{r} |
Results | Modified results | |
---|---|---|---|---|---|---|

Subcase (c) | 1 | 1 | 0 | 0 | 3 | 1 |

1 | 0 | 0 | 0 | 1 | 1 | |

0 | 1 | 1 | 1 | 0 | 0 | |

0 | 0 | 1 | 1 | −2 | 0 | |

Subcase (d) | 0 | 0 | 0 | 0 | 0 | 0 |

0 | 1 | 0 | 0 | 2 | 1 | |

0 | 0 | 1 | 0 | −1 | 0 | |

0 | 0 | 0 | 1 | −1 | 0 | |

1 | 0 | 1 | 0 | 0 | 0 | |

1 | 0 | 0 | 1 | 0 | 0 | |

0 | 1 | 1 | 0 | 1 | 1 | |

0 | 1 | 0 | 1 | 1 | 1 | |

1 | 1 | 1 | 0 | 2 | 1 | |

1 | 0 | 1 | 1 | −1 | 0 | |

1 | 1 | 0 | 1 | 2 | 1 | |

1 | 1 | 1 | 1 | 1 | 1 |

The mathematical formulation of Case 2 is:

Discrete Binary Senior Gaining and Sharing Stage: The working mechanism of the discrete binary senior gaining and sharing stage is the same as that of the binary junior gaining and sharing stage with value of _{f}

Case 1.

_{best} |
_{worst} |
_{middle} |
Results | Modified results | |
---|---|---|---|---|---|

Subcase (a) | 0 | 0 | 0 | 0 | 0 |

0 | 0 | 1 | 1 | 1 | |

1 | 1 | 0 | 0 | 0 | |

1 | 1 | 1 | 1 | 1 | |

Subcase (b) | 1 | 0 | 0 | 1 | 1 |

1 | 0 | 1 | 2 | 1 | |

0 | 1 | 0 | −1 | 0 | |

0 | 1 | 1 | 0 | 0 |

Subcase (a): If _{best}

Subcase (b): If _{best}_{best}

Case 1 can be mathematically formulated in the following way:

Case 2.

_{best} |
_{t} |
_{worst} |
_{middle} |
Results | Modified results | |
---|---|---|---|---|---|---|

Subcase (c) | 1 | 1 | 0 | 0 | 3 | 1 |

1 | 0 | 0 | 0 | 1 | 1 | |

0 | 1 | 1 | 1 | 0 | 0 | |

0 | 0 | 1 | 1 | −2 | 0 | |

Subcase (d) | 0 | 0 | 0 | 0 | 0 | 0 |

0 | 1 | 0 | 0 | 2 | 1 | |

0 | 0 | 1 | 0 | −1 | 0 | |

0 | 0 | 0 | 1 | −1 | 0 | |

1 | 0 | 1 | 0 | 0 | 0 | |

1 | 0 | 0 | 1 | 0 | 0 | |

0 | 1 | 1 | 0 | 1 | 1 | |

0 | 1 | 0 | 1 | 1 | 1 | |

1 | 1 | 1 | 0 | 2 | 1 | |

1 | 0 | 1 | 1 | −1 | 0 | |

1 | 1 | 0 | 1 | 2 | 1 | |

1 | 1 | 1 | 1 | 1 | 1 |

Subcase (c): When _{best}_{best}

Subcase (d): If any case arises other than (c), then the obtained results are equal to _{t}

The mathematical formulation of Case 2 is given as:

The flow chart of DBGSK is presented in

The problem is handled by using the proposed novel DBGSK algorithm. The used parameters are presented in

Parameters of DBGSK | Considered values |
---|---|

NP | 800 |

k | 10 |

_{r} |
0.9 |

p | 0.1 |

_{f} |
1 |

Maximum number of iterations | 200 |

DBGSK was run on a personal computer with Intel^{TM} i5-7200U CPU@2.50 GHz and 4 GB RAM and coded on MATLAB R2015a. To get the optimal solutions, 30 independent runs were completed. The obtained statistics are provided in

Algorithm | Best (Maximum) | Median | Average | Worst (Minimum) | Standard deviation |
---|---|---|---|---|---|

DBGSK | 14.50 | 14.50 | 14.50 | 14.50 | 0.00 |

The route provided by the optimum solution can be seen in

The main conclusions for this paper can be summarized as follows:

An optimum distribution of safeguarding substances in the context of the huge danger of a pandemic is presented. The objective is to achieve the maximum total amount of the distributed safeguarding substances to consumers in a specific time shift.

A nonlinear integer constrained mathematical programming model is formulated for the given problem, which is hard to be solved using exact algorithms, especially in large dimensions.

The mathematical model and the solution method are used to solve a real application case study for five hospitals located in El-Gharbia Governorate in Egypt.

The proposed model of the case study is solved by a novel Discrete Binary Gaining Sharing Knowledge-based optimization algorithm, which involves two main stages: discrete binary junior and senior gaining and sharing stages with knowledge factor _{f}

DBGSK has the ability of finding the solutions of the introduced problem, and the obtained results demonstrate the robustness and convergence of DBGSK toward the optimal solutions.

The suggestions for future research are as follows:

To apply the same procedure for other Governorates, other regions in the country, and other countries.

To apply the same problem formulation to other similar fields, such as industry, agriculture, business, education, telecommunications, investing, quality assurance, social and community services, pollution, medical, tourism, marketing, sales, advertising, sports, arts, cooking, and others.

To check the performance of the DBGSK approach in solving different complex optimization problems.

Other problems can be investigated by the extension of DBGSK with different kinds of constraint handling methods.

The authors are grateful to the Deanship of Scientific Research, King Saud University, KSA, for funding through the Vice Deanship of Scientific Research Chairs.