Boundedness of fractional integrals on grand weighted Herz spaces with variable exponent

1.   Introduction

War is among the deadliest disasters for humans at all levels. It accounts for about a third of all hazards globally and affects more people than other natural disasters. Wars also lead to massive loss of life and the destruction of infrastructure and resources. Diseases and famine in war zones spread quickly due to a lack of services. It is a fertile ground for the emergence of terrorist groups and armed militias and human rights are violated in these areas and people are deprived of their most basic right, which is to live in peace free of fear, hunger and disease. Therefore, humans' preservation, development and prosperity are the basis for forming sustainable development goals. Hence the goal of international organizations is to develop humanitarian plans to confront disasters, the latest of which is the COVID-19 pandemic. The need for humanitarian aid increases in places where there is more than one disaster, such as areas of conflict, famine and disease. Various emergency supplies are needed and the quantity is usually high. One common obstacle is the lack of supplies in the first phase of the rescue. Moreover, the duration of the rescue from the war is prolonged. The demand for the type and quantity of emergency supplies depends on the stage of the rescue danger.

On the other hand, Yemen, one of the poorest countries in the world, has been suffering from a complex civil war since 2015, in which many people have been killed and Yemen was recently classified as going through the worst humanitarian disaster in half a century ^[1].

The ACLED dataset indicates that from Jan 2015 to December 2021, the armed conflicts caused 155,123 fatalities in Yemen; where the number of fatalities were caused by battles (93,493 fatalities), by explosions/remote violence (58,893 fatalities), by violence against civilians (2,274 fatalities), by strategic developments (341 fatalities), by riots (77 fatalities) and by protests (45 fatalities).

According to the report issued by OCHA ^[2] on Yemen, the estimated population of Yemen is 30.8 M, where the estimated number of people in need is 20.7 M (67%). Moreover, the estimated number of people in acute need is 12.1 M (39%). The humanitarian situation in Yemen can be summarized in the following two tables:

Table 1 reports the needy people by sector, as the most significant number of people need health, then food and after that they need protection, wash, sanitation and hygiene. Most of the neediest groups are boys and girls, then women and men, respectively. Figure 1 illustrates the spread and clusters of the needy people at the governorate level.

Table 2 summarizes the data related to the humanitarian situation in Yemen (2016–2022). It can be noted that the peak of the deterioration of the situation in Yemen was 2019, but on the other hand, the funding coverage was 87% of the needs. The situation has slightly improved in 2020 and 2021 compared to 2019, but the situation is still deteriorating. The funding coverage decreased to a large extent so that the year 2021 had the least the funding coverage (57%) compared to the previous five years.

This work is inspired by some of the difficulties challenged by donor countries and humanitarian organizations in the distribution of aids and relief to affected areas in Yemen. The main concerns of donors are to ensure that relief is distributed equitably, impartially and transparently to all affected areas. Another concern is to meet all needs of people by urging donor countries to increase funds and fulfill their pledges. From this point of view, the contributions of this work are as follows:

❖ We propose a new bi-level optimization model that aims to maximize the funds sent by donors to affected people and minimize the unmet demand.

❖ We consider four dimensions in the proposed mathematical model: donors, sectors, intermediary humanitarian organizations and beneficiaries.

❖ We consider the desired level of equity and efficiency of distribution among the affected areas or at the level of sectors to ensure effective and equitable distribution among all beneficiaries.

❖ We derivate the single-level model from the proposed model using the Karush-Kuhn-Tucker (KKT) conditions.

❖ We apply the proposed model to discuss a real case study on the Yemen Humanitarian Response Plan 2021.

The following is how the rest of the article is structured: A brief review of related literature is in Section 2. The proposed mathematical model is presented in Section 3. A description of the data is provided in Section 4. The results are reported and discussed in Section 5. Finally, Section 6 presents the conclusion with the limitations and directions for future research.

2.   Related background and literature review

a.   Bi-level programming

A bi-level program is a mathematical program in which decision-making takes place at a hierarchical level and with the interaction of two decision-makers. So, the decision-maker at the upper level (the leader) seeks to find the optimal solution for the objectives under a set of constraints while considering the optimal solution for the decision-maker (the follower) at the lower-level model. The basic form of the bi-level programming is proposed by Bracken J ^[3], as follows;

where $F, f:{R}^{m}\times {R}^{n}\to R, \; G:{R}^{m}\times {R}^{n}\to {R}^{p}$ and $g:{R}^{m}\times {R}^{n}\to {R}^{q}$ are continuous and twice differentiable functions.

b.   Literature review

Humanitarian logistics is defined as the activities of planning, implementing and controlling the storage and flow of goods and information between the origin and consumption point to satisfy the needs of beneficiaries. Logistics and humanitarian relief distributions in disasters and armed conflicts face some difficulties, such as security risks, unmet needs and distribution bias. So, met demand and equity distribution are two essential goals of relief distribution during large-scale disasters. Hence, decision-makers must work to overcome these difficulties to meet the needs of the beneficiaries in all affected areas in an urgent, fair and satisfactory fashion. Moreover, particularly in conflict zones, humanitarian relief operations face obstacles, including restrictions on imports, visas and movement permits and aid delivery to many communities that need it most.

Equity distribution is one of the difficulties that are difficult to achieve in conflict areas, in which equity refers to fairness in the distribution of aids among recipients ^[4]. Equity is measured by the maximal ratio between the proportions of the satisfied demand of each pair of demand points. An equitable solution is achieved by a set of constraints, achieving the minimal percentages by certain parameters. Several studies addressed the fairness distribution in mathematical models. Some studies included fairness in the objective functions, while others included it in the constraints. Shehadeh and Snyder ^[5] reviewed the different measures of equity and then studied the static and mobile healthcare facility locations under uncertainty and fairness restrictions. A similar study in ^[6] reviewed literature that dealt with equity and analyzed some mathematical formulas for how to introduce equity into the objective functions of models, comparing and evaluating them. The authors in ^[7] measured the equity by comparing the fulfillment rates, arrival times and deprivation times. Afterward, they balanced between equity and efficiency. Then, the model was applied in the Haiti earthquake case. In ^[8], they proposed a mixed-integer model for minimizing the total cost of distributing food donations and wastage cost while maintaining maximum equitability, efficiency and effectiveness in the distribution. The authors in ^[1] designed a framework of humanitarian supply chains in conflict zones subject to the inherent risks. They applied the proposed framework in the Yemen case. In ^[9], the authors proposed a stochastic model that addressed uncertain demand and disturbances during transportation. The goals of this model were to maximize efficiency and equity. Also, genetic algorithm (GA) was used to solve the proposed model on real data obtained from the Kartal district of Istanbul. Xiaoping Li et al. in ^[10] proposed a mathematical model for distributing gasoline fairly and efficiently during natural disasters. Hurricane Sandy in New Jersey was used to test the model. Equity is achieved through defining a constraint to maximize the minimum ratio between the total quantity of outputs to the region's needs. Noham and Tzur ^[11] proposed a mathematical model that hybridizes between the design of a relief network and the study of the effect of incentives for improving humanitarian relief operations in line with the humanitarian behavioral aspects working in the network. Also, a vision was presented for how to ensure balance, equity and efficiency. The authors in ^[12] and ^[13] proposed mathematical models to maximize the amount of donated food from the food bank, a distribution that ensures equality and effectiveness among all beneficiaries. They also provided a management vision for capacity investment in collaboration with local agencies to improve the food bank's ability to achieve these equity and efficiency goals. Mohammad Firouz et al. ^[14] developed a flexible, robust model that considers efficiency and equity to achieve equity. The proposed model was tested in a food bank, which gives an administrative vision for charitable works, helping the stakeholders make optimal decisions. A similar study in ^[15] introduced a flexible, robust model considering three axes of efficiency, efficacy and equity. Then, they applied the proposed model to the home healthcare problem. In ^[16], they suggested a mathematical model for minimizing the total unmet demand for those affected by a disaster. The proposed model was formulated as a weighted total of the unsaturated demand for all affected, for all relief items and overall time periods. Constraints have been proposed to impose a minimum level of service and through which equity among disaster victims is achieved over all time periods. Enayati and Özaltın ^[17] proposed a mathematical model for an optimal distribution of influenza vaccines that ensures the quality and fairness of the distribution. Through this model, the number of doses distributed is minimized and, in turn, the outbreak of the disease is eliminated in its early stages. Mathematical models have been proposed in ^[18,19] for distributing vaccines in developing countries. The proposed models can achieve equitable distribution of vaccines. Moreover, it can select manufacturers, plan capacity, allocate orders and manage waiting time. Z. Liu et al. in ^[20] proposed a two-stage fuzzy random mixed integer optimization model using a hybrid intelligent algorithm to solve facility location problems under an uncertain environment. M.M. Miah et al. in ^[21] solved the uncertain multi-objective transportation problems. While S. Kousar et al. in ^[22] proposed a neutrosophic fuzzy multi-objective optimization and they applied the model to solve a crop production problem. S. Shiripoura and N.M. Amirib in ^[23] formulated an integer nonlinear programming (INLP) model to solve a location-allocation-routing problem for the distribution of the injured in disaster response scenarios. The authors in ^[24] developed a robust stochastic model that considers the locations of the facility and inventory and the equitable distribution. The proposed model includes two stages, the stage of determining the optimal location and capacity and the stage of scheduling the distribution that aims for fairness and for minimizing the costs of logistics services. Mollah et al. ^[25] addressed the humanitarian logistics and relief distributions during floods. The main objective is the total cost which is the sum of the cost for transporting population and relief-kits and penalty cost associated with the un-evacuated in-need population. Two methodologies are developed for the problem based on mixed-integer programming techniques and genetic algorithms. Both of the algorithms are run on the hypothetically developed data as well as real-life data and the results are compared. GA achieves a much better result. Chen et al. ^[26] addressed the relief material allocation problem based on bi-level programming including two objectives. The first objective is to minimize the weighted distribution time to deliver all relief materials, which represents the upper level. The second objective is to maximize the minimum fulfillment rate of all affected sites required for every kind of relief material, which represents the lower level. An improved differential evolution (IDE) algorithm is used to solve this model. The numerical results are compared with several conventional differential evolution algorithms. Safaei et al. ^[27] developed a robust bi-level optimization model for a supply–distribution relief network under uncertainty in demand and supply parameters. In the upper-level of the hierarchy, the number and location of transfer depots and the amount of victims' demand for relief commodities are determined with the aim of minimizing logistics costs and maximizing the satisfaction at demand points. Whereas the lower-level of the hierarchy identifies convenient suppliers with the lower risk and determines the optimal order. It aims to minimize the supply risk and satisfy demand under disaster scenarios. Saranwong S and Likasiri ^[28] developed a robust bi-level optimization model for an integrated model of distribution and production processes. Optimizing the distribution centers (DC) locations and allocating supplies to minimize the total cost represents the upper-level model, while minimizing the total transportation cost for all customers represents the lower-level model. Five hybrid (meta) heuristic methods are proposed to solve each level of the problem. Safaei et al. ^[29] reformulated the bi-level programming as a single-level linear problem and used the goal programming for solving the model. The upper level aims to minimize total operational cost and total unsatisfied demand considering the effect of distribution locations of relief supplies, while the lower-level aims to minimize the total supply risk. Camacho-Vallejo et al. ^[30] proposed a bilevel model to minimize both the total response time and the total cost. Moreover, they reduced the model into a nonlinear single-level mathematical model to solve it. Shokr et al. ^[31] proposed a robust bi-level model to minimize both relief chain costs and unmet demand. Also, they solved the model using the developed Benders decomposition algorithm and applied the model using a real-world example. Cao et al. ^[32] proposed a fuzzy bi-level model for multi-period post-disaster relief distribution. Three functions were minimized in the upper-level model, namely the unmet demand rate, potential environmental risks and emergency costs. Survivors' perceived satisfaction was maximized on the lower level. Xuehong Gao in ^[33] proposed a bi-level stochastic mixed-integer nonlinear model where the aim of the upper level is to minimize the total dissatisfaction level, while the aim of the lower level is to minimize transportation time. Xueping Li et al. ^[34] proposed a mathematical model to maximize the size of relief items in disaster areas subject to the cost constraints and distribution facilities in order to cover the needs of the most affected people. The authors in ^[35,36] proposed bi-objective stochastic optimization models considering multi-commodity to minimize the total transportation time and maximize the fairness by minimizing the unmet demand.

An interesting study in ^[37] proposed a bi-level multi-objective scenario-based model that takes into account public donations, efficiency, supply risks, optimal selection of suppliers, coverage of the demand and the optimal facility locations. Hezam in ^[38] proposed an optimization model to maximize the funds and minimize the unmet demand in COVID-19 global humanitarian response plan with equity constraints.

c.   Research gaps

Reviewing the literature, it appears that the bi-level optimization model that aims to minimize the unmet demand by maximizing the funds sent from donors and considering multiple sectors fairly and effectively has not yet been studied. Herein, we propose a new bi-level model considering the amount of funds sent from donor countries and the extent to which they meet the demand for each affected region from several sectors. Four dimensions were taken in this study: donors, intermediary humanitarian organizations, sectors, as well as a number of affected areas. It is also desired that the proposed model distributes funds fairly and effectively at the level of regions and sectors.

3.   Proposed model

During wars, infrastructure is destroyed, the economy is disrupted and sources of income are cut off, which directly causes great harm to all people in these areas. This situation calls for the intervention of humanitarian organizations to provide rapid relief assistance to the affected people.

The humanitarian response plan consists of four phases. In the first phase: in order for humanitarian organizations to carry out their whole duty, they must know the extent of the disaster through field surveys and identify the needs of each region from each sector. In the second phase: humanitarian organizations launch appeals for fundraising from donor countries. Furthermore, the donor countries, in turn, pledge to support organizations with the funds. However, sometimes there are some difficulties as:

● There is a lack of sufficient funds (Due to such a scenario, we will try to minimize the unmet demand).

● There is a failure to fulfill pledges (Motivated by such a circumstance, we will try to maximize the funds sent from donor countries and humanitarian organizations and appeal to them to increase the funds).

● Only certain regions are supplied for regional, political, or other considerations (As a result of such scenarios, we will set a percentage to ensure fairness and effectiveness of distribution among all governorates).

In the third phase: the humanitarian organizations, after receiving the funds, send them to the affected areas through local agents. In the final phase, the beneficiaries receive their needs, which alleviate their suffering and achieve sustainable development goals.

In this model, we will assume that we have a number of donor countries indexed by $i\in I$, and the funds will be sent to a number of humanitarian organizations indexed by $j\in J$ to cover a number of sectors indexed by $l\in L$ for covering the needs of a number of regions indexed by $k\in K$. Let ${P}_{i}$ be the maximum funds that can be sent by the donor country $i$, and let ${H}_{il}$ is the maximum funds that can be sent by the donor country $i$ for covering the sector $l$ (We need this because some donor countries only support some sectors and not others, such as the health sector and confronting COVID-19). We assume that ${QP}_{l}$ is the maximum funds that can be sent for the sector $l$. We denote to the requirements of the humanitarian organization $j$ from the sector $l$ by ${Q}_{jl}$. Let ${D}_{kl}$ denote to the requirements of the affected area $k$ from the sector $l$, let ${D}_{k}$ denote to the requirements of the affected area $k$ from all sectors and let ${D}_{l}$ denote to the requirements of the sector $l$ for all affected areas. We define ${x}_{ijl}$ as the nonnegative decision variable which represents the amount funds sent by donor country $i$ to humanitarian organization $j$ for covering the sector $l$. Also, we define ${y}_{jkl}$ is the nonnegative decision variable which represents the amount funds received by affected area $k$ from the humanitarian organization $j$ for covering the sector $l$. The corresponding network of the humanitarian supply chain is illustrated in Figure 2.

The list of all nomenclatures is defined below:

Nomenclature.

Sets:

$I$      Set of all the donor countries, indexed as $i\in I$;

$J$     Set of all the humanitarian organization, indexed as $j\in J$;

$K$     Set of all the governorates of Yemen, indexed as $k\in K$.

$L$     Set of all the sectors type, indexed as $l\in L$.

Decision variables:

${x}_{ijl}$     The mount funds for sector $l$ from the donor country $i$ to the humanitarian organization $j$;

${y}_{jkl}$     The mount funds for sector $l$ from the humanitarian organization $j$ to the governorate $k$;

Parameters:

${w}_{ijl}$     The weight of priority to send fund for sector $l$ from the donor country $i$ to the humanitarian organization $j$;

${w{'}}_{jkl}$     The weight of priority to send fund for sector $l$ from the humanitarian organization $j$ to the governorate $k$;

${D}_{kl}$     The demand of the governorate $k$ for each sector $l$ where each governorate's demand is proportional to its population;

${D}_{l}$     The total demand of the sector $l$ for all governorates;

${D}_{k}$     The total demand of the governorate $k$ for all sectors;

${P}_{i}$     The maximum funds from the donor country$i$;

${F}_{l}$     The minimum funded for the sector $l$;

${QP}_{l}$     The maximum funds for the sector $l$;

${DF}_{j}$     The Total funding for the humanitarian organization $j$;

$\beta$     Parameter of deviation from the rate of total demand;

$\theta$     Parameter of deviation from the needs of each other governorates;

${\omega }_{min}$     The minimum level of governorate $k$ satisfaction;

${\sigma }_{min}$     The minimum level of sector $l$ satisfaction;

${\tau }_{min}$     The minimum level of governorate $k$ satisfaction of the sector $l$;

${\pi }_{min}$     The minimum level of the utilization rate, where ${\pi }_{min} = 1$ at the perfect efficiency

$\mu$     Lagrange multipliers

$\bar{G}$     The percentage between satisfied demand at the total demand for all governorates

${G}_{k}$     The proportion between met demand at the total demand for each governorate $k$.

The proposed model:

Upper-Level Model:

Lower-Level Model:

The bi-level problem is defined by Constraints (2)–(17). In (2), the objective function of the upper level appears and it shows the leader wanting to minimize the unmet demand. Constraint (3) ensures that humanitarian organizations $j$ can not send more than the funding obtained from donor countries.

Constraint (4) indicates that the difference between the met demand rate for each governorate $k$ and the total met demand rate does not exceed $\beta$, while constraint (5) specifies that the absolute difference in the ratio of demand fulfilled between any two governorates. That means the difference between the met demand rate of the governorate $k$ and the met demand rate of governorate ${k}^{{'}}, k\ne k{'}$does not exceed the $\theta$.

To simplify, we can rewrite the constraints (4), (5) as:

which can also be simplified more as:

At perfect equity these constraints become

This means that each governorate must receive the same fraction of its needs at the perfect equity point.

Constraint (6) imposes a minimum percentage (${\omega }_{min}$) for the total funds sent to the governorate $k$, while constraint (7) sets a minimum percentage (at least ${\sigma }_{min}\%$) covering sector $l$. Also, constraint (7) specifies a minimum percentage (at least ${\tau }_{min}\%$) of needs governorate $k$ from the sector $l$. Constraints (4)–(8) present the equity constraints. The equity was specified as the maximum of the minimum ratio of total met demand over the total demand.

Expressions (9)–(17) makes this problem a bi-level programming model; hence, Expression (9) is called the objective function of the lower level that indicates the desire to maximize the funds send by the donors to the humanitarian organizations and then to governorates of Yemen. Constraint (10) states that the total funds received by humanitarian organizations $j$ from the sector $l$ are same as the total funds sent to affected areas. Constraint (11) reflects the minimum funding of the sector $l$, while constraint (12) reflects the maximum requirements of the sector $l$. Constraint (13) guarantees a donor $i$ cannot send more than the available funding. Constraint (14) is related to the efficiency level, where the perfect efficiency rate equals one. Constraint (15) ensures that the donor countries must send to the humanitarian organizations $j$ more than the receive in funding. Finally, constraints (16, 17) indicate the non-negativity for the decision variables related to the fundings. On the other hand, and due to the bi-level mathematical model being an NP-hard problem and classified as a complex model, the computational complexity of bilevel programming problems is exceptionally high especially when solving large-scale and high-dimensional practical applications, such as with the humanitarian relief distribution. The authors in ^[39,40] showed that the natural complexity of the bilevel problem is $\sum _{k}^{P}hard$, where $k$ is the k^th level of the polynomial hierarchy.

Derivation the single-level model

The Lagrangian function associated with the lower model (9)-(17) can be defined as:

Both necessary and sufficient KKT conditions for the optimality in the lower model can be used to convert the bi-level model to its single-level model, which is easy to solve. Hence, the following four KKT conditions are replaced by the lower-level model.

Stationarity constraints: this kind of constraint is directly derived from the Lagrangian function (26). Here, the gradient of the Lagrangian function concerning the lower-level decision variables must be equal to zero.

Primal feasibility constraints: The KKT primal feasibility conditions imply that the lower-level constraints should be satisfied with the optimal value of the variables. These consist of constraints (10)–(17).

Complementary slackness conditions: these conditions define the general relationship between primal constraints and their associated Lagrange multipliers, in which the multiplication of the slack variables in the primal constraints and the respective multipliers are equal to zero. We formulated the primal constraints (11)–(15) as constraints (28) to (32).

Dual feasibility constraints: The KKT dual feasibility conditions ensure the feasibility of the optimal solution to the dual problem. Hence, the Lagrange multipliers associated with greater than or equal to zero constraints must be defined as in (33), while the Lagrange multipliers associated with other constraints are unrestricted in sign.

Therefore, the single-level formulation is obtained by:

The upper-level objective function (Equation (2)).

Subject to:

● The upper-level constraints (Equations (3) to (8))

● The primal feasibility constraints (Equations (10) to (17))

● The stationarity constraints (Equation (27))

● The complementary slackness constraints (Equations (28) to (32))

● The dual feasibility constraints (Equation (33))

4.   Case study

Before the investigation of the case study, we will implement the proposed model on a simple test example to ensure the validity and effectiveness of the proposed model. In this example, there are no unmet demands. Therefore, it is expected that all affected areas will receive their full requirements, which makes the satisfying rates of equity and efficiency distribution among aid recipients equal to one.

4.1.   Simple example

In this subsection, our goal is to test a simple example for illustrating the validity and performance of the proposed method. In this example, we assumed two donors, two humanitarian organizations, two sectors and three affected areas.

Set ${P}_{1} = 200, {P}_{2} = 100$, ${DF}_{1} = 150, {DF}_{2} = 150$, ${QP}_{1} = 200, Q{P}_{2} = 150$, ${F}_{1} = 180, {F}_{2} = 120$. Further, we assumed the other related parameter values as given in Table 3.

The results of the optimal distribution plan for this example are shown in Tables 4 and 5.

We can see clearly from the obtained results that equitably and effective distribution was achieved because the unmet demand in this example equals zero and the donors' funds covered all requirements of the affected areas. Hence, the results of this simple test example confirm the robustness and performance of the proposed model.

4.2.   Data description of the case study

Although the proposed model is applicable to various humanitarian response plans, with some minor adaptations, here we apply it to the humanitarian response plans in Yemen 2021. We now provide a case study on Yemen to validate the model. Yemen is a country that is situated at the southern end of the Arabian Peninsula in Western Asia. It has a total area of 527,970 sq. km. and an estimated population of 30,041,712. Yemen consists of twenty-two governorates are Abyan, Aden, Al Bayda, Al Dhalee, Al Hudaydah, Al Jawf, Al Maharah, Al Mahwit, Amanat Al Asimah, Amran, Dhamar, Hadramaut, Hajjah, Ibb, Lahj, Marib, Raymah, Sa'ada, Sana'a, Shabwah, Socotra and Taizz. Data are analyzed that were gathered in 2021 from a financial tracing service (https://fts.unocha.org/appeals/1024/summary) FTS.

We consider the largest ten sources of the response plan, namely: United States of America, Saudi Arabia, Germany, United Arab Emirates, European Commission, World Bank, United Kingdom, Japan, Central Emergency Response Fund and Canada, and we collect the funds from other donors in one named as "other donors".

Table 6 presents the largest donors and their funding.

Moreover, Table 7 summarizes the overall funding, funded, Required fund, and the unmet requirement for Yemen 2021.

The donor countries send the funds to non-profit international or national organizations; more than 250 such organizations are working in Yemen. In this work, we considered the five top organizations and the rest of the organizations are listed under the name "others". Table 8 lists the details of the humanitarian organizations.

We also consider nine categories of sectors: Food Security and Agriculture; Nutrition; Health; WASH, Sanitation and Hygiene; Education; Protection; Shelter and NFI; Camp Coordination & Camp Management; Refugees and Migrants Multisector; and other clusters/sectors (shared). Table 9 reports the requirements for each sector and the existing funding with its percentage coverage.

Since the actual demand in each governorate is challenging to identify due to many factors affecting demand, the reasonable assumption is to consider the needs in each governorate to be correlated with the estimated population of that governorate. So, the governorate's demand was calculated in proportion to its population from the total need for this sector. Table 10 lists the estimated demand of each governorate from each sector.

5.   Data analysis and results

We assume that all weights ${w}_{ijl} = {w\mathrm{{'}}}_{jkl} = 1$ and the minimum levels of ${\sigma }_{min}, {\tau }_{min}$ and ${\omega }_{min}$ are equal to 40%; we found by experimentation that this value is the best percentage. The mathematical model was implemented by LINGO 18 software. We can see clearly from Table 11 that each governorate received at least 40% of its needs. It can be noted that the governorates with the smallest needs obtained the highest rates, such as the governorates of Al Maharah, Raymah and Socotra, which got a rate of 73%. Meanwhile, the large governorates with the highest needs got low percentages. For example, Hajjah got only 40% of its needs, Amanat Al-Asimah and Aden got 45% of its needs and the rest of the governorates got different rates, between 40% and 73%. Moreover, we can see that all governorates received 58% of the total needs.

Figure 3 depicts the obtained results for the Yemen case. Further, Tables 15–25 (in Appendix A) show the funds sent by each donor to humanitarian organizations for all sectors, and we notice that the total amount sent by each donor is equal to the funds granted by the donor, and this means that they sent all the funds they donated, which indicates that the efficacy rate is 100%.

Tables 26–47 (in Appendix B) report the funds received by each governorate from humanitarian organizations to cover each sector. Due to the lack of funding, some sectors were not covered in some governorates, so we can redistribute the governorate's share to include the most necessary sectors. This allows flexibility for the decision-maker to redistribute the funds in the governorate to spend in the necessary sectors because, in some governorates, sectors are more important than others, unlike other governorates. For example, coverage of the displaced sector in Ma'rib governorate is very important to cover due to the large number of displaced people in this governorate.

Table 12 shows the relationship between the results, the requirements and the funding for each sector.

We can clearly see that the model's results covered 58% of the requirements as a total, and in return, the funds sent, according to the model's results, increased by 30% compared with the actual funding. All sectors increased by up to 50% relative to the actual funding, except for the food security and agriculture sector, which increased by 17%. On the other hand, there is a disparity between sectors compared to their requirements due to that available funds only covered 58%.

Sensitivity analysis

To analyze how the minimum levels of ${\sigma }_{min}, {\tau }_{min}$ and ${\omega }_{min}$ values affect the distribution ratio in each affected area, six values of ${\sigma }_{min}, {\tau }_{min}$ and ${\omega }_{min}$ were tested and Table 13 shows the best minimum level values of ${\sigma }_{min}, {\tau }_{min}$ and ${\omega }_{min}$ are 40%, where all affected areas received at least 40% and the total received for all affected was the highest among other ratios.

Moreover, if we raise the percentage to 60% or 70%, or reduce it to 25% or 30%, the distribution will be uneven between the governorates, as some of them will get higher percentages, especially the governorates with the slightest need. This uneven distribution is due to the lack of funds sent by donor countries. Hence, we found that the best percentage is 40% to reduce disparities and achieve a balance between governorates.

Overall, we can conclude the following main points:

❖ Equity can be achieved more as we increase the values of equity parameters.

❖ It requires sending more funds from donor countries the more equity parameters we increase.

❖ Unmet demand decreases as donors increase funds.

❖ The proposed model ensures that 40% of each governorate's needs are met. The minimum rate of equity is dynamic so that the decision-maker can change it according to the availability of funds.

❖ The weights can be adjusted based on other factors such as the interests and competencies of any humanitarian organization, such as the WHO, which will have a higher weight in the field of health than others, as well as the WFP will contribute to meeting the food sector more and therefore requires receiving more aid for this sector. The weights of each governorate depend on the security level and humanitarian situation, the displaced, the stability of the local government and other factors.

❖ The model results indicate the importance of reducing unmet demand and increasing funding from donor countries.

❖ Despite the importance of available sufficient funding to cover all needs, we also point out that the distributive fairness is fundamental, especially in the most deserving regions.

6.   Conclusions

Our study proposed a novel bi-objective optimization model for examining the Yemen Humanitarian Response Plan 2021, where the actual data have been collected from FTS. Both level models aim to achieve fairness and effective distribution among the Yemeni governorates, minimize the unmet demand and maximize the funds granted by donor countries and intermediary UN organizations. Our results provide minimum distributional fairness of 40% to satisfy the governorates' demand based on the purely humanitarian aspect, away from political or regional tendencies. Furthermore, we have noted some limitations of our study that may help shape future research directions. Finally, we hope that our study provides insight into the importance of equitable and efficient distribution, meeting unmet demand and understanding the humanitarian response plan, which will better reflect on the effectiveness of humanitarian relief efforts.

6.1.   Limitations and further research directions

The humanitarian response plan, in general, is complex and depends on many overlapping factors, the most prominent of which is an appeal to donor countries to increase grants and fulfill their pledges. The plan also faces security, humanitarian and political risks, especially in war zones, which calls for a rapid response to such risks and more cooperative efforts between humanitarian organizations.

Therefore, reality cannot be simulated with a mathematical model that can be easily solved. Nevertheless, the proposed model presents a vision and broad lines for equitable distribution among the governorates. Hence, we recommend that donors and international and local organizations take advantage of the proposed mathematical models to achieve the minimum level of fairness, effectiveness and flexibility based on feedback for funding the urgent sectors.

Some main Limitations are considered as directions of future studies as:

❖ Although the proposed model is general and can be applied to most similar cases with some minor modifications, we used data specific to Yemen, and for this reason, the model can be applied to similar cases, especially since data are available for most countries to achieve a certain level of fair and effective distribution and minimize the unmet demand.

❖ We considered only ten donors, six non-profit humanitarian organizations, ten sectors and twenty-two governates. So, more than these nodes of the network model can consider in the future works.

❖ We considered the demand of each governate according to its populations only. We do not take considering the security factor and other factors. Recently, Yemen has been ruled by many conflicting governments in different areas of Yemen, so the regions differ from each other in terms of security and economics, the availability of job and salary opportunities, the difficulty of moving between them and the presence of refugees and displaced persons in some areas. Therefore, these factors are essential to be considered in future studies in determining the actual needs for each area. However, the study was relied upon by ^[1]. Also, there are difficulties in measuring and identifying these factors easily, and the ratio of the estimate to the actual estimate will be inaccurate.

❖ We considered certain data, and some parameters can be considered as uncertain data. As we referred to before, the difficulties in estimating each sector's actual demand for each governorate. So, the building mathematical model considering the uncertainty (robust, fuzziness, rough, etc.) is important to future works.

❖ The present model minimized unmet demand as the objective function in the upper-level model and maximized the sent funds in the lower-level model. Hence, adding more objective functions into both level models as minimizing the delivery times and minimizing the emergency risks will be promising topics in future works.

❖ The proposed model had been solved by LINGO 18 software "Hyper version". So, introducing an efficient solution approach as a metaheuristic approach for larger instances will be interesting work in future work.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Data Availability Statement

The data presented in this study are available on https://fts.unocha.org/appeals/1024/summary).

Acknowledgements

The authors extend their appreciation to the Deputyship for Research & Innovation, Ministry of Education in Saudi Arabia for funding this research work through the project no. (IFKSUOR3–037–1).

We should like to thank the Editors of the journal as well as the anonymous reviewers for their valuable suggestions that make the paper stronger and more consistent.

Conflicts of Interest

The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

Appendix A

Appendix B