# Healthcare’s Sustainable Resource Planning Using Neutrosophic Goal Programming

Authors: Ibrahim M. Hezam; Sarah A. H. Taher; Abdelaziz Foul; Adel Fahad Alrasheedi
Journal: Journal of Healthcare Engineering
 (2022)

---

## Abstract

We develop neutrosophic goal programming models for sustainable resource planning in a healthcare organization. The neutrosophic approach can help examine the imprecise aspiration levels of resources. For deneutrosophication, the neutrosophic value is transformed into three intervals based on the truth, falsity, and indeterminacy-membership functions. Then, a crisp value is derived. Moreover, multi-choice goal programming is also used to get a crisp value. The proposed models seek to draw a strategic plan and long-term vision for a healthcare organization. Accordingly, the specific aims of the proposed flexible models are meant to evaluate hospital service performance and to establish an optimal plan to meet the growing patient needs. As a result, sustainability’s economic and social goals will be achieved so that the total cost would be optimized, patients’ waiting time would be reduced, high-quality services would be offered, and appropriate medical drugs would be provided. The simplicity and feasibility of the proposed models are validated using real data collected from the Al-Amal Center for Oncology, Aden, Yemen. The results obtained indicate the robustness of the proposed models, which would be valuable for planners who could guide healthcare staff in providing the necessary resources for optimal annual planning.


---

## Body

## 1. Introduction

Optimizing existing resources in health organizations is critical to meeting the needs of patients. At the same time, organizations must develop a future strategic plan of the existing resources commensurate with the predicted growth in the number of patients. Optimization approaches can support planning and decision-making at all levels. One of these approaches is goal programming in resources planning. Goal programming is a multi-objective optimization tool that helps a solution to move toward an ideal goal. In recent years, the use of goal programming has become more widespread, especially for analyzing and evaluating healthcare organizations. Numerous authors have considered goal programming to optimize the resources in health organizations. Parra et al. (1997) [1] proposed a goal programming model to evaluate the performance of a surgical service at a local general hospital. The authors aimed to improve the service under the available resources—for example, spatial occupation, staff availability, and financial support. Munoz et al. (2018) [2] improved a mathematical model based on goal programming to evaluate proposals in order to help in the selection of a mix of proposals. The main function of goal programming is the incorporation of strategic goals that support the vision and objectives of institutes. The model is applied using real data obtained from the Clinical and Translational Science Institute (CTSI) at Pennsylvania State University. Blake and Carter (2002) [3] proposed two linear goal programming models. The first model is used to determine case mix and volume for physicians under fixed service costs conditions. The second model addresses case-mix decisions as a commensurate set of practice changes for physicians. The trade-offs between case-mix and case costs are balanced using the proposed models and minimizing disturbance in order to preserve physician income. Rifai and Pecenka (1990) [4] applied goal programming to allocate resources in healthcare planning. Minimizing idle capacity and maximizing the profit are the main goals in this work. Kwak and Lee (2002) [5] proposed a multi-criteria mathematical programming model to evaluate strategic planning in a business process. This model is also based on goal programming. The goal levels are identified and prioritized using the analytic hierarchy process. Similarly, Lee and Kwak (1999) [6] presented a goal programming model for designing and evaluating effective information resource planning in a healthcare system. The proposed model addressed the multiple conflicting goals of a healthcare system and the multi-dimensional aspects of resource allocation planning. Also, the model allows flexibility of decision-making in resource allocation. The goals are decomposed and prioritized with respect to the corresponding criteria using the analytic hierarchy process. Grigoroudis and Phillis (2013) [7] proposed a nonlinear programming model to improve the health level of a population under several constraints. The system of systems approach is used to model the hierarchical structure of healthcare systems as well as for dynamic budget allocation for a national healthcare system in order to develop optimal policies. Lee and Kwak (2011) [8] developed a multi-criteria decision-making model for strategic enterprise resource planning adoption by considering both financial and nonfinancial business factors. Turgay and Taşkın (2015) [9] developed a fuzzy mixed goal programming model to optimize the healthcare organization’s resource allocation problem in an uncertain environment.The use of neutrosophic concept theory in goal programming was first introduced by Hezam et al. (2016) [10]. For more information on the works related to neutrosophic goal programming, see Munoz et al. (2018), Islam and Kundu (2018), Maiti et al. (2019), Sarma et al. (2019), Dey and Roy (2017), Pramanik and Banerjee (2018), and Al-Quran and Hassan (2017) [2, 11–17].At the same time, in conflict zones and poor communities, health organizations face difficulty meeting the population’s needs [18]. There is an urgent need to provide excellent and comprehensive high-quality service for all patients under limited resources. In this regard, cancer is one of the most common diseases globally. Its treatment requires specialized experts, medical drugs, as well as exorbitant costs and time. In recent years, the number of people needing oncology services has increased significantly, especially in Yemen. Particularly, the country’s growing population is facing health threats from the hazardous habit of chewing khat as well as pesticides used for growing the khat plant. The lack of health education, as well as early detection of tumors, also exacerbates the issue. Moreover, Yemen is a poor country that is suffering from ongoing conflicts. It has a dearth of human, material, and financial resources. Hence, the number of specialized oncology centers in Yemen is limited and, thus, does not meet the needs of oncology patients. As a result, most patients travel abroad for treatment. The Al-Amal Center for Oncology is one such specialized center for the treatment of tumors in Yemen. A large number of patients are treated by these health centers, which have limited resources. These complexities are reflected in the health facilities, making our data inaccurate, lacking, or ambiguous. Hence, we use the neutrosophic concept in this study.In this article, we propose neutrosophic goal programming models for evaluating and optimizing the existing resources of health organizations and for optimal future planning. The main strategic goals to design the proposed models are (a) optimizing the center’s resources; (b) matching the center service with the requirements of patients by providing high-quality services and appropriate medical drugs as well as reducing the waiting time; and (c) planning to meet the ideal center requirements by increasing the center capacity according to the predicted growth of patients. The real data are obtained from the Al-Amal Center for Oncology, Aden, Yemen. We use the data to validate the proposed models.The remaining paper is organized as follows. Section2 provides an overview of goal programming, dynamic goal programming, and multi-choice goal programming. Besides, it presents neutrosophic concepts and deneutrosophication. In Section 3, we formulate the proposed models, while a case study is presented in Section 4. Section 5 reports the results and discussion, and then Section 6 concludes the study.


## 2. Theory

### 2.1. Goal Programming

Goal Programming is the most known method in multiple-criteria decision-making, proposed by Charnes and Cooper (1961). Goal programming is a generalization of linear programming that handles multiple conflicting objective measures, where a target is set for each measure to be achieved. The new objective function, or the “achievement function,” seeks to minimize unwanted deviations from aspiration levels or a set of target values.We can introduce two types of constraints in goal programming: hard and soft constraints. Hard constraints are the system constraints that cannot be violated (e.g., system resources and relational model constraints). In contrast, soft constraints are associated with prespecified targets. Both negative and positive deviational variables are added to these constraints and the undesired deviations are included in the objective function to be minimized.The priority of goal programming is to satisfy the hard constraints before soft constraints. The preference structures to minimize the undesired deviations require different methods: preemptive, nonpreemptive, and Tchebycheff. The preemptive variant is used when there is a natural priority structure to the decision-maker(s) preferences, the nonpreemptive variant is used when each unwanted deviation has a relative weight (which can be equal), and Tchebycheff goal programming is used when the maximum deviation from the target is minimized.Consider the following model:(1)z=min∑l=1Lwlnl+pl,(2)flx+nl−pl=gl,∀l∈L,(3)nl×pl=0,∀l∈L,(4)Ax≤b,(5)x≥0,nl,pl≥0,∀l∈L,where gl is specific goal of the objective function flx∀l∈L.wl is the penalty weight. nl and pl are the under- and upper achievements of the lth goal, respectively. Equation (1) represents the objective function that minimizes the sum of the positive/negative deviations for each goal. Equation (2) is related to the decision-maker’s goals and computes the respective positive and negative deviations from each goal. Equation (3) ensures that at least one of the deviations must be equal to zero. Equation (4) relates to the system constraints in the decision space. Equation (5) ensures that all decision variables are nonnegative.


### 2.2. Dynamic Goal Programming

Dynamic goal programming allows for a target value to be dynamic. This approach is used to evaluate along a planning period. References [19–21] addressed the dynamic goal programming where the target values are changed as per the planning period:(6)z=min∑l=1L∑t=1Twtnlp+plp,flx+nlt−plt=glt,∀l∈L,∀t∈T,nlt×plt=0,∀l∈L,∀t∈T,Ax≤b,x≥0,nlt,plt≥0,∀l∈L,∀t∈T,where glt is specific goal of the objective function flx∀l∈L per period t.


### 2.3. Multi-Choice Goal Programming

Multi-choice goal programming was proposed by Chang (2007) [22]. In this case, decision-makers are allowed to define multi-choice aspiration levels for each target. The decision-maker can use the multi-choice model as a decision aid to make better decisions for a given problem. The mathematical model is as follows:(7)min∑l=1Lflx−gl1orgl2or…orglm.Subject to(8)X∈FFis a feasible set.Here, there are multi aspiration levelsgl1orgl2or…orglm. This model can be reformulated as:(9)minnl+pl,flx+nl−pl=∑j=1mbljSljB,∀l∈L,SljB∈Rlx,∀l∈L,nlt×plt=0,∀l∈L,∀t∈T,Ax≤b,x≥0,nlt,plt≥0,∀l∈L,∀t∈T,where blj is the jth aspiration level of the lth goal and SljB indicates the function of the binary serial number; Rlx is the function of resources boundaries. For three aspiration levels, the first constraint of model (9) can be reformulated as the following constraints:(10)flx+nl−pl=gl1b1b2+gl2b11−b2+gl31−b1b2,b1+b2>0,b1,b2∈0,1.Constraint (10) makes at least b1 or b2 not equal zero. Therefore, only three choices—gl1, gl2, or gl3— are yielded.


### 2.4. Neutrosophic Concept

In real applications, the uncertainty of the parameters is common in mathematical computations. Uncertainty arises owing to imprecise and inconsistent data. Zadeh proposed the fuzzy theory in 1965 to deal with these kinds of data [23]. However, fuzzy sets consider only the truth-membership function that is unable to efficiently represent accurate information. A new membership function, called falsity-membership function, was later developed by Atanassov (1986) [24], who introduced the intuitionistic fuzzy set. Nevertheless, this technique too was limited by drawbacks in decision-making. In 1998, Smarandache [25] introduced a new concept named “neutrosophic” that considers three memberships functions: truth, indeterminacy, and falsity.


#### 2.4.1. Neutrosophic Set

Definition 1.
LetX≠∅ be a universe set. A neutrosophic set A in X is characterized by a truth-membership function μA˜N, an indeterminacy-membership function σAN˜, and a falsity-membership function νAN˜:(11)A˜N=x,μA˜Nx,σA˜Nx,υA˜Nx;x∈X,where μA˜N:X⟶0−,1+, σA˜N:X⟶0−,1+, and υA˜N:X⟶0−,1+ represent the degrees of the truth-, indeterminacy-, and falsity-membership functions, respectively. No restriction exists on the sum of μA˜N, σA˜N, and νA˜N. Thus, 0−≤μA˜Nx+σA˜Nx+νA˜Nx≤3+ for x∈XDefinition 2.
A setα,β,γ−cuts, generated by A˜N, where α,β,γ∈0,1 are a fixed number such that α+β+γ≤3 is defined as:(12)A˜α,β,γN=x,μA˜Nx,σA˜Nx,υA˜Nx;x∈X,μA˜Nx≥α,σA˜Nx≤β,υA˜Nx≤γ;α,β,γ∈0,1,where α,β,γ−cuts, denoted by A˜α,β,γN, is defined as the crisp set of elements x that belong to A˜N at least to the degree α and that belongs to A˜N at most to the degree β and γ.


#### 2.4.2. Generalized Triangular Neutrosophic Number

A generalized triangular neutrosophic number (GTNN)τ˜aN=a,lμ,rμ;wa,a,lσ,rσ;ua,a,lν,rν;ya is a special neutrosophic set on a real number set ℜ whose degree of truth, indeterminacy, and falsity are given by:(13)μτ˜aNx=x−a+lμlμa−lμ≤x<awax=aa+rμ−xrμa<x≤a+rμ0otherwise,στ˜aNx=a−x+uax−a+lσlσa−lσ≤x<auax=ax−a+uaa+rσ−xrσa<x≤a+rσ1otherwise,ντ˜aNx=a−x+yax−a+lνlνa−lν≤x<ayax=ax−a+yaa+rν−xrνa<x≤a+rν1otherwise,where lμ, rμ, lσ, rσ, lν, and rν are called the spreads of the truth-, indeterminacy-, and falsity-membership functions, respectively; and a is the mean value. wa represents the maximum degree of the truth-membership function, while ua and ya represent the minimum degrees of the indeterminacy- and falsity-membership functions, respectively, such that they satisfy the conditions below:(14)0≤wa≤1,0≤ua≤1,0≤ya≤1,0≤wa+ua+ya≤3.


#### 2.4.3.α,β,γ−Cut Set of GTNN

Definition 3.
Anα,β,γ−cut set of GTNN τ˜aN=a,lμ,rμ;wa,a,lσ,rσ;ua,a,lν,rν;ya is a crisp subset of ℜ, which is defined as:(15)τ˜Naα,β,γ=x:μτ˜Nax≥α,στ˜Nax≤β,υτ˜Nax≤γ,where 0≤wa≤1,0≤ua≤1,0≤ya≤1, and 0≤wa+ua+ya≤3.
Anα−cut set of a GTNN τ˜aN is a crisp subset of ℜ, which is defined as:(16)τ˜Naα=x:μτ˜Nax≥α,where 0≤α≤wa.
According to the definition of GTNN, it can be easily shown thatτ˜Naα=x:μτ˜Nax≥α, is a closed interval, defined by τ˜Naα=aLα,aRα where aLα=a−lμa+αlμa/wa and aRα=a+rμa−αrμa/wa. The mean of τ˜Naα is(17)amidα=a+rμa−lμa21−αwa.
Similarly, aβandγ−cut set of GTNN τ˜aN is a crisp subset of ℜ, which is defined as(18)τ˜Naβ=x:στ˜Nax≤β,τ˜Naγ=x:υτ˜Nax≤γ.where ua≤β≤1,ya≤γ≤1
It follows from the definition thatτ˜Naβ and τ˜Naγ are closed intervals, denoted by τ˜Naβ=aLβ,aRβ and τ˜Naγ=aLγ,aRγ, which can be calculated as:aLβ=a−lσa+1−βlσa/1−ua and aRβ=a+rσa−1−βrσa/1−uaaLγ=a−lνa+1−γlνa/1−ya and aRγ=a+rνa−1−γrνa/1−ya
Thus, the means ofτ˜Naβandτ˜Naγ are:(19)amidβ=a+rσa−lσa2β−ua1−ua,(20)amidγ=a+rνa−lνa2γ−ya1−ya.


#### 2.4.4. De-Neutrosophication

In this work, the neutrosophic parameters will be treatment using two methods. In the first method, the crisp value will be the average of the mean of the three intervals obtained from theα,β,γ−cut set of GTNN. The crisp value can be calculated as equation (21).(21)τNaα,β,γ=13amidα+amidβ+amidγ,=a+rμa−lμa61−αwa,+rσa−lσa6β−ua1−ua,+rνa−lνa6γ−ya1−ya.It can be easily proven that forτ˜aN=a,lμ,rμ;wa,a,lσ,rσ;ua,a,lν,rν;ya∈GTNNℜ and for any α∈0,wa, β∈ua,1,γ∈ya,1, where 0≤α+β+γ≤3:(22)τ˜Naα,β,γ=τ˜Naα∧τ˜Naβ∧τ˜Naγ,where the symbol ∧ denotes the minimum among τ˜Naα,τ˜Naβandτ˜NaγFigure1 illustrates the membership functions for a generalized triangular neutrosophic number (NN).Figure 1
Graphical representation of membership functions for NN.In the second method of the deneutrosophication, we employ multi-choice goal programming to select a crisp value in the three obtained intervals that led to the truth-, indeterminacy-, and falsity-membership functionsaLα,aRα, aLβ,aRβ, and aLγ,aRγ, respectively. Thus, the multi-choice model will be employed to select one value from the three mean values that have been obtained using equations (17), (19) and (20).


## 2.1. Goal Programming

Goal Programming is the most known method in multiple-criteria decision-making, proposed by Charnes and Cooper (1961). Goal programming is a generalization of linear programming that handles multiple conflicting objective measures, where a target is set for each measure to be achieved. The new objective function, or the “achievement function,” seeks to minimize unwanted deviations from aspiration levels or a set of target values.We can introduce two types of constraints in goal programming: hard and soft constraints. Hard constraints are the system constraints that cannot be violated (e.g., system resources and relational model constraints). In contrast, soft constraints are associated with prespecified targets. Both negative and positive deviational variables are added to these constraints and the undesired deviations are included in the objective function to be minimized.The priority of goal programming is to satisfy the hard constraints before soft constraints. The preference structures to minimize the undesired deviations require different methods: preemptive, nonpreemptive, and Tchebycheff. The preemptive variant is used when there is a natural priority structure to the decision-maker(s) preferences, the nonpreemptive variant is used when each unwanted deviation has a relative weight (which can be equal), and Tchebycheff goal programming is used when the maximum deviation from the target is minimized.Consider the following model:(1)z=min∑l=1Lwlnl+pl,(2)flx+nl−pl=gl,∀l∈L,(3)nl×pl=0,∀l∈L,(4)Ax≤b,(5)x≥0,nl,pl≥0,∀l∈L,where gl is specific goal of the objective function flx∀l∈L.wl is the penalty weight. nl and pl are the under- and upper achievements of the lth goal, respectively. Equation (1) represents the objective function that minimizes the sum of the positive/negative deviations for each goal. Equation (2) is related to the decision-maker’s goals and computes the respective positive and negative deviations from each goal. Equation (3) ensures that at least one of the deviations must be equal to zero. Equation (4) relates to the system constraints in the decision space. Equation (5) ensures that all decision variables are nonnegative.


## 2.2. Dynamic Goal Programming

Dynamic goal programming allows for a target value to be dynamic. This approach is used to evaluate along a planning period. References [19–21] addressed the dynamic goal programming where the target values are changed as per the planning period:(6)z=min∑l=1L∑t=1Twtnlp+plp,flx+nlt−plt=glt,∀l∈L,∀t∈T,nlt×plt=0,∀l∈L,∀t∈T,Ax≤b,x≥0,nlt,plt≥0,∀l∈L,∀t∈T,where glt is specific goal of the objective function flx∀l∈L per period t.


## 2.3. Multi-Choice Goal Programming

Multi-choice goal programming was proposed by Chang (2007) [22]. In this case, decision-makers are allowed to define multi-choice aspiration levels for each target. The decision-maker can use the multi-choice model as a decision aid to make better decisions for a given problem. The mathematical model is as follows:(7)min∑l=1Lflx−gl1orgl2or…orglm.Subject to(8)X∈FFis a feasible set.Here, there are multi aspiration levelsgl1orgl2or…orglm. This model can be reformulated as:(9)minnl+pl,flx+nl−pl=∑j=1mbljSljB,∀l∈L,SljB∈Rlx,∀l∈L,nlt×plt=0,∀l∈L,∀t∈T,Ax≤b,x≥0,nlt,plt≥0,∀l∈L,∀t∈T,where blj is the jth aspiration level of the lth goal and SljB indicates the function of the binary serial number; Rlx is the function of resources boundaries. For three aspiration levels, the first constraint of model (9) can be reformulated as the following constraints:(10)flx+nl−pl=gl1b1b2+gl2b11−b2+gl31−b1b2,b1+b2>0,b1,b2∈0,1.Constraint (10) makes at least b1 or b2 not equal zero. Therefore, only three choices—gl1, gl2, or gl3— are yielded.


## 2.4. Neutrosophic Concept

In real applications, the uncertainty of the parameters is common in mathematical computations. Uncertainty arises owing to imprecise and inconsistent data. Zadeh proposed the fuzzy theory in 1965 to deal with these kinds of data [23]. However, fuzzy sets consider only the truth-membership function that is unable to efficiently represent accurate information. A new membership function, called falsity-membership function, was later developed by Atanassov (1986) [24], who introduced the intuitionistic fuzzy set. Nevertheless, this technique too was limited by drawbacks in decision-making. In 1998, Smarandache [25] introduced a new concept named “neutrosophic” that considers three memberships functions: truth, indeterminacy, and falsity.


### 2.4.1. Neutrosophic Set

Definition 1.
LetX≠∅ be a universe set. A neutrosophic set A in X is characterized by a truth-membership function μA˜N, an indeterminacy-membership function σAN˜, and a falsity-membership function νAN˜:(11)A˜N=x,μA˜Nx,σA˜Nx,υA˜Nx;x∈X,where μA˜N:X⟶0−,1+, σA˜N:X⟶0−,1+, and υA˜N:X⟶0−,1+ represent the degrees of the truth-, indeterminacy-, and falsity-membership functions, respectively. No restriction exists on the sum of μA˜N, σA˜N, and νA˜N. Thus, 0−≤μA˜Nx+σA˜Nx+νA˜Nx≤3+ for x∈XDefinition 2.
A setα,β,γ−cuts, generated by A˜N, where α,β,γ∈0,1 are a fixed number such that α+β+γ≤3 is defined as:(12)A˜α,β,γN=x,μA˜Nx,σA˜Nx,υA˜Nx;x∈X,μA˜Nx≥α,σA˜Nx≤β,υA˜Nx≤γ;α,β,γ∈0,1,where α,β,γ−cuts, denoted by A˜α,β,γN, is defined as the crisp set of elements x that belong to A˜N at least to the degree α and that belongs to A˜N at most to the degree β and γ.


### 2.4.2. Generalized Triangular Neutrosophic Number

A generalized triangular neutrosophic number (GTNN)τ˜aN=a,lμ,rμ;wa,a,lσ,rσ;ua,a,lν,rν;ya is a special neutrosophic set on a real number set ℜ whose degree of truth, indeterminacy, and falsity are given by:(13)μτ˜aNx=x−a+lμlμa−lμ≤x<awax=aa+rμ−xrμa<x≤a+rμ0otherwise,στ˜aNx=a−x+uax−a+lσlσa−lσ≤x<auax=ax−a+uaa+rσ−xrσa<x≤a+rσ1otherwise,ντ˜aNx=a−x+yax−a+lνlνa−lν≤x<ayax=ax−a+yaa+rν−xrνa<x≤a+rν1otherwise,where lμ, rμ, lσ, rσ, lν, and rν are called the spreads of the truth-, indeterminacy-, and falsity-membership functions, respectively; and a is the mean value. wa represents the maximum degree of the truth-membership function, while ua and ya represent the minimum degrees of the indeterminacy- and falsity-membership functions, respectively, such that they satisfy the conditions below:(14)0≤wa≤1,0≤ua≤1,0≤ya≤1,0≤wa+ua+ya≤3.


### 2.4.3.α,β,γ−Cut Set of GTNN

Definition 3.
Anα,β,γ−cut set of GTNN τ˜aN=a,lμ,rμ;wa,a,lσ,rσ;ua,a,lν,rν;ya is a crisp subset of ℜ, which is defined as:(15)τ˜Naα,β,γ=x:μτ˜Nax≥α,στ˜Nax≤β,υτ˜Nax≤γ,where 0≤wa≤1,0≤ua≤1,0≤ya≤1, and 0≤wa+ua+ya≤3.
Anα−cut set of a GTNN τ˜aN is a crisp subset of ℜ, which is defined as:(16)τ˜Naα=x:μτ˜Nax≥α,where 0≤α≤wa.
According to the definition of GTNN, it can be easily shown thatτ˜Naα=x:μτ˜Nax≥α, is a closed interval, defined by τ˜Naα=aLα,aRα where aLα=a−lμa+αlμa/wa and aRα=a+rμa−αrμa/wa. The mean of τ˜Naα is(17)amidα=a+rμa−lμa21−αwa.
Similarly, aβandγ−cut set of GTNN τ˜aN is a crisp subset of ℜ, which is defined as(18)τ˜Naβ=x:στ˜Nax≤β,τ˜Naγ=x:υτ˜Nax≤γ.where ua≤β≤1,ya≤γ≤1
It follows from the definition thatτ˜Naβ and τ˜Naγ are closed intervals, denoted by τ˜Naβ=aLβ,aRβ and τ˜Naγ=aLγ,aRγ, which can be calculated as:aLβ=a−lσa+1−βlσa/1−ua and aRβ=a+rσa−1−βrσa/1−uaaLγ=a−lνa+1−γlνa/1−ya and aRγ=a+rνa−1−γrνa/1−ya
Thus, the means ofτ˜Naβandτ˜Naγ are:(19)amidβ=a+rσa−lσa2β−ua1−ua,(20)amidγ=a+rνa−lνa2γ−ya1−ya.


### 2.4.4. De-Neutrosophication

In this work, the neutrosophic parameters will be treatment using two methods. In the first method, the crisp value will be the average of the mean of the three intervals obtained from theα,β,γ−cut set of GTNN. The crisp value can be calculated as equation (21).(21)τNaα,β,γ=13amidα+amidβ+amidγ,=a+rμa−lμa61−αwa,+rσa−lσa6β−ua1−ua,+rνa−lνa6γ−ya1−ya.It can be easily proven that forτ˜aN=a,lμ,rμ;wa,a,lσ,rσ;ua,a,lν,rν;ya∈GTNNℜ and for any α∈0,wa, β∈ua,1,γ∈ya,1, where 0≤α+β+γ≤3:(22)τ˜Naα,β,γ=τ˜Naα∧τ˜Naβ∧τ˜Naγ,where the symbol ∧ denotes the minimum among τ˜Naα,τ˜Naβandτ˜NaγFigure1 illustrates the membership functions for a generalized triangular neutrosophic number (NN).Figure 1
Graphical representation of membership functions for NN.In the second method of the deneutrosophication, we employ multi-choice goal programming to select a crisp value in the three obtained intervals that led to the truth-, indeterminacy-, and falsity-membership functionsaLα,aRα, aLβ,aRβ, and aLγ,aRγ, respectively. Thus, the multi-choice model will be employed to select one value from the three mean values that have been obtained using equations (17), (19) and (20).


## 2.4.1. Neutrosophic Set

Definition 1.
LetX≠∅ be a universe set. A neutrosophic set A in X is characterized by a truth-membership function μA˜N, an indeterminacy-membership function σAN˜, and a falsity-membership function νAN˜:(11)A˜N=x,μA˜Nx,σA˜Nx,υA˜Nx;x∈X,where μA˜N:X⟶0−,1+, σA˜N:X⟶0−,1+, and υA˜N:X⟶0−,1+ represent the degrees of the truth-, indeterminacy-, and falsity-membership functions, respectively. No restriction exists on the sum of μA˜N, σA˜N, and νA˜N. Thus, 0−≤μA˜Nx+σA˜Nx+νA˜Nx≤3+ for x∈XDefinition 2.
A setα,β,γ−cuts, generated by A˜N, where α,β,γ∈0,1 are a fixed number such that α+β+γ≤3 is defined as:(12)A˜α,β,γN=x,μA˜Nx,σA˜Nx,υA˜Nx;x∈X,μA˜Nx≥α,σA˜Nx≤β,υA˜Nx≤γ;α,β,γ∈0,1,where α,β,γ−cuts, denoted by A˜α,β,γN, is defined as the crisp set of elements x that belong to A˜N at least to the degree α and that belongs to A˜N at most to the degree β and γ.


## 2.4.2. Generalized Triangular Neutrosophic Number

A generalized triangular neutrosophic number (GTNN)τ˜aN=a,lμ,rμ;wa,a,lσ,rσ;ua,a,lν,rν;ya is a special neutrosophic set on a real number set ℜ whose degree of truth, indeterminacy, and falsity are given by:(13)μτ˜aNx=x−a+lμlμa−lμ≤x<awax=aa+rμ−xrμa<x≤a+rμ0otherwise,στ˜aNx=a−x+uax−a+lσlσa−lσ≤x<auax=ax−a+uaa+rσ−xrσa<x≤a+rσ1otherwise,ντ˜aNx=a−x+yax−a+lνlνa−lν≤x<ayax=ax−a+yaa+rν−xrνa<x≤a+rν1otherwise,where lμ, rμ, lσ, rσ, lν, and rν are called the spreads of the truth-, indeterminacy-, and falsity-membership functions, respectively; and a is the mean value. wa represents the maximum degree of the truth-membership function, while ua and ya represent the minimum degrees of the indeterminacy- and falsity-membership functions, respectively, such that they satisfy the conditions below:(14)0≤wa≤1,0≤ua≤1,0≤ya≤1,0≤wa+ua+ya≤3.


## 2.4.3.α,β,γ−Cut Set of GTNN

Definition 3.
Anα,β,γ−cut set of GTNN τ˜aN=a,lμ,rμ;wa,a,lσ,rσ;ua,a,lν,rν;ya is a crisp subset of ℜ, which is defined as:(15)τ˜Naα,β,γ=x:μτ˜Nax≥α,στ˜Nax≤β,υτ˜Nax≤γ,where 0≤wa≤1,0≤ua≤1,0≤ya≤1, and 0≤wa+ua+ya≤3.
Anα−cut set of a GTNN τ˜aN is a crisp subset of ℜ, which is defined as:(16)τ˜Naα=x:μτ˜Nax≥α,where 0≤α≤wa.
According to the definition of GTNN, it can be easily shown thatτ˜Naα=x:μτ˜Nax≥α, is a closed interval, defined by τ˜Naα=aLα,aRα where aLα=a−lμa+αlμa/wa and aRα=a+rμa−αrμa/wa. The mean of τ˜Naα is(17)amidα=a+rμa−lμa21−αwa.
Similarly, aβandγ−cut set of GTNN τ˜aN is a crisp subset of ℜ, which is defined as(18)τ˜Naβ=x:στ˜Nax≤β,τ˜Naγ=x:υτ˜Nax≤γ.where ua≤β≤1,ya≤γ≤1
It follows from the definition thatτ˜Naβ and τ˜Naγ are closed intervals, denoted by τ˜Naβ=aLβ,aRβ and τ˜Naγ=aLγ,aRγ, which can be calculated as:aLβ=a−lσa+1−βlσa/1−ua and aRβ=a+rσa−1−βrσa/1−uaaLγ=a−lνa+1−γlνa/1−ya and aRγ=a+rνa−1−γrνa/1−ya
Thus, the means ofτ˜Naβandτ˜Naγ are:(19)amidβ=a+rσa−lσa2β−ua1−ua,(20)amidγ=a+rνa−lνa2γ−ya1−ya.


## 2.4.4. De-Neutrosophication

In this work, the neutrosophic parameters will be treatment using two methods. In the first method, the crisp value will be the average of the mean of the three intervals obtained from theα,β,γ−cut set of GTNN. The crisp value can be calculated as equation (21).(21)τNaα,β,γ=13amidα+amidβ+amidγ,=a+rμa−lμa61−αwa,+rσa−lσa6β−ua1−ua,+rνa−lνa6γ−ya1−ya.It can be easily proven that forτ˜aN=a,lμ,rμ;wa,a,lσ,rσ;ua,a,lν,rν;ya∈GTNNℜ and for any α∈0,wa, β∈ua,1,γ∈ya,1, where 0≤α+β+γ≤3:(22)τ˜Naα,β,γ=τ˜Naα∧τ˜Naβ∧τ˜Naγ,where the symbol ∧ denotes the minimum among τ˜Naα,τ˜Naβandτ˜NaγFigure1 illustrates the membership functions for a generalized triangular neutrosophic number (NN).Figure 1
Graphical representation of membership functions for NN.In the second method of the deneutrosophication, we employ multi-choice goal programming to select a crisp value in the three obtained intervals that led to the truth-, indeterminacy-, and falsity-membership functionsaLα,aRα, aLβ,aRβ, and aLγ,aRγ, respectively. Thus, the multi-choice model will be employed to select one value from the three mean values that have been obtained using equations (17), (19) and (20).


## 3. Proposed Models

In this section, we formulate three goal programming models. First, we construct the basic goal programming model. We extend this model to include the neutrosophic concepts, whose parameters will be treated using the two methods in Section 2.4.4.The nomenclature of the parameters and the variables we use herein are defined below:NomenclatureSets:n=1,2,…,n: Set of all the staff kinds, indexed by i;m=1,2,…,m: Set of all the medical device types, indexed by j;K=1,2,…,K: Set of all the medical drug types, indexed by k.Decision variablesxi: The number of i staff;yj: The number of j medical devices;zk: The number of k medical drugs;p,n: The positive and the negative deviational variables;pxi,pyj,pzk: The positive deviational variables associated with the corresponding variable i,j,k;nxi,nyj,nzk: The negative deviational variables associated with the corresponding variable i,j,k;δj∈0,1: A binary variable, where δj=1ifyj>TNEj0ifyj≤TNEj∀jbr: A binary variable ∀r;M: A large number.Parametersci: The cost of i staff;cmj: The cost of j medical devices;cdk: The cost of k medical drugs;Wl: The weights of priority;TB: The total budget;TBS: The total budget for staff resources;TBM: The total budget of the medical device resources;TBD: The target budget of the medical drugs;TNSi: The target number of i staff;TNMj: The target number of the j medical devices;QDk: The quantity demand of the k medical drugs;SH: The number of shifts;t: The number of years;XPday: The estimated number of the patients per day;XRi: The optimal ratio between the number of i stuff and the patients;YRj: The optimal ratio between the number of j medical device and the patients;GR: The estimated growth rate of patients.


### 3.1. Goal Programming Model (GP)

The goal programming model can be formulated as follows:(23)minz=∑l=14pl+nl,+∑i=1nnxi+pxi,K+∑j=1mnyj+pyj,+∑k=1Knzk+pzk.Subject to(24)∑i=1ncixi+∑j=1mcmjyj+∑k=1Kcdkzk+n1−p1≤TB,(25)∑i=1ncixi+n2−p2≤TBS,(26)∑j=1mcmjyj+n3−p3≤TBM,(27)∑k=1Kcdkzk+n4−p4≤TBD,(28)zk+nzk−pzk≥QDk∀k,(29)xi+nxi−pxi≥TNSi,∀i,(30)yj+nyj−pyj≥TNMj,∀j,(31)yj>TNMj−1−δjM,(32)yj<TNMj+δjM,(33)TBS+cmj−1−δjM≤∑i=1ncixi≤TBS+cmj+1−δjM,(34)TBS−δjM≤∑i=1ncixi≤TBS+δjM,(35)xi,yj,zk∈ℤ+,∀i,j,k.The objective function in equation (23) is to minimize unwished deviations from the targets. Unwished deviations in these models are the negative deviational variables. Positive deviations are also added to the objective function to avoid obtaining large, exaggerated deviations. This way, decision-makers can provide the obtained budgets. The constraint mentioned in equation (24) ensures that the total available budget is not exceeded. In the flexible model, the budget is dynamic, that is, it changes according to the annual growth rate of the number of patients. Thus, equation (36) is used to calculate the targeted total budget over t number of years.(36)TB=TB+t×GR×TB.As the number of patients increases with time, the total budget must be increased to cover all costs of the optimal staff, medical devices, and medical drugs that will optimally satisfy patients’ demand. To achieve the maximum budget, that is, to attain the aspiration level total budget TB, the undesired variablen1 must be minimized. Constraints (25)–(27), respectively, guarantee the sub-budgets for the staff, medical devices, and medical drugs each, such that they are not violated.Similarly, in the flexible model, the sub-budgets change according to the growth of the number of patients. Thus, the dynamic sub-budgets of the staff, medical devices, and medical drugs change per year according to the following equations:(37)TBS=TBS+t×GR×TBS,TBM=TBM+t×GR×TBM,TBD=TBD+t×GR×TBD.Constraint (28) allows for increasing the amount of the k medical drug coinciding with the number of patients annually. In the same way, the amount of the k medical drug will change with time based on the following dynamic equation:(38)QDk=QDk+t×GR×QDk.Constraints (29) and (30) relate to the optimal ratio between the number of patients to the number of staff and medical devices. For example, the optimal ratio between the number of patients to number of oncologists for optimal care is about 1 : 15. On the other hand, the number of patients can be predicted from the previous data using the prediction techniques. Thus, we can estimate the number of patients daily. Therefore, the number of oncologists should be not less than(39)x1≥TNS1=SHXR1XPday.Similarly, for all other staff and medical advices that must be not less than the required number according to constraints (29) and (30), the target number of staff and medical devices can be calculated by the following equations:(40)TNSi=SHXRiXPday,∀i,TNMj=SHYRjXPday,∀j.We assume the importance of the availability of staff in the healthcare system and that medical devices have long-term durability. Therefore, constraints (31)–(34) transfers the surplus of the budget of the medical devices to the staff budget. This constraint investigates whether the medical device is available, and then shifts the budget of this type of device to the staff budget. Constraint (35) is related to the type of variables that must be integer variables.


### 3.2. Neutrosophic Goal Programming Model (NGP)

The state of a country’s economy can be expressed through its economic growth, stability, and stagnation. Thus, the total budget of any health center is affected by these economic states. Consequently, the sub-budgets will increase or decrease according to the economic state. Hence, the most suitable mathematical concept that expresses these states is the neutrosophic concept. In this subsection, we propose neutrosophic goal programming for healthcare planning. Three degrees are introduced in this case: acceptable, indeterminacy, and rejection.The neutrosophic goal programming model is the extension of the goal programming model, where constraints (24)–(27) and (31)–(34) are rewritten as follows:(41)∑i=1ncixi+∑j=1mcmjyj+∑k=1Kcdkzk+n1−p1≤TB˜N,∑i=1ncixi+n2−p2≤TBS˜N,∑j=1mcmjyj+n3−p3≤TBM˜N,∑k=1Kcdkzk+n4−p4≤TBD˜N,yj>TNMj−1−δjM,yj<TNMj+δjM,TBS˜N+cmj−1−δjM≤∑i=1ncixi≤TBS˜N+cmj+1−δjM,TBS˜N−δjM≤∑i=1ncixi≤TBS˜N+δjM,TB˜N=TB,lμ,rμ;wTB,TB,lσ,rσ;uTB,TB,lν,rν;yTB,TBS˜N=TBS,lμ,rμ;wTBS,TBS,lσ,rσ;uTBS,TBS,lν,rν;yTBS,TBM˜N=TBM,lμ,rμ;wTBM,TBM,lσ,rσ;uTBM,TBM,lν,rν;yTBM,TBD˜N=TBD,lμ,rμ;wTBD,TBD,lσ,rσ;uTBD,TBD,lν,rν;yTBD,where A˜N denotes the neutrosophic number defined using constraint (41).


### 3.3. Neutrosophic Multi-Choices Goal Programming Model (NMCGP)

In this model, we use multi-choice goal programming to deal with values obtained from the neutrosophic set. Hence, constraints (24)–(27) are rewritten as follows:(42)∑i=1ncixi+∑j=1mcmjyj+∑k=1Kcdkzk+n1−p1≤TB1Nb1b2+TB2Nb11−b2+TB3N1−b1b2,∑i=1ncixi+n2−p2≤TBS1Nb3b4+TBS2Nb31−b4+TBS3N1−b3b4,∑j=1mcmjyj+n3−p3≤TBM1Nb5b6+TBM2Nb51−b6+TBM3N1−b5b6,∑k=1Kcdkzk+n4−p4≤TBD1Nb7b8+TBD2Nb71−b8+TBD3N1−b7b8,br+br+1>0,br,br+1∈0,1,r∈1,2,,…,7,where TB1N,TB2N,TB3N,TBS1N,TBS2N,TBS3N,TBM1N,TBM2N,TBM3N,TBD1N,TBD2N,andTBD3N can be obtained using equations (17), (19) and (20).


## 3.1. Goal Programming Model (GP)

The goal programming model can be formulated as follows:(23)minz=∑l=14pl+nl,+∑i=1nnxi+pxi,K+∑j=1mnyj+pyj,+∑k=1Knzk+pzk.Subject to(24)∑i=1ncixi+∑j=1mcmjyj+∑k=1Kcdkzk+n1−p1≤TB,(25)∑i=1ncixi+n2−p2≤TBS,(26)∑j=1mcmjyj+n3−p3≤TBM,(27)∑k=1Kcdkzk+n4−p4≤TBD,(28)zk+nzk−pzk≥QDk∀k,(29)xi+nxi−pxi≥TNSi,∀i,(30)yj+nyj−pyj≥TNMj,∀j,(31)yj>TNMj−1−δjM,(32)yj<TNMj+δjM,(33)TBS+cmj−1−δjM≤∑i=1ncixi≤TBS+cmj+1−δjM,(34)TBS−δjM≤∑i=1ncixi≤TBS+δjM,(35)xi,yj,zk∈ℤ+,∀i,j,k.The objective function in equation (23) is to minimize unwished deviations from the targets. Unwished deviations in these models are the negative deviational variables. Positive deviations are also added to the objective function to avoid obtaining large, exaggerated deviations. This way, decision-makers can provide the obtained budgets. The constraint mentioned in equation (24) ensures that the total available budget is not exceeded. In the flexible model, the budget is dynamic, that is, it changes according to the annual growth rate of the number of patients. Thus, equation (36) is used to calculate the targeted total budget over t number of years.(36)TB=TB+t×GR×TB.As the number of patients increases with time, the total budget must be increased to cover all costs of the optimal staff, medical devices, and medical drugs that will optimally satisfy patients’ demand. To achieve the maximum budget, that is, to attain the aspiration level total budget TB, the undesired variablen1 must be minimized. Constraints (25)–(27), respectively, guarantee the sub-budgets for the staff, medical devices, and medical drugs each, such that they are not violated.Similarly, in the flexible model, the sub-budgets change according to the growth of the number of patients. Thus, the dynamic sub-budgets of the staff, medical devices, and medical drugs change per year according to the following equations:(37)TBS=TBS+t×GR×TBS,TBM=TBM+t×GR×TBM,TBD=TBD+t×GR×TBD.Constraint (28) allows for increasing the amount of the k medical drug coinciding with the number of patients annually. In the same way, the amount of the k medical drug will change with time based on the following dynamic equation:(38)QDk=QDk+t×GR×QDk.Constraints (29) and (30) relate to the optimal ratio between the number of patients to the number of staff and medical devices. For example, the optimal ratio between the number of patients to number of oncologists for optimal care is about 1 : 15. On the other hand, the number of patients can be predicted from the previous data using the prediction techniques. Thus, we can estimate the number of patients daily. Therefore, the number of oncologists should be not less than(39)x1≥TNS1=SHXR1XPday.Similarly, for all other staff and medical advices that must be not less than the required number according to constraints (29) and (30), the target number of staff and medical devices can be calculated by the following equations:(40)TNSi=SHXRiXPday,∀i,TNMj=SHYRjXPday,∀j.We assume the importance of the availability of staff in the healthcare system and that medical devices have long-term durability. Therefore, constraints (31)–(34) transfers the surplus of the budget of the medical devices to the staff budget. This constraint investigates whether the medical device is available, and then shifts the budget of this type of device to the staff budget. Constraint (35) is related to the type of variables that must be integer variables.


## 3.2. Neutrosophic Goal Programming Model (NGP)

The state of a country’s economy can be expressed through its economic growth, stability, and stagnation. Thus, the total budget of any health center is affected by these economic states. Consequently, the sub-budgets will increase or decrease according to the economic state. Hence, the most suitable mathematical concept that expresses these states is the neutrosophic concept. In this subsection, we propose neutrosophic goal programming for healthcare planning. Three degrees are introduced in this case: acceptable, indeterminacy, and rejection.The neutrosophic goal programming model is the extension of the goal programming model, where constraints (24)–(27) and (31)–(34) are rewritten as follows:(41)∑i=1ncixi+∑j=1mcmjyj+∑k=1Kcdkzk+n1−p1≤TB˜N,∑i=1ncixi+n2−p2≤TBS˜N,∑j=1mcmjyj+n3−p3≤TBM˜N,∑k=1Kcdkzk+n4−p4≤TBD˜N,yj>TNMj−1−δjM,yj<TNMj+δjM,TBS˜N+cmj−1−δjM≤∑i=1ncixi≤TBS˜N+cmj+1−δjM,TBS˜N−δjM≤∑i=1ncixi≤TBS˜N+δjM,TB˜N=TB,lμ,rμ;wTB,TB,lσ,rσ;uTB,TB,lν,rν;yTB,TBS˜N=TBS,lμ,rμ;wTBS,TBS,lσ,rσ;uTBS,TBS,lν,rν;yTBS,TBM˜N=TBM,lμ,rμ;wTBM,TBM,lσ,rσ;uTBM,TBM,lν,rν;yTBM,TBD˜N=TBD,lμ,rμ;wTBD,TBD,lσ,rσ;uTBD,TBD,lν,rν;yTBD,where A˜N denotes the neutrosophic number defined using constraint (41).


## 3.3. Neutrosophic Multi-Choices Goal Programming Model (NMCGP)

In this model, we use multi-choice goal programming to deal with values obtained from the neutrosophic set. Hence, constraints (24)–(27) are rewritten as follows:(42)∑i=1ncixi+∑j=1mcmjyj+∑k=1Kcdkzk+n1−p1≤TB1Nb1b2+TB2Nb11−b2+TB3N1−b1b2,∑i=1ncixi+n2−p2≤TBS1Nb3b4+TBS2Nb31−b4+TBS3N1−b3b4,∑j=1mcmjyj+n3−p3≤TBM1Nb5b6+TBM2Nb51−b6+TBM3N1−b5b6,∑k=1Kcdkzk+n4−p4≤TBD1Nb7b8+TBD2Nb71−b8+TBD3N1−b7b8,br+br+1>0,br,br+1∈0,1,r∈1,2,,…,7,where TB1N,TB2N,TB3N,TBS1N,TBS2N,TBS3N,TBM1N,TBM2N,TBM3N,TBD1N,TBD2N,andTBD3N can be obtained using equations (17), (19) and (20).


## 4. Case Study

For the case study, we collected and mined real data from the Al-Amal Center for Oncology in Aden, Yemen. Then, the number of patients was predicted daily, and the goals were set according to the predicted data.The center was established in 2014 and covers an area of 50,000km2. The center includes several departments and sections, such as outpatient clinics, inpatient section, laboratories and radiology department, early detection center, research center, intensive care, emergency section, a medical college, administrative, training, and staff housing.Al-Amal is one of the few specialized centers for oncology in Yemen. Owing to the increasing rates of cancer, this center faces immense challenges in providing the necessary supplies for staff, medical devices, and medical drugs. Table1 shows the increasing growth of the number of patients for the period between July 2015 and December 2017.Table 1
Number of patients (7/2015–12/2017).
MonthsJAN.FEB.MAR.APR.MAYJUN.JUL.AUG.SEP.OCT.NOV.DEC.Total2015——————26327727829633436118092016366382102195230104201203309400453644358920176506816896967307337377557707887948388861Based on the given data, the linear regression equation to predict the numbers of patients isXP=1227+3526t, where XP is the estimated number of patients and t is the number of years. Hence, the number of patients increases gradually with an average annual growth rate of 8%. That is, the estimated number of patients on a working day is XPday≅5+10t. Hence, the increasing number of patients daily under the existing budget requires more optimal use of resources. Moreover, we hope to increase the number of staffs, medical devices, and medical drugs as well, which would require increasing the budget. In the next step, increasing the total budget and sub-budgets would be a target, which would help meet the needs of growing patient numbers.Table2 shows the number of each kind of staff and the respective cost. The staff include oncologists, general doctors, radiologists, pharmacists, lab technicians, X-ray technicians, nurses, and other staff (other staff include remaining staff such as those in the administrative department).Table 2
The number of staffs with corresponding costs.
Staffx1x2x3x4x5x6x7x8TotalOncologistsGeneral doctorsRadiologistsPharmacistsLab technicianX-ray technicianNursesOther staffNumber of staffs22122162036Cost ($)96003840576033603360312024003600Total cost ($)19200768057606720672031201440072000135600XRi1520101001030410Table3 shows the number of each type of medical device and the respective cost. The medical devices include blood testing apparatus, chemistry apparatus, oncology indications, as well as ultrasonic, X-ray, mammogram, and other equipment (e.g., blood pressure- and blood sugar-testing devices as well as medical stethoscopes). Table 4 shows the list of the medical drugs, including their name, size, price, type, quantities used, and doses.Table 3
The number of machines with corresponding costs.
Machinesy1y2y3y4y5y6y7TotalBlood TestChemistry apparatusOncology indicationsUltrasonicX-rayMammogramOther equipmentNumber of machines111111200206Cost4200480066609600100Total cost ($)42004800666096002000045260YRj1030303030503Table 4
Medical drug data.
Medical drug namesThe size of the medicine (cm3)Drug Price ($)Type of medicationQuantity 2016Dose of medicationz1Capecitabin2431.18Tab13503500 mgz2Cisplatin -1-1281.67Vial8010 mgz3Cisplatin -2-212.56.04Vial48450 mgz4Cyclophosphamide -1-1121.1Vial1061500 mgz5Cyclophosphamide -2-540.7Vial1049200 mgz6Cyclophosphamide-3-212.51.61Vail3661000 mgz7Docitaxel -1-1509.69Vial61020 mgz8Docitaxel -2-282.618.98Vial69480 mgz9Epirubicin -1-37.54.66Vial15510 mgz10Epirubicin -2-11218.7Vial60550 mgz11Letrozole tabl.178.50.25Tab45352.5 mgz12Paclitaxel -1-12831.45Vial210150 mgz13Paclitaxel -2-68.2514.8Vial275100 mgz14Paclitaxel -3-546.66Vial15530 mgz15Fluorouracil (5-fu)-1-2000.66Vail110250 mgz16Fluorouracil (5-fu)-2-262.51.38Vail226500 mgz17Tamoxifen950.12Tab637420 mgz18Thalidomide80.3251.35Tab778100 mgz19Doxorubcin -1-91.8755.26Vial49050 mgz20Doxorubcin -2-58.51.66Vial75610 mgz21Ifosphamid211.754.85Vial1521000 mgz22Zoledronic acid58.515.82Vial2134 mgz23Imatinib441.4Tab2575400 mgz24Biclutamide33.80.82Tab200250 mgz25Carboplatin -1-212.535.7Vial281450 mgz26Carboplatin -2-12816.83Vial262150 mgz27Gemcitabin -1-12829.06Vial4111000 mgz28Dacarbazin151.87513.98Vail158500 mgz29Pazopanib332.7515.68Tab913400 mgz30Irenotican4516.92Vail11040 mgz31Ca-folinat452.21Vial12350 mgz32Mesna391.23Vial786200 mgz33Vincristine -2-34.3751.1Vial3321 mgz34Etopside1122Vail170100 mgz35Bleomycin4511.78Vial24015 mgz36Vinblastin -1-58.55.7Vial28610 mgz37Filgrastim 30U.i11610.34Inj210810 mgz38Fludarabine12668Vial4150 mgz39Zoladex495420Inj5210.8 mgz40Liposomol Doxorubcin210137.53Vial16820 mgThe existing budget of the center for the staff, medical devices, and medical drug are US$135,600, US$45,260, and US$390,000, respectively, or US$570,860 in total.


## 5. Results and Discussion

In this section, we implement the three proposed models using LINGO 18.0 software. We use these models for planning the sustainable development of the Al-Amal center for Oncology, Aden, Yemen. The proposed models would help decision-makers to plan optimality, as it allows us to determine the optimal number of required staffs, devices, and medical drugs for every period. The optimal ratio between the number of patients to the number of staff/medical devices/quantity of the medical drugs are assumed and the proposed models are applied to eight periods, from 2018 to 2048.


### 5.1. Results of the Goal Programming Model

The results of the first model are illustrated in Tables5–10, where Table 5 shows the optimal budgets for each year and the ratio of each sub-budget of the total budget. Four budgets in this study are considered: total budget, staff budget, medical devices budget. and medical drugs budget. In the last column of this table, the growth rate for each budget is presented. The results indicate the need to increase the budget of the staff numbers, medical devices, and medical drugs by 13%, 6%, 9%, and 32%, respectively, annually. Figure 2 illustrates the growth of the total budget as well as the other budgets.Table 5
The obtained budgets using GP model.
Year201820202024202820322036204020442048GR (%)TBCost316705.7426305.1585093.9920536.31263664158618719213832258103258057913TBSCost876001574402592004946407204809432001178640140112016238406(%)283744545759616263TBMCost260003280057400922001448001794002142002636002982009(%)8810101111111212TBDCost203105.7236065.1268493.9333696.3398384.2463587.2528543.1593383658539.232(%)645546363229282626Table 6
The number and corresponding cost of staff obtained using GP model.
Staffx1x2x3x4x5x6x7x8TotalOncologistsGeneral doctorsRadiologistsPharmacistsLab technicianX-ray technicianNursesOther staff2018No.2232324321Ratio (%)1010141014101914100Cost192007680172806720100806240960010800876002020No.43515313539Ratio (%)1081331383313100Cost384001152028800336016800936031200180001574402024No.65919323965Ratio (%)981421453514100Cost576001920051840336030240936055200324002592002028No.1291721764317123Ratio (%)1071421453514100Cost115200345609792067205712018720103200612004946402032No.17132532596325180Ratio (%)971421453514100Cost16320049920144000100808400028080151200900007204802036No.221733433118333236Ratio (%)971421453514100Cost2112006528019008013440110880343201992001188009432002040No.2821415411410341294Ratio (%)1071421453514100Cost26880080640236160168001377604368024720014760011786402044No.3325495491712349350Ratio (%)971411453514100Cost31680096000282240168001646405304029520017640014011202048No.3829576571914357406Ratio (%)971411453514100Cost36480011136032832020160191520592803432002052001623840GR (%)68635611366Table 7
The number with the corresponding cost of medical devices obtained using GP model.
Medical devicesy1y2y3y4y5y6y7TotalBlood testChemistry apparatusOncology indicationsUltrasonicX-rayMammogramOther devices2018No.111111410Ratio (%)10101010101040100Cost420048006660320032003200800260602020No.2111111724Ratio (%)84444471100Cost8400480066603200320032003400328602024No.3222213042Ratio (%)75555271100Cost126009600133206400640032006000575202028No.5333325776Ratio (%)74444375100Cost21000144001998096009600640011400923802032No.75555384114Ratio (%)64444374100Cost29400240003330016000160009600168001451002036No.966664110147Ratio (%)64444375100Cost378002880039960192001920012800220001797602040No.1177775137181Ratio (%)64444376100Cost462003360046620224002240016000274002146202044No.1399995164218Ratio (%)64444275100Cost546004320059940288002880016000328002641402048No.15101010106190251Ratio (%)64444276100Cost63000480006660032000320001920038000298800GR (%)7111111111824Table 8
The quantities of medical drugs obtained using the GP model.
Medical drug names201820202024202820322036204020442048GR (%)z113503156641782422145264663078735108394294375032z2809310613215718320823426032z3484562639794949110412591414156932z410611231140117412080242027593099343832z510491217138517212057239227283064339932z63664254846017188359521069118632z761070880610011196139115861782197732z869480691711391361158318052027224932z915518020525530435440345350332z106057027999931186138015731767196132z11453552615987743888891034011791132431469432z1221024427834541247954661468132z1327531936345153962771580389132z1415518020525530435440345350332z1511012814618121625128632235732z1622626329937144351658866073332z1763747394841410454124941453316573186132065232z18778903102712761525177420232272252132z19790917104312961549180220542307256032z2075687799812401482172419662208245032z2115217720125029834739644449332z2221324828235041848655462269132z2325752987339942235047587166957519834332z2420022323264332843924456552065846648732z2528132637146155164173182191132z2626230434643051459868276684932z2741147754367580693810691201133232z2815818420926031036141146251232z299131060120614981790208223742666295932z3011012814618121625128632235732z3112314316320224228132036039932z32786912103812901541179320442296254732z33332386439545651757864970107632z3417019822527933438844249755132z3524027931739447154862470177832z3628633237847056165374483692732z3721082446278334584132480754816156683032z3841485568819410712013332z395261698610211913615216932z4016819522227633038443749154532Sum441995128858361725098664610079311492912908114322332Table 9
The negative deviational variables obtained using GP model.
Negative deviations201820202024202820322036204020442048N14800000000000N2000000000N3000.2000000N44800000000000Table 10
The positive deviational variables obtained using the GP model.
Positive deviations201820202024202820322036204020442048P10288160415.8562485.6965498.41344271175526721484612527308P201448020827225645470463403282608010051681184496P300017973.656090.476207.2103107.3138125.4158316.4P401448020827225645470463403282608010051681184496PX11.3333330.66666700.6666670.33333300.6666670.3333330PX21.50.50.50.50.50.50.50.50.5PX3200000000PX41.90.50.10.30.50.70.90.10.3PX5200000000PX61.6666671.33333300.3333330.66666700.3333330.6666670PX71.50.50.50.50.50.50.50.50.5PX8200000000PY10.750.750.750.750.750.750.750.750.75PY20.8333330.1666670.50.1666670.8333330.50.1666670.8333330.5PY30.8333330.1666670.50.1666670.8333330.50.1666670.8333330.5PY40.8333330.1666670.50.1666670.8333330.50.1666670.8333330.5PY50.8333330.1666670.50.1666670.8333330.50.1666670.8333330.5PY60.90.50.10.30.50.70.90.10.3PY70.6666670.33333300.3333330.66666700.3333330.6666670Figure 2
Budget growth using the GP model.Table6 shows the optimal staff numbers that should be employed annually. Each row in this table indicates the years and the corresponding optimal staff number, each ratio of the total, and the corresponding cost for each. In the last row of this table, we observe an increase in the varying annual growth rates for each kind of staff, according to the corresponding optimal rate with respect to the number of patients. The growth rate for the total staff is 6%.Figure3 illustrates the optimal number of different staffs. Evidently, the number of nurses increases the most with time. Similarly, Tables 7 and 8 show the increasing number of medical devices and medical drugs, respectively. Figure 4 illustrates the growth of the number of medical devices; we find that other equipment increases the most with time. The growth rate for the total number of medical devices is 4%, which is less than the total number of staff growth owing to converted constraints (31)–(34).Figure 3
The staff growth using the GP model.Figure 4
Medical devices growth using GP model.Table9 shows the negative deviational variables. Almost all variables equal to zero except N1 and N4, which are equal to US$4,800. That is, in 2018, the optimal budget of the medical drugs must be decreased to US$385,200. All the remaining negative deviations are equal to zero and, hence, there is no need to record them in the table. In contrast, the nonzero positive deviational variables are reported in Table 10.We find that the total budgets, staff, and medical devices should be increased further in order to provide high-quality service in expectation of the annual increase in patients. The positive deviations of medical drugs are equal to zero—its dynamic budget can sufficiently cover needs.


### 5.2. Results of the Neutrosophic Goal Programming Model

There are three states of an economy: growth, stability, and stagnation. The budgets are assumed in accordance with these states. That is, the budgets are increased in the optimism state; are left unchanged or changed only marginally in the stability state; and are decreased in the pessimism state. This allows us to represent the budgets using the neutrosophic concept. Table11 shows the values of the neutrosophic parameters. Then, equation (21) is used to give the deneutrosophic treatment to the obtained values.Table 11
Values of the neutrosophic variables.
τ˜aNaa,lμ,rμ;wa,a,lσ,rσ;ua,a,lν,rν;yaαβγTB˜N570860100000,200000;0.9,30000,40000;0.3,800000,100000;0.50.10.90.8TBS˜N13560013000,100000;0.9,18000,15000;0.3,20000,30000;0.50.10.90.8TBM˜N452601000,5000;0.9,9000,10000;0.3,10000,15000;0.50.10.90.8TBD˜N39000010000,100000;0.9,20000,30000;0.3,15000,20000;0.50.10.90.8Table12 shows the optimal different budgets for each year. The results indicate that the budgets’ growth rate is 15% annually. There is a marginal increase in the growth rate in the goal programming model. Table 13 shows that the optimal staff number should be determined yearly using this model. The growth rate for the total staff is 8%. There is a small decrease in the growth rate in the goal programming model. In the same way, Tables 14 and 15 show the increase in medical devices and medical drugs. We find that other equipment increases the most with time. The negative deviational variables are also equal to zero except N3, which is equal US$20,064 for 2018, as reported in Table 16. In contrast, the nonzero positive deviational variables are mentioned in Table 17.Table 12
Budgets results of NGP model.
Year201820202024202820322036204020442048GR (%)TBCost359415.6407990.3672852.8920536.31196277157947119213832288403261362415TBSCost1209601519203261604946406840009408001178640142584016516808(%)343748545760616263TBMCost2660028000620009220012200017920021420026400029860010(%)779101011111211TBDCost211855.6228070.3284692.8333696.3390277.3459470.6528543.1598563.466334433(%)595642363329282625Table 13
The staff numbers obtained using the NGP model.
Staffx1x2x3x4x5x6x7x8TotalOncologistsGeneral doctorsRadiologistsPharmacistsLab technicianX-ray technicianNursesOther staff2018No.3342427429Ratio (%)10101471472414100Cost288001152023040672013440624016800144001209602020No.42616310436Ratio (%)1161731782811100Cost38400768034560336020160936024000144001519202024No.86112114281181Ratio (%)1071421453514100Cost7680023040633606720369601248067200396003261602028No.1291721764317123Ratio (%)1071421453514100Cost115200345609792067205712018720103200612004946402032No.16122432486024171Ratio (%)971421453514100Cost15360046080138240100808064024960144000864006840002036No.221733433118233235Ratio (%)971421453514100Cost2112006528019008013440110880343201968001188009408002040No.2821415411410341294Ratio (%)1071421453514100Cost26880080640236160168001377604368024720014760011786402044No.3425505501712450355Ratio (%)1071411453514100Cost32640096000288000168001680005304029760018000014258402048No.3929586582014458412Ratio (%)971411453514100Cost37440011136033408020160194880624003456002088001651680GR (%)811835811588Table 14
The medical devices numbered obtained using NGP model.
Medical devicesy1y2y3y4y5y6y7TotalBlood TestChemistry apparatusOncology indicationsUltrasonicX-rayMammogramOther devices2018No.111111713Ratio (%)88888854100Cost4200480066603200320032001400266602020No.1111111420Ratio (%)55555570100Cost4200480066603200320032002800280602024No.3222223750Ratio (%)64444474100Cost126009600133206400640064007400621202028No.5333325776Ratio (%)74444375100Cost21000144001998096009600640011400923802032No.64444380105Ratio (%)64444376100Cost25200192002664012800128009600160001222402036No.966664109146Ratio (%)64444375100Cost378002880039960192001920012800218001795602040No.1177775137181Ratio (%)64444376100Cost462003360046620224002240016000274002146202044No.1399995166220Ratio (%)64444275100Cost546004320059940288002880016000332002645402048No.15101010106192253Ratio (%)64444276100Cost63000480006660032000320001920038400299200GR7111111111846Table 15
The medical drugs quantities obtained using NGP model.
Medical drug names201820202024202820322036204020442048GR (%)z114044151241890522145259263051735108397534407433z2849011213215418120823626233z3504543678794930109412591425158033z411041189148617412038239827593124346433z510911175146917212015237127283089342433z63814105136017038289521078119533z763568485410011172137915861796199233z872277897211391333156918052044226633z916217421725529835140345750633z106306788479931162136815731782197533z11471750806349743887081025011791133521480333z1221923629434540447554661968633z1328630838545152862271581089833z1416217421725529835140345750633z1511512415418121224928632436033z1623625431737143451158866673833z1766297139892410454122391440616573187662080533z18810872109012761494175920232291254033z19822885110612961517178620542326257933z20787847105912401452170919662226246833z2115917121325029234439644849733z2222223929935040948255462869633z2326782884360542234944582066957581840533z2420832243280332843844452552065894653533z2529331539446154063673182891833z2627329436743050459368277285633z2742846157667579092910691210134233z2816517722226030435841146651633z299501023127914981753206423742688298133z3011512415418121224928632436033z3112813817320223727832036340233z32818881110112901510177720442314256633z33346372465545638751864978108433z3417719123827932738544250155533z3525026933639446154362470778433z3629832140147055064774484293433z3721932361295234584048476554816206688133z3843465868799310712113433z395559738610011813615417034z4017518923627632338043749554933Sum45989495226189372509848829991111492913014114428633Table 16
The negative deviational variables obtained using the NGP model.
Negative deviations201820202024202820322036204020442048N1000000000N2000000000N32006400000000N4000000000Table 17
The positive deviational variables obtained using the NGP model.
Positive deviations201820202024202820322036204020442048P1096276448.6691594.3888565.61352026196015621907132575798P22006448136320249634.3423648634344790216.410266341209082P3003808.57115674.1241269.5783338.0599665.71137445.5157635P4048136320249634.3423648634344790216.410266341209082PX11.6666671.3333330.6666670.66666700.3333330.6666670.9333330.6PX2200.50.500.750.50.20.2PX3220000.500.40.4PX41.80.60.90.30.60.750.94.00E-020.24PX5220000.500.40.4PX61.3333331.6666670.3333330.33333300.1666670.3333330.4666670.8PX7200.50.500.750.500PX8200000.500.40.4PY10.500.250.7500.8750.750.60.6PY20.6666670.3333330.1666670.16666700.5833330.1666670.7333330.4PY30.6666670.3333330.1666670.16666700.5833330.1666670.7333330.4PY40.6666670.3333330.1666670.16666700.5833330.1666670.7333330.4PY50.6666670.3333330.1666670.16666700.5833330.1666670.7333330.4PY60.80.60.90.30.60.750.94.00E-020.24PY70.3333330.6666670.3333330.33333300.6666670.3333330.6666670


### 5.3. Results of the Neutrosophic Multi-Choices Goal Programming Model

In this model, the multi-choice goal programming was applied to randomly select one value from the three values obtained from the neutrosophic membership functions. Table18 shows the three means for each budget obtained using equations (17), (19) and (20) with the same parameters reported in Table 11.Table 18
Values of the neutrosophic variables.
τ˜gNgg1,g2,g3TB˜N570860615304.4,575145.7143,590860TBS˜N135600174266.7,134314.3,131600TBM˜N4526047037.78,45688.57,47260TBD˜N390000430000,394285.7,392000Tables19–22 show the obtained values for all variables. Similarly, Figures 5–7 illustrate the growth in the budgets, staff, medical devices, and medical drugs. Notably, we observe the aliasing of the curves, in contrast to Figure 2, indicating the random selection of the values of binary variable b using multi-choice goal programming.Table 19
The obtained budgets using NMCGP model.
Year201820202024202820322036204020442048GR (%)TBCost337882.6512647.8585093.9874117.31299211158618719401972164600261803814TBSCost907202083202592004612807476009432001193760134712016540806(%)153632505759616263TBMCost2720052000574008740014540017940021440024020029880010(%)108791111111212TBDCost219962.6252327.8268493.9325437.3406211.4463587.2532037.4577280.5665157.634(%)765661413229282626Table 20
The staff number obtained using the NMCGP model.
Staffx1x2x3x4x5x6x7x8OncologistsGeneral doctorsRadiologistsPharmacistsLab technicianX-ray technicianNursesOther staffTotal2018No.2231318323Ratio (%)991341343513100Cost1920076801728033601008031201920010800907202020No.54717318752Ratio (%)1081321363513100Cost480001536040320336023520936043200252002083202024No.65919323965Ratio (%)981421453514100Cost576001920051840336030240936055200324002592002028No.1181621664016115Ratio (%)1071421453514100Cost10560030720921606720537601872096000576004612802032No.18132632696526186Ratio (%)1071421453514100Cost17280049920149760100808736028080156000936007476002036No.221733433118333236Ratio (%)971421453514100Cost2112006528019008013440110880343201992001188009432002040No.2821425421410442298Ratio (%)971421453514100Cost26880080640241920168001411204368024960015120011937602044No.3224475471611847336Ratio (%)1071411453514100Cost30720092160270720168001579204992028320016920013471202048No.3929586582014558413Ratio (%)971411453514100Cost37440011136033408020160194880624003480002088001654080GR (%)6861866666Table 21
The medical devices numbered obtained using NMCGP model.
Medical devicesy1y2y3y4y5y6y7TotalBlood testChemistry apparatusOncology indicationsUltrasonicX-rayMammogramOther devices2018No.1111111016Ratio (%)66666663100Cost4200480066603200320032002000272602020No.2222212435Ratio (%)66666369100Cost84009600133206400640032004800521202024No.3222213042Ratio (%)75555271100Cost126009600133206400640032006000575202028No.4333325472Ratio (%)64444375100Cost16800144001998096009600640010800875802032No.75555387117Ratio (%)6%4%4%4%4%3%74%100%Cost29400240003330016000160009600174001457002036No.966664110147Ratio (%)64444375100Cost378002880039960192001920012800220001797602040No.1177775138182Ratio (%)64444376100Cost462003360046620224002240016000276002148202044No.1288885157206Ratio (%)64444276100Cost504003840053280256002560016000314002406802048No.15101010106193254Ratio (%)6%4%4%4%4%2%76%100%Cost63000480006660032000320001920038600299400GR (%)7111111111867Table 22
Medical drugs quantities obtained using the NMCGP model.
Medicament name201820202024202820322036204020442048GR (%)z114584167441782421605270063078735324383494418234z28710010612816018321022826234z3523601639775968110412671375158434z411461316140116982122242027763014347234z511331301138516792098239227452980343334z63964544845867328359581040119834z76597578069761220139115961733199634z875086191711111388158318161971227134z916819320524831035440644150834z106547517999681210138015831719198034z11489856245987725690701034011864128801483934z1222726127833642047955059768834z1329734136344055062772078190034z1416819320524831035440644150834z1511913714617622025128831336034z1624528129936245251659264274034z1768847904841410199127481453316675181032085634z18841965102712451556177420362210254634z19854980104312641580180220672244258534z2081793899812101512172419782148247434z2116518920124430434739843249834z2223126528234142648655860569734z2327813193339941205150587167377313842634z2421632483264332044004456552385686655134z2530434937145056264173679992034z2628332534642052459868674585834z2744451054365882293810761168134534z2817119620925331636141444951734z299871133120614611826208223892593298834z3011913714617622025128831336034z3113315316319724628132235040334z32849975103812581572179320572233257234z33359412439532664757869943108734z3418421122527234038844548355734z3526029831738448054862868278634z3630935537845857265374981393634z3722772614278333734216480755155987689834z3845515566829410811713535z395765698410411913714817135z4018220922226933638444047855034Sum477535482558361707308839810079311564712554614463734Figure 5
Budget growth using the NMCGP model.Figure 6
The staff growth using the NMCGP model.Figure 7
The medical devices growth using the NMCGP model.Overall, Table23 summarizes the comparison between the three proposed models. The growth rates of the total budget and total staff numbers using the neutrosophic goal programming model were the highest, whereas the multi-choice model shows the highest growth rate of the number of medical devices. The growth rate of medical drugs for all proposed models is almost equal. The multi-choice model yields the least deviations. We now discuss the results obtained in 2048 as an example. The total budget, total staff, and total medical devices obtained using the multi-choice model are higher than those obtained using the other models. Similarly, in the same year, the multi-choice model yields the lowest summation of deviations. Figures 8–10 illustrate a comparison between the proposed models for staff growth, the increasing medical devices, and the increase in the demand for medicines for the period between 2018 and 2048, respectively.Table 23
Summary of the comparison between the proposed models.
201820202024202820322036204020442048GR (%)Total BudgetGP316705.7426305.1585093.9920536.31263664158618719213832258103258057913NGP359415.6407990.3672852.8920536.31196277157947119213832288403261362415MCNGP337882.6512647.8585093.9874117.31299211158618719401972164600261803814Total StaffGP2139651231802362943504066NGP2936811231712352943554128MCNGP2352651151862362983364136Total DevicesGP102442761141471812182514NGP132050761051461812202536MCNGP163542721171471822062547Total MedicinesGP441995128858361725098664610079311492912908114322332NGP45989495226189372509848829991111492913014114428632MCNGP477535482558361707308839810079311564712554614463732Objective FunctionGP96019.55581.753208361124976193100526885483510540429693050546212NGP40147.1202.2552902.21206541177713227040613640260438143251516021MCNGP4.656.8529499.45200538.3525486.9808013.91166643138287918405690Figure 8
Comparison of staff growth between the proposed models.Figure 9
Comparison of increase in devices between the proposed models.Figure 10
Comparison of the growing demand for medicines between the proposed models.The case study shows that, in the three proposed models, the ratio of staff’s budget to the total budget increases annually. In contrast, the ratio of the medical drugs’ budget to the total budget decreases annually. Tables5, 12, and 19 show that the ratios in 2018 are 28%, 34%, and 15% of the staff’s budget from the total budget and 64%, 59%, and 76% of medical drugs’ budget from the total budget, while the ratios in 2048 are 63%, 63%, and 63% of the staff’s budget from the total budget and 26%, 25%, and 26% of medical drugs’ budget from the total budget. The obtained results of the case study are almost similar. This convergence indicates the stability and robustness of the mathematical models.On the other hand, the large numbers, in this case, have made significant differences unclear. In general, the results obtained using both neutrosophic goal programming models are more realistic and flexible than the results obtained without the neutrosophic approach. We thus introduced several scenarios for each period, allowing the decision-maker more flexibility to choose the most appropriate model that corresponds to other uncontrolled factors in this study. For example, the economy grew rapidly in a specific period, which would enable the decision-maker to choose the largest budget then; the converse holds true during periods of recession. Therefore, the proposed models in this study provide sustainable planning for several future periods.


## 5.1. Results of the Goal Programming Model

The results of the first model are illustrated in Tables5–10, where Table 5 shows the optimal budgets for each year and the ratio of each sub-budget of the total budget. Four budgets in this study are considered: total budget, staff budget, medical devices budget. and medical drugs budget. In the last column of this table, the growth rate for each budget is presented. The results indicate the need to increase the budget of the staff numbers, medical devices, and medical drugs by 13%, 6%, 9%, and 32%, respectively, annually. Figure 2 illustrates the growth of the total budget as well as the other budgets.Table 5
The obtained budgets using GP model.
Year201820202024202820322036204020442048GR (%)TBCost316705.7426305.1585093.9920536.31263664158618719213832258103258057913TBSCost876001574402592004946407204809432001178640140112016238406(%)283744545759616263TBMCost260003280057400922001448001794002142002636002982009(%)8810101111111212TBDCost203105.7236065.1268493.9333696.3398384.2463587.2528543.1593383658539.232(%)645546363229282626Table 6
The number and corresponding cost of staff obtained using GP model.
Staffx1x2x3x4x5x6x7x8TotalOncologistsGeneral doctorsRadiologistsPharmacistsLab technicianX-ray technicianNursesOther staff2018No.2232324321Ratio (%)1010141014101914100Cost192007680172806720100806240960010800876002020No.43515313539Ratio (%)1081331383313100Cost384001152028800336016800936031200180001574402024No.65919323965Ratio (%)981421453514100Cost576001920051840336030240936055200324002592002028No.1291721764317123Ratio (%)1071421453514100Cost115200345609792067205712018720103200612004946402032No.17132532596325180Ratio (%)971421453514100Cost16320049920144000100808400028080151200900007204802036No.221733433118333236Ratio (%)971421453514100Cost2112006528019008013440110880343201992001188009432002040No.2821415411410341294Ratio (%)1071421453514100Cost26880080640236160168001377604368024720014760011786402044No.3325495491712349350Ratio (%)971411453514100Cost31680096000282240168001646405304029520017640014011202048No.3829576571914357406Ratio (%)971411453514100Cost36480011136032832020160191520592803432002052001623840GR (%)68635611366Table 7
The number with the corresponding cost of medical devices obtained using GP model.
Medical devicesy1y2y3y4y5y6y7TotalBlood testChemistry apparatusOncology indicationsUltrasonicX-rayMammogramOther devices2018No.111111410Ratio (%)10101010101040100Cost420048006660320032003200800260602020No.2111111724Ratio (%)84444471100Cost8400480066603200320032003400328602024No.3222213042Ratio (%)75555271100Cost126009600133206400640032006000575202028No.5333325776Ratio (%)74444375100Cost21000144001998096009600640011400923802032No.75555384114Ratio (%)64444374100Cost29400240003330016000160009600168001451002036No.966664110147Ratio (%)64444375100Cost378002880039960192001920012800220001797602040No.1177775137181Ratio (%)64444376100Cost462003360046620224002240016000274002146202044No.1399995164218Ratio (%)64444275100Cost546004320059940288002880016000328002641402048No.15101010106190251Ratio (%)64444276100Cost63000480006660032000320001920038000298800GR (%)7111111111824Table 8
The quantities of medical drugs obtained using the GP model.
Medical drug names201820202024202820322036204020442048GR (%)z113503156641782422145264663078735108394294375032z2809310613215718320823426032z3484562639794949110412591414156932z410611231140117412080242027593099343832z510491217138517212057239227283064339932z63664254846017188359521069118632z761070880610011196139115861782197732z869480691711391361158318052027224932z915518020525530435440345350332z106057027999931186138015731767196132z11453552615987743888891034011791132431469432z1221024427834541247954661468132z1327531936345153962771580389132z1415518020525530435440345350332z1511012814618121625128632235732z1622626329937144351658866073332z1763747394841410454124941453316573186132065232z18778903102712761525177420232272252132z19790917104312961549180220542307256032z2075687799812401482172419662208245032z2115217720125029834739644449332z2221324828235041848655462269132z2325752987339942235047587166957519834332z2420022323264332843924456552065846648732z2528132637146155164173182191132z2626230434643051459868276684932z2741147754367580693810691201133232z2815818420926031036141146251232z299131060120614981790208223742666295932z3011012814618121625128632235732z3112314316320224228132036039932z32786912103812901541179320442296254732z33332386439545651757864970107632z3417019822527933438844249755132z3524027931739447154862470177832z3628633237847056165374483692732z3721082446278334584132480754816156683032z3841485568819410712013332z395261698610211913615216932z4016819522227633038443749154532Sum441995128858361725098664610079311492912908114322332Table 9
The negative deviational variables obtained using GP model.
Negative deviations201820202024202820322036204020442048N14800000000000N2000000000N3000.2000000N44800000000000Table 10
The positive deviational variables obtained using the GP model.
Positive deviations201820202024202820322036204020442048P10288160415.8562485.6965498.41344271175526721484612527308P201448020827225645470463403282608010051681184496P300017973.656090.476207.2103107.3138125.4158316.4P401448020827225645470463403282608010051681184496PX11.3333330.66666700.6666670.33333300.6666670.3333330PX21.50.50.50.50.50.50.50.50.5PX3200000000PX41.90.50.10.30.50.70.90.10.3PX5200000000PX61.6666671.33333300.3333330.66666700.3333330.6666670PX71.50.50.50.50.50.50.50.50.5PX8200000000PY10.750.750.750.750.750.750.750.750.75PY20.8333330.1666670.50.1666670.8333330.50.1666670.8333330.5PY30.8333330.1666670.50.1666670.8333330.50.1666670.8333330.5PY40.8333330.1666670.50.1666670.8333330.50.1666670.8333330.5PY50.8333330.1666670.50.1666670.8333330.50.1666670.8333330.5PY60.90.50.10.30.50.70.90.10.3PY70.6666670.33333300.3333330.66666700.3333330.6666670Figure 2
Budget growth using the GP model.Table6 shows the optimal staff numbers that should be employed annually. Each row in this table indicates the years and the corresponding optimal staff number, each ratio of the total, and the corresponding cost for each. In the last row of this table, we observe an increase in the varying annual growth rates for each kind of staff, according to the corresponding optimal rate with respect to the number of patients. The growth rate for the total staff is 6%.Figure3 illustrates the optimal number of different staffs. Evidently, the number of nurses increases the most with time. Similarly, Tables 7 and 8 show the increasing number of medical devices and medical drugs, respectively. Figure 4 illustrates the growth of the number of medical devices; we find that other equipment increases the most with time. The growth rate for the total number of medical devices is 4%, which is less than the total number of staff growth owing to converted constraints (31)–(34).Figure 3
The staff growth using the GP model.Figure 4
Medical devices growth using GP model.Table9 shows the negative deviational variables. Almost all variables equal to zero except N1 and N4, which are equal to US$4,800. That is, in 2018, the optimal budget of the medical drugs must be decreased to US$385,200. All the remaining negative deviations are equal to zero and, hence, there is no need to record them in the table. In contrast, the nonzero positive deviational variables are reported in Table 10.We find that the total budgets, staff, and medical devices should be increased further in order to provide high-quality service in expectation of the annual increase in patients. The positive deviations of medical drugs are equal to zero—its dynamic budget can sufficiently cover needs.


## 5.2. Results of the Neutrosophic Goal Programming Model

There are three states of an economy: growth, stability, and stagnation. The budgets are assumed in accordance with these states. That is, the budgets are increased in the optimism state; are left unchanged or changed only marginally in the stability state; and are decreased in the pessimism state. This allows us to represent the budgets using the neutrosophic concept. Table11 shows the values of the neutrosophic parameters. Then, equation (21) is used to give the deneutrosophic treatment to the obtained values.Table 11
Values of the neutrosophic variables.
τ˜aNaa,lμ,rμ;wa,a,lσ,rσ;ua,a,lν,rν;yaαβγTB˜N570860100000,200000;0.9,30000,40000;0.3,800000,100000;0.50.10.90.8TBS˜N13560013000,100000;0.9,18000,15000;0.3,20000,30000;0.50.10.90.8TBM˜N452601000,5000;0.9,9000,10000;0.3,10000,15000;0.50.10.90.8TBD˜N39000010000,100000;0.9,20000,30000;0.3,15000,20000;0.50.10.90.8Table12 shows the optimal different budgets for each year. The results indicate that the budgets’ growth rate is 15% annually. There is a marginal increase in the growth rate in the goal programming model. Table 13 shows that the optimal staff number should be determined yearly using this model. The growth rate for the total staff is 8%. There is a small decrease in the growth rate in the goal programming model. In the same way, Tables 14 and 15 show the increase in medical devices and medical drugs. We find that other equipment increases the most with time. The negative deviational variables are also equal to zero except N3, which is equal US$20,064 for 2018, as reported in Table 16. In contrast, the nonzero positive deviational variables are mentioned in Table 17.Table 12
Budgets results of NGP model.
Year201820202024202820322036204020442048GR (%)TBCost359415.6407990.3672852.8920536.31196277157947119213832288403261362415TBSCost1209601519203261604946406840009408001178640142584016516808(%)343748545760616263TBMCost2660028000620009220012200017920021420026400029860010(%)779101011111211TBDCost211855.6228070.3284692.8333696.3390277.3459470.6528543.1598563.466334433(%)595642363329282625Table 13
The staff numbers obtained using the NGP model.
Staffx1x2x3x4x5x6x7x8TotalOncologistsGeneral doctorsRadiologistsPharmacistsLab technicianX-ray technicianNursesOther staff2018No.3342427429Ratio (%)10101471472414100Cost288001152023040672013440624016800144001209602020No.42616310436Ratio (%)1161731782811100Cost38400768034560336020160936024000144001519202024No.86112114281181Ratio (%)1071421453514100Cost7680023040633606720369601248067200396003261602028No.1291721764317123Ratio (%)1071421453514100Cost115200345609792067205712018720103200612004946402032No.16122432486024171Ratio (%)971421453514100Cost15360046080138240100808064024960144000864006840002036No.221733433118233235Ratio (%)971421453514100Cost2112006528019008013440110880343201968001188009408002040No.2821415411410341294Ratio (%)1071421453514100Cost26880080640236160168001377604368024720014760011786402044No.3425505501712450355Ratio (%)1071411453514100Cost32640096000288000168001680005304029760018000014258402048No.3929586582014458412Ratio (%)971411453514100Cost37440011136033408020160194880624003456002088001651680GR (%)811835811588Table 14
The medical devices numbered obtained using NGP model.
Medical devicesy1y2y3y4y5y6y7TotalBlood TestChemistry apparatusOncology indicationsUltrasonicX-rayMammogramOther devices2018No.111111713Ratio (%)88888854100Cost4200480066603200320032001400266602020No.1111111420Ratio (%)55555570100Cost4200480066603200320032002800280602024No.3222223750Ratio (%)64444474100Cost126009600133206400640064007400621202028No.5333325776Ratio (%)74444375100Cost21000144001998096009600640011400923802032No.64444380105Ratio (%)64444376100Cost25200192002664012800128009600160001222402036No.966664109146Ratio (%)64444375100Cost378002880039960192001920012800218001795602040No.1177775137181Ratio (%)64444376100Cost462003360046620224002240016000274002146202044No.1399995166220Ratio (%)64444275100Cost546004320059940288002880016000332002645402048No.15101010106192253Ratio (%)64444276100Cost63000480006660032000320001920038400299200GR7111111111846Table 15
The medical drugs quantities obtained using NGP model.
Medical drug names201820202024202820322036204020442048GR (%)z114044151241890522145259263051735108397534407433z2849011213215418120823626233z3504543678794930109412591425158033z411041189148617412038239827593124346433z510911175146917212015237127283089342433z63814105136017038289521078119533z763568485410011172137915861796199233z872277897211391333156918052044226633z916217421725529835140345750633z106306788479931162136815731782197533z11471750806349743887081025011791133521480333z1221923629434540447554661968633z1328630838545152862271581089833z1416217421725529835140345750633z1511512415418121224928632436033z1623625431737143451158866673833z1766297139892410454122391440616573187662080533z18810872109012761494175920232291254033z19822885110612961517178620542326257933z20787847105912401452170919662226246833z2115917121325029234439644849733z2222223929935040948255462869633z2326782884360542234944582066957581840533z2420832243280332843844452552065894653533z2529331539446154063673182891833z2627329436743050459368277285633z2742846157667579092910691210134233z2816517722226030435841146651633z299501023127914981753206423742688298133z3011512415418121224928632436033z3112813817320223727832036340233z32818881110112901510177720442314256633z33346372465545638751864978108433z3417719123827932738544250155533z3525026933639446154362470778433z3629832140147055064774484293433z3721932361295234584048476554816206688133z3843465868799310712113433z395559738610011813615417034z4017518923627632338043749554933Sum45989495226189372509848829991111492913014114428633Table 16
The negative deviational variables obtained using the NGP model.
Negative deviations201820202024202820322036204020442048N1000000000N2000000000N32006400000000N4000000000Table 17
The positive deviational variables obtained using the NGP model.
Positive deviations201820202024202820322036204020442048P1096276448.6691594.3888565.61352026196015621907132575798P22006448136320249634.3423648634344790216.410266341209082P3003808.57115674.1241269.5783338.0599665.71137445.5157635P4048136320249634.3423648634344790216.410266341209082PX11.6666671.3333330.6666670.66666700.3333330.6666670.9333330.6PX2200.50.500.750.50.20.2PX3220000.500.40.4PX41.80.60.90.30.60.750.94.00E-020.24PX5220000.500.40.4PX61.3333331.6666670.3333330.33333300.1666670.3333330.4666670.8PX7200.50.500.750.500PX8200000.500.40.4PY10.500.250.7500.8750.750.60.6PY20.6666670.3333330.1666670.16666700.5833330.1666670.7333330.4PY30.6666670.3333330.1666670.16666700.5833330.1666670.7333330.4PY40.6666670.3333330.1666670.16666700.5833330.1666670.7333330.4PY50.6666670.3333330.1666670.16666700.5833330.1666670.7333330.4PY60.80.60.90.30.60.750.94.00E-020.24PY70.3333330.6666670.3333330.33333300.6666670.3333330.6666670


## 5.3. Results of the Neutrosophic Multi-Choices Goal Programming Model

In this model, the multi-choice goal programming was applied to randomly select one value from the three values obtained from the neutrosophic membership functions. Table18 shows the three means for each budget obtained using equations (17), (19) and (20) with the same parameters reported in Table 11.Table 18
Values of the neutrosophic variables.
τ˜gNgg1,g2,g3TB˜N570860615304.4,575145.7143,590860TBS˜N135600174266.7,134314.3,131600TBM˜N4526047037.78,45688.57,47260TBD˜N390000430000,394285.7,392000Tables19–22 show the obtained values for all variables. Similarly, Figures 5–7 illustrate the growth in the budgets, staff, medical devices, and medical drugs. Notably, we observe the aliasing of the curves, in contrast to Figure 2, indicating the random selection of the values of binary variable b using multi-choice goal programming.Table 19
The obtained budgets using NMCGP model.
Year201820202024202820322036204020442048GR (%)TBCost337882.6512647.8585093.9874117.31299211158618719401972164600261803814TBSCost907202083202592004612807476009432001193760134712016540806(%)153632505759616263TBMCost2720052000574008740014540017940021440024020029880010(%)108791111111212TBDCost219962.6252327.8268493.9325437.3406211.4463587.2532037.4577280.5665157.634(%)765661413229282626Table 20
The staff number obtained using the NMCGP model.
Staffx1x2x3x4x5x6x7x8OncologistsGeneral doctorsRadiologistsPharmacistsLab technicianX-ray technicianNursesOther staffTotal2018No.2231318323Ratio (%)991341343513100Cost1920076801728033601008031201920010800907202020No.54717318752Ratio (%)1081321363513100Cost480001536040320336023520936043200252002083202024No.65919323965Ratio (%)981421453514100Cost576001920051840336030240936055200324002592002028No.1181621664016115Ratio (%)1071421453514100Cost10560030720921606720537601872096000576004612802032No.18132632696526186Ratio (%)1071421453514100Cost17280049920149760100808736028080156000936007476002036No.221733433118333236Ratio (%)971421453514100Cost2112006528019008013440110880343201992001188009432002040No.2821425421410442298Ratio (%)971421453514100Cost26880080640241920168001411204368024960015120011937602044No.3224475471611847336Ratio (%)1071411453514100Cost30720092160270720168001579204992028320016920013471202048No.3929586582014558413Ratio (%)971411453514100Cost37440011136033408020160194880624003480002088001654080GR (%)6861866666Table 21
The medical devices numbered obtained using NMCGP model.
Medical devicesy1y2y3y4y5y6y7TotalBlood testChemistry apparatusOncology indicationsUltrasonicX-rayMammogramOther devices2018No.1111111016Ratio (%)66666663100Cost4200480066603200320032002000272602020No.2222212435Ratio (%)66666369100Cost84009600133206400640032004800521202024No.3222213042Ratio (%)75555271100Cost126009600133206400640032006000575202028No.4333325472Ratio (%)64444375100Cost16800144001998096009600640010800875802032No.75555387117Ratio (%)6%4%4%4%4%3%74%100%Cost29400240003330016000160009600174001457002036No.966664110147Ratio (%)64444375100Cost378002880039960192001920012800220001797602040No.1177775138182Ratio (%)64444376100Cost462003360046620224002240016000276002148202044No.1288885157206Ratio (%)64444276100Cost504003840053280256002560016000314002406802048No.15101010106193254Ratio (%)6%4%4%4%4%2%76%100%Cost63000480006660032000320001920038600299400GR (%)7111111111867Table 22
Medical drugs quantities obtained using the NMCGP model.
Medicament name201820202024202820322036204020442048GR (%)z114584167441782421605270063078735324383494418234z28710010612816018321022826234z3523601639775968110412671375158434z411461316140116982122242027763014347234z511331301138516792098239227452980343334z63964544845867328359581040119834z76597578069761220139115961733199634z875086191711111388158318161971227134z916819320524831035440644150834z106547517999681210138015831719198034z11489856245987725690701034011864128801483934z1222726127833642047955059768834z1329734136344055062772078190034z1416819320524831035440644150834z1511913714617622025128831336034z1624528129936245251659264274034z1768847904841410199127481453316675181032085634z18841965102712451556177420362210254634z19854980104312641580180220672244258534z2081793899812101512172419782148247434z2116518920124430434739843249834z2223126528234142648655860569734z2327813193339941205150587167377313842634z2421632483264332044004456552385686655134z2530434937145056264173679992034z2628332534642052459868674585834z2744451054365882293810761168134534z2817119620925331636141444951734z299871133120614611826208223892593298834z3011913714617622025128831336034z3113315316319724628132235040334z32849975103812581572179320572233257234z33359412439532664757869943108734z3418421122527234038844548355734z3526029831738448054862868278634z3630935537845857265374981393634z3722772614278333734216480755155987689834z3845515566829410811713535z395765698410411913714817135z4018220922226933638444047855034Sum477535482558361707308839810079311564712554614463734Figure 5
Budget growth using the NMCGP model.Figure 6
The staff growth using the NMCGP model.Figure 7
The medical devices growth using the NMCGP model.Overall, Table23 summarizes the comparison between the three proposed models. The growth rates of the total budget and total staff numbers using the neutrosophic goal programming model were the highest, whereas the multi-choice model shows the highest growth rate of the number of medical devices. The growth rate of medical drugs for all proposed models is almost equal. The multi-choice model yields the least deviations. We now discuss the results obtained in 2048 as an example. The total budget, total staff, and total medical devices obtained using the multi-choice model are higher than those obtained using the other models. Similarly, in the same year, the multi-choice model yields the lowest summation of deviations. Figures 8–10 illustrate a comparison between the proposed models for staff growth, the increasing medical devices, and the increase in the demand for medicines for the period between 2018 and 2048, respectively.Table 23
Summary of the comparison between the proposed models.
201820202024202820322036204020442048GR (%)Total BudgetGP316705.7426305.1585093.9920536.31263664158618719213832258103258057913NGP359415.6407990.3672852.8920536.31196277157947119213832288403261362415MCNGP337882.6512647.8585093.9874117.31299211158618719401972164600261803814Total StaffGP2139651231802362943504066NGP2936811231712352943554128MCNGP2352651151862362983364136Total DevicesGP102442761141471812182514NGP132050761051461812202536MCNGP163542721171471822062547Total MedicinesGP441995128858361725098664610079311492912908114322332NGP45989495226189372509848829991111492913014114428632MCNGP477535482558361707308839810079311564712554614463732Objective FunctionGP96019.55581.753208361124976193100526885483510540429693050546212NGP40147.1202.2552902.21206541177713227040613640260438143251516021MCNGP4.656.8529499.45200538.3525486.9808013.91166643138287918405690Figure 8
Comparison of staff growth between the proposed models.Figure 9
Comparison of increase in devices between the proposed models.Figure 10
Comparison of the growing demand for medicines between the proposed models.The case study shows that, in the three proposed models, the ratio of staff’s budget to the total budget increases annually. In contrast, the ratio of the medical drugs’ budget to the total budget decreases annually. Tables5, 12, and 19 show that the ratios in 2018 are 28%, 34%, and 15% of the staff’s budget from the total budget and 64%, 59%, and 76% of medical drugs’ budget from the total budget, while the ratios in 2048 are 63%, 63%, and 63% of the staff’s budget from the total budget and 26%, 25%, and 26% of medical drugs’ budget from the total budget. The obtained results of the case study are almost similar. This convergence indicates the stability and robustness of the mathematical models.On the other hand, the large numbers, in this case, have made significant differences unclear. In general, the results obtained using both neutrosophic goal programming models are more realistic and flexible than the results obtained without the neutrosophic approach. We thus introduced several scenarios for each period, allowing the decision-maker more flexibility to choose the most appropriate model that corresponds to other uncontrolled factors in this study. For example, the economy grew rapidly in a specific period, which would enable the decision-maker to choose the largest budget then; the converse holds true during periods of recession. Therefore, the proposed models in this study provide sustainable planning for several future periods.


## 6. Conclusions

In this paper, we proposed three models for solving healthcare planning problems. The proposed models were applied to a realistic case study of the Al-Amal Center for Oncology in Aden, Yemen. We used dynamic goal programming to predict the optimal solution for each variable in every period addressed in this article. Three models were eventually proposed: crisp, neutrosophic, and neutrosophic multi-choice goal programming. The goals addressed in the proposed models are related to the budget, the number of staffs and materials to perform the tasks efficiently. The proposed models yielded the optimum budget as well as the optimal number of staff and other medical supplies required to provide high-quality service for patients. Our results and insights thereof would be valuable for planners who could guide healthcare staff in providing the necessary resources for optimal annual planning. The diversity in the results obtained from the proposed models gives decision-makers the flexibility to make optimal decisions based on the state of the economy in each period. Although the proposed models were applied to healthcare planning, our approaches can be implemented on a large-scale healthcare system. Moreover, metaheuristics algorithms can be used to solve the models.



---

*Source: 3602792-2022-01-07.xml*