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Dr. Mohammad Khan
Saudi Electronic University

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0 Decision-Making
0 Mathematical Programming
0 Reliability
0 Stochastic Programming
0 goal programming

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Journal article
Published: 23 July 2021 in Sustainability
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The demand for cost-efficient and clean power energy cannot be overemphasised, especially in a developing nation like India. COVID-19 has adversely affected many nations, power sector inclusive, and resiliency is imperative via flexible and sustainable power generation sources. Renewable energy sources are the primary focus of electricity production in the world. This study examined and assessed the optimal cost system of electricity generation for the socio-economic sustainability of India. A sustainable and flexible electricity generation model is developed using the concept of flexible fuzzy goal programming. This study is carried out with the aim of achieving the government’s intended nationally determined contribution goals of reducing emission levels, increasing the capacity of renewable sources and the must-run status of hydro and nuclear, and technical and financial parameters. The result shows an optimal cost solution and flexibility in how increased electricity demand would be achieved and sustained via shifting to renewable sources such as solar, wind and hydro.

ACS Style

Mohammad Khan; Asif Pervez; Umar Modibbo; Jahangir Chauhan; Irfan Ali. Flexible Fuzzy Goal Programming Approach in Optimal Mix of Power Generation for Socio-Economic Sustainability: A Case Study. Sustainability 2021, 13, 8256 .

AMA Style

Mohammad Khan, Asif Pervez, Umar Modibbo, Jahangir Chauhan, Irfan Ali. Flexible Fuzzy Goal Programming Approach in Optimal Mix of Power Generation for Socio-Economic Sustainability: A Case Study. Sustainability. 2021; 13 (15):8256.

Chicago/Turabian Style

Mohammad Khan; Asif Pervez; Umar Modibbo; Jahangir Chauhan; Irfan Ali. 2021. "Flexible Fuzzy Goal Programming Approach in Optimal Mix of Power Generation for Socio-Economic Sustainability: A Case Study." Sustainability 13, no. 15: 8256.

Journal article
Published: 04 March 2021 in IEEE Access
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In this article, a multiobjective multiproduct production planning (MOMPP) problem discussed for a hardware firm. The hardware firm produces various types of hardware locks and other items in a production run. The firm manager’s objectives are minimizing the production cost and inventory holding cost while maximizing the net profit subject to some system constraints. The multiproduct production planning is solved with the last production run information precisely known to the decision-maker, and finally, the model is solved using the intuitionistic and neutrosophic programming approaches, respectively. Also, the multiproduct production planning problem is discussed for situations when the product information is vague. The interval-valued trapezoidal neutrosophic numbers used to define this Vagueness. The multiobjective multiproduct production planning problem under fuzziness is solved using the neutrosophic compromise programming. The stepwise solution procedures are discussed using the case study.

ACS Style

Mohammad Faisal Khan; Ahteshamul Haq; Aquil Ahmed; Irfan Ali. Multiobjective Multi-Product Production Planning Problem Using Intuitionistic and Neutrosophic Fuzzy Programming. IEEE Access 2021, 9, 37466 -37486.

AMA Style

Mohammad Faisal Khan, Ahteshamul Haq, Aquil Ahmed, Irfan Ali. Multiobjective Multi-Product Production Planning Problem Using Intuitionistic and Neutrosophic Fuzzy Programming. IEEE Access. 2021; 9 ():37466-37486.

Chicago/Turabian Style

Mohammad Faisal Khan; Ahteshamul Haq; Aquil Ahmed; Irfan Ali. 2021. "Multiobjective Multi-Product Production Planning Problem Using Intuitionistic and Neutrosophic Fuzzy Programming." IEEE Access 9, no. : 37466-37486.

Research article
Published: 06 November 2020 in Journal of Mathematics
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Some algebraic properties of Cesáro ideal convergent sequence spaces, C I and C 0 I , are studied in this article and some inclusion relations on these spaces are established.

ACS Style

Mohammad Faisal Khan. Some Results on Strongly Cesáro Ideal Convergent Sequence Spaces. Journal of Mathematics 2020, 2020, 1 -4.

AMA Style

Mohammad Faisal Khan. Some Results on Strongly Cesáro Ideal Convergent Sequence Spaces. Journal of Mathematics. 2020; 2020 ():1-4.

Chicago/Turabian Style

Mohammad Faisal Khan. 2020. "Some Results on Strongly Cesáro Ideal Convergent Sequence Spaces." Journal of Mathematics 2020, no. : 1-4.

Journal article
Published: 19 September 2020 in Symmetry
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Fuzzy goal programming (FGP) is applied to solve fuzzy multi-objective optimization problems. In FGP, the weights are associated with fuzzy goals for the preference among them. However, the hierarchy within the fuzzy goals depends on several uncertain criteria, decided by experts, so the preference relations are not always easy to associate with weight. Therefore, the preference relations are provided by the decision-makers in terms of linguistic relationships, i.e., goal A is slightly or moderately or significantly more important than goal B. Due to the vagueness and ambiguity associated with the linguistic preference relations, intuitionistic fuzzy sets (IFSs) are most efficient and suitable to handle them. Thus, in this paper, a new fuzzy goal programming with intuitionistic fuzzy preference relations (FGP-IFPR) approach is proposed. In the proposed FGP-IFPR model, an achievement function has been developed via the convex combination of the sum of individual grades of fuzzy objectives and amount of the score function of IFPRs among the fuzzy goals. As an extension, we presented the linear and non-linear, namely, exponential and hyperbolic functions for the intuitionistic fuzzy preference relations (IFPRs). A study has been made to compare and analyze the three FGP-IFPR models with intuitionistic fuzzy linear, exponential, and hyperbolic membership and non-membership functions. For solving all three FGP-IFPR models, the solution approach is developed that established the corresponding crisp formulations, and the optimal solution are obtained. The validations of the proposed FGP-IFPR models have been presented with an experimental investigation of a numerical problem and a banking financial statement problem. A newly developed distance measure is applied to compare the efficiency of proposed models. The minimum value of the distance function represents a better and efficient model. Finally, it has been found that for the first illustrative problem considered, the exponential FGP-IFPR model performs best, whereas for the second problem, the hyperbolic FGP-IFPR model performs best and the linear FGP-IFPR model shows worst in both cases.

ACS Style

Abdul Razzaq Abdul Ghaffar; Gulzarul Hasan; Zubair Ashraf; Mohammad Faisal Khan. Fuzzy Goal Programming with an Imprecise Intuitionistic Fuzzy Preference Relations. Symmetry 2020, 12, 1548 .

AMA Style

Abdul Razzaq Abdul Ghaffar, Gulzarul Hasan, Zubair Ashraf, Mohammad Faisal Khan. Fuzzy Goal Programming with an Imprecise Intuitionistic Fuzzy Preference Relations. Symmetry. 2020; 12 (9):1548.

Chicago/Turabian Style

Abdul Razzaq Abdul Ghaffar; Gulzarul Hasan; Zubair Ashraf; Mohammad Faisal Khan. 2020. "Fuzzy Goal Programming with an Imprecise Intuitionistic Fuzzy Preference Relations." Symmetry 12, no. 9: 1548.

Journal article
Published: 03 June 2020 in Symmetry
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Goal programming (GP) is a powerful method to solve multi-objective programming problems. In GP the preferential weights are incorporated in different ways into the achievement function. The problem becomes more complicated if the preferences are imprecise in nature, for example ‘Goal A is slightly or moderately or significantly important than Goal B’. Considering such type of problems, this paper proposes standard goal programming models for multi-objective decision-making, where fuzzy linguistic preference relations are incorporated to model the relative importance of the goals. In the existing literature, only methods with linear preference relations are available. As per our knowledge, nonlinearity was not considered previously in preference relations. We formulated fuzzy preference relations as exponential membership functions. The grades or achievement function is described as an exponential membership function and is used for grading levels of preference toward uncertainty. A nonlinear membership function may lead to a better representation of the achievement level than a linear one. Our proposed models can be a useful tool for different types of real life applications, where exponential nonlinearity in goal preferences exists. Finally, a numerical example is presented and analyzed through multiple cases to validate and compare the proposed models. A distance measure function is also developed and used to compare proposed models. It is found that, for the numerical example, models with exponential membership functions perform better than models with linear membership functions. The proposed models will help decision makers analyze and plan real life problems more realistically.

ACS Style

Mohammad Faisal Khan; Gulzarul Hasan; Abdul Quddoos; Armin Fügenschuh; Syed Suhaib Hasan. Goal Programming Models with Linear and Exponential Fuzzy Preference Relations. Symmetry 2020, 12, 1 .

AMA Style

Mohammad Faisal Khan, Gulzarul Hasan, Abdul Quddoos, Armin Fügenschuh, Syed Suhaib Hasan. Goal Programming Models with Linear and Exponential Fuzzy Preference Relations. Symmetry. 2020; 12 (6):1.

Chicago/Turabian Style

Mohammad Faisal Khan; Gulzarul Hasan; Abdul Quddoos; Armin Fügenschuh; Syed Suhaib Hasan. 2020. "Goal Programming Models with Linear and Exponential Fuzzy Preference Relations." Symmetry 12, no. 6: 1.

Journal article
Published: 01 June 2020 in Advances in Dynamical Systems and Applications
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ACS Style

Vakeel A. Khan; Abdullah A. H. Makharesh; Mohammad Faisal Khan; Sameera A. A. Abdullah; Kamal M. A. S. Alshlool. On Intuitionistic Fuzzy n–Normed I –Convergence of Sequence spaces Defined by Orlicz Function. Advances in Dynamical Systems and Applications 2020, 15, 1 .

AMA Style

Vakeel A. Khan, Abdullah A. H. Makharesh, Mohammad Faisal Khan, Sameera A. A. Abdullah, Kamal M. A. S. Alshlool. On Intuitionistic Fuzzy n–Normed I –Convergence of Sequence spaces Defined by Orlicz Function. Advances in Dynamical Systems and Applications. 2020; 15 (1):1.

Chicago/Turabian Style

Vakeel A. Khan; Abdullah A. H. Makharesh; Mohammad Faisal Khan; Sameera A. A. Abdullah; Kamal M. A. S. Alshlool. 2020. "On Intuitionistic Fuzzy n–Normed I –Convergence of Sequence spaces Defined by Orlicz Function." Advances in Dynamical Systems and Applications 15, no. 1: 1.

Journal article
Published: 08 May 2018 in Advances in Difference Equations
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Here, the concept of a new and interesting Riemann–Liouville type fractional derivative operator is exploited. Treatment of a fractional derivative operator has been made associated with the extended Appell hypergeometric functions of two variables and Lauricella hypergeometric function of three variables. With a view on analytic properties and application of new Riemann–Liouville type fractional derivative operator, we have obtained new fractional derivative formulas for some familiar functions and for Mellin transformation formulas. For the sake of justification of our new operator, we have established some presumably new generating functions for an extended hypergeometric function using the new definition of fractional derivative operator.

ACS Style

M. Shadab; Mohammad Khan; J. Luis Lopez-Bonilla. A new Riemann–Liouville type fractional derivative operator and its application in generating functions. Advances in Difference Equations 2018, 2018, 167 .

AMA Style

M. Shadab, Mohammad Khan, J. Luis Lopez-Bonilla. A new Riemann–Liouville type fractional derivative operator and its application in generating functions. Advances in Difference Equations. 2018; 2018 (1):167.

Chicago/Turabian Style

M. Shadab; Mohammad Khan; J. Luis Lopez-Bonilla. 2018. "A new Riemann–Liouville type fractional derivative operator and its application in generating functions." Advances in Difference Equations 2018, no. 1: 167.

Research article
Published: 10 March 2016 in Advances in Fuzzy Systems
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We consider an allocation problem in two-stage stratified Warner’s randomized response model and minimize the variance subject to cost constraint. The costs (measurement costs and total budget of the survey) in the cost constraint are assumed as fuzzy numbers, in particular triangular and trapezoidal fuzzy numbers due to the ease of use. The problem formulated is solved by using Lagrange multipliers technique and the optimum allocation obtained in the form of fuzzy numbers is converted into crisp form using -cut method at a prescribed value of . An illustrative numerical example is presented to demonstrate the proposed problem.1. IntroductionSample survey is a method of drawing an inference about the characteristic of a population or universe by observing only a part of the population. In modern complex surveys it is not possible to obtain true measurements on all the characteristics of interest on all the units in the sample because they are affected by two types of errors, that is, sampling errors and nonsampling errors. Nonsampling error is further classified into two types, response and nonresponse errors. Reduction in the reliability of measurements results in response error which can be minimized over repeated measurements. Whereas nonresponse errors are due to the nonavailability of information about some selected units for one or the other reason.A major source of nonresponse errors in sample surveys is the difficulty to obtain true responses when respondents are asked questions of highly personal or controversial nature, for example, questions on accumulated savings, intentional tax evasion, consumption of illegal drugs, and extramarital affairs. To avoid providing the requisite information or to avoid embarrassment, some respondents may refuse to answer or may intentionally give wrong answers. Thus the estimates obtained from a direct survey on such topics would be subject to high bias and any inference drawn from these would be erroneous. In order to solve this problem Warner [1] introduced a randomized response technique (RRT) which was developed subsequently by different authors.Some other authors who introduce other randomized response techniques are Mangat and Singh [2], Chua and Tsui [3], Padmawar and Vijayan [4], Chang and Huang [5], Chaudhuri [6], and so forth. Kim and Warde [7] suggested a stratified randomized response using optimum allocation. Mangat and Singh [2] proposed a two-stage randomized response model.Traditional decision making problems are handled either by the deterministic approach or by probabilistic approach. Deterministic approach completely avoiding the uncertainty provides an approximate solution while probabilistic approach on an assumption represents any uncertainty as a probability distribution. Both of these approaches only partially capture reality. Uncertainty also is involved in decision problems due to vagueness or impreciseness associated with linguistic information; then in this case optimization using fuzzy mathematical theories becomes more relevant. The idea of fuzzy decision making problems was proposed by Bellmann and Zadeh [8] and this idea was used in problems of mathematical programming by Tanaka and Asai [9]. Many authors use fuzzy data/fuzzy numbers in decision making problems such as Mahapatra and Roy [10], Pramanik and Roy [11], Abbasbandy and Hajjari [12], Kaur and Kumar [13], Ebrahimnejad [14], Sen et al. [15], and Gupta and Bari [16].In this paper, the deterministic problem formulated by Ghufran et al. [17] is extended by considering it into an uncertain environment. Here we consider the measurement cost and total budget for the survey as fuzzy numbers and formulate a fuzzy nonlinear programming problem. Then, the fuzzy nonlinear problem is solved by Lagrange multipliers technique after converting it into crisp problem using -cut. For demonstrating the proposed problem an illustrative example is presented.2. PreliminaryBefore formulating the problem of interest, we should know the basic definitions of fuzzy sets, fuzzy numbers, and so forth, which are reproduced here, from Bector and Chandra [18], Mahapatra and Roy [10], Hassanzadeh et al. [19], and Aggarwal and Sharma [20] as follows.Fuzzy Set. A fuzzy set in a universe of discourse is defined as the following set of pairs . Here is a mapping called the membership function of the fuzzy set and is called the membership value or degree of membership of in the fuzzy set . The larger the value of , the stronger the grade of membership in .-Cut. The -cut for a fuzzy set is shown by and for is defined to bewhere is the universal set.Upper and lower bounds for any -cut are given by and , respectively.Fuzzy Number. A fuzzy set in is called a fuzzy number if it satisfies the following conditions:(i)is convex and normal.(ii) is a closed interval for every .(iii)The support of is bounded.Triangular Fuzzy Number (TFN). A fuzzy number is said to be a triangular fuzzy number if its membership function is given byTrapezoidal Fuzzy Number (TrFN). A fuzzy set on real numbers is called a trapezoidal fuzzy number with membership function as follows:3. Statement of the Problem of Two-Stage Randomized Response ModelConsider a stratified population of size partitioned into disjoint strata of size ; and . Let denote stratum weights, denotes sample size, and is the total sample size for the stratum .In the first stage an individual respondent in the sample is instructed to use the randomization device which consists of the following two statements.(i) “I belong to the sensitive group” and (ii) “Go to the randomization device in the second stage” with known probabilities and , respectively.In the second stage the respondents are instructed to use the randomization device which consists of the following two statements.(i) “I belong to the sensitive group” and (ii) “I do not belong to the sensitive group” with known probabilities and , respectively.Assuming that the “Yes” or “No” reports are made truthfully for different outcomes and and are set by the interviewer, then the probability of a “Yes” answer in stratum is given byand is the proportion of respondents belonging to the sensitive group from stratum . The maximum likelihood estimate of iswhere is the estimated proportion of “Yes” answers which follows a binomial distribution . It can be seen that the estimator is unbiased for with varianceIf the suffix “” is removed then the expressions 1, 2, and 3 will be reduced in Mangat and Singh’s expressions.Since are drawn independently from each stratum, the estimators for individual strata can be added to obtain the estimator for the whole population. Thus an unbiased estimate of is given byUsing (5)with a sampling varianceorTo find the optimum allocation we either maximize the precision for fixed budget or minimize the cost for fixed precision.A linear cost function which is an adequate approximation of the actual cost incurred will bewhere is per unit cost of measurement in the th stratum and is overhead cost.In view of (4) to (11) the problem of finding optimum allocation is formulated as nonlinear programming problem (NLPP) as follows:The restrictions and are placed to have the representation of every stratum in the sample and to avoid the oversampling, respectively.4. Fuzzy Formulation of Two-Stage Randomized Response ProblemGenerally, real-world situations involve a lot of parameters such as cost and time, whose values are assigned by the decision makers and in the conventional approach, they are required to fix an exact value to the aforementioned parameters. However decision-makers frequently do not precisely know the value of those parameters. Therefore, in such cases it is better to consider those parameters or coefficients in the decision-making problems as fuzzy numbers. The mathematical modeling of fuzzy concepts was presented by Zadeh in [21]. Therefore, the fuzzy formulation of problem (12) with fuzzy cost constraint is given by considering two cases of fuzzy numbers, that is, triangular fuzzy number (TFN) and trapezoidal fuzzy number (TrFN).4.1. Case 1: Nonlinear Problem with TFNConsiderwhereand is triangular fuzzy numbers with membership functionSimilarly, the membership function for available budget can be expressed as4.2. Case 2: Nonlinear Problem with TrFNConsiderwhereand is trapezoidal fuzzy numbers with membership functionSimilarly, the membership function for available budget can be expressed as5. Lagrange Multipliers TechniqueIn problem (13), after ignoring the restrictions and taking equality in cost constraint the NLPP with TFNs is solved by Lagrange multipliers technique (LMT) as follows.The Lagrangian function can be defined asDifferentiating (21) with respect to and and equating to zero, we get the following sets of equations:orAlso,which givesorNow using (23) and (26), we obtainIn similar manner, the optimum allocation of NLPP (17) with trapezoidal fuzzy numbers can be obtain asThe allocations obtained in (27) and (28) are fuzzy in nature, so we have to convert fuzzy allocations into a crisp allocation by -cut method at a prescribed value of .6. Procedure for the Conversion of Fuzzy NumbersTo convert the fuzzy allocation into crisp allocation -cut method is used as follows.Let be a TFN. An -cut for , is computed aswhere is the corresponding -cut (see Figure 1).Figure 1: Triangular fuzzy number with an -cut.The allocation obtained in (27) is in the form of triangular fuzzy number; therefore by using (29) the equivalent crisp allocation is given bySimilarly, if is a TrFN, then the -cut for , is computed aswhere is the corresponding -cut (see Figure 2).Figure 2: Trapezoidal fuzzy number with an -cut.In similar manner, the crisp allocation corresponding to (28) is given byThe allocations obtained by (30) and (32) provide the solution to NLPP (13) and (17) if it satisfies the restriction ; . The allocations obtained in (30) and (32) may not be integer allocations, so to get integer allocations, rou

ACS Style

Mohammad Khan; Neha Gupta; Irfan Ali. Two-Stage Stratified Randomized Response Model with Fuzzy Numbers. Advances in Fuzzy Systems 2016, 2016, 1 -8.

AMA Style

Mohammad Khan, Neha Gupta, Irfan Ali. Two-Stage Stratified Randomized Response Model with Fuzzy Numbers. Advances in Fuzzy Systems. 2016; 2016 ():1-8.

Chicago/Turabian Style

Mohammad Khan; Neha Gupta; Irfan Ali. 2016. "Two-Stage Stratified Randomized Response Model with Fuzzy Numbers." Advances in Fuzzy Systems 2016, no. : 1-8.

Journal article
Published: 01 January 2014 in American Journal of Operations Research
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ACS Style

Shafiullah   Mohammad Faisal Khan; Irfan Ali. Fuzzy Geometric Programming in Multivariate Stratified Sample Surveys in Presence of Non-Response with Quadratic Cost Function. American Journal of Operations Research 2014, 04, 173 -188.

AMA Style

Shafiullah &nbsp, Mohammad Faisal Khan, Irfan Ali. Fuzzy Geometric Programming in Multivariate Stratified Sample Surveys in Presence of Non-Response with Quadratic Cost Function. American Journal of Operations Research. 2014; 04 (03):173-188.

Chicago/Turabian Style

Shafiullah   Mohammad Faisal Khan; Irfan Ali. 2014. "Fuzzy Geometric Programming in Multivariate Stratified Sample Surveys in Presence of Non-Response with Quadratic Cost Function." American Journal of Operations Research 04, no. 03: 173-188.

Original articles
Published: 10 January 2013 in Communications in Statistics - Simulation and Computation
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In this paper we consider the problem of determining the optimum number of repairable and replaceable components to maximize a system's reliability when both, the cost of repairing the components and the cost of replacement of components by new ones, are random. We formulate it as a problem of non-linear stochastic programming. The solution is obtained through Chance Constrained programming. We also consider the problem of finding the optimal maintenance cost for a given reliability requirement of the system. The solution is then obtained by using Modified E-model. A numerical example is solved for both the formulations.

ACS Style

Irfan Ali; Yashpal Singh Raghav; Mohammad Khan; Abdul Bari. Selective Maintenance in System Reliability with Random Costs of Repairing and Replacing the Components. Communications in Statistics - Simulation and Computation 2013, 42, 2026 -2039.

AMA Style

Irfan Ali, Yashpal Singh Raghav, Mohammad Khan, Abdul Bari. Selective Maintenance in System Reliability with Random Costs of Repairing and Replacing the Components. Communications in Statistics - Simulation and Computation. 2013; 42 (9):2026-2039.

Chicago/Turabian Style

Irfan Ali; Yashpal Singh Raghav; Mohammad Khan; Abdul Bari. 2013. "Selective Maintenance in System Reliability with Random Costs of Repairing and Replacing the Components." Communications in Statistics - Simulation and Computation 42, no. 9: 2026-2039.