ISSN:
1547-5816
eISSN:
1553-166X
All Issues
Journal of Industrial & Management Optimization
January 2015 , Volume 11 , Issue 1
Select all articles
Export/Reference:
2015, 11(1): 1-11
doi: 10.3934/jimo.2015.11.1
+[Abstract](2654)
+[PDF](344.4KB)
Abstract:
Based on the feature of small profits but quick turnover, group-buying, an emerging e-commerce model, benefits both retailers and customers. In this paper, we explore the optimal price discount, order quantity and minimum quantity with a fixed selling price of the product to maximize the sellers' profit. The traditional newsvendor model framework is used in view of the shortened life cycle of most products. The demand of customers is assumed to be in addition form and product form, respectively, and the impacts of demand parameters are examined numerically. It is revealed that in some cases the profit cannot be improved significantly through price discount because of unconspicuous increase in demand. However, when the demand changes obviously with price discount, group-buying can bring more profit through price discount and inspire vendors to order more goods. Through numerical results, it is shown that the influence of demand in the product form is more evident than that in the addition form under the strategy of group-buying. Furthermore, the profit-based minimum quantity and the probability of selling nothing during the group time are also shown in this paper.
Based on the feature of small profits but quick turnover, group-buying, an emerging e-commerce model, benefits both retailers and customers. In this paper, we explore the optimal price discount, order quantity and minimum quantity with a fixed selling price of the product to maximize the sellers' profit. The traditional newsvendor model framework is used in view of the shortened life cycle of most products. The demand of customers is assumed to be in addition form and product form, respectively, and the impacts of demand parameters are examined numerically. It is revealed that in some cases the profit cannot be improved significantly through price discount because of unconspicuous increase in demand. However, when the demand changes obviously with price discount, group-buying can bring more profit through price discount and inspire vendors to order more goods. Through numerical results, it is shown that the influence of demand in the product form is more evident than that in the addition form under the strategy of group-buying. Furthermore, the profit-based minimum quantity and the probability of selling nothing during the group time are also shown in this paper.
A new approach for uncertain multiobjective programming problem based on $\mathcal{P}_{E}$ principle
2015, 11(1): 13-26
doi: 10.3934/jimo.2015.11.13
+[Abstract](2347)
+[PDF](351.3KB)
Abstract:
On the basis of the uncertainty theory, this paper is devoted to the uncertain multiobjective programming problem. Firstly, several principles are provided to define the relationship between uncertain variables. Then a new approach is proposed for obtaining Pareto efficient solutions in uncertain multiobjective programming problem based on $\mathcal{P}_{E}$ principle, which involves transforming the uncertain multiobjective problem into a problem with only one uncertain objective function, and its validity has been proved. Due to the complexity of this problem, it is very suitable for the use of genetic algorithm. Finally, a numerical example is presented to illustrate the novel approach proposed, and the genetic algorithm is adopted to solve it.
On the basis of the uncertainty theory, this paper is devoted to the uncertain multiobjective programming problem. Firstly, several principles are provided to define the relationship between uncertain variables. Then a new approach is proposed for obtaining Pareto efficient solutions in uncertain multiobjective programming problem based on $\mathcal{P}_{E}$ principle, which involves transforming the uncertain multiobjective problem into a problem with only one uncertain objective function, and its validity has been proved. Due to the complexity of this problem, it is very suitable for the use of genetic algorithm. Finally, a numerical example is presented to illustrate the novel approach proposed, and the genetic algorithm is adopted to solve it.
2015, 11(1): 27-40
doi: 10.3934/jimo.2015.11.27
+[Abstract](2862)
+[PDF](465.1KB)
Abstract:
This paper is concerned with a maximum principle for a new class of non-zero sum stochastic differential games. Compared with the existing literature, the game systems in this paper are forward-backward systems in which the control variables consist of two components: the continuous controls and the impulse controls. Necessary optimality conditions and sufficient optimality conditions in the form of maximum principle are obtained respectively for open-loop Nash equilibrium point of the foregoing games. A fund management problem is used to shed light on the application of the theoretical results, and the optimal investment portfolio and optimal impulse consumption strategy are obtained explicitly.
This paper is concerned with a maximum principle for a new class of non-zero sum stochastic differential games. Compared with the existing literature, the game systems in this paper are forward-backward systems in which the control variables consist of two components: the continuous controls and the impulse controls. Necessary optimality conditions and sufficient optimality conditions in the form of maximum principle are obtained respectively for open-loop Nash equilibrium point of the foregoing games. A fund management problem is used to shed light on the application of the theoretical results, and the optimal investment portfolio and optimal impulse consumption strategy are obtained explicitly.
2015, 11(1): 41-63
doi: 10.3934/jimo.2015.11.41
+[Abstract](2841)
+[PDF](1333.5KB)
Abstract:
In this paper we deal with the distributed optimization of problems where the objective/constraints are related to the whole variables of the system. In this kind of problems we need to bring into play all the distributed agents of the system simultaneously to guarantee that the solution is feasible and of a good quality. We propose an efficient agents coordination protocol which avoids centralizing control and computation to one agent. It overcomes the shortcoming of tree search methods related to the agents order and accelerates the search process. The proposed protocol is applied to the distributed emergency vehicle management problem that consists in taking, in a distributed way, the decisions about the locations of a set of emergency vehicles in order to solve the dispatching and covering issues simultaneously. The protocol is compared to the Synchronous Limited Discrepancy Search method. It is also compared to other distributed and centralized methods. The obtained results showed the efficiency of the proposed protocol in finding good solutions in short times and its capacity to lessen the effect of the agents ordering on the solution quality.
In this paper we deal with the distributed optimization of problems where the objective/constraints are related to the whole variables of the system. In this kind of problems we need to bring into play all the distributed agents of the system simultaneously to guarantee that the solution is feasible and of a good quality. We propose an efficient agents coordination protocol which avoids centralizing control and computation to one agent. It overcomes the shortcoming of tree search methods related to the agents order and accelerates the search process. The proposed protocol is applied to the distributed emergency vehicle management problem that consists in taking, in a distributed way, the decisions about the locations of a set of emergency vehicles in order to solve the dispatching and covering issues simultaneously. The protocol is compared to the Synchronous Limited Discrepancy Search method. It is also compared to other distributed and centralized methods. The obtained results showed the efficiency of the proposed protocol in finding good solutions in short times and its capacity to lessen the effect of the agents ordering on the solution quality.
2015, 11(1): 65-81
doi: 10.3934/jimo.2015.11.65
+[Abstract](2468)
+[PDF](462.9KB)
Abstract:
In this paper, we analyze the convergence properties of a nonlinear Lagrangian method based on Log-Sigmoid function for nonconvex semidefinite programming (NCSDP) problems. It is different from other convergence analysis, because the subproblem in our algorithm is inexactly solved. Under the constraint nondegeneracy condition, the strict complementarity condition and the second order sufficient conditions, it is obtained that the nonlinear Lagrangian algorithm proposed is locally convergent by choosing a proper stopping criterion and the error bound of solution is proportional to the penalty parameter when the penalty parameter is less than a threshold.
In this paper, we analyze the convergence properties of a nonlinear Lagrangian method based on Log-Sigmoid function for nonconvex semidefinite programming (NCSDP) problems. It is different from other convergence analysis, because the subproblem in our algorithm is inexactly solved. Under the constraint nondegeneracy condition, the strict complementarity condition and the second order sufficient conditions, it is obtained that the nonlinear Lagrangian algorithm proposed is locally convergent by choosing a proper stopping criterion and the error bound of solution is proportional to the penalty parameter when the penalty parameter is less than a threshold.
2015, 11(1): 83-104
doi: 10.3934/jimo.2015.11.83
+[Abstract](2331)
+[PDF](490.5KB)
Abstract:
This paper investigates the M/M/1 warm-standby machine repair problem with multiple vacations and working breakdowns. We first apply a matrix-analytic method to obtain the steady-state probabilities. Next, we construct the total expected profit per unit time and formulate an optimization problem to find the maximum profit. The particle swarm optimization (PSO) algorithm is implemented to determine the optimal number of warm standbys and two variable service rates simultaneously at the optimal maximum profit. We compare the searching results of the PSO algorithm with those of Genetic algorithm (GA) and Exhaustive Search Method (ESM) to ensure the superior searching quality of the PSO algorithm. Sensitivity analysis with numerical illustrations is also provided to improve the design quality of system engineers.
This paper investigates the M/M/1 warm-standby machine repair problem with multiple vacations and working breakdowns. We first apply a matrix-analytic method to obtain the steady-state probabilities. Next, we construct the total expected profit per unit time and formulate an optimization problem to find the maximum profit. The particle swarm optimization (PSO) algorithm is implemented to determine the optimal number of warm standbys and two variable service rates simultaneously at the optimal maximum profit. We compare the searching results of the PSO algorithm with those of Genetic algorithm (GA) and Exhaustive Search Method (ESM) to ensure the superior searching quality of the PSO algorithm. Sensitivity analysis with numerical illustrations is also provided to improve the design quality of system engineers.
Statistical process control optimization with variable sampling interval and nonlinear expected loss
2015, 11(1): 105-133
doi: 10.3934/jimo.2015.11.105
+[Abstract](2286)
+[PDF](712.6KB)
Abstract:
The optimization of a statistical process control with a variable sampling interval is studied, aiming in minimization of the expected loss. This loss is caused by delay in detecting process change and depends nonlinearly on the sampling interval. An approximate solution of this optimization problem is obtained by its decomposition into two simpler subproblems: linear and quadratic. Two approaches to the solution of the quadratic subproblem are proposed. The first approach is based on the Pontryagin's Maximum Principle, leading to an exact analytical solution. The second approach is based on a discretization of the problem and using proper mathematical programming tools, providing an approximate numerical solution. Composite solution of the original problem is constructed. Illustrative examples are presented.
The optimization of a statistical process control with a variable sampling interval is studied, aiming in minimization of the expected loss. This loss is caused by delay in detecting process change and depends nonlinearly on the sampling interval. An approximate solution of this optimization problem is obtained by its decomposition into two simpler subproblems: linear and quadratic. Two approaches to the solution of the quadratic subproblem are proposed. The first approach is based on the Pontryagin's Maximum Principle, leading to an exact analytical solution. The second approach is based on a discretization of the problem and using proper mathematical programming tools, providing an approximate numerical solution. Composite solution of the original problem is constructed. Illustrative examples are presented.
2015, 11(1): 135-144
doi: 10.3934/jimo.2015.11.135
+[Abstract](2527)
+[PDF](372.9KB)
Abstract:
In this paper, new results are derived for the $(Q,r)$ stochastic inventory model. We derive approximate formulas for the optimal solution for the particular case of an exponential demand distribution. The approximate solution is within 0.29% of the optimal value. We also derive simple formulas for a Poisson demand distribution. The original expression involves double summation. We simplify the formula and are able to calculate the exact value of the objective function in $O(1)$ time with no need for any summations.
In this paper, new results are derived for the $(Q,r)$ stochastic inventory model. We derive approximate formulas for the optimal solution for the particular case of an exponential demand distribution. The approximate solution is within 0.29% of the optimal value. We also derive simple formulas for a Poisson demand distribution. The original expression involves double summation. We simplify the formula and are able to calculate the exact value of the objective function in $O(1)$ time with no need for any summations.
2015, 11(1): 145-170
doi: 10.3934/jimo.2015.11.145
+[Abstract](2379)
+[PDF](1324.8KB)
Abstract:
In this paper we model the working of local community finances. As a result of this first step, we obtain a systemic model that is used to formalize the problem of Alternative Financial Solutions Seeking, which consists in building a collection of Alternative Multi-Year Prospective Budgets from two Multi-Year Prospective Budgets built by a finance expert. The modeling and formalization steps are led in a way that allows us to implement a software code for Alternative Financial Solutions Seeking based on a Genetic Like Algorithm.
In this paper we model the working of local community finances. As a result of this first step, we obtain a systemic model that is used to formalize the problem of Alternative Financial Solutions Seeking, which consists in building a collection of Alternative Multi-Year Prospective Budgets from two Multi-Year Prospective Budgets built by a finance expert. The modeling and formalization steps are led in a way that allows us to implement a software code for Alternative Financial Solutions Seeking based on a Genetic Like Algorithm.
2015, 11(1): 171-183
doi: 10.3934/jimo.2015.11.171
+[Abstract](2471)
+[PDF](420.9KB)
Abstract:
A complex matrix $P$ is called Hermitian and $\{k+1\}$-potent if $P^{k+1}=P=P^*$ for some integer $k\geq 1$. Let $P$ and $Q$ be $n\times n$ Hermitian and $\{k+1\}$-potent matrices, we say that complex matrix $A$ is $\{P,Q,k+1\}$-reflexive (anti-reflexive) if $PAQ=A$ ($PAQ=-A$). In this paper, the solvability conditions and the general solutions of the left and right inverse eigenvalue problem for $\{P,Q,k+1\}$-reflexive and anti-reflexive matrices are derived, and the minimal and maximal rank solutions are given. Moreover, the associated optimal approximation problem is also considered. Finally, numerical example is given to illustrate the main results.
A complex matrix $P$ is called Hermitian and $\{k+1\}$-potent if $P^{k+1}=P=P^*$ for some integer $k\geq 1$. Let $P$ and $Q$ be $n\times n$ Hermitian and $\{k+1\}$-potent matrices, we say that complex matrix $A$ is $\{P,Q,k+1\}$-reflexive (anti-reflexive) if $PAQ=A$ ($PAQ=-A$). In this paper, the solvability conditions and the general solutions of the left and right inverse eigenvalue problem for $\{P,Q,k+1\}$-reflexive and anti-reflexive matrices are derived, and the minimal and maximal rank solutions are given. Moreover, the associated optimal approximation problem is also considered. Finally, numerical example is given to illustrate the main results.
2015, 11(1): 185-198
doi: 10.3934/jimo.2015.11.185
+[Abstract](2497)
+[PDF](753.6KB)
Abstract:
Online scheduling on identical machines is investigated in the setting where jobs arrive over time. The goal is to minimize the total completion time. A waiting strategy based online algorithm is designed and is proved to be $2.28$-competitive. The result improves the current best online algorithm from the worse-case prospective.
Online scheduling on identical machines is investigated in the setting where jobs arrive over time. The goal is to minimize the total completion time. A waiting strategy based online algorithm is designed and is proved to be $2.28$-competitive. The result improves the current best online algorithm from the worse-case prospective.
2015, 11(1): 199-216
doi: 10.3934/jimo.2015.11.199
+[Abstract](2685)
+[PDF](936.1KB)
Abstract:
We consider a multi-agent control problem using PDE techniques for a novel sensing problem arising in the leakage detection and localization of offshore pipelines. A continuous protocol is proposed using parabolic PDEs and then a boundary control law is designed using the maximum principle. Both analytical and numerical solutions of the optimality conditions are studied.
We consider a multi-agent control problem using PDE techniques for a novel sensing problem arising in the leakage detection and localization of offshore pipelines. A continuous protocol is proposed using parabolic PDEs and then a boundary control law is designed using the maximum principle. Both analytical and numerical solutions of the optimality conditions are studied.
2015, 11(1): 217-230
doi: 10.3934/jimo.2015.11.217
+[Abstract](2279)
+[PDF](353.7KB)
Abstract:
In this work a generalization of the notion of exhauster is considered. Exhausters are new tools in nonsmooth analysis introduced in works of Demyanov V.F., Rubinov A.M., Pshenichny B.N. In essence, exhausters are families of convex compact sets, allowing to represent the increments of a function at a considered point in an $\inf\max$ or $\sup\min$ form, the upper exhausters used for the first representation, and the lower one for the second representation. Using this objects one can get new optimality conditions, find descent and ascent directions and thus construct new optimization algorithms. Rubinov A.M. showed that an arbitrary upper or lower semicontinuous positively homogenous function bounded on the unit ball has an upper or lower exhausters respectively. One of the aims of the work is to obtain the similar result under weaker conditions on the function under study, but for this it is necessary to use generalized exhausters - a family of convex (but not compact!) sets, allowing to represent the increments of the function at a considered point in the form of $\inf\sup$ or $\sup\inf$. The resulting existence theorem is constructive and gives a theoretical possibility of constructing these families. Also in terms of these objects optimality conditions that generalize the conditions obtained by Demyanov V.F., Abbasov M.E. are stated and proved. As an illustration of obtained results, an example of $n$-dimensional function, that has a non-strict minimum at the origin, is demonstrated. A generalized upper and lower exhausters for this function at the origin are constructed, the necessary optimality conditions are obtained and discussed.
In this work a generalization of the notion of exhauster is considered. Exhausters are new tools in nonsmooth analysis introduced in works of Demyanov V.F., Rubinov A.M., Pshenichny B.N. In essence, exhausters are families of convex compact sets, allowing to represent the increments of a function at a considered point in an $\inf\max$ or $\sup\min$ form, the upper exhausters used for the first representation, and the lower one for the second representation. Using this objects one can get new optimality conditions, find descent and ascent directions and thus construct new optimization algorithms. Rubinov A.M. showed that an arbitrary upper or lower semicontinuous positively homogenous function bounded on the unit ball has an upper or lower exhausters respectively. One of the aims of the work is to obtain the similar result under weaker conditions on the function under study, but for this it is necessary to use generalized exhausters - a family of convex (but not compact!) sets, allowing to represent the increments of the function at a considered point in the form of $\inf\sup$ or $\sup\inf$. The resulting existence theorem is constructive and gives a theoretical possibility of constructing these families. Also in terms of these objects optimality conditions that generalize the conditions obtained by Demyanov V.F., Abbasov M.E. are stated and proved. As an illustration of obtained results, an example of $n$-dimensional function, that has a non-strict minimum at the origin, is demonstrated. A generalized upper and lower exhausters for this function at the origin are constructed, the necessary optimality conditions are obtained and discussed.
2015, 11(1): 231-240
doi: 10.3934/jimo.2015.11.231
+[Abstract](2029)
+[PDF](444.6KB)
Abstract:
Based on the analysis of force and action points of an UAV (unmanned aerial vehicle), we propose a concept called static stability degree deviation (SSDD) factor, which is related to the focus position, and can be used to modify the data for control law design. Furthermore, a SSDD-based method is presented to avoid the flight oscillation caused by the data deviation of aerodynamic focus. By using the attitude angle difference between real fight data and simulation data as an optimization index, the identification of SSDD factor and the data reproduction of the real flight data are achieved. The identification results are then used to modify aerodynamic blowing data. Based on the modified model, the augmentation control is designed by applying the altitude angle rate feedback to improve the equivalent damping ratio and frequency; thus the iteration design of the control law is performed.
Based on the analysis of force and action points of an UAV (unmanned aerial vehicle), we propose a concept called static stability degree deviation (SSDD) factor, which is related to the focus position, and can be used to modify the data for control law design. Furthermore, a SSDD-based method is presented to avoid the flight oscillation caused by the data deviation of aerodynamic focus. By using the attitude angle difference between real fight data and simulation data as an optimization index, the identification of SSDD factor and the data reproduction of the real flight data are achieved. The identification results are then used to modify aerodynamic blowing data. Based on the modified model, the augmentation control is designed by applying the altitude angle rate feedback to improve the equivalent damping ratio and frequency; thus the iteration design of the control law is performed.
2015, 11(1): 241-264
doi: 10.3934/jimo.2015.11.241
+[Abstract](3127)
+[PDF](1205.5KB)
Abstract:
In this paper we develop a numerical approach to a fractional-order differential Linear Complementarity Problem (LCP) arising in pricing European and American options under a geometric Lévy process. The LCP is first approximated by a nonlinear penalty fractional Black-Scholes (fBS) equation. We then propose a finite difference scheme for the penalty fBS equation. We show that both the continuous and the discretized fBS equations are uniquely solvable and establish the convergence of the numerical solution to the viscosity solution of the penalty fBS equation by proving the consistency, stability and monotonicity of the numerical scheme. We also show that the discretization has the 2nd-order truncation error in both the spatial and time mesh sizes. Numerical results are presented to demonstrate the accuracy and usefulness of the numerical method for pricing both European and American options under the geometric Lévy process.
In this paper we develop a numerical approach to a fractional-order differential Linear Complementarity Problem (LCP) arising in pricing European and American options under a geometric Lévy process. The LCP is first approximated by a nonlinear penalty fractional Black-Scholes (fBS) equation. We then propose a finite difference scheme for the penalty fBS equation. We show that both the continuous and the discretized fBS equations are uniquely solvable and establish the convergence of the numerical solution to the viscosity solution of the penalty fBS equation by proving the consistency, stability and monotonicity of the numerical scheme. We also show that the discretization has the 2nd-order truncation error in both the spatial and time mesh sizes. Numerical results are presented to demonstrate the accuracy and usefulness of the numerical method for pricing both European and American options under the geometric Lévy process.
2015, 11(1): 265-289
doi: 10.3934/jimo.2015.11.265
+[Abstract](3631)
+[PDF](476.7KB)
Abstract:
With an increasing number of large-scale natural and man-created disasters over the last decade, there is growing focus on the application of operations research techniques for humanitarian relief in the emerging field of emergency evacuation. Even though a large diversity of models have been developed, many rely on solving network-flow problems on appropriate graphs. In this survey, we give a systematic collection of network flow models used in emergency evacuation and their applications. We especially focus on results interrelating these models. Considered models include max flows and min cost flows, lexicographic flows, quickest flows, and earliest arrival flows, as well as contraflows and time-dependent problems.
With an increasing number of large-scale natural and man-created disasters over the last decade, there is growing focus on the application of operations research techniques for humanitarian relief in the emerging field of emergency evacuation. Even though a large diversity of models have been developed, many rely on solving network-flow problems on appropriate graphs. In this survey, we give a systematic collection of network flow models used in emergency evacuation and their applications. We especially focus on results interrelating these models. Considered models include max flows and min cost flows, lexicographic flows, quickest flows, and earliest arrival flows, as well as contraflows and time-dependent problems.
2015, 11(1): 291-305
doi: 10.3934/jimo.2015.11.291
+[Abstract](2220)
+[PDF](378.3KB)
Abstract:
In this paper, we consider the warehouse-retailer network design game based on the warehouse-retailer network design problem (WRND) proposed by Teo and Shu (2004). By carefully defining the generalized distance function, we present a cost-sharing method for the warehouse-retailer network design game. We show that the proposed cost-sharing scheme is cross-monotonic, competitive, and $3$-approximate cost recovery.
In this paper, we consider the warehouse-retailer network design game based on the warehouse-retailer network design problem (WRND) proposed by Teo and Shu (2004). By carefully defining the generalized distance function, we present a cost-sharing method for the warehouse-retailer network design game. We show that the proposed cost-sharing scheme is cross-monotonic, competitive, and $3$-approximate cost recovery.
2015, 11(1): 307-328
doi: 10.3934/jimo.2015.11.307
+[Abstract](1876)
+[PDF](515.4KB)
Abstract:
In this paper, combining the method of quasi-strongly sub-feasible directions (MQSSFD) and the working set technique, a new QP-free algorithm with an arbitrary initial iteration point for solving inequality constrained optimization is proposed. At each iteration, the algorithm solves only two systems of linear equations with a same uniformly nonsingular coefficient matrix to obtain the search direction. Furthermore, the positive definiteness assumption on the Hessian estimate is relaxed. Under some necessary assumptions, the new algorithm not only possesses global and strong convergence, but also ensures that the iteration points can get into the feasible set after finite iterations. Finally, a series of preliminary numerical results are reported to show that the algorithm is promising.
In this paper, combining the method of quasi-strongly sub-feasible directions (MQSSFD) and the working set technique, a new QP-free algorithm with an arbitrary initial iteration point for solving inequality constrained optimization is proposed. At each iteration, the algorithm solves only two systems of linear equations with a same uniformly nonsingular coefficient matrix to obtain the search direction. Furthermore, the positive definiteness assumption on the Hessian estimate is relaxed. Under some necessary assumptions, the new algorithm not only possesses global and strong convergence, but also ensures that the iteration points can get into the feasible set after finite iterations. Finally, a series of preliminary numerical results are reported to show that the algorithm is promising.
2015, 11(1): 329-343
doi: 10.3934/jimo.2015.11.329
+[Abstract](2640)
+[PDF](366.3KB)
Abstract:
In this paper, an optimal control problem for a class of hybrid systems is considered. By introducing a new time variable and transforming the hybrid optimal control problem into an equivalent problem, second order sufficient optimality conditions for this hybrid problem are derived. It is shown that sufficient optimality conditions can be verified by checking the Legendre-Clebsch condition and solving some Riccati equations with certain boundary and jump conditions. An example is given to show the effectiveness of the main results.
In this paper, an optimal control problem for a class of hybrid systems is considered. By introducing a new time variable and transforming the hybrid optimal control problem into an equivalent problem, second order sufficient optimality conditions for this hybrid problem are derived. It is shown that sufficient optimality conditions can be verified by checking the Legendre-Clebsch condition and solving some Riccati equations with certain boundary and jump conditions. An example is given to show the effectiveness of the main results.
2019 Impact Factor: 1.366
Authors
Editors
Referees
Librarians
Email Alert
Add your name and e-mail address to receive news of forthcoming issues of this journal:
[Back to Top]