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Questions related to Linear Programming
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In the file that I attached below there is a line upon the theta(1) coefficient and another one exactly below C(9). In addition, what is this number below C(9)? There is no description
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I asked that question to the person who code that package, and he said C(9) coefficient does not have any meaning here, just ignore. It comes up because the package is written for the old version of Eviews and has not been updated that is why.
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How can I get a MATLAB code for solving multi objective transportation problem and traveling sales man problem?
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The link that Mohamed-Mourad Lafifi mentioned is useful. In addition, to finding code kindly check GitHub:
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Consider a linear program of the form
min <c,x>: Ax <= b.
Suppose that b depends on a parameter y, such that b(y) is concave. Then the value of the LP is convex in this parameter y. This is easy to prove via duality and should be widely known. However, I do not find an appropriate reference. The only case I have found is when b is linear, this is treated in a recent book of Dantzig and Thapa. Could anyone help with a reference for the general concave case?
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See the following comprehensive references:
1) Linear Programming and Network Flows
Book by Hanif D. Sherali, John J. Jarvis, and M. S. Bazaraa
2) Linear Programming, 1983
Book by Katta G. Murty
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I am working on a 33 bus distribution network and I am trying yo maximise the number of critical loads restored and am using Dist flow for my power flow studies which I incorporated as one of the constraints but i cant seem to get the constraints formulated correctly in MATLAB
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Hi Sibabalo,
Do you get an answer to your question? I have your problem now.
Thanks
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I want to learn MARKAL and TIMES energy modelling software. May I know if there are any videos/ tutorials/ study material available for the same?
Thank you
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Power Network Expansion Planning is the problem of deciding the new transmission lines that should be added to an existing transmission network in order to satisfy system objectives efficiently. It is one of the main strategic decisions in power systems and has a deep, long-lasting impact on the operation of the system. Several challenges such as deregulation, renewable penetration, large-scale generation projects, market integration, and regional planning are discussed in the literature to some extent.
In the context of the smart grid, what can be the potential future challenges in terms of different scenarios, applications, modeling, solution, and novel devices in the network?
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Two main problems in the perspective of SG:
*Integration of Microgrid, while considering uncertain factors and limitations of the present grid.
*Integration of charging stations for EVs taking into view users' profiles and traditional grid capacity.
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Transportation problem
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In general, it is not a better metaheuristic than the other ones. But there are general approaches for facing the practical problems: in your cases it is needed to make evolve one or more solutions populations. In the work attached you could find a useful approach for facing your problem. Neverless, would be useful using more than one metaheuristic and to compare them by different indicators
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For an Integer Linear Programming problem (ILP), an irreducible infeasible set (IIS) is an infeasible subset of constraints, variable bounds, and integer restrictions that becomes feasible if any single constraint, variable bound, or integer restriction is removed. It is possible to have more than one IIS in an infeasible ILP.
Is it possible to identify all possible Irreducible Infeasible Sets (IIS) for an infeasible Integer Linear Programming problem (ILP)?
Ideally, I aim to find the MIN IIS COVER, which is the smallest cardinality subset of constraints to remove such that at least one constraint is removed from every IIS.
Thanks for your time and consideration.
Regards
Ramy
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Dear Experts,
I am looking for a solution for a convex optimization problem (linear programming) and I am thinking about evolutionary algorithms. Do you think it would be helpful?
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Most LPs arising in practice can be solved quickly using simplex. For very large and/or degenerate instances, interior point methods (IPMs) may be faster. Many solvers (e.g., CPLEX, Gurobi, Xpress, MOSEK) allow the user to try several methods.
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When seeking a solution to a linear programming problem with fuzzy coefficients in the objective function, some technique must be applied to transform to crisp values, or to translate these fuzzy elements to real values. What techniques do you suggest?
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I am just exploring different methods for this phase, but as you said, once the defuzzification is done I will find the solution. I particularly want to explore alpha-cuts based methods.
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I want the following help in linear fashion:
  • I have a vector: P(n)=[1,2,3,4,5,...,n]
  • I have binary variables with length of P: y(1),y(2).....y(n). These variables, y(i) can only take 0 or 1.
A new vector Q(n) is needed, such that, if y(i) is 1, then the ith element in Q(n) should be 0. And in rest of the places, the P(n) should get repeated.
  • Example : if n=6, and y(1) and y(4) are 1 ,then Q(1) and Q(4), should be 0. The vector, Q will be as follows :
Q=[0 1 2 0 1 2].
  • Another example : if y(3) is 1, then Q should be as follows:
Q=[1 2 0 1 2 3].
Basically, beside the zeros on ith position, the vector P should repeat itself in non-zero positions of Q
Can you please help me formulating this problem?
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In case of MATLAB you can try this:
CLC; clear;
P=[1:10]; %can work for any size or elements of P
Y=[1 1 0 1 1 1 0 1 1 1];
Q=P;
for i=1:length(p)
If y(i) ==0
Q(i)=0;
Q(i+1:end)=P(1:end-i)
end
end
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Mixed integer linear programming
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In addition we have a few special algorithms for large-scale problems, such as
Benders decomposition, Dantzig-Wolfe decomposition, and more varieties of
decomposition tools.
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Need to know about the softwares available for linear programming.
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If you have access to multiple solvers you can try Pyomo framework (https://pyomo.readthedocs.io/en/stable/#). Using Pyomo you can test multiple solvers for example AMPL, PICO, CBC, CPLEX, IPOPT, Gurobi and GLPK; then you can decide which solver is better for the problem you want to solve.
Also, in Wikipedia you can find a list of solvers https://en.wikipedia.org/wiki/List_of_optimization_software
My best regards,
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Does someone have an excel file for formulating feeds using linear programming or a free online tool where this can be found?
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Yes, I have some files of solution for linear programming with solver in excel, I can share with you
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My basic objective is to solve epsilon insensitive SVR using linear programming . Therefore I need the base code for LPSVR.
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What is the average time complexity of the simplex method for solving linear programming? Is it polynomial or logarithmic?
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The complexity of the simplex algorithm is an exponential-time algorithm. In 1972, Keely and Minty proved that the simplex algorithm is an exponential-time algorithm by one example. On the other hand, the simplex algorithm is behaving in the polynomial-time algorithm for solving real-life problems.
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Dear all,
As we know, interval matrices are matrices with 0 and 1 entries with the property that the ones in each column (row) are contiguous. Interval matrices are totally unimodular (TU). Hence, the integer programming (IP) problems with such matrices of technical coefficients and can be solved as linear programming problems.
However, in the consecutive-ones with wrap around, the ones are wrapped.
For instance, in the following matrix, the ones are wrapped in columns 4 and 5:
1 0 0 1 1
1 1 0 0 1
1 1 1 0 0
0 1 1 1 0
0 0 1 1 1
This example is not TU with a sub-matrix with determinant 2 (deleting rows and columns 2 and 4).
Two questions:
1- When wrapping does not violate TU property?
2- Is there a general approach to solve IP problems with consecutive ones and wrapping around matrix of technical coefficients, efficiently?
Thank you for your kind help/
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If all righthand sides are 1, then you can reduce the problem to a series of shortest-path problems:
If not, then you can use the method in this paper:
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Any recommendation from a scientific journal to submit a paper on operations research applying linear programming and vehicle routing (VRP) using the B&B algorithm?
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You can ask your thesis advisor about what journal they think would be best to submit your work. It is hard to suggest a journal for you without seeing the actual paper.
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The routing problem can be easily solved using ILP or mixed ILP, why metaheuristic algorithms are required to solve this problem?
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Some useful contextualization for the simplest variant of the vehicle routing problem (the CVRP):
• The Christofides, Mingozzi, and Toth [CMT] vehicle routing instances with up to 200 clients have been released in 1979, and it took around 35 years of research and progress on exact methods to solve them optimally. In particular, simple MILP formulation for the CVRP (e.g., two-index vehicle flow formulation) can only solve some problems with a few dozes of clients.
• For the newer Uchoa et al. [X] instances with 100 to 1000 clients, there are monetary prizes for anyone able to consistently solve them to optimality (http://vrp.atd-lab.inf.puc-rio.br/index.php/en/cvrp-challenge). So far, this prize remains unclaimed, and many instances with as few as 300 customers remain unsolved.
• Practical applications often involve >5,000 clients (!!!) and *many* complicating constraints, additional decisions, and objectives called "attributes". Application cases with over 50,000 stops/clients are also quite common in courier delivery and refuse collection. Opting for an optimal solution through MILP in these situations is a guaranteed failure, and it usually reflects a lack of practical experience in the field. Even good lower bounds can be hard to get when problem size grows.
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where matrices A and B are known while X and Y are to be solved.
For example,
A=[a 0;
0 b];
B=[a a 0;
0 0 b;
0 0 b;];
a and b are known elements.
It is easy to see that the solution is
X=t*[1 1 0;
0 0 1;];
Y=1/t*[1 0;
0 1;
0 1;];
where t is any non-zero real number.
But how to derive this solution step by step in a systimatical way?
It would be better if there is a programmable approach.
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Dear Liming, at a first glance the problem seems not to have a unique solution in general. Assuming for example that all the matrices are nxn, you need to solve n2 equations (corresponding to the elements of B) in 2n2 unknowns (the elements of X and Y.
If what matters is to find a solution, even if not unique, a possible approach could be to look for a minimum of the norm ||XAY-B|| with respect to the entries xij and yij. To get a differentiable function, one could take the Frobenius norm of the residual XAY-B.
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I noticed that many studies which treat Electric Vehicle charging (in a V2X scenario) as an MILP (Mixed-Integer Linear Problem) problem, consider a constraint to prevent that an EV (Electric Vehicle) could simultaneous charge and discharge during the same time slot (the duration of this inteval varies from a study to another; the most used are: 15 minutes, 30 minutes; 1 hour).
I'm agree in considering that charging and discharging an EV at the same continuous time instant is not a realistic scenario, but which could be the reason of preventing this situation during a time inteval which lasts 15 minutes or 1 hour?
PS: I guess that it's done to simplify the complexity of the problem, but I'm wondering if there are other reasons.
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Dear All,
I'm totally agree with colleague Paul Oke.
Best regards
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Dear Colleagues,
I am running a time series regression (OLS) based on stationary dependent variables and log form of explanatory variables. Very few of logged exp. variables are stationary. When I took 1st difference, I could not get any significant results, plus, also the 1st difference did not make much sense to me when I analyzed it graphically. My question is, will my regression results be untrusted if I report such an analysis. I asked a similar question and got some replies that even if you log your variables, still you have to test it for a unit root; however, I am observing several papers with log variables where stationarity was not taken into account. Thank you beforehand.
Best
Ibrahim
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To answer to criticism of spurious regression, if it is time trended. Causality is still a bigger issue than just uncovering stationarity of time dated series. We need the theory to guide us and the use of statistics in experimental research design to handle causality is still not resolve.
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Multi-Market-Models were much famous a decade ago to carryout policy simulations. Are they still famous in economics now? Scope for publication in the present time?
Thanks in advance
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Yes. Still studies are going on in some countries and areas. Some studies are concentrating on Agricultural Policy Impact Analysis with Multi-Market Models.
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I need to model a construction and demolition waste management network in order to maximize waste recycling. I read some works that use MILP. Among the software used are GAMS, JULIA, MATLAB and R. Are there other software that can be used?
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I have used most of the tools mentioned before. While each of them has its strengths and weaknesses, nowadays I have been constantly relying on the same tool for solving all kinds of optimization problems.
This tool is a combination of Julia programming language with the Julia package JuMP which defines (in my opinion) the most elegant and intuitive modeling language that makes it easy to try different solvers on the same problem by simply changing the solver name in the code. It also allows easy access to more advanced features of MILP solvers such as generic support for callback functions that can be used to modify the problem while solving it with the addition of valid inequalities and user-defined heuristics.
As an example, solving the TSP using the DFJ (Danzig-Fulkerson-Johnson) formulation that has an exponential number of subtour elimination constraints is straightforward with JuMP. You can simply set up a "lazy constraint callback" function that will check if subtours are present in the solution and if not, add constraints to break these subtours. This is particularly easy when considering only integer solutions, but you can also use the "user callback" function to cut off fractional subtours. JuMP has turorials of several important modeling paradigms that can be used as a basis for your model. Here are the relevant links:
You can download Julia programming language from
Website for the JuMP package
Tutorials for modeling with JuMP
More example problems solved with JuMP
Best regards
Juho
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Hi All,
Are there any opinions and experiences of the LocalSolver solver?
Comparing for example accuracy, speed, etc. to other solvers, etc.
Interesting to hear about them ...
/Lars
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Dear Lars
I agreed 100 % with you in that a Solver must identify unfeasibility, but my question is: How many solvers or MCDM methods do you know that have that capacity?
Only one: Linear Programming
The procedure is very simple: It compares criteria independent values, and if a solution satisfies these values. If only one criterion is not satisfied the project is unfeasible.
Nowadays, problems are 'solved' assuming that a problem is feasible, not taking into account that t hat circumstance may not exist.
I have read hundreds of comments and papers from our colleagues. How many of them posed this problem?
Nobody.
I wrote in RG almost a year ago about this problem, you can see it in my profile under the number 304, and again, in May 2020 under the number 318. Both have had some moderate reading but nobody came forward to acknowledge and discuss it. You are the only person that addresses that issue.
Regarding LocalSolver I know what it, is but my experience on it is null.
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Please suggest a collection of examples of Zimmerman approach to solve fuzzy linear programming problems?
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Thanks a lot sir for sharing this good content.
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Any bibliographic recommendations on the problem of routing vehicles with multiple deposits, homogeneous capacities? less than 10 nodes
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A multi-depot VRP with less than 10 nodes should be almost enumerable, as there exist less than 1024 possible subsets of customers. Given this fact, perhaps the simplest solution approach is to generate all feasible routes from each depot, discard those that are not TSP-optimal, and directly solve a set partitioning formulation based on these routes. Now, if you face larger problems (e.g., 15 nodes or more), you should use the formulations suggested by Adam and Noha, or even go for sophisticated branch-and-price approaches as described in
since the code associated with this paper is freely accessible at
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Hi, I have an optimization problem (primal problem) which is solved by the duality theorem. So I have constraints of the dual and its variable's value. it is worth mentioning the problem is linear. how can I calculate primal variables indirectly and by the dual answer?
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Thank you all friends. I added all the primal constraints with the strong duality constraint which connects primal and dual variables together. it worked and primal and dual variables were calculated simultaneously by the solver.
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I am formulating a multi-objective linear programming model. It is a pure linear program and I plan to only use scalars to convert multiple objectives into one single objective. Before then, I need to make sure the Pareto front is in a convex form. I know for MIP problems, it could be non-convex. Is the linear programming model a sufficient condition for the convex Pareto front? Thanks.
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I appreciate all above answers, and they are very helpful! Ralf Gollmer Gary Paul Martin Simpson Nuno Miguel Marques de Sousa Mohamed-Mourad Lafifi
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If I'm not wrong, there can be non linear programming problems solved by iteratively calling the simplex algorithm on a modified sub-problem.
Are there optimization problems which require both the primal and dual simplex?
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If you have solved one of the primal or dual LP problems, you can derive the solution to the other without having solved another problem - IF the problems are not degenerate - and that seldom appears in practice. But when they do occur, you have a simple solution to that bug also.
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Hi,
I am working on a MIP problem, my program runs too long or does not find the result as I change my parameters. I have been dealing for a long time, although I have made many changes to the code, I don't think of a solution anymore.
If you are interested, I can send my model and code.
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Hello,
You need to debug your code first. You may need to test each part of your code to see if you get any feasible results. For example, instead of the objective function, use a dummy function to check if the equations are correct. Moreover, disable all the constraints to see if your problem is defined correctly. Remove each equation one by one and check the effect on your results.
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Hello,
Recently, I was reading about several techniques that solves Unconstrained Mixed Integer Linear Programs (UM-ILP) using a meta-heuristic algorithm called simulated annealing. I was thinking about the Constrained zero-one ILP. I have a linear objective function with a linear set of equality/inequality constraints and I'm thinking about reformulating the problem using the kkt/Lagrangian function. However, I'm not sure if it is even the right approach given that my optimization problem have binary variables and it is linear and hence, solutions such as the penalty method and barrier log would work best for me. My goal is to transform a constrained problem to non-constrained then apply an optimization method like simulated annealing to solve my new formulation of the problem but i'm not really sure which method is even the right approach? Also from what I read the problem will not be a simple minimization problem anymore but instead a saddle point problem.
Thank you all for your time and replies.
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O.k., of course you can use any heuristic method. It will give you SOME result - and you'll never kow whether tar solution is even close to optimal.
I strongly support Michaelk Patriksson's reply: try a really good MIP solver like CPLEX or Gurobi (which are free for academic use). And only if these take too long or are even unable to find a feasible solution the usage of a heuristics is justified.
Btw. many of the heuristics have problems fulfilling linear constraints being no simple bounds....
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I want to run a MILP problem that is time dependent for a given duration (i.e. something like a for loop situation). I have parameters that change with time and the results from the previous simulation affect the next simulation. I am using CPLEX solver in MATLAB how can I achieve this.
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Good topic can please provide more information
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Some years ago I wrote some command line using linear programming to optimize production for table water company using table water and sachet water as two product and using production hours of the two and number workers with profit made as constraints, I need some help in changing my command lines to software algorithms in order to develop an useful and helpful App.
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You can simply use MSExcel data analysis tool pack to develop your optimization problem and solve. You can find lots of videos in YouTube. Good luck.
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A machine manufactures spare parts at the rate of 20, 000 per month. A second machine uses those spare parts at the rate 50 00 per month and remainder put into stock. It cost ₦100, 000 to set up the machine the company establish their stock hold costs 20% per annum of the average stock value each parts costs ₦250 to make. Required What batch size should be produced on the first machine and what frequency.
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Hey guys thanks, I later noticed it was an inventory problem and not an LPP. The student I was supposed to solve for said he wanted to solve an LPP but I later did more research and found it was an Inventory Problem. Thanks anyways.
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I have a question regarding making the dual of the MILP model. More precisely, I am working on the job-shop scheduling problem, and I want to have a dual problem of that.
How can I create the dual constraints for integer variables?
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Temitayo Bankole I am afraid that this procedure only works for LP problems, while I am dealing with the MILP problem, job-shop scheduling.
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I understand the idea of Best Worst Method for multi criteria decision making
and i know that there is a solver to get the weights but i need to understand the mathematical equation and know how to solve it with my own self.
Can any one help me?
{|𝑤𝐵 − 𝑎𝐵𝑗𝑤𝑗|} ≤ 𝜉L for all j
{|𝑤𝑗 − 𝑎𝑗𝑊𝑤𝑊|} ≤ 𝜉𝐿 for all j
Σ𝑤𝑗 = 1, 𝑤𝑗 ≥ 0 for all j
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Dear Nouran,
You can find the answer in this paper:
Best regards,
Sarbast
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I have formulated an Integer programming problem with around variables and 30 constraints. I observe that the relaxed LP takes same time to execute (around 5 minutes) as the original IP. How is it possible when Linear programming problem is polynomial time solvable and IP is not?
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Perhaps your IP is equivalent to an LP - for example single-commodity flows (including shortest path) are solvable as LP.
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Simplex method in linear programming.
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The Simplex algorithm is a mathematical tool primarily. Some statistical fitting problems can be cast as a linear program, but I consider this incidental.
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Based on your expertise, what is the better optimizer tool between GAMS ( https://www.gams.com/ ) and Gurobi ( https://www.gurobi.com/ )?
Please also let me know your field of research.
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GAMS itself is no solver, it is a modeling system with interfaces to a number of solvers - Gurobi being one of them.
So there is no point in trying to compare GAMS with any solver.
Gurobi is a solver for linear and (convex) quadratic mixed-integer problems.
GAMS has interfaces to solvers for different problem classes, too (like more general nonlinear optimization problems).
But if your problem is a MILP, you might compare the modeling via GAMS with the modeling via the Gurobi Python API.
I have worked with both (and with other modeling systems like AMPL, OPL and other solvers, especially CPLEX) in research on discrete stochastic optimization. The choice of the modeling interface depends simply on your preferences and the tasks to be performed (like modifying the problem according to the solution and solving a series of problems). If you are used to the GAMS language you might prefer it to others. GAMS and AMPL are not free software and you have to pay for many of the solvers for them, too.
Best regards
Ralf
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It will be helpful if you can give me some references if there are MINLP optimization problems solved by PSO.
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I think that rounding approach is the best, whether before evaluating the objective function or after evaluating the objective function because in both cases the reason for the same result occurs.
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What is the best commercially available non-linear optimization problem evolutionary solver/ algorithm?
Must be freely or easily obtainable.
Not necessarily free.
relatively Easy to install and use.
Anything better than excel solver's evolutionary algorithm?
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This question, like many others you have posted in the last few days, can all be answered in the same way: it depends on the problems you are wanting to solve.
I would suggest looking at the results of the Black Box Optimisation Benchmarks (BBOB).
There are many different algorithms that are compared and many are freely available via github, or from the authors of the codes. The tremendous advantage is that you can see how they perform on many different types of problems, some of which might be similar to what are wanting to solve in your own work. Furthermore, you won't have to trawl through reams of cloying self-assessment and self-appraisal of many commercial products which, not surprisingly, are always fulsome in praise of their own product.
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What impact does the initial solution have in the case of non-linear optimization problems and (solved by) evolutionary algorithms?
For instance, initial solutions have an impact with linear programming.
Would it have any effect to rerun a non-linear optimization problem after a certain or good initial solution was found (and the evolutionary algorithm stopped)? Particularly/ for example in cases with many possible local solutions, or local solutions rather close to each other.
Would the latest known initial/ best solution affect something like solution space sampling or local solution sampling?
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As recommended by Michael Patriksson, I also recommend that you attempt to solve (or get bounds for) at least small instances of your problem with mathematical programming algorithms such as COUENNE (https://projects.coin-or.org/Couenne) or other similar solvers. Beyond the insights on problem structure that you may learn in the process, this will also help to assess the solution quality of your metaheuristic on small datasets.
Now, regarding the impact of initial solutions in population-based search (e.g., genetic algorithms), you should have two general goals in mind: (1) diversity, (2) quality. For some problems, spending a lot of (coding and computational) effort to generate clever solutions is not worth it, as in a fraction of seconds your population-based algorithm may retrieve solutions of equally good or better quality. In such situations, ad-hoc construction techniques may even lead to reduce your population diversity without a clear benefit. For example, for some combinatorial optimization problems arising in transportation and logistics (vehicle routing problem), we had much more success with a randomized initial population subject to local search (equivalent to gradient descent in continuous space) as this would much better sample our search space. Moreover, diversity is important in the initial stages of the search as well as later on, and therefore should be actively preserved to avoid phenomenons of premature convergence.
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With non-linear optimization, when you are interested in the different possible local solutions, because it can help you towards the optimal (global) solution, and to identify the optimal solution, particularly when you have a hint or idea what it would or would not look like, what would be an effective way to surface possible local solutions?
Do you simply note and cross out the local solutions found, as they are found?
Is it simply a matter of your initial solution sample or set?
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You do not have to implement it by yourself. Many solvers have already built-in multistart routines. For example:
  • Matlab uses the scatter search algorithm to sample initial points (Glover, F. “A template for scatter search and path relinking.” Artificial Evolution (J.-K. Hao, E.Lutton, E.Ronald, M.Schoenauer, D.Snyers, eds.). Lecture Notes in Computer Science, 1363, Springer, Berlin/Heidelberg, 1998, pp. 13–54.) See: https://fr.mathworks.com/help/gads/how-globalsearch-and-multistart-work.html#bsfvjv8
  • Excel, Pyomo or Aimms have such routines.
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what is the best way to search or interrogate an initial solution of a non-linear optimization problem you (strongly) believe is close to the optimal solution?
is there a better approach or strategy than to incrementally or iteratively increase a band of floors and ceilings around the initial solution, until this no longer has any effect? this approach should be better than starting with a wide band around the initial solution from the onset?
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Not for the chance of plugging a paper written me and a colleague, but the link below is a thorough analysis of what optimality means:
The above is the last preprint - here is the permanent link to Oper Res:
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What is the best way to choose initial solutions for a non-linear optimization problem sensitive to the initial solution?
If you have a significant number of optimization problem coefficients (50 in this case), and your optimization problem is such (non-linear with multiple possible local solutions) that it is sensitive to the initial solution, what are the best ways to choose (a set of) initial solutions?
Will the excel solver evolutionary algorithm do this?
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Optimization algorithms are really pretty simple. Just keep climbing up the hill until you get to the top. Take some precautions to make sure your step size isn't too big and try many different starting points far away form the optimum to make sure you have found the true optimum an not just a local optimum.
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With a non-linear optimization problem, can insensitivity to the initial solution be interpreted as a global solution, or (high) probability of a global solution?
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As I indicated in my answer to your previous similar question, you have to demonstrate insensitivity to initial solution (or starting search points) for at least a few dozens of them or more, not just a few of them. Your initial solutions (starting search points) must be random uniformly distributed points in the multidimensional solution domain. Again, there is no rigorous proof that your solution found this way is the global one. You may only state that there is a high probability that it is the global (but not 100%). The more initial solutions were use the higher that probability.
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PLEASE Refer to Document containing Data Set
Suppose that the minimum number of security staff required at different hours are at the ith hour (i=1,2,…,24) are outlined as follows (refer to doc)
I'm not able to figure out how to develop a model for this issue and I need assistance in developing a linear programming model to minimize total cost.
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This may be a repeat of a previous question i asked, but i want to make such a statement in a paper, and want to evaluate if it can be refuted beforehand:
Are there (non-linear) optimization problems where it is not/ no longer possible to walk from one local solution, to a better local solution?
And can one define the global solution in this way - the inability to walk from a local solution to a better local solution?
Then, Can one base this on higher order partial derivatives?
I am currently using higher order partial derivatives to indicate if steps are still possible, post excel solver NLP (and evolutionary), and to identify such steps.
If i obtain steps still possible this way, i am walking to better solutions.
If i no longer am able to identify steps through higher order partial derivatives, can i say i have reached the global solution?
A higher order partial derivative would be the delta in objective function in response to a delta change in 3 or more coefficients. It differs from 2nd order partial derivatives, and this is evident when you consider the actual calculation.
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I would not make such a statement in a paper. It can be easily be refuted.
If an algorithm is unable to walk from a local solution to a better local solution, it only means that it locally converges. The global solution could be far away from that local one.
In general, I would use a set of uniformly random starting points for finding a set of local optimal solutions. If these local solutions found from different starting points are close enough to each other (or the same), then it is likely that this is the global solution. The more random starting points you use (the bigger the density of the starting points in the solution domain) the higher the probability that the solution is global. However, there is no rigorous proof of that. This will be just a heuristic judgement.
I would refrain from the use of higher order partial derivatives (higher than 2). Numerical calculation of high order partial derivatives is computationally unstable, lose the numerical accuracy very fast, up to total non-usability for deciding on the new step in search of the solution.
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Why would an (the excel) evolutionary algorithm be better at, or simply be able to, solve what seems to be a linear problem, when the LP (and also NLP) algorithm struggle to, or simply fail to.
The problem is of the form:
Max X; X = x1 + x2 + x3...
x1 = (k1 × z1) + (k2 x z2) + (k3 x z3)...
x2 = (k1 x y1) + (k2 x y2) + (k3 x y3)...
x3 = (k1 x q1) + (k2 x q2) + (k3 x q3)...
k are the coefficients of the problem to optimize; z, y, q are factors and are given/ constants.
The evolutionary algorithm can find a (better) solution (to the initial solution). The LP algorithm occasionally finds a solution, and not better (mostly worse) than the evolutionary algorithm. The NLP algorithm does not really find a solution at all.
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You problem is equivalent to (if I understand your notation):
X=k1(z1+y1+q1)+k2(z2+y2+q2)+k3(z3+y3+q3)->max
If z, y, q are given constants, then their sums are also given constants, say K1, K2, K3
X=k1*K1 + k2*K2 +k3*K3 ->max
There are no constraints for the decision variables k1, k2, k3.
This is a linear form that is maximized as these variables take unlimited max values......, hence there is no unique solution. Something is missing in your problem statement.
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Can higher order partial derivatives be used to move non-linear programming problems from local solutions towards the global solution?
That is, Partial derivatives of order greater than 2nd order.
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I would suggest to make your initial point (initial solution) better. Probably your solver from current initial point only finds local optima.
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Suppose we have a linear programming model in the below:
Model:
Max = 3*x1+5*x2
x1 <= 4
3*x1 + 2* x2 <= 18
x1, x2 >=0
Calc:
! I would like to do some calculation here using to the optimization result obtain from the above model
For example: Y = x1+ x2
EndCalc
End
Any answer will be appreciated, thanks
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It's called an accounting row. It is an equation that simply adds things up.
x1 + x2 - Y = 0
Later, these things that are added up can also be constrained.
Y < 100
or
Y > 10
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Few days earlier on a project presentation on Stochastic Programming Real life applications, i constructed 3 real life scenario based Stochastic Models: A Farmer's Problem, Container Allotment Problem and another on Stochastic Arc Routing. Also solved them for particular scenario.
As stochastic linear programs are lengthy programs with a lot of constraints, it is long-time process to solve a stochastic linear program. And therefore i used LINDO solver to solve the problem. I have a L-shaped algorithm based example too.
But the examiner said me that, why you didn't used the general solving procedure to solve these LP problems? I explained about the long programs and complexity. In reply, I found complement that all credit goes to the LINDO solver, not you.
I am wondering that advances in Science could make our works easier and faster. Shouldn't we take these type of advances in our daily life?
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Yes, there are a number of softwares which can solve linear programming problem in a click giving the optimal solution or an indication of in-feasibility or unbounded solution. But, that is second stage. First stage is to learn the basics of linear programming solution procedures i.e. graphical method and Simplex method. The students will have in-depth knowledge of such procedures by manually solving them. Once got acquainted with the procedure, they can use software like Lindo, Matlab etc. to solve the complex LP problems which can not be handled manually.
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Is linear programming one part of convex optimization?
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Convex optimization involves minimizing a convex objective function (or maximizing a concave objective function) over a convex set of constraints.
Linear programming is a special case of convex optimization where the objective function is linear and the constraints consist of linear equalities and inequalities.
Nonlinear programming concerns optimization where at least one of the objective function and constraints is nonlinear.
(Adapted from Mathematical optimization: Major subfields on Wikipedia.)
Therefore, convex optimization overlaps both linear and nonlinear programming, being a proper superset of the former and a proper subset of the latter. However, note that nonlinear programming, while technically including convex optimization (and excluding linear programming), can be used to refer to situations where the problem is not known to be convex (see Boyd and Vandenberghe, p. 9, below). Hence, it may be more useful in practice to think of a hierarchy: linear - convex - nonlinear. Another useful view is given by the following quote, kindly supplied by littleO: "The great watershed in optimization isn't between linearity and nonlinearity, but convexity and nonconvexity." -- R. Tyrrell Rockafellar, in SIAM Review, 1993
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Deterministic Global optimization relies on convex relaxation of the non-convex problems. Certain nonlinearities are duly converted into linear forms underestimators to be solved by efficient MILP solvers (e.g. signomial functions/ bilinear terms).
Most nonlinearityies are approximated to linear functions by piece-wise linearizations. However, I am wondering if this linearizations guarantees that the approximations are understimators of the original nonconvex problem (i.e. for all x in Domf, f(x) >= u(x) where u is the understimator)
because otherwise the understimator may miss the global optimum during the branch and bound process.
Can the solver still converge even if the relaxation is not an understimator?
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That's right. More precisely, if an inequality is of "<=" type, we use convex under-estimators for the left-hand side. If the inequality is of ">=" type, we use concave over-estimators (or you can just multiply the inequality by -1).
McCormick's original paper (from 1972) explains this very well. So does this paper:
Ryoo, H. S., & Sahinidis, N. V. (1996). A branch-and-reduce approach to global optimization. Journal of global optimization, 8(2), 107-138.
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Is there essentially a difference? Which one is optimal or low complexity ? is here a relation with the rank?
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Please have a look at this attached chapter and hopefully you find your answer
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How many methods we have for multi-criteria Classification (sorting) problems? Could you please name them?
As I understood we have some methods in the below approaches:
1. Multi-Attribute decision making (ELECTRE-TRI, FlowSort, Promethee IV)
2. Multi-objective decision making
3. Goal programming
4. Linear programming (Integer programming)
5. Supervised methods (UTADIS/Decision tree)
6- Clustering (K-means/K-medoids/2steps/c-means)
Could you please name some more methods which can be applied for multi-criteria classification problems?
Thank you in advance.
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Hello,
I am doing a comparison of different algebraic modeling langues (AMPL, AIMMS, GAMS, Pyomo) in both theoretical and practical terms. As a practical experiment I am trying to measure problem model instance creation time. I.e. taking optimization problem defined in one of the above languages I am trying to load the model and it export/convert it to LP solver input format. Model instance creation time is being measured and model instance characteristics are being examined.
So far I have only used sample problems provided by the authors of AMLs which are quite small. In order to have a meaningful benchmark I would like to find test data to build much larger problem model instance.
Maybe someone could guide me where I could find larger data sets for the models of linear programming problems defined in the AMLs mentioned above?
Regards,
Vaidas Jusevčius
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http://netlib.org/lp/ has a nice collection of large-scale LP problems.
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I have been reading about performing sensitivity analysis of the solution of Linear Programming problem (calculating shadow prices, reduced costs and intervals within which the basic solution remains valid). It is clearly described on academical problems with 2 or 3 variables, but in fact, when tried to apply the same logic for real-life, scalable problem, I didn't get promising results. This is because only a few of variables values matters for me, while other are rather placed for another purposes (like changing hard constraints to soft ones etc). But all of them are taken into account when checking if basic solution has changed, hence the interval that is returned by a solver is a way more narrow than I want it to be.
Where can I find an example of real applied sensitivity analysis, if there is any?
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Thank you Katarzyna
I believe that we have a misunderstanding.
I agree with your definition of variables, but I DON’T CALL as ‘criterion’ an objective function. NOT in LP, although YES in SIMUS, because in this method criteria and objective functions are interchangeable.
But in general, in MCDM, my definition of criterion or constraint is exactly the same as yours, as well as your definition of performance values. However, if it is true that criteria ‘constrain’ the alternatives, I prefer to use the term ‘constrain’ for the values which are limits or thresholds of criteria, that is, the RHSs, because they constrain the range of validity of criteria. In reality, criteria ‘force’ the alternatives to respect some conditions, which in turn may be high or low, and bounded by the RHSs.
You mentioned before that you had 2500 variables or alternatives, and you say that there are 3 production units; does it mean that each production unit may have many alternatives?
Can a particular alternative, be replicated in any number of production units? For instance, alternative 789 can be in units A and C, or even in A, B and C, while other alternatives may only be in one of them?
When you say that other variables are placed, are you referring to artificial and slack variables? If this is so, it is strange, since the solver adds them automatically, or maybe I did not understand your statement.
I assume that you want to know how production of unit B changes when you modify the RHS of some constants, for instance # 1.
What do you mean by ‘I display’ constraint #1 up, constraint 1, and constrain # 1 down? I understand that you want to say is that you can put low and high limits for constraint # 1. Is this correct?
I have two questions for this:
1. Why you consider a constrain 1, when if you have low and high limits defined by ≥ and ≤ operators respectively, constraint 1 is included in the range? In my opinion that arrangement may cause you to get an infeasible solution, because if the model select a value between the two ends, how does it manage to also comply with constraint 1, which I imagine has the ‘=’ operator?
2. How do you know that constraint 1 or any other is the one you have to work with? It could very well be that said constraint does not have any influence in the alternatives selected. That is, you have to work with the criteria that are responsible for the selection, maybe one or several. Once determined, you will have the certainty that these will change the solution found. If you work with constraint that are not relevant, you can increment and decrement by changing their RHSs, and you will see that there will be no change in the selection.
This information, as you know, is in the dual, or you can see directly in the Solver screen for sensitivity analysis (SA). Of course, you know all of this, but my comment derives because you don’t explain why criterion 1 is selected for SA.
No, I was not referring to the objective function Z coefficients. I am referring to changes in the RHSs.
To your question, my answer is that yes, you can. Each time you modify the RHS of a binding constraint and use the Simplex, you can immediately see the new values of the alternatives.
It is extremely useful to work at the same time with the dual, and once you get the new values for the alternatives go to ‘sensitivity’ (in Solver you have to run the Solver twice for each RHS change, the first gives you the new alternative values, and then you have to run it again in order to get the sensitivity analysis screen), that gives a lot of information, such as the shadow prices, the relevant criteria, the validity range for each shadow price, as well as the reduced costs.
For my software, don’t worry, you can have it any time you want.
I understand that there is a mistake when you say that the slope is determined by the reduced cost value.
The slope is given by the shadow prices. The reduced cost tells you how much you have to modify a coefficient in the objective function in order that the corresponding alternative enters in the solution. Observe that the reduced cost for selected variables is zero.
Outside the range nothing happens, because the criteria that you were using is no longer binding, and the alternative keeps it last value.
In your next paragraph, you say: ‘But if I change RHS to some value outside of the range, I can see that indeed, basis have been changed as well, but the change is related to, for example, a variable indicating the production in unit A jumped from 50 to 0’.
Yes, you are right, if the base changes, there will be another variable instead of A, and this one will be zero.
The jump from 50 to 0, means that when the upper/lower limit of the range is exceeded, that variable no longer belongs to the solution, and since Z is equal to de sum of the products between the aijs and the solution found, and A is no longer in the solution, its value is zero, but the variable keeps it last value..
I am attaching a worksheet for an example that I did years ago, with two alternatives A1 and A2, subject two five criteria and which objective was to maximize production.
The first optimal result indicates that A3=0.16 and A6 = 1.22. This result is showing a Z=52,000 and that the corresponding criteria are C2 and C3.
Observe that:
· C2 has as shadow price (λ2) = 0.07 and C3 a shadow price of (λ3) = 17.95.
· Solver also shows that maximum increment for λ3 is 429, while the maximum increment for λ2 is 48.
Consequently, the upper limit for C3 = 335 (Its RHS) + 429 = 764 and for C2= 559 (Its RHS) + 48 =607.
· If we consider only C3 and we increase its RHS, as shown in the Excel spreadsheet (we can increase in any appropriate amount even if not equal). Observe how the A3 value progressively rises up to 2.74, which corresponds to the upper limit of C3, that is 764.
· Both λ3 and λ2 keep constant along the whole procedure
· Simultaneously the value or score of A6 progressively decreases from 1.22 up to 0, for RHS3 = 764.
· The original upper limit of RHS3 decreases from 420 to 0, for RHS3 = 764.
Correspondingly the original upper limit of RHS2 increases from 48 to 825, for RHS3= 764.
· If the value of 764 is surpassed, say we put 765, alternative A3 keeps its value, since C3 does not have influence on it, however A6 = 0, meaning that this alternative it is no longer a solution, however, notice that λ3= 0 while λ2= 24.61, after the 764 limit is reached.
In fact, we should change RHS3 and RHS2 simultaneously, each one with its own increments/or decrements, and then the performance curve will be the result of both acting at the same time, as in real-world scenarios.
The graphic shows Z performance curve with the λ3. It appears as a broken straight line because I was using different amounts of increments for RHS3, since
I believe that this elemental example shows you how the range is diminishing in the selected alternative
Now, if you want to keep the original production of B unchanged, why don’t you express it as a new constraint, using the binary format, that is in this criterion all alternatives are zero, except B which is 1.
If you have a value in mind for B, say for instance 820,you put it as the RHS, and indicate with the ≥, ≤, or = operators if you want to get a result for B larger than 820, or lower than 820, or equal to 820, respectively.
However, I would stay away from the equality, because it imposes very hard restrictions to the problem.
I sincerely hope that my comments help you, and indeed, I am very interested if you can keep me informed me if it works
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Hi,
In minimizing the difference between two variables inside an absolute term e.g., Min |a-b| . How to make the term linear so that can be solved by LP or MILP . Where a and b are free integer variable (they take positive and negative values).
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To minimise |V-1|, where V is a positive continuous variable, just minimise x subject to x >= V-1 and x >= 1- V.
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Please give their appropriate cases.
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we can chose linear or dual dependent to conditional constrants
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The constraints include both linear constraints and nonlinear constraints. The essential issue lies in to how to deal with the nonlinear constraints.
It would be better if this algorithm can transform these nonlinear constraints into the equivalent linear ones.
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Being Scandinavian I have read and seen countless of times some representation of the children's story about the emperor's new clothes - by Hans Christian Andersen.
As you might know the emperor's new costume is soooo fine that it is weightless, and no-one - but a child - dares to speak up to the emperor and exclaim that he in fact is naked! Thanks to the child, the curtain opens, and we truly see that he indeed is naked.
I find metaheuristics - to some degree - to be that, too, within the mathematical optimization domain: quite shallow, devoid of solid theory, and often (but not always) a game of draw, guess, and jump (I do not know the exact English translation of the name of the children's game). I have no problem with it when we are dealing with very complicated combinatorial/integer/bilevel problems in industry, especially when we do not have an explicit formulation, such as when we need to deal with the use of simulations within the optimization, or uncertain coefficients. But then we are talking about industrial mathematics, which is something else than mathematics - which is an exact science.
The theme of metaheuristics was, I hope, originally an attempt to find "reasonable" (however that is defined) feasible solutions to the most difficult and large problems, especially for nonlinear integer models in industry, with the explicit sign that with these techniques we might hope get a fairly good solution if we are lucky, or we may not - as that is actually how it works: metaheuristics are NOT globally convergent in general, and they were - make a mental note of this - NOT EVEN CONSTRUCTED TO BE.
Yet there are plenty of scholars - especially in this forum, for a reason I do not fully understand - that insist on applying their favourite metaheuristic(s) on just anything. Yesterday I think it was when I at RG found a paper on a metaheuristic used to "solve" a very, very simple linear program with one (1) linear constraint. I blew my top, as they say. WTF is going on? I blew my top because I know how to solve to guaranteed optimality such a problem in under a 1/1000 of a second on a slow computer. It's a problem of complexity O(n) - hence the easiest problem on this planet.
Can any sane person closer to that field address this, please? It is irrational, to begin with; or it is simply the fact that the world of scientific endeavours no longer are defined by codes of conduct? I am really troubled by more and more often seeing this unscientific methodology being used, and I sure hope it never will be seriously compared with mathematical optimization. Well, in fact, it is compared every day in industry, and math always wins.
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A friend just told me not to answer (even if it is only now and then). He supposed that even good answers do not change people's mind. Anyway, we had these discussions again and again. But these discussions seem to get lost every now and then. So, let me bring it to one or two threads again, may be using different words.
Like in every field there may be good research and bad research. So, do not put all of it together into one pot with possibly even wrong comprehensions.
Regarding the metaheuristics community, there is some awareness at least in parts of it that not everything that seems shiny is shining. A good reference to explore this view is by our friend Kenneth Sörensen:
Metaheuristics—the metaphor exposed
And another good pointer is to the word matheuristics which we framed to investigate the interoperation of metaheuristics and mathematical programming techniques. Is it allowed to point to a wikipedia page? May be yes: https://en.wikipedia.org/wiki/Matheuristics
Have fun exploring and learning how to separate the wheat from the chaff :-)
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Dear colleagues,
I would like to know of applications of fuzzy (and fully fuzzy) linear programming in optimal control.
Thank you.
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Excellent answers
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I want to blend phosphate ore. Please consult the attached form.
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For example:
max -1/3x1+x2
subject to 1) -x1+x2<=-0.5
2) -0.5x1+x2=0.5
3) 0.5x1+x2<=1.5
In Matrix form
A=[-1 1; -0.5 1; 0.5 1]; b=[-0.5; 0.5; 1.5]; so A*x<=b
In this case constraint number 2) is not needed. The solution will be the same when inequality number 2) is omitted.
In my problem the size of the A matrix is 264x100. Is there a way to find out which constraints or inequalities are not needed?
Basically a way to find unnecessary inequalities for defining a problem?
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To check if an inequality is redundant, set up another LP in which you try to maximise the violation of the given inequality, subject to the other inequalities. If the violation is zero, the given inequality is redundant.
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I have a project about operatios research. In my case I have several-vary vehicles but one source and one target. Vehicles have const and they must assign some areas. Like vehicle 1 must carry a type product , vehicle 2 must carry b type product etc.. But all products stored same place. I can not find problem type for this case.
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Dear Barış Karakum, in the first OR courses, we usually teach the students, that the important thing is to solve the problems, not apply models.
Thus, unless it is for theoretical purposes, the model that is applying is not important. The important thing is that the model you constructed solves the problem situation. Whether or not it belongs to a certain type is irrelevant.
In any case, even if it has only one source and only one destination, if it has several types of vehicles, especially if they have different characteristics, it can be considered as a transportation problem. Vehicles are transformed into sources or destinations, as appropriate. Here in Research Gate there are several articles of ours that discuss the problem of multiple transports, which may be useful to you.
We hope many successes in your research. Best regards,
José Hernández.
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Hi,
I am interested in techniques that can prove that an integer linear program has no solutions. I am just looking at feasibility and have no objective function to maximize etc.
This is part of the best known algorithm for calculation optimal addition chains. The system I want to check for infeasibility is quite simple. It's just the Frobenius diophantine equation with an upper bound on the variables.
$\sum_{i=1}^{z}a_{i}x_{i}=n$
With $1\leq x_{i}\leq u$. $a_i$ and $n$ are constants determined by a partial search for an addition chains.
The code that uses this will try to solve billions of small problems like this.
I currently use branch and bound to try and prove there are no solutions or reject the few there are with additional problem constraints.
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Checking feasibility of an equality-constrained knapsack problem is a classic NP-complete problem, which has applications in cryptography.
If the right-hand side is reasonably small, you can solve the problem with dynamic programming.
If that is not the case, you could try the lattice basis reduction methods developed by Karen Aardal and others.
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Mixed-Integer Linear Programming,
Exact algorithm
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Try find and read recommendation about project portfolio planning from PMI Standard for Portfolio Management (overview is available at https://www.pmi.org/learning/library/pmi-standard-portfolio-management-8216).
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An optimization problem must be expressed mathematically with the least number of variables or the mathematical formulation that allows to demonstrate its optimal solution, however it requires to increase the number of variables. How should it be expressed?
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Rafael: It is not necessarily the case at all that the optimization model that you construct to represent your real problem should have, as you state, "the least number of variables". In fact, in combinatorial optimization - which is a rather large subset of the reality-based models that we want to solve - there are many successful approaches based on what is known as "column generation" methods. Such methods will simply - well, not that simply - enable fruitful variables in the original space of variables to be generated, and used to derive an optimal solution to the explicit optimization problem. Such methods have been involved in many successful applications during the last three or so decades. Just "google" the term and perhaps the problem type that you are interested in, and see if you get a hit! :-)
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I'D wish to use nutrisurvey to generate formulations for a food product but I'd wish to use the nutrient contents ie Vitamin A and iron as constraint. Do I put the levels of iron and vitamin A I desire to achieve as minimum or maximum considering that the product is meant for the whole population, no special group?
If you have a detailed work on this, please share with me.
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Thank you for the answer
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Dear all,
It would be very appreciated if I have any hints or reading recommendations regarding the following question.
Can we use Machine Learning to optimize a capacitated network topology (routing and association between a set of clients and a set of servers), where I have the problem modeled as a Mixed Integer Linear Program (MILP).
Can I apply some of the Machine Learning techniques in order to meet a set of constraints and tune some variables to maximize a scalar objective value.
Note: I am still new to Machine Learning, but I want to have some ideas to decide if I can go in this direction or not.
All ideas and reading suggestions are more than welcomed.
Thanks.
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Thank you for your response Mr. Phil B Brubaker, kindly note that my question is about employing machine learning in solving such optimization problems (regardless the linearity).
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In transportation problem, which method gives the best result: North-West, Row Minima, Column Minima, Least Cost or Vogel’s Approximation (VAM)?
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MODI and VAM will provide near optimal result.
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In converting an optimization problem with nonlinear obj function with linear constraints into an LP or a MIP. What standard techniques are out there for doing such, Is a first orderTaylor series expansion sufficient? if yes, at what point do you linearize? Do we randomly sample the obj func over search space and fit with a hyperplane?
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First, look up the Frank-Wolfe algorithm. It is based precisely on a first-order approximation of the objective function, resulting in a linear function that is then minimized. (Because of the linearity of the approximation, the subproblem solution is one of the extreme points of the feasible set.) A line search in the original objective function over the line segment between the current iterate and the extreme point just found yields the next iteration, and so on. A more detailed description is here:
It's definitely NOT an efficient method, but for small examples it is ok, and it's a very good illustration how an iterative method converges.
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