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# Integer Programming - Science topic

Explore the latest questions and answers in Integer Programming, and find Integer Programming experts.

Questions related to Integer Programming

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.

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

I am solving Bi-objective integer programming problem using this scalarization function ( F1+ epslon F2). I have gotten all my result correct but it says Cplex can not give an accurate result with this objective function. It says cplex may give approximate non-dominated solution not exact. As I said before, I am very sure that my result is right because I already checked them. Do I need to prove that cplex give right result in my algorithm even sometimes it did mistake in large instance?

Thanks in advance.

Hi all, I'm using CPLEX for solving VRPTW (vehicle routing problem with time window) and observe there is a huge computing time difference even when I change the problem size by just 1. By "problem size", I mean number of nodes inside problem.

For example, for 20 nodes, it took only 20 secs to solve. However, it took more than 1 hour to solve 19 nodes instance. I understand VRPTW is NP-hard and so such phenomenon is expected to happen.

The gap is still too big, I wonder if there is any technique to make computing time more consistent with problem size?

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/

I am doing my PhD in Mathematics. My research problem is optimization in supply chain management. I am using Mixed Integer programming for framing the model and many constraints are involved.

I am not getting an idea on coding related to Genetic Algorithm for the framed model since multiple constraints are involved.

I have an understanding of basic MATLAB coding.

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?

I'm dealing with a problem of findings specific graphs with cycles of a given length and there are many isomorphic graphs which makes it difficult for the Integer Programming Model to find the optimal solutions. We have tried modifying the objective function to break symmetry between equivalent solutions and we prohibit certain arcs while searching to narrow the search, but with not enough success.

I am currently working on solving an ILP model using a branch and bound method which will be implemented by Matlab and in each node I need to determine an upper bound using Lagrangian relaxation method so I need to solve the dual problem using Cplex and the iterative procedure(to update multipliers) will be also implemented by Matlab.

I will be grateful if anyone had experience integrating Cplex with Matlab in a similar way, could help me?

Hello everyone,

We have the following integer programming problem with two integer decision variables, namely x and y:

Min F(f(x), g(y))

subject to the constraints

x <= x

_{b},y <= y

_{b},x, y non-negative integers.

Here, the objective function F is a function of f(x) and g(y). Both the functions f and g can be computed in linear time. Moreover, the function F can be calculated in linear time. Here, x

_{b}and y_{b}are the upper bounds of the decision variables x and y, respectively.How do we solve this kind of problem efficiently? We are not looking for any metaheuristic approaches.

I appreciate any help you can provide. Particularly, it would be helpful for us if you can provide any materials related to this type of problem.

Regards,

Soumen Atta

I am interested to solve a mathematical problem (MILP) using evolutionary algorithms but confuse about which one to choose as a beginner in the programming languages. Suggest an algorithm easy to implements with better results.

Thanks

I want to share my latest working in Grasshopper which may come handy to the new beginners of Grasshopper. First of all Grasshopper comes built in within the latest versions of Rhino i.e. Rhino 6 and Rhino WIP (Work In Progress A.K.A Rhino 7).

While working in Grasshopper in Rhino 5 i came to know that if you connect the panel object having a certain number value to the expressions object you need to double quote the numbers in expression tab as shown in attached figure, but if you use the number slider you do not need to to enclose the numbers in expression object in double quotes to be shown an error free result.

I know it works definitely like this for me for sure due to the trial and error failure attempts :D

Why is is this way and is it the same in Rhino 6 and Rhino 7 as well share your thoughts?

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?

I am looking to solve a Two-Stage Stochastic Mixed-Integer optimization problem in GAMS for a pre- and - post-disaster resource allocation problem.

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?

I am looking for references to show that GA solutions do not necessarily converge to optimal solution to defend the use of an integer program, exact solution. I want to criticize heuristic and meta-heuristic algorithm, especially GA.

I think a book might be a good reference but not sure which one to use!

My problem consists:

1. More than thousand Constraints and Variables

2. It is purely 0-1 programming i.e. all variables are binary.

3. Kindly note that I am not a good programmer.

Please provide me some links of books or videos discussing application of GA in Matlab for solving 0-1 programming with large number of variables and constraints.

I have gone through many YouTube videos but they have taken examples with only two or three variables without integer restrictions.

I'm trying to identify which approach would work best to select a set of elements that have different features that minimise a certain value. To be more specific, I might have a group of elements with Feature 1, 2, 3, 4 and another group with Feature 2, 3, 4, 5.

I'm trying to minimise the overall value of Feature 2 and 3, and I also need to pick a certain number of elements of each group (for instance 3 from the first group and 1 from the second).

From the research I did it seems that combinatorial optimization and integer programming are the best suited for the job. Is there any other option I should consider? How should I set up the problem in terms of cost function, constraints, etc.?

Many thanks,

Marco

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.

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?

Hi All,

I have modeled an MILP model using two different formulations, one of the formulation uses three indexes, while the other formulation uses five indexes. Comparing the solution speed of two formulations using the same solver (Gurobi, CPLEX), it turns out that the formulation with five indexes is solved faster by the solver. Not sure why this is happening, has anyone had this experience or are any studies related to this problem available. Please let me know.

Thanks,

Bhawesh

As I know, the conventional cutting stock problem can be easily solved by column generation.

Now I want to carry these cuts by truck carriers and this time we want to minimize the number of trucks for transport. ( of course, less waste of stock leads to less number of trucks)

How to formulate this problem in one ILP? Which meets the orders for cuts and also minimize used truck carriers.

Any paper or other resources to help me with this problem?

As we know that any MILP/MINLP problem is feasible only on some points in its search space. Consequently, it is not possible get its JACOBIAN as well as HESSIAN matrices, as I think. As a result, for MILP/MINLP problems it is not important to know its convexity. Further, as MILP/MINLP problems are having their feasible search space in form of a set of some discrete points so these problems are NON-CONVEX.

How can you justify my observations? Am I right? or Am I missing something very important?

You comments about the above observations are highly appreciable.

With sincere regards,

M. N. Alam

There are N tasks and M workers.

- For every tuple task-worker the efficiency is known;
- For every task one worker must be assigned;
- For every worker at least one task must be assigned;
- For every worker multiple tasks can be assigned;
- Tasks must be grouped (e.g. by location), and for every group the number of workers is fixed. Every worker must be in exactly one group.

Can you suggest an algorithm or approach for optimal (or suboptimal) assignment (maximal efficiency)?

As my knowledge goes:

- Without 4. and 5. this problem can be stated as the “Assignment Problem”, for which there are algorithms with polynomial complexity;
- Without 4. this problem can be addressed as “Generalized Assignment Problem” which is NP-hard;
- Without 4. and if M = 1 this problem can be addressed as “0-1 Knapsack Problem”.

I can’t see how to use any of the mentioned to address my problem.

Dear Friends and colleagues

I have an optimization in which I have a nonlinear term in the following form:

x(t)* a(k)

where, x and a are variables. a is a binery variable and the sets in which each of the variables are defined is not the same. Would you please suggest me a method that I can use to handle this term and transfer my model to a mixed integer linear programming?

Thank you for your suggestions.

I have seen many scholars use CPLEX solver in GAMS as they can solve the problem with ILOG CPLEX software. So in this case,they should possesses same results?

I would like to change the following linear programming model to restrict the decision variables to two integers, namely a and b (a<b):

minimize (1,1,...,1)' e

(Y-Zx) > -e

-(Y-Zx) > -e

where Y is a n-dimensional vector, Z is a n \times k matrix and x is a k-dimensional vector. e represents a n-dimensional vector of errors which need to be minimized. In order to make sure that x's only can have values equal to "a" or "b", I have added the following constraints keeping the original LP formulation:

-a/(b-a) - (1/2)' + I/(b-a) x > -(E/(b-a) +(1/2)')

-(-a/(b-a) - (1/2)' + I/(b-a) x ) > -(E/(b-a) +(1/2)')

where I stands for a k \times k identity matrix and E is a k-dimensional vector of deviations which needs to be minimized (subsequently, the objective would be minimize (1,1...,1)' (e; E)).

But, yet there is no guarantee that the resulting optimal vector only consists in a and b. Is there any way to fix this problem? Is there any way to give a higher level of importance to two latter constraints than to the two former's?

I have a 30*40 matrix. Lets say the components in the matrix are specified with "P" and the related number of the row and column of each "P" is specified by "X" and "Y" accordingly. I have a model that the output should give us the P, X and Y. How can I define constraints (for solving a simplex) which connect P with it's exact X and Y? I want to say for example:

if X=1 and Y=1 then P= 0.1

if X=1 and Y=2 then P= 0.5

if X=1 and Y=3 then P= 0.8 and so on.

I don't want the model to return a P that does not match it's location in the matrix. How can I achieve this?

Everything is known, except for P(k), X and Y

Is it possible to transform a binary-variable ({0,1}) mixed-integer linear programming to a linear programming with continuous variable ([0,1])?

To make it clear, I give an example.Optimal decision making problems often pose binary variables in optimization, in form of an action (1) or no action (0). Now, think of not a discrete decision variable, rather a probability of making an action which can take any values between zero and one. The discrete variable {0,1} leads to an integer linear programming while the second one is a linear programming. Is is possible to to transform the integer variable to the continuous one, and alternatively solve an LP instead of MIP?

Thanks a lot in advance for your answers.

Vahid

I am working on small project which is to apply the Operation research knowledge into daily life. Do you have fun or brilliant ideas ? Please share!

Thank you

In Branch and Bound Algorithm, if the linear relaxation of the problem provides more than one fractional values (for more than one decision variables), then which decision variable should be considered for the next step?

If arbitrarily any one of the fractional values is chosen, does it guarantee the optimal solution finally?

Looking through the literature, I realized all the proofs for NP- hardness of QIP are based on the claim that Binary Quadratic Integer Programming is NP- hard. Is that true?

The structure of this problem is similar (not equal) to other problems that admits simple solutions. Maybe, the colleagues of this community could help me in identifying a solution to this problem.

I was interested in listing all the possible integer solutions to

**f(n/10)-f(n/11) = 1 (eq1)**

Where f(x)=floor(x) is the

**floor function**, relating each real number**x**to the greatest integer**z**less or equal to x.The floor function wasn't so easy to deal with, as it seems at first sight. I replaced n/10 = x.. and took 10x/11, having a similar equation:

**f(x)-f(10x/11) = 1**

The solution set was then easily verified, X = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}..

But didn't mean the possible solutions to (eq1) would resume to:

Y= 10*X = {10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110}.

**I computed every solution to the equation (shown in the figure) with the support of an algorithm.**

I realized then, the relation with another equation by taking a look thru the floor function identities...

**–11x + 10y = 110 – n ,**

where

**x = n mod 10**, and**y = n mod 11**.Seems like a diophantine approximation involved. Of course there are theorems to help with solutions of diophantine equations... But...

What if we have an equation:

**f(x/a) + f(x/b) = c**, where x is the variable and a, b, c are positive integers, where f(x)=floor(x) is once again the floor function.

How can we compute the possible solutions for x integer?

Is there any property that we can operate with floor functions? I suppose not cuz the function isn't continuous.

This subject just caught my attention. Maybe it's easier than seems. Any clue/tips?

I am looking for solvers for mixed-integer nonlinear programs (MINLP). Are there solvers that instead of exact solutions, provide faster but approximate solutions with some error bound?

I am wondering if GA is capable of solving large-scale(say, 10000 design variables) 0-1programming problems. If not, what is the potential alternatives?

I'm trying to solve the vehicle routing problem with the column generation algorithm, but it does not converge and generates too many columns in a very long time.

Thanks!

Is there any condition (similar to K.K.T condition in convex optimization) can be used to get the analytical solution.

I want to get some theorems form the optimization model, and analyze the relationship between the variables and the parameters.

How about the Multi-modules (used to prove the optimality of the model) ?

I'm trying to identify which approach would work best to get optimal decision among three layers in multilayer network.

From the research I did it seems that combinatorial optimization and integer programming are the best suited for the job. Is there any other option I should consider? How should I set up the problem considering parameters index and performance metrics to take optimal decision

Many thanks in Advance

Rashmi

I have a bi-objective MIP model. I proposed some valid inequalities to tighten the model. I know in a single objective problem, I can check the linear relaxation solution to evaluate the performance of valid inequalities. In your opinion, is it correct to use the linear relaxation criterion in a bi-objective problem or use other criteria?

I am using NSGA-II for carrying out a study. For this I intend to stabilize the algorithm. Now when I am testing algorithm with ZDT-4 test function, I am not getting satisfactory results. I am using "https://in.mathworks.com/matlabcentral/fileexchange/10429-nsga-ii--a-multi-objective-optimization-algorithm" as source code and I have done some modifications to it according to my need. ZDT-4 is converging to local pareto optimal solution. How to get global optimal solution? All other functions are working fine.

I'm interested in 'rounding algorithms for Linear Programs'.

Unfortunately I can only find the already mentioned once on the internet.

What I search, is basically a algorithm which converts fractional solutions into integral solutions. Thank you :)

Monotone 0-1 IP's are integer programs with 0-1 variables with two non zero coefficients per variable of opposite signs.

I am stabilizing NSGA-II algorithm using test functions. For this I need points on ideal Pareto curve of ZDT functions to calculate hypervolume. Where can I find these points?

actually i am willing to model a classic problem of association rule by a standard constraint optimization which can be nonlinear and non-convex too but with minimum number of constraints and optimization variables.

Suppose we obtain some non- dominated fronts in a multi-objective problem solved using NSGA-II

F1= {5 ,6 ,9}

F2={11,1,3,8}

and so on.

where number represents the solution number.

What does it interpret?

Is there any (approximate) way to convert a complementarity constraint (i.e. x.y greater than 0) to a linear constraint (i.e. a.x+b.y+c greater than 0)?

Hi,

I am doing MIP optimization on conveying capacity increase of pellets by proper silo routing and unloading. I need to specify a constraint saying that once a silo is selected it needs to be filled continuously at each time interval.

But currently, the silo is able to hold and capacity is constant for certain period of time which is not the case in real life. Hence, there exists many number of combinations in which silo can hold the capacity and the model is slow.

Can anyone formulate a constraint which tells the program to fill the silo continuously with certain flow rate and not hold back.

Regards

Pradyumna Krishnan

Assuming that ther is a function f(x) where x is the vector [n1,n2,...nm] where ni is the number of balls in the box i={1,..m}, and sum(ni) = n. f(x) being a non linear, non convex function.

What's the complexity of the problem of finding the distribution of balls that maximize f(x) ?

Also what's a good algorithm for solving this kind of problems ? GA, PSO, etc??

In the literature only i found the hedonic model for maximation of housing Project.

Dear respected scientists

I am currently working on solving an MIP model using Lagrangian relaxation method. I propose to solve the dual problem in iterated manner. Initially, the multipliers are set to zero and thereafter, they will be updated in each iteration until a good enough solution is obtained. I think about solving the dual problem by B&B method using Cplex While the iterative procedure(to update multipliers) will be implemented by Matlab. My question is " does any one experience integrating Cplex with Matlab in a similar way? or is this a good idea in term of CPU time and solution quality?" i will appreciate any help.

best regards

Ahmed Karam

Let two positive integers n and m be given. How many pairs of integers (x,y) can one find in the range 1..n such that (i) all x's are different, (ii) all y's are different, (iii) all x+y's are different, and (iv) x+y <= m.

I came across this problem (with n=7, m=11) in a safety discussion of a stream cypher (trinomials that have to fit into an opportunity window). I can only solve it by an

*extremely*lengthy analysis, while I'm hoping there is an elegant (or at least a not too lengthy) solution.If a third coordinate z is defined as m+1-x-y, the problem becomes a kind of magic-square thing: fill each of three rows with distinct numbers in 1..n, such that each column has constant sum m+1. How long can the rows be? This problem (which is the actual translation of the cryptological question) is not exactly equivalent with the pairs-problem; it may have slightly smaller solutions.

Can this perhaps be linked to a known problem?

Does any one know about the meaning of offset and the formula of this link: http://oeis.org/A002898/internal ?

I would like to first know if there is a general form for scheduling problem that can be reformulate in terms of a constraint optimization problem.

for example job shop problem is one of the famous scheduling problems however i couldn't find any related closed form integer programming or mixed integer programming.

There is the MINOPT library, but its code is not open.

Hello everyone,

I am working on a case study which is about project planning with limited resources. Activities can be completed either in normal duration or in its crash duration. It is required to finish project as soon as possible with minimum achievable cost. I think this problem can be modelled by using integer programming. So, I need some modelling examples on network/project scheduling with limited resources problem. If there are some could you please provide me ? Thanks in advance.

Can anyone tell me how to convert my waveform or signal which i have in floating point numbers into integer sequence?I need to encode my signal residue using arithnmetic coding for which i need the input as integer sequence. Kindly help

Hello everybody,

My task is to write a UMAT for nonlinear viscoelastic material based on the attached paper. For now, I have written the UMAT (1D formulation) and I am able to generate the output. My problems are

**Validity of the code**I have attached the stress vs time graph for a relaxation test on 1D Truss element. The behaviour looks normal (correct me if you have to) to me.- If i change the input parameters, say I play with the
**initial time increment of Steps**or change the**displacement from 5 units to 10 units,**the job keeps running and does not complete. I have to manually delete**.lck file**from the folder and again run the job. I am not able to understand what is going wrong. Would be really helpful if somebody could throw some light on it.

Thanks in advance for your time

Shree

**P.S**I have

**attached the input file**and the

**UMAT**for your reference

The question may be stupid but it really confuses me for a long time.

I read a lot of papers in wireless sensor network. Many researchers model their problems into the form of ILP. However, ILP is NP-Complete so it is not efficient for solving a problem.

**So why people write their problems into the form of ILP? Do they do that to make their problem clear to see and easy to understand? Or do I make some mistakes understanding the relations between ILP and NPC?**

I am really appreciated that you can help me to solve this question.

here thousands of variables or involved.and i implement this problem on GAMS software and i got some answer so I want to know that this solution is correct or not but how to check this?Any idea plz help as soon as possible.

I am currently using intlinprog (Mixed Integer Linear Programming) to solve LRP(location routing problem).I am now stuck at subtour elimination constraint and I don't know how to compute this on MATLAB. The subtour elimination method that I will be using is Miller-Tucker-Zemlin (MTZ) constraints.

Can anyone help me? The constraints are:

for all i qi<=ui<=C

for all i-j client ui-uj+xij*C<=C-qj

are there Matlab codes availables showing how to set up and solve a mixed-integer linear programming problem (parameters and Variables as matrix).i found some ones in mathWork but I'm looking for others with more constraints.I am trying to solve a model for location routing problem (problem combining Facility location Problem FLP and vehicule routing problem VRP) using MATLAB

Thanks

I've already completed the algorithms in vehicle routing problems, that is the last-mile-problem. But I wonder, is there any do's or don'ts in developing my own system to solve the vehicle routing problem, with the algorithm developed by myself.

Thank you for the response in advance.

The problem below is Integer and linear problem (ILP). The idea is to solve it using linear (and continuous) programing (LP) techniques to obtain a relaxed solution. Then, a

**rounding algorithm**should be used to obtain an integer solution that consists in an approximative solution to the optimal solution of the original ILP. My question is about an efficient and (possibly) approximative algorithm for the**rounding**procedure.I would appreciate any

**direct and succinct**text about this subject.There are various method to find prime numbers but not efficient to calculate the large prime integers of 40 digit in fraction of time. Even many system are not supporting more than 11-18 digit integer. I tried to find it through C/C++ and Matlab even I used JAVA using biginteger but I am unable to find this large primes.

Some 40 digits prime integer are available on net and some other prime calculator are there which generates prime integers immediately but those are manual. For my work I need some special prime which are derived from primes, so that i can recheck the derived integer from a prime integer whether a prime or not. So I need a powerful tool or program for generating and checking large prime integer within second of time. If somebody knows please kindly post the solution.

An MSc student of mine is currently working on operation room scheduling problem. She is struggling on how to convert the related IP model into Set Modeling of Lingo optimization software. Any help (an example, a modelling done in Lingo, ...) would be highly appreciated.

Can someone suggest something?

I actually use my own desktop computer, for modeling and solve problems using the solve cplex, but the large scale problems is very expensive in time (days of calculus) to reach optimal solutions, I wish explore the option of a cloud service to these large scale problems

I have used the gamultiobj function in Matlab, but it ignores the problem constraints if the variables are binary. I was wondering if some of you have used a different solver for this type of problem.

Hello all.

Using rule "RandomInteger[{2^30,2^31}]", we can produce one random integer with 31 bit ? In fact by execute this rule , produce one integer with about 60 bits !!!!!! Yes, this is a bug in Mathematica 6.0. But , however , mathematica is one best and fast softwares that I know.

I use rule RandomInteger[]+2*RandomInteger[{2^29, 2^30}] . Is this right?

Thank you all.

As I know, most mixed integer programming problems with block diagonal structure are suitable to be solved with Dantzig-Wolfe decomposition and column generation. Analogously, which kinds of problems or which structures are preferable to be solved with Bender decomposition. Can Bender decomposition also be feasible for general MIP problem, and which one can get better performance for general MIP, Bender decomposition or Dantzig-Wolfe decomposition?

I need to find a good method for solving a MIP problem that has a big size. I hope you could present some suggestion about that (some references please)