Questions tagged [modeling]

For questions related to the process of converting a real-world problem into a mathematical model. Can include questions related to linearization, logical constraints, tightness of formulations, and so on.

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11
votes
2answers
916 views

In an integer program, how can I “activate” a constraint only if a decision variable has a certain value?

Suppose we have the constraint $$a_1x_1 + \cdots + a_nx_n \gtreqless b,$$ where $a_i$ and $b$ are constants and $x_i$ are decision variables. Suppose also that we want the constraint to hold if $y=1$ (...
10
votes
1answer
360 views

I've formulated my optimization model; now what?

I've formulated my linear/nonlinear/integer/mixed-integer optimization problem in algebraic form (possibly with the help of the folks on this site). Now what? How do I solve it?
12
votes
1answer
612 views

Comparison of Algebraic modelling languages and general programming languages

Some optimization software/frameworks (commercial or open-source) such as AMPL, GAMS, Cplex, ... have a specific Algebraic modelling language. Some of them have another type of programming that uses ...
17
votes
6answers
2k views

Infeasibility in mathematical optimization models

Sometimes, when solving mathematical optimization models (especially MIPs), they may be infeasible. Is there any comprehensive method to deal with the infeasibility conditions? (especially in complex ...
24
votes
5answers
1k views

Examples of machine learning applied to operations research?

Can someone give me a few examples, if they exist, of problems in operations research that could be solved using machine learning. I am aware that machine learning examples are data-driven and do not ...
12
votes
2answers
502 views

Expressing a chain of boolean ORs using ILP

How to express a chain of OR operations in an ILP in which each expression is a less than or equal constraint and the left hand side variable in all inequalities is always the same? All the variables ...
10
votes
4answers
413 views

How to linearize a constraint with a maximum of binary variables times some coefficient in the right-hand-side

I have the following constraint that I'd like to linearize: $P$ is a given set $b_p \in \{0,1\} , \forall p \in P$ a binary variable associated with each element of $P$ $c_p \in \mathbb{R}^+$, a ...
31
votes
8answers
1k views

Modeling floor function exactly

Suppose we want to enforce a constraint $$ y=\lfloor{x}\rfloor $$ where $x$ is some continuous variable. One option is to use $$ x-1\leq{y}\leq{x},\quad y\in\mathbb{Z}, $$ which fails on the edge case ...
11
votes
1answer
295 views

Expressing a chain of boolean ORs using ILP involving different variables

How can I express a chain of OR operations in an ILP, given that each operand is an inequality between two binary variables? I have asked a similar question here: Chain of Boolean ORs. In that ...
9
votes
3answers
453 views

How could we simplify solving the large scale MIPs without using any advanced methods like decompositions?

Many practical optimization models (specially MIPs) are NP-Hard and solving them need much time even with the modern solvers like CPLEX or GUROBI. One of the best way (but not easy) is using ...
6
votes
5answers
2k views

Algorithms vs LP or MIP

Is there a way of writing an algorithm with if-, while-statements to find an optimal solution without using linear-programming (LP)/MIP? If so, what would the benefits be against the LP/MIP? Is it ...
6
votes
1answer
156 views

TSP subtour elimination by assigning distance traveled

Given a set $S$ which we need to travel following TSP rules. I was wondering if this sub tour elimination method is good enough or not? Let $b_{i,j}$ denote edge from $i$ to $j$ is taken or not and $...
4
votes
1answer
188 views

How to set a limit for a switch to 0 of a variable

I would like to know how to define a constraint to set a limit for switching to 0 for a decision variable? So I have a linear variable $x(t)$ which quantifies the modulation degree of a heating device....
21
votes
5answers
2k views

Validation and verification of mathematical models

Within the subject of simulation I have found some literature on validation and verification (e.g. Sargent's paper). My question is, what techniques do you use to validate and verify your mathematical ...
26
votes
1answer
362 views

The rationale to improve MTZ?

Currently I need to solve a quite specific problem involving symmetric TSP as a sub-problem (i.e., a Hamiltonian cycle is a necessary condition for optimizing some problem-specific variables that ...
13
votes
2answers
838 views

How to choose between high number of binary variables or fewer number of integer (not only 0 and 1) variables in a IP formulation?

When I have to write the formulation of an IP, I usually have the choice between writing $i\times j$ binary variables with two indices such as $ x_{i,j} $ or, writing $j$ integer variables $x_i$. Is ...
11
votes
2answers
1k views

Linear programming: objective function with “buckets”

I had a linear programming problem with the following objective function $$f(x) = \sum_{j}x_jq_jp_j - \sum_{i}\left(\sum_{j}x_jq_jC_{ij} \right) c_i$$ Where $q, p, C, c$ are known. This problem was ...
4
votes
2answers
174 views

Is there efficient way to deal with division by zero?

I am trying to solve an optimization problem in which there is an objective function in the following form: \begin{equation}f(x, y)=x+\left(\frac{1}{y}\right)\end{equation} Where, $x,y$ are positive ...
21
votes
5answers
317 views

Tightness of an LP relaxation without using objective function

How can we measure the tightness of a linear programming relaxation for a mixed integer linear program without using the objective value? I would like to get a measure in terms of the feasible set and ...
18
votes
2answers
272 views

Mathematically creating the 'perfect' permutation for reservations in a hostel

I am working at a hostel which uses a reservation system for each room and the beds in the room (e.g. $14$ beds in one room, bed numbers $1-14$.) When we get bookings for multiple people, we assign ...
12
votes
2answers
383 views

Debugging cplex model

I implemented a cplex model and I am convinced that the model should allow a better solution on a specific instance. However, when I impose the variable values of the solution onto the model, it ...
12
votes
2answers
342 views

Expressing an implication as ILP where each implication term comprises a chain of boolean ORs

Consider an implication of the form $A \implies B$ where both $A, B$ comprises a chain of Boolean OR variables. For example, $(a_1 \lor a_2 \lor a_3) \implies (b_1 \lor b_2 \lor b_3)$. How can this ...
10
votes
5answers
498 views

Dealing with non-overlapping constraints

Let us consider the following problem: Let $T$ be a set of tasks. Each task $t \in T$ has a duration $d_t$ and a target start time $s_t$. No two tasks can be executed in parallel. The objective is to ...
8
votes
2answers
133 views

Bounding arrival time at a node in a resource-constrained shortest path problem

Given a city map (a graph) $G$, $b_{i,j}$ is a Boolean variable for whether or not edge $i$,$j$ is allocated, $d_{i,j}$ denotes the distance between $i$,$j$. The objective is to move from $s$ to $e$ ...
7
votes
2answers
639 views

Finding an Objective Function for Assigning Employees to Sequence Dates

I am using a mixed-integer-program to schedule employees to projects. These projects can have a time window to get completed from a few weeks to a few months. At the moment I am working in a ...
7
votes
0answers
266 views

OR-TOOLS : delivery node with multiple possible pickup nodes

I am using ortools to model a VRP with pickup and delivery constraints, where pickups can be done at different nodes. For example, if node A has a demand, it must be picked at node B or C. Here is how ...
6
votes
2answers
219 views

Good sources on developing mathematical models [closed]

What are good sources (literature, internet, etc) on learning to model real-world problems using mathematical formulations? In particular, I would like to know sources which establish the relationship ...
5
votes
2answers
295 views

How to model If $A \le B$ then $Y = 1$, otherwise $Y = 0$

Somehow I don't get it right. I would like to model the following conditional: If $A\le B$ then $Y=1$ otherwise $Y=0$ where $A, B$ are reals and $Y$ is binary. I can model as follows: $Y \cdot A \le B$...
4
votes
0answers
192 views

How can I formulate this multi-objective optimization problem?

Now, for each system $X$ $(X=A,B,C,E)$, my objective is $$\max\min\frac{s_{x_u}}{d_{x_u}}$$ here, $x=a$ for system A, $x=b$ for system B and follows... and for the whole system, my objective is $$\max\...
3
votes
2answers
244 views

Find all Combinations of a Matrix

I have a $16\times11$ matrix and want to find all eligible* combinations of this matrix including always entities from all 11 columns. A simple example from a $2\times3$ matrix would be the following:...
2
votes
1answer
183 views

Controlling the constraint execution sequence in docplex

I have tried to implement my question asked here with the help of Github code link. Below is the linearized version of the MTZ like constraint as per the answer given here. Constraints below assign ...
2
votes
0answers
110 views

Montoya et al. 2017 benchmark Results

I am testing Montoya et al. 2017 dataset and trying to replicate the Result from this paper The electric vehicle routing problem with capacitated charging stations. I am unable to replicate the exact ...
2
votes
1answer
149 views

Working with large models

I am working with a variant of TSP, number of nodes that I need to test are in between 2500 to 3000 nodes, I am using docplex for modelling, I have a 8 gb Ram but it gets filled with only 400 nodes. ...
1
vote
1answer
247 views

How to mathematically formulate the optimization problem?

I have a system with $S$ service points. There are also $U$ users in the system. We have $$U>S>G$$ One group can have maximum $M$ service points, but there is no restrictions on the number of ...
1
vote
3answers
261 views

How can I linearize this IF-THEN constraint?

Let $P_{t,u}; t=1,2,\ldots,T, u=1,2,\ldots,U$ be known values $\alpha$ is also a known parameter $X_{t,u}$ an optimization variable I have the following constraint: IF $P_{t,u}\geq\alpha$, THEN $X_{...
-6
votes
2answers
211 views

How to decide to write an objective function?

I'm working on this problem: In the Njaba river basin, the available water was allocated for the purposes of consumption, irrigation, and electric power supply among three communities. The water ...