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Questions tagged [combinatorial-optimization]

For questions about optimization over a discrete solution space.

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What are the efficient ways to model this scheduling problem and ways to improve running time

There are $N - \{1 \ldots N\}$ jobs, each with processing time $p_j$, to be scheduled on $M - \{1 \ldots M\}$ machines over span of $D - \{1 \ldots D\}$ days and while working $T - \{1 \ldots E \ldots ...
Amogh Bhosekar's user avatar
0 votes
3 answers
598 views

Using big M values for a constraint

I want to enforce $x_{i,j}=x_{k,j}\implies z_i \neq z_k$ where $k = i-1$ so I used \begin{align}z_k + 1 - (x_{i,j} - x_{k,j})) \leq z_i \leq z_k - 1 - (x_{i,j} - x_{k,j})\quad\text{for each $j$}\end{...
OR Junior's user avatar
  • 543
2 votes
1 answer
189 views

Does this problem fall into any common problem definition....Knapsack maybe?

I am struggling to find a representative problem formulation for this optimization challenge. I have implemented a MILP in Matlab, but the run time is taking more then a day. My goal is to see if it ...
S moran's user avatar
  • 335
1 vote
1 answer
132 views

Derived variables of when a decision variable appears?

I am dealing with a multi-travelling passenger problem. $x_{i,j}$ is a binary variable that allocate a passenger $i$ to a vehicle $j$, every vehicle can carry only $n_{pv}$ passenger where $i \in \{1,...
OR Junior's user avatar
  • 543
3 votes
2 answers
310 views

Could DOcplex.CP recognize that it solves the graph coloring minimization problem?

I created a graph coloring DOcplex.CP model inspired by this example. However, I do not know the number of colors in advance. The goal is to minimize the number of colors (i.e., get as close as ...
Alexander Pozdneev's user avatar
2 votes
1 answer
618 views

Same values constraint and grouping of variables

In a linear program, I would like some variables to: 1. Take the same values 2. Group some variables i.e. some variables should take same values or lie within certain percentage. 3. All different ...
Sam's user avatar
  • 161
7 votes
1 answer
872 views

Minimum vertex cover and linear programming

Suppose we have a graph G. Consider the minimum vertex cover problem of G formulated as a linear programming problem, that is for each vertex $v_{i}$ we have the variable $x_{i}$, for each edge $v_{i}...
Mario Giambarioli's user avatar
5 votes
2 answers
697 views

Where can I find resources to learn mathematical modelling for real life operation research problems like combinatorial optimization?

I find it hard to form math models for real life operations research problems, how can I learn this? Any books, tutorials available?
21vs's user avatar
  • 489
8 votes
1 answer
247 views

Heuristic Search Planning Tree Leading to Worse TSP Solutions than Naive Greedy

I'm doing a Traveling Salesman Problem (TSP) homework for a coursera optimization course. My first attempt was a regular naive greedy approach, from each point, moving to the closest node (that hadn't ...
Max Power's user avatar
6 votes
1 answer
94 views

Maximum weight b-matching with global cardinality constraint

Suppose $A$ is an $n$-by-$n$ symmetric matrix whose entries are all nonnegative. $A_{ii} = 0$ for all $i$. We want to find an $n$-by-$n$ binary ($0/1$ valued) matrix $X$ that maximizes $$\sum_{ij} A_{...
user306101's user avatar
4 votes
1 answer
86 views

How to model $A_i=B_i$ for only one $i$

I would like to model the following: Only one of the following equalities can hold. $$(A_1 = B_1)\quad\text{OR}\quad(A_2 = B_2)\quad\text{OR}\quad\dots\quad\text{OR}\quad(A_k = B_k)$$ I can ...
Clement's user avatar
  • 2,252
5 votes
2 answers
565 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: $Y \cdot A \le B$...
Clement's user avatar
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12 votes
2 answers
260 views

Restoring a list from differences

Given a list of (absolute valued) pair differences ordered and with duplicates removed, how can we recover/reconstruct the list that generated these differences? We do not know anything about the ...
hakank's user avatar
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9 votes
2 answers
400 views

Current Issues of Interest

What are some current issue of interest in Operations Research? I am interested in current topics that experts in the field are interested in researching.
9 votes
1 answer
170 views

Equivalence of formulations

I have a simple model such as: \begin{align}\max&\quad z=X_1+X_2+X_3+X_4\\\text{s.t.}&\quad y_1+y_2+y_3+y_4=2\\&\quad X_1 \leq y_1\\&\quad X_2 \leq y_1+y_2\\&\quad X_3 \leq y_2+...
Evren Guney's user avatar
26 votes
5 answers
2k views

What's the current status of the Vehicle Routing Problem in the logistics industry?

After a bit of reading I think I've been able to conclude that state-of-the-art VRP can get solutions for 100~500 stops. My question is around how this actually affects logistics (like Amazon for ...
Alexander Soare's user avatar
9 votes
3 answers
538 views

Equipment replacement problem

I have a question on the Equipment Replacement Problem, where the following is taken (with some syntactic modifications) from IB2070 Mathematical Programming II (MP2), Warwick Business School. ...
Slim Shady's user avatar
7 votes
1 answer
205 views

How to convert 3D bin packing problem to 2D bin packing approximation?

I'm trying to approximately solve a 3D container loading problem. Is it possible to use 2D bin packing algorithms? If so, how do we make the transformation? What are the conditions needed to make the ...
Surya's user avatar
  • 73
12 votes
1 answer
451 views

Two-stage $k$-means clustering

The problem I am facing is clustering problem, needed for a Vehicular Routing Problem (VRP) I'm tackling. It is a heterogeneous VRP with Time Window (TW) and a capacity utilization constraint, i.e. a ...
Dimitris Boukosis's user avatar
8 votes
2 answers
556 views

Is there a greedy heuristic approach to the MILP problem?

I have the following optimization problem which is an MILP. I can solve it with an MILP solver. \begin{alignat}{1}\max_{x_n,t}\,&\quad t\quad\\\text{s.t.}&\quad\sum_{n=1}^{N} x_n \,&= M\\...
KGM's user avatar
  • 2,377
11 votes
3 answers
866 views

Modeling the Choose function

In statistics, one often encounters the choose function ${x \choose y}$ which encodes the number of ways of choosing $y$ items from a set of $x$ items. How would one go about modeling a choose ...
Josh Allen's user avatar
9 votes
2 answers
380 views

How to formulate a MIP that can minimize the costs with a combination of subsets given a set?

I am trying to solve the following problem. I have a set $\{1,2,3\}$, which gives the following subsets with its costs: $\{1\}=8$, $\{2\}=9$, $\{3\}=7$, $\{1,2\}=9$, $\{1,3\}=18$, $\{2,3\}=15$ and $\{...
Harry van t Kamp's user avatar
18 votes
1 answer
3k views

Duality in mixed integer linear programs

I know that the standard duality theory for the linear programming problem does not hold for mixed integer linear programming problems. I was wondering why an integer program does not have a dual ...
rasul's user avatar
  • 2,150
10 votes
2 answers
194 views

Optimal set order to maximize stochastic reward

You have a ticket allowing you to visit up to $n$ of $M$ carnival booths offering games of chance. At each booth you have probability $p_{i}$ of winning a reward with average value $r_{i}$. Each booth ...
sedge's user avatar
  • 103
10 votes
2 answers
184 views

Heuristic methods for optimising complex black box function over permutations/ranks?

Suppose I have a set $S=\{1,2,\dots,500\}$ and some function $f(\sigma)$ from the permutations $\operatorname{Perm}(S) \rightarrow \mathbb{R}$ to be minimized. The function is complex (simulation ...
Brendan Hill's user avatar
18 votes
3 answers
1k views

Variable fixing based on a good feasible solution

Suppose you have a combinatorial optimization problem that is formulated as a mixed integer linear program (minimization). The problem size is denoted $n$ and the expected $n$ is around $100$. The ...
rasul's user avatar
  • 2,150
11 votes
2 answers
1k views

What is the difference between job shop scheduling and resource constrained project scheduling?

I read here https://slideplayer.com/slide/3353960/ that RCPS is a generalized version of job shop scheduling. I'm new to this area and I'm trying to classify a specific variation of these types of ...
cdrootsudormstardashr's user avatar
9 votes
2 answers
171 views

CVRP With Unconstrained Fleet Size: Upper Bound on Optimal Fleet Size

Given a CVRP where the number of trucks is not constrained, is there an upper bound on the number of trucks used in an optimal solution in terms of number of customers, some distances, capacities, and ...
Bo Jones's user avatar
6 votes
0 answers
86 views

What are the top three applications (in terms of number of citations) of the "reverse search" algorithm of David Avis?

I can see that this algorithm is quite popular, and that one of the original papers now has 820 citations on Google Scholar. However, what are the most highly cited applications of it? If in Google ...
Nike Dattani's user avatar
  • 1,278
15 votes
2 answers
2k views

State-of-the-art algorithms for solving linear programs

Průša and Werner (2019) show that the general linear programming problem reduces in nearly linear time to the LP relaxations of many classical NP-hard problems (assuming sparse encoding of instances)....
rasul's user avatar
  • 2,150
10 votes
1 answer
580 views

How to get solver time from CPLEX when using the NEOS server through Pyomo?

I've been using CPLEX on the NEOS server, via Pyomo, to solve a binary program I'm working on. NEOS is amazing, but the documentation is somewhat lacking on the Pyomo side, so I haven't been able to ...
domdomdom's user avatar
  • 421
13 votes
7 answers
963 views

What are the examples (applications) of the MIPs in which the objective function has nonzero coefficients for only continuous variables?

I'm specifically looking for real applications of the following form of MIP: $$\max\,Cx$$ subject to: \begin{align}Ax +By &= D\\Ax &= E\\By &= F\\ x &\ge 0\\ y &\in \mathbb{...
Junior MIP's user avatar
8 votes
1 answer
170 views

What class of scheduling problem models jobs which require multiple machines simultaneously?

In the Flow/Job Shop problems, and other related scheduling problems, a common assumption is that at any given time, a particular job will be being processed on at most one machine (usually... none). ...
Brendan Hill's user avatar
6 votes
2 answers
1k views

How to modify EMSR when capacity for each fare class is different

In the normal EMSRa and EMSRb (Expected Marginal Seat Revenue) algorithms, each fare class is utilizes exactly 1 unit of capacity (for example, one seat on a plane). But I have a similar problem for ...
dg428's user avatar
  • 231
14 votes
3 answers
2k views

A variant of the Multiple Traveling Salesman Problem

I am trying to find a reference (or a reformulation) of a variant of the multiple Traveling Salesman Problem, where multiple agents need to visit each vertex in a graph with minimal cost. Most of the ...
kemalduldul's user avatar
8 votes
1 answer
450 views

Finding minimum time for vehicle to reach to its destination

Given a set of Vehicles with source and destination I need to find the minimum time of travel for all the vehicles, there are also some charging stations and its necessary for vehicles to charge 1 ...
ooo's user avatar
  • 1,589
7 votes
1 answer
191 views

How to interpret the random solution pick by Lévy flight on cuckoo search

I am working on an implementation of Cuckoo Search for a set covering problem. After reading some papers I cannot understand how choosing a random solution (new cuckoo) works. What I see is that ...
Cristofer Fuentes's user avatar
7 votes
1 answer
419 views

How to reformulate (linearize/convexify) a budgeted assignment problem?

I have a scheduling problem at hand. In my system, there is a service station with $M$ service outlets, therefore, the service station can serve $M$ users at a time. But, there are $N$ users $N>M$ ...
KGM's user avatar
  • 2,377
12 votes
2 answers
226 views

Benchmark problems for combinatorial multi-objective optimisation

Does anyone know of any good benchmark problems for combinatorial multi-objective optimisation? Something where pareto frontiers are known for example would be very useful.
domdomdom's user avatar
  • 421
31 votes
5 answers
5k views

Optimization terminology: "Exact" v. "Approximate"

In optimization literature, I frequently see solution methods termed "exact" or "approximate". (I use the term "method" here because I suspect exactness, or its lack, is a function of both algorithm ...
prubin's user avatar
  • 39.6k
7 votes
2 answers
499 views

How can I linearize or convexify this binary quadratic optimization problem?

I have an optimization problem as below. I am having a hard time with the last constraint. $\max \eta$ subject to ${\bf U}(:,m)^T{\bf A}{\bf U}(:,m)=0,m=1,2,\cdots,M$ here $\bf{A}$ is a Binary ...
KGM's user avatar
  • 2,377
15 votes
4 answers
926 views

Optimization models for portfolio optimization

What are the mainstream models for portfolio optimization? We have Markowitz mean-variance model and CVaR-based models (e.g., max return subject to a CVaR constraint). What else is out there in terms ...
Daniel Duque's user avatar
  • 1,355
11 votes
2 answers
500 views

Finding an optimal set without forbidden subsets

Given $n$ items, I want to select a set items $S\subseteq\{1,2,\dots,n\}$ that maximize profit. The profit of item $i\in\{1,2,\dots,n\}$ is given by $p_i$ and may be assumed to be non-negative. ...
Kevin Dalmeijer's user avatar
18 votes
3 answers
3k views

What are some real-world applications of QUBO?

QUBO (Quadratic Unconstrained Binary Optimization) is the minimization of a quadratic function of binary variables. It has been used for computer vision, Ramsey numbers, factoring numbers, the ...
Nike Dattani's user avatar
  • 1,278
16 votes
3 answers
1k views

Bin Packing with Relational Penalization

There are $ N $ bins with equal capacity $ C $. In addition, there are $ N $ objects $x_1, x_2, \dots, x_N $ that need to be accommodated using the least amount of bins. Each object $x_i$ has a volume ...
Duns's user avatar
  • 303
5 votes
0 answers
124 views

Maximum eigenvalue across subsamples

I have an $N$-dimensional vector of data, say $X_{t}$, with $1 \leq t \leq T$. Of this vector $X_{t}$, I want to consider sub-vectors, say $X_{t}^{b}$, which are $m$-dimensional combinations of ...
Lorenzo Trapani's user avatar
13 votes
3 answers
3k views

Are there any efficient algorithms to solve the longest path problem in networks with cycles?

I have a directed social network and as a preprocessing step I need to calculate the longest path lengths for each node. Longest path problem is NP-hard as far as I know but I've seen dynamic ...
Evren Guney's user avatar
10 votes
1 answer
193 views

Algorithmic gap for Hochbaum's (greedy) algorithm for (metric) uncapacitated facility location

In Jain et al. (2003)1, at the bottom of page 801, they construct an instance of (metric) uncapacitated facility location for which they claim the greedy (Hochbaum's) algorithm has gap $\Omega\left(\...
ydubey7's user avatar
  • 579
10 votes
1 answer
218 views

Relationship between the Assignment Problem and the Stable Marriage Problem

Suppose I'm solving a minimum-weight matching problem in a bipartite graph with sets $\mathcal{I}$ and $\mathcal{J}$, where $w_{ij}$ denotes the weight of matching item $i$ to $j$. I can model the ...
tuba's user avatar
  • 101
19 votes
1 answer
491 views

Combinatorial Optimization: Metaheuristics, CP, IP -- "versus" or "and"?

"Recently" someone asked on Twitter whether "people still use genetic algorithms for integer programs". The "majority answer", i.e., 1 out of 1, was: "Yes" . So,...
fbahr's user avatar
  • 1,026