Miller Tucker Zemlin Research Articles

Travel plan for tourists

This article presents the proposal for a model travel plan for tourists in the Jalapão State Park [PEJ - Parque Estadual do Jalapão], located in the State of Tocantins, Brazil. The research shows the use of the Gurobi Optimizer library in Python Software associated with using Miller-Tucker-Zemlin (MTZ) constraints to ensure a viable route circuit. Through the Traveling Salesman Problem (TSP), two viable optimal routes are presented for two research problems: i) minimize the distance of access to the PEJ from the city of Palmas -TO and ii) find an optimal route path for tourists considering some of the most relevant points of the PEJ. The study presents a viable solution to route problems and contributes with an actual model, showing that TSP and the use of restrictions MTZ can be adequate to solve these problems and others to be solved in PEJ.

Relaxed Connected Dominating Set Problem for Power System Cyber–Physical Security

Motivated by power system cyber–physical security applications, this article investigates a variant of the connected dominating set problem called the relaxed connected dominating set (RCDS) problem, where between every two nodes in the solution dominating set there exists a path not traversing two consecutive nodes not in the set. The RCDS problem is NP-hard. To facilitate solving, presolving and problem reduction techniques are developed. In addition, three integer program formulations are obtained using single commodity flow, Dantzig–Fulkerson–Johnson and Miller–Tucker–Zemlin ideas to enforce graph connectedness. For a realistic power network repository (MATPOWER 7) including graphs with 70k nodes, our presolving and reduction procedures simplify problem instances to on average 30% of the original sizes. All reduced instances, when modeled by our formulations, can be solved with optimality gap less than 0.5% within 20 min using a PC with Gurobi 9.

A conditional-logic interpretation for Miller–Tucker–Zemlin inequalities and extensions

A conditional-logic interpretation for Miller–Tucker–Zemlin inequalities and extensions

A Matheuristic Approach for the No-Wait Flowshop Scheduling Problem with Makespan Criterion

The No-wait Flowshop Scheduling Problem (NWFSP) has always been a research hotspot because of its importance in various industries. This paper uses a matheuristic approach that combines exact and heuristic algorithms to solve it with the objective to minimize the makespan. Firstly, according to the symmetry characteristics in NWFSP, a local search method is designed, where the first job and the last job in the symmetrical position remain unchanged, and then, a three-level neighborhood division method and the corresponding rapid evaluation method at each level are given. The two proposed heuristic algorithms are built on them, which can effectively avoid al-ready searched areas, so as to quickly obtain the local optimal solutions, and even directly obtain the optimal solutions for small-scale instances. Secondly, using the equivalence of this problem and the Asymmetric Traveling Salesman Problem (ATSP), an exact method for solving NWFSP is constructed. Importing the results of the heuristics into the model, the efficiency of the Mil-ler-Tucker-Zemlin (MTZ) model for solving small-scale NWFSP can be improved. Thirdly, the matheuristic algorithm is used to test 141 instances of the Tailard and Reeves benchmarks, and each optimal solution can be obtained within 134 s, which verifies the stability and effectiveness of the algorithm.

A Connectivity Constrained MILP Model for Optimal Transmission Switching

This letter formulates network connectivity as Miller-Tucker-Zemlin (MTZ) constraints and incorporates them into the mixed-integer linear programming (MILP) model for the optimal transmission switching (OTS) problem. The connectivity constraints are linear and for a power network with n buses, m branches, and d loads in pre-contingency or each post-contingency state there are approximately <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O</i> (n+5m+d) constraints, and <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O</i> (n) continuous and <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O</i> (2 m) binary variables, which is much smaller than those in the existing formulations. The MILP OTS model with the proposed connectivity constraints can be readily solved by well-developed MILP solvers. Case studies on the PJM 5-bus system, IEEE 300-bus system, and French 1888-bus system validate the effectiveness of the proposed model.

Comparison of the Sub-Tour Elimination Methods for the Asymmetric Traveling Salesman Problem Applying the SECA Method

There are many sub-tour elimination constraint (SEC) formulations for the traveling salesman problem (TSP). Among the different methods found in articles, usually three apply more than others. This study examines the Danzig–Fulkerson–Johnson (DFJ), Miller–Tucker–Zemlin (MTZ), and Gavish–Graves (GG) formulations to select the best asymmetric traveling salesman problem (ATSP) formulation. The study introduces five criteria as the number of constraints, number of variables, type of variables, time of solving, and differences between the optimum and the relaxed value for comparing these constraints. The reason for selecting these criteria is that they have the most significant impact on the mathematical problem-solving complexity. A new and well-known multiple-criteria decision making (MCDM) method, the simultaneous evaluation of the criteria and alternatives (SECA) method was applied to analyze these criteria. To use the SECA method for ranking the alternatives and extracting information about the criteria from constraints needs computational computing. In this research, we use CPLEX 12.8 software to compute the criteria value and LINGO 11 software to solve the SECA method. Finally, we conclude that the Gavish–Graves (GG) formulation is the best. The new web-based software was used for testing the results.

A note on the lifted Miller–Tucker–Zemlin subtour elimination constraints for routing problems with time windows

We propose lifted versions of the Miller–Tucker–Zemlin subtour elimination constraints for routing problems with time windows (TW). The constraints are valid for problems such as the travelling salesman problem with TW, the vehicle routing problem with TW, the generalized travelling salesman problem with TW, and the general vehicle routing problem with TW. They are corrected versions of the constraints proposed by Desrochers and Laporte (1991).

Accelerating the Miller–Tucker–Zemlin model for the asymmetric traveling salesman problem

In this article, we present an easy to implement algorithmic approach that improves the computational performance of the Miller–Tucker–Zemlin (MTZ) model for the asymmetric traveling salesman problem (ATSP) by efficiently generating valid inequalities from fractional solutions. Computational experiments show that the proposed approach enhances considerably the performance of MTZ-based formulations reported in the literature. By adding facet-defining inequalities of the underline ATSP-polytope, the number of nodes in the branch-and-bound tree is drastically reduced, and the convergence of the MTZ-type formulations is accelerated. We also extend this idea to solve the multiple asymmetric traveling salesman problem (mATSP). This approach can help practitioners to solve real-life problems to near optimality using a standard optimization solver and may be useful to solve a variety of routing problems that use MTZ-type of subtour elimination constraints.

Protection Placement for Power System State Estimation Measurement Data Integrity

In this article, a protection placement problem for power network measurement system data integrity is considered. The placement problem is motivated from [G. Dan and H. Sandberg, “Stealth attacks and protection schemes for state estimators in power systems,” in Proc. 1st IEEE Int. Conf. Smart Grid Commun ., 2010, pp. 214–219], where supervisory-control-and-data-acqui-sition and phasor measurement unit measurements can be selectively encrypted and protected to prevent false data injection attacks. The problem is shown to be NP-hard and an integer linear programming formulation is provided based on the topological observability condition by [G. Krumpholz, K. Clements, and P. Davis, “Power system observability: A practical algorithm using network topology,” IEEE Trans. Power App. Syst. , vol. PAS99, no. 4, pp. 1534–1542, Jul. 1980.]. The description of the observability condition requires a graph connectivity constraint, which is found to be most effectively described by the Miller–Tucker–Zemlin conditions. For a specialization without line power flow measurements, the protection placement problem can be modeled by a domination-type integer linear program that is much easier to solve than the general formulation requiring graph connectivity. Numerical studies with IEEE benchmark and other large power systems with more than 2000 buses indicate that by using the proposed formulations, the protection placement problem can be solved in a negligible amount of time in realistic application settings.

Multi-depot vehicle routing problem: a comparative study of alternative formulations

ABSTRACTThis paper proposes a two-commodity flow formulation for the MDVRP considering a heterogeneous vehicle fleet and maximum routing time. Its computational performance is compared to a traditional formulation, the so called three-index formulation, which is adapted to fit the same problem. With the creation of subtours being a problem within such formulations, four alternative sets of constraints are considered to eliminate them: the Dantzig–Fulkerson–Johnson constraint, the Miller–Tucker–Zemlin constraint, the transit load constraint, and the arrival time constraint. The resulting mixed-integer linear programming models are then applied to a number of benchmark instances and the obtained performance results are compared.

Two dependency constrained spanning tree problems

AbstractWe introduce two spanning tree problems with dependency constraints, where an edge can be chosen only if at least one or all edges in its dependency set are also chosen, respectively. The dependencies on the input graph G are described by a digraph D whose vertices are the edges of G, and the in‐neighbors of a vertex are its dependency set. We show that both problems are NP‐hard even if G is an outerplanar chordal graph with diameter 2 or maximum degree 3, and D is a disjoint union of arborescences of height 2. We also prove that the problems are inapproximable to a factor, unless , and that they are W[2]‐hard. As an attempt to narrow the hardness gap, we present two polynomial cases. We test integer linear programming formulations based on directed cut (DCUT) and Miller–Tucker–Zemlin (MTZ) constraints. Computational experiments are reported.

Facility Location with Tree Topology and Radial Distance Constraints

Let Gd = (V, Ed) be an input disk graph with a set of facility nodes V and a set of edges Ed connecting facilities in V. In this paper, we minimize the total connection cost distances between a set of customers and a subset of facility nodes S⊆V and among facilities in S, subject to the condition that nodes in S simultaneously form a spanning tree and an independent set according to graphs and Gd, respectively, where is the complement of Gd. Four compact polynomial formulations are proposed based on classical and set covering p‐Median formulations. However, the tree to be formed with S is modelled with Miller–Tucker–Zemlin (MTZ) and path orienteering constraints. Example domains where the proposed models can be applied include complex wireless and wired network communications, warehouse facility location, electrical power systems, water supply networks, and transportation networks, to name a few. The proposed models are further strengthened with clique valid inequalities which can be obtained in polynomial time for disk graphs. Finally, we propose Kruskal‐based heuristics and metaheuristics based on guided local search and simulated annealing strategies. Our numerical results indicate that only the MTZ constrained models allow obtaining optimal solutions for instances with up to 200 nodes and 1000 users. In particular, tight lower bounds are obtained with all linear relaxations, e.g., less than 6% for most of the instances compared to the optimal solutions. In general, the MTZ constrained models outperform path orienteering ones. However, the proposed heuristics and metaheuristics allow obtaining near‐optimal solutions in significantly short CPU time and tight feasible solutions for large instances of the problem.

An exact and heuristic approach for the d-minimum branch vertices problem

Given a connected graph \(G=(V,E)\), the d-Minimum Branch Vertices (d-MBV) problem consists in finding a spanning tree of G with the minimum number of vertices with degree strictly greater than d. We developed a Miller–Tucker–Zemlin based formulation with valid inequalities for this problem. The results obtained for different values of d show the effectiveness of the proposed method, which has solved several instances faster than previous methods. Also, an heuristic is proposed for this problem, that was tested on several instances of the Minimum Branch Vertices problem, which is the d-MBV problem, when \(d = 2\).

Optimal production, replenishment, delivery, routing and inventory management policies for products with perishable inventory

This paper presents a generalized production-inventory-routing model with perishable inventory. We analyze the optimal integrated decisions of when and how much to deliver and sell products with varying manufacturing periods. We discuss main inventory management policies to demonstrate the applicability of the model in real-world applications for production routing problems (PRPs) with perishable inventory. Furthermore, an exact branch-and-cut algorithm is developed and discussed. We introduce new families of logical, strengthened lot-sizing and lifted Miller–Tucker–Zemlin subtour elimination constraints for the PRP with perishable inventory. Finally, we test the performance of the algorithm. We also implement and compare 8 suboptimal delivery and selling priority policies with an optimized policy to develop managerial implications.

Data for: Optimal production, replenishment, delivery, routing and inventory management policies for products with perishable inventory

Abstract This paper presents a generalized production-inventory-routing model with perishable inventory. We analyze the optimal integrated decisions of when and how much to deliver and sell products with varying manufacturing periods. We discuss main inventory management policies to demonstrate the applicability of the model in real-world applications for production routing problems (PRPs) with perishable inventory. Furthermore, an exact branch-and-cut algorithm is developed and discussed. We introduce new families of logical, strengthened lot-sizing and lifted Miller–Tucker–Zemlin subtour elimination constraints for the PRP with perishable inventory. Finally, we test the performance of the algorithm. We also implement and compare 8 suboptimal delivery and selling priority policies with an optimized policy to develop managerial implications.

Short combinatorial proof that the DFJ polytope is contained in the MTZ polytope for the Asymmetric Traveling Salesman Problem

For the Asymmetric Traveling Salesman Problem (ATSP), it is known that the Dantzig–Fulkerson–Johnson (DFJ) polytope is contained in the Miller–Tucker–Zemlin (MTZ) polytope. The analytic proofs of this fact are quite long. Here, we present a proof which is combinatorial and significantly shorter by relating the formulation to distances in a modified graph.

A note on the Miller-Tucker-Zemlin model for the asymmetric traveling salesman problem

Abstract An enhancement of the Miller-Tucker-Zemlin (MTZ) model for the asymmetric traveling salesman problem is presented by introducing additional constraints to the initial formulation. The constraints account for ordering of boundary nodes as well as all successive nodes in the salesman tour. The enhanced MTZ subtour elimination constraints are computationally compared with the basic MTZ constraints and the version of MTZ lifted by Desrochers and Laporte. The proposed enhancement shows improved performance on a number of asymmetric TSPLIB instances.

Steiner Ring Star 문제를 해결하기 위한 새로운 Mixed-Integer Programming Modeling

In this paper, we deal with a Steiner Ring Star (SRS) problem arising from the design of survivable telecommunication networks. We develop two mixed integer programming formulations for the SRS problem by implementing MillerTucker-Zemlin (MTZ) and Sarin-Sherali-Bhootra (SSB) subtour elimination constraints, and then apply the reformulationlinearization technique (RLT) to enhance the lower bound obtained by the LP relaxation. By exploiting the ring-star structure of underlying network, we devise some valid inequalities that tighten the LP relaxation. Computational results demonstrate the effectiveness of the proposed solution procedure. Keyword:Steiner Ring Star Problem, Mixed Integer Programming, Reformulation-Linearization Technique 한국경영과학회지 제39권 제1호 2014년 3월

Looking for edge-equitable spanning trees

This paper studies the Ordered Spanning Tree Problem, where the weights of the edges of the tree are sorted and then linearly combined using a previously given coefficients vector. Depending on the coefficients, several objectives can be incorporated to the problem. We pay special attention to the search of spanning trees with balanced weights, i.e., where the differences among the weights are, in some sense, minimized. To solve the problem, we propose two Integer Programming formulations, one based on flow and the other one on the Miller–Tucker–Zemlin constraints. We analyze several potential improvements for both the formulations whose behaviors are checked by means of a computational experiment. Finally, we show how both the formulations can be adapted to incorporate to the objective non-linear functions of the weights.

Requiem for the Miller–Tucker–Zemlin subtour elimination constraints?

The Miller–Tucker–Zemlin (MTZ) Subtour Elimination Constraints (SECs) and the improved version by Desrochers and Laporte (DL) have been and are still in regular use to model a variety of routing problems. This paper presents a systematic way of deriving inequalities that are more complicated than the MTZ and DL inequalities and that, in a certain way, “generalize” the underlying idea of the original inequalities. We present a polyhedral approach that studies and analyses the convex hull of feasible sets for small dimensions. This approach allows us to generate generalizations of the MTZ and DL inequalities, which are “good” in the sense that they define facets of these small polyhedra. It is well known that DL inequalities imply a subset of Dantzig–Fulkerson–Johnson (DFJ) SECs for two-node subsets. Through the approach presented, we describe a generalization of these inequalities which imply DFJ SECs for three-node subsets and show that generalizations for larger subsets are unlikely to exist. Our study presents a similar analysis with generalizations of MTZ inequalities and their relation with the lifted circuit inequalities for three node subsets.