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It is a well known fact that finite time optimal controllers, such as MPC does not necessarily result in closed loop stable systems. Within the MPC community it is common practice to add a final state constraint and/or a final state penalty in order to obtain guaranteed stability. However, for more advanced controller structures it can be difficult to show stability using these techniques. Additionally in some cases the final state constraint set consists of so many inequalities that the complexity of the MPC problem is too big for use in certain fast and time critical applications. In this paper we instead focus on deriving a tool for a-postiori analysis of the closed loop stability for linear systems controlled with MPC controllers. We formulate an optimisation problem that gives a sufficient condition for stability of the closed loop system and we show that the problem can be written as a Mixed Integer Linear Programming Problem (MILP)

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Bibliography: leaf 37

The development is documented of a heuristic method for the solution of pure integer linear programming problems. The procedure draws its methodology from the ideas of Hooke and Jeeves type 1 and 2 exploratory searches, greedy procedures, and neighborhood searches. It uses an efficient rounding method to obtain its first feasible integer point from the optimal continuous solution obtained via the simplex method. Since this method is based entirely on simple addition or subtraction of one to each variable of a point in n-space and the subsequent comparison of candidate solutions to a given set of constraints, it facilitates significant complexity improvements over existing techniques. It also obtains the same optimal solution found by the branch-and-bound technique in 44 of 45 small to moderate size test problems. Two example problems are worked in detail to show the inner workings of the method. Furthermore, using an established weighted scheme for comparing computational effort involved in an algorithm, a comparison of this algorithm is made to the more established and rigorous branch-and-bound method. A computer implementation of the procedure, in PC compatible Pascal, is also presented and discussed.

A survey of the methods of solving the integer program, max summation from j=1 to j=n of the quantity (c sub j x sub j) subject to summation, j=1 to j=n of the quantity (a sub ij x sub j) = b sub i, i=1,...,m, and x sub j = or 0 and integer (j=1,...,n) is presented. Emphasis is placed on methods developed since 1960 with many as yet unpublished methods presented. Examples are given for the unpublished methods.

The application of Linear Programming (LP) models in the resource allocation process of organizations with workload partitioned in discrete projects was examined by developing two integer and one mixed-integer, large-scale LP models. These models have been implemented to seek optimal five-year project plans for Naval-Air-Test-Center (NATC) , involving a workforce numbered in hundreds and a budget of millions of dollars. In allocating resources, the optimal solution is the solution which yields the 'best' value, or the most desirous return to the organization; this return can also be called the solution benefit. Solutions should also be feasible in terms of the limitations on the availability of the resources by location and by time. MODEL-1 and MODEL-2 use a static workforce distribution and MODEL-3 allows limited reallocation of personnel to improve the solution; when reallocation is ordered, both reduction in labor efficiency and a penalty in the project benefit are introduced by MODEL-3. All three models have been implemented successfully using real data from NATC. The implementation is described and the solutions are compared with the solution given by NATC without models. A proposal is made to use the models in practice so that NATC can achieve more optimal five-year plans and also to improve the existing workforce distribution by location and by time.

Igor Ferst https://www.pyohio.org/2019/presentations/86 How do airlines choose which planes service which routes? How does a hospital optimize the shift schedule for hundreds of doctors and nurses? How do you choose the optimal location for a group of fulfillment centers, or oil derricks, or cell towers? These kinds of problems (and many others!) can be solved with integer linear programming (ILP), a powerful and decades-old framework for solving optimization problems. In this talk we will give a brief introduction to ILP and describe it's uses, strengths, and weaknesses. We will also show how to solve a real-world vehicle routing problem using Google's open-source python library for ILP. Trigger warning: this talk will contain high-school level math. Integer linear programming (ILP) is a powerful framework for solving optimization problems related to scheduling, resource allocation, vehicle routing, and many other areas. This talk will give a brief introduction to ILP and show how to solve a real-world vehicle routing problem using Google's open-source python library for ILP. === https://pyohio.org A FREE annual conference for anyone interested in Python in and around Ohio, the entire Midwest, maybe even the whole world. Produced by NDV: https://youtube.com/channel/UCQ7dFBzZGlBvtU2hCecsBBg?sub_confirmation=1 Sun Jul 28 14:30:00 2019 at Hays Cape

Thesis (M.S. in Operations Research and M.S. in Comp. Sci.)--Naval Postgraduate School, 1979

This paper focuses on the problem of optimal placement of the PMU (Phasor Measurement Unit) in a large power network. Optimal PMU placement has multiple solutions depending upon the consideration of constraints and therefore consideration of field based topological and operational criteria can give better and actual result. This has been carried out using integer linear programming with the consideration of different types of criteria such as, zero injection measurement, cost constraint, voltage level, future expansion, bus location in the network, owner of the substation, centre of load point, availability of communication link, connection with generating station and evaluation based on system operating condition. Different indices like BOI (Bus Observability Index), SORI (System observability Redundancy Index) and System Operating conditions are also being considered for determination of location at which PMU can be installed to achieve full network observability. It has been clearly visualized change in location of PMU in cases of with and without consideration of said criteria.

New branching rules and new methods to control the size of LP's resulted in a world's record for the solution of large traveling salesman problems.

Much research has centered on the problem of finding shortest paths in graphs. It is well known that there is a direct correspondence between the single-source shortest-paths problem and the following simple linear programming problem. Let S be a set of linear inequalities of the form x sub j - x sub i or - a sub ij, where the x sub i are unknowns and the a sub ij are given rea constants. Determine a set of values for the x sub i such that the inequalities in S are satisfied, or determine that no such values exist. This paper considers the mixed-integer linear programming variant of this problem in which some (but not necessarily all) of the x sub i are required to be integers. The problem arises in the context of synchronous circuit optimization, but it has applications to PERT scheduling and VLSI layout compaction as well.

Recent proposals by Gomory and others for solving linear programs involving integer-valued variables appear sufficiently promising that it is worthwhile to systematically review and classify problems that can be reduced to this class and thereby solved. Historically, non-linear, nonconvex and combinatorial problems are areas where classical mathematics almost always fails. It is therefore significant that the reduction can be made for problems involving multiple dichotomies and k-fold alternatives which include problems with discrete variables, non-linear separable minimizing functions, conditional constraints, global minimum of general concave functions and combinatorial problems such as the fixed charge problem, traveling salesman problem, orthogonal latin square problems, and map coloring problems.

This report describes an exact algorithm for the pure, general integer linear programming problem (ILP). Common applications of this model occur in capital budgeting (project selection), resource allocation and fixed- charge (plant location) problems. The central theme of our algorithm is to enumerate a subset of all solutions called feasible 1-ceiling points. A feasible 1-ceiling point may be thought of as an integer solution lying on or near the boundary of the feasible region for the LP-relaxation associated with (ILP). Precise definitions of 1-ceiling points and the role they play in an integer linear program are presented in a recent report by the authors. One key theorem therein demonstrates that all optimal solutions for an (ILP) whose feasible region is non-empty and bounded are feasible 1-ceiling points. Consequently, such a problem may be solved by enumerating just its feasible 1-ceiling points. Our approach is to implicitly enumerate 1-ceiling points with respect to one constraint at a time while simultaneously considering feasibility. Computational results from applying this incumbent-improving Exact Ceiling Point Algorithm to 48 test problems taken from the literature indicate that this enumeration scheme may hold potential as a practical approach for solving problems with certain types of structure.

A mixed-integer linear programming formulation is developed for minimizing delay to traffic in a signal controlled road network. Offsets, splits of green time and a common cycle time for the network are considered as decision variables simultaneously. The traffic flow pattern is modeled as a periodic platoon, and a link performance function is derived in the form of a piecewise linear convex surface representing the delay incurred by these platoons. Stochastic effects are accounted for by a saturation deterrence function representing the expected overflow queue on each link and are included as an additive component in the objective function. Computational results, using the MPSX system, are given for an arterial with 11 signals in Waltham, Mass., and a portion of the UTCS network in Washington, D.C. containing 20 nodes, 63 links and 21 loops.

We present a bounding technique for use in implicit enumeration algorithms for solving the integer linear programming problem with binary variables. The main assumptions used by this technique are the associated linear program, obtained by dropping the integrality constraints on the variables, possesses a unique optimal solution, this optimal solution is not binary, and a good feasible solution to the original problem is available. An alternative to the last assumption which is weaker is also presented. We show that joint bounds can be obtained on the values of a subset of the variables. In addition we given an efficient method to implement this bounding technique. Finally, a class of problems particularly well-suited to this bounding procedure is specified. (Author)

This report examines the role played by several kinds of ceiling points in solving the pure, general integer linear programming problem (ILP). While no assumptions are made concerning the structure or signs of the data of the problem, it is assumed that the feasible region for (ILP) is non-empty and bounded. A ceiling point with respect to a single constraint maybe thought of as an integer solution on or close to the boundary of the feasible region defined by the constraint. The definition of a ceiling point with respect to a single constraint is extended to take multiple constraints into consideration simultaneously, defining what is called a feasible ceiling point. It is shown that the set all feasible ceiling points contains at least one optimal solution for (ILP). A related class of solutions called feasible 1-ceiling points is also characterized and shown to contain all optimal solutions for (ILP). Moreover, 1- ceiling points are computationally easier to identify than ordinary ceiling points and may be sought with respect to one constant at a time. It is also demonstrated that solving (ILP) requires only enumerating feasible 1-ceiling points with respect to a subset of all functional constraints. Keywords: Integer variables; Enumeration algorithms.

The paper discusses duality for linear and nonlinear programs in which some of the variables are arbitrarily constrained. The most important class of such problems is that of mixed-integer (linear and nonlinear) programs. The paper introduces the duality constructions and discusses algorithms based on them.

The paper discusses duality for linear and nonlinear programs in which some of the variables are arbitrarily constrained. The most important class of such problems is that of mixed-integer (linear and nonlinear) programs. The paper introduces the duality constructions and discusses algorithms based on them.

This paper focuses on the problem of optimal placement of the PMU (Phasor Measurement Unit) in a large power network. Optimal PMU placement has multiple solutions depending upon the consideration of constraints and therefore consideration of field based topological and operational criteria can give better and actual result. This has been carried out using integer linear programming with the consideration of different types of criteria such as, zero injection measurement, cost constraint, voltage level, future expansion, bus location in the network, owner of the substation, centre of load point, availability of communication link, connection with generating station and evaluation based on system operating condition. Different indices like BOI (Bus Observability Index), SORI (System observability Redundancy Index) and System Operating conditions are also being considered for determination of location at which PMU can be installed to achieve full network observability. It has been clearly visualized change in location of PMU in cases of with and without consideration of said criteria.

The program solves integer linear programs of the form (P) Minimize cx subject to b + Ax = or 0, x sub j = 0 or 1 where c and x are n-vectors, b is an m-vector, and A is m by n. Any bounded integer linear program can be written in this form, using elementary manipulations if necessary. Familiarity with 2 previous papers in the series is assumed.

In this report the author discusses the strength of surrogate constraints in general and presents a heuristic procedure for iteratively constructing stronger surrogates beginning with the dual multiplier surrogate.

Telescope networks are gaining traction due to their promise of higher resource utilization than single telescopes and as enablers of novel astronomical observation modes. However, as telescope network sizes increase, the possibility of scheduling them completely or even semi-manually disappears. In an earlier paper, a step towards software telescope scheduling was made with the specification of the Reservation formalism, through the use of which astronomers can express their complex observation needs and preferences. In this paper we build on that work. We present a solution to the discretized version of the problem of scheduling a telescope network. We derive a solvable integer linear programming (ILP) model based on the Reservation formalism. We show computational results verifying its correctness, and confirm that our Gurobi-based implementation can address problems of realistic size. Finally, we extend the ILP model to also handle the novel observation requests that can be specified using the more advanced Compound Reservation formalism.

Thesis (MS)?Naval Postgraduate School, 1968

This document demonstrates that the efficient approach for diagnosis of Petri nets via integer linear programming may be unable to detect a fault even if the system is diagnosable.

This document demonstrates that the efficient approach for diagnosis of Petri nets via integer linear programming may be unable to detect a fault even if the system is diagnosable.

This document demonstrates that the efficient approach for diagnosis of Petri nets via integer linear programming may be unable to detect a fault even if the system is diagnosable.

Much research has centered on the problem of finding shortest paths in graphs. It is well known that there is a direct correspondence between the single source shortest-paths problem and the following simple linear programming problems: Let S be a set of linear inequalities of the form x sub j - x sub i or = (a sub ij, where the x sub i are unknowns and the a sub ij are given real constants. Determine a set of values for the x sub i such that the inequalities in S are satisfied, or determine that no such values exist. This paper considers the mixed-integer linear programming variant of this problem in which some (but not necessarily all) of the x sub i are required to be integers. The problem arises in the context of synchronous circuit optimization but it has applications to PERT scheduling and VLSI layout compaction as well. Keywords: Algorithms, Combinatorial optimization.

Much research has centered on the problem of finding shortest paths in graphs. It is well known that there is a direct correspondence between the single source shortest-paths problem and the following simple linear programming problems: Let S be a set of linear inequalities of the form x sub j - x sub i or = (a sub ij, where the x sub i are unknowns and the a sub ij are given real constants. Determine a set of values for the x sub i such that the inequalities in S are satisfied, or determine that no such values exist. This paper considers the mixed-integer linear programming variant of this problem in which some (but not necessarily all) of the x sub i are required to be integers. The problem arises in the context of synchronous circuit optimization but it has applications to PERT scheduling and VLSI layout compaction as well. Keywords: Algorithms, Combinatorial optimization.

We consider so-called squaring the square-puzzles where a given square (or rectangle) should be dissected into smaller squares. For a specific instance of such problems we demonstrate that a mathematically rigorous solution can be quite involved. As an alternative to exhaustive enumeration using tailored algorithms we describe the general approach of formulating the problem as an integer linear program.

The Johnson-Lindenstrauss lemma allows dimension reduction on real vectors with low distortion on their pairwise Euclidean distances. This result is often used in algorithms such as $k$-means or $k$ nearest neighbours since they only use Euclidean distances, and has sometimes been used in optimization algorithms involving the minimization of Euclidean distances. In this paper we introduce a first attempt at using this lemma in the context of feasibility problems in linear and integer programming, which cannot be expressed only in function of Euclidean distances.

The Johnson-Lindenstrauss lemma allows dimension reduction on real vectors with low distortion on their pairwise Euclidean distances. This result is often used in algorithms such as $k$-means or $k$ nearest neighbours since they only use Euclidean distances, and has sometimes been used in optimization algorithms involving the minimization of Euclidean distances. In this paper we introduce a first attempt at using this lemma in the context of feasibility problems in linear and integer programming, which cannot be expressed only in function of Euclidean distances.

We present a system for the semantic role labeling task. The system combines a machine learning technique with an inference procedure based on integer linear programming that supports the incorporation of linguistic and structural constraints into the decision process. The system is tested on the data provided in CoNLL-2004 shared task on semantic role labeling and achieves very competitive results.

Some heuristic procedures for seeking a good approximate solution of any pure integer linear programming problem are evaluated. It was found that the procedures are extremely efficient, being computationally feasible for problems having hundreds of variables and constraints. Furthermore, they proved to be very effective in identifying good solutions, often obtaining optimal ones. Thus, the procedures provide a way of dealing with the frequently encountered integer programming problems that are beyond the computational capability of existing algorithms. For smaller problems, they also provide an advanced start for accelerating certain primal algorithms, including the author's Bound-and-Scan algorithm and Faaland and Hillier's Accelerated Bound-and-Scan algorithm. In addition, Jeroslow and Smith have found that imbedding the first part of one of these procedures inside the iterative step of a branch-and-bound algorithm can greatly improve the latter's efficiency in locating solutions whose objective function value is within a specified percentage of that for the optimal solution.

This report gives documentation for the computer code HEUR for finding good approximate solutions to integer linear programming problems using Hillier's heuristic procedure. A listing of the program with sample input and output is included. (Author)

This report gives documentation for the computer code HEUR for finding good approximate solutions to integer linear programming problems using Hillier's heuristic procedure. A listing of the program with sample input and output is included. (Author)

Bibliography: pt. 1, p. 46-47; pt. 2, p. 48

Ralph Gomory has recently aroused interest in a special type of knapsack problem in which the constraint coefficients and constant term are elements of a finite additive group. The significance of this problem lies in the fact that it is closely related to the general integer linear programming problem, resulting by removing the nonnegativity restrictions on those variables in the general problem that lie in an optimal basis for the associated linear program. Gomory has shown how to solve the special knapsack problem by adapting a dynamic programming recursion originally designed for the ordinary knapsack problem, and has identified sufficient conditions under which the solution of the special knapsack problem will satisfy the nonnegativity requirements in the general integer program, thereby yielding an optimal solution to that problem as well. In this paper the author presents an algorithm for solving the special knapsack problem that is capable of accommodating a variety of constraints in addition to the special knapsack constraint. The purpose in doing this is to expand the range of problems for which the optimal solution for the special problem will also provide an optimal solution to the general integer program from which it was derived.

We propose a new exact approach for solving integer linear programming (ILP) problems which we will call projective splitting algorithms (PSAs). Unlike classical methods for solving ILP problems, PSAs conduct the search for the optimal solution by generating candidate solutions tailored to specific values of the objective function. As a consequence of this strategy, the number of variables in the original ILP problem is systematically reduced without adding any additional constraint to the initial formulation. This is the first of a two-part series on PSAs. In this paper we focus on the resolution of pure integer linear programming (PILP) problems, leaving the treatment of mixed integer linear programming (MILP) formulations to the second part of this series. The proposed algorithm was tested against the IBM ILOG CPLEX [2] optimizer on instances of the 0-1 Multidimensional Knapsack Problem (0-1MKP), showing satisfactory results on instances with a large number of variables.

We consider the feasibility problem of integer linear programming (ILP). We show that solutions of any ILP instance can be naturally represented by an FO-definable class of graphs. For each solution there may be many graphs representing it. However, one of these graphs is of path-width at most 2n, where n is the number of variables in the instance. Since FO is decidable on graphs of bounded path- width, we obtain an alternative decidability result for ILP. The technique we use underlines a common principle to prove decidability which has previously been employed for automata with auxiliary storage. We also show how this new result links to automata theory and program verification.

This document is chiefly a bibliography listing titles such as: An Efficient Interactive Solution Framework for Bicriteria Integer Programming, An Improved Interactive Multicriteria Integer Programming Algorithm, An Interactive Multicriteria Linear Programming Method Using Constraints and Convex Cones.

"NPS-55-82-008"--Cover

We transform join ordering into a mixed integer linear program (MILP). This allows to address query optimization by mature MILP solver implementations that have evolved over decades and steadily improved their performance. They offer features such as anytime optimization and parallel search that are highly relevant for query optimization. We present a MILP formulation for searching left-deep query plans. We use sets of binary variables to represent join operands and intermediate results, operator implementation choices or the presence of interesting orders. Linear constraints restrict value assignments to the ones representing valid query plans. We approximate the cost of scan and join operations via linear functions, allowing to increase approximation precision up to arbitrary degrees. Our experimental results are encouraging: we are able to find optimal plans for joins between 60 tables; a query size that is beyond the capabilities of prior exhaustive query optimization methods.

A multiple objective fuzzy liner programming approach to the capital budgeting problem is developed. Since much of the available data in any capital budgeting decision situation is either of an imprecise or ill-defined nature, a mathematical optimization technique is required that is capable of incorporating this inherent uncertainty. Fuzzy linear programming provides an effective methodology for this analysis. Specifically, a mathematical model is developed which utilizes fuzzy linear programming as a solution technique for the research and development program or project selection problem. In addition an exchange heuristic, a modified form of C. C. Peterson's exchange algorithm, is presented. A limited bibliography of works in multiple objective optimization is presented. Two computer codes are included. The first utilizes the IBM MPSX/Mixed Integer Programming procedures to solve the (0-1) linear integer programming problem. The second is a FORTRAN program to solve the exchange heuristic algorithm discussed previously. (Author)

This report describes a heuristic algorithm for the pure, general integer linear programming problem (ILP). In attempting to quickly obtain a near-optimal solution (with-out concern for establishing optimality), the algorithm searches for a feasible 1-ceiling point. A feasible 1-ceiling point may be thought of as an integer solution lying on or near the boundary of the feasible region for the LP-relaxation associated with the ILP. Precise definitions of 1 -ceiling points and the role they play in an integer linear program are presented in a recent report by the authors. One key theorem therein demonstrates that all optimal solutions for an ILP whose feasible region is non- empty and bounded are feasible 1-ceiling points. Consequently, such a problem may be solved by enumerating just its feasible 1-ceiling points. Our heuristic approach is based upon the idea that a feasible 1-ceiling point found relatively near the optimal solution for the LP-relaxation is apt to have a high (possibly even optimal) objective function value. Having applied this Heuristic Ceiling Point Algorithm to 48 test problems taken from the literature, it appears that searching for such 1-ceiling points usually does provide a very good solution with a moderate amount of computational effort. Keywords: Subroutines, General integer variables.

This article is from Nucleic Acids Research , volume 40 . Abstract Deregulation of cell signaling pathways plays a crucial role in the development of tumors. The identification of such pathways requires effective analysis tools that facilitate the interpretation of expression differences. Here, we present a novel and highly efficient method for identifying deregulated subnetworks in a regulatory network. Given a score for each node that measures the degree of deregulation of the corresponding gene or protein, the algorithm computes the heaviest connected subnetwork of a specified size reachable from a designated root node. This root node can be interpreted as a molecular key player responsible for the observed deregulation. To demonstrate the potential of our approach, we analyzed three gene expression data sets. In one scenario, we compared expression profiles of non-malignant primary mammary epithelial cells derived from BRCA1 mutation carriers and of epithelial cells without BRCA1 mutation. Our results suggest that oxidative stress plays an important role in epithelial cells of BRCA1 mutation carriers and that the activation of stress proteins may result in avoidance of apoptosis leading to an increased overall survival of cells with genetic alterations. In summary, our approach opens new avenues for the elucidation of pathogenic mechanisms and for the detection of molecular key players.

Load disaggregation based on aided linear integer programming (ALIP) is proposed. We start with a conventional linear integer programming (IP) based disaggregation and enhance it in several ways. The enhancements include additional constraints, correction based on a state diagram, median filtering, and linear programming-based refinement. With the aid of these enhancements, the performance of IP-based disaggregation is significantly improved. The proposed ALIP system relies only on the instantaneous load samples instead of waveform signatures, and hence does not crucially depend on high sampling frequency. Experimental results show that the proposed ALIP system performs better than the conventional IP-based load disaggregation system.

We describe an approach for accomplishing the high-level mission planning required for a heterogeneous team of autonomous vehicles performing mine countermeasure (MCM) survey missions in multiple areas. The high-level mission scheduling and waterspace management requires sequencing the order and location of lower-level MCM tasks to be completed by each vehicle in the heterogeneous team: unmanned surface vessels (USVs) and unmanned underwater vehicles (UUVs). We propose solving this complex sequencing operation by leveraging unique information processing, communication, refueling, and planning windows that form constraints within the system within a formal scheduling optimization framework known as mixed-integer linear programming. We pose the problem using a mixed-integer linear programming optimization framework, compare several complexity reduction heuristics to the full optimization, and include methods to account for relative uncertainty in the duration of planned tasks in such a manner to balance the risk of schedule slips or conservative and sub-optimal schedule.

Considering the intermittency of renewable energy systems, a sizing and scheduling model is proposed for a finite number of static electric loads. The model objective is to maximize solar energy utilization with and without storage. For the application of optimal load size selection, the energy production of a solar photovoltaic is assumed to be consumed by a finite number of discrete loads in an off-grid system using mixed-integer linear programming. Additional constraints are battery charge and discharge limitations and minimum uptime and downtime for each unit. For a certain solar power profile the model outputs optimal unit size as well as the optimal scheduling for both units and battery charge and discharge (if applicable). The impact of different solar power profiles and minimum up and down time constraints on the optimal unit and battery sizes are studied. The battery size required to achieve full solar energy utilization decreases with the number of units and with increased flexibility of the units (shorter on and off-time). A novel formulation is introduced to model quasi-dynamic units that gradually start and stop and the quasi-dynamic units increase solar energy utilization. The model can also be applied to search for the optimal number of units for a given cost function.

This paper presents Multi-Objective Integer Non-Linear Programming (MOINLP) involving Negative Total Float (NTF) for improving the basic model of Multi-Objective Programming (MOP) in case the optimization of the additional cost for Project Scheduling Compression (PSC). Using the basic MOP to solve the more complex problems is a challenging task. We suspect that Negative Total Float (NTF) having an indication to make the basic MOP to solve the more general case, both simple and complex of PSC. The purpose of this research is identifying the conflicting objectives in PSC problem using NTF and improving MOINLP by involving the NTF parameter to solve the PSC problem. The Solver Application, which is an add-in of MS Excel, is used to perform optimization process to the model developed. The results show that NTF has an important role to identify the conflicting objectives in PSC. We define NTF is an automatic maximum value of the activity duration reduction to achieve due date of PSC. Furthermore, the use of NTF as a constraint in MOINLP can solve the more general case for both simple and complex PSC problem. Base on the condition, we state that the basic MOP is still significant to solve the PSC complex problems using MOINLP as a sophisticated MOP technique. 

Thesis advisor(s): Dell, Robert F