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Lecture from a Business Mathematics course (2005 - 6) by KIT

This papers deals with the constrained discounted control of piecewise deterministic Markov process (PDMPs) in general Borel spaces. The control variable acts on the jump rate and transition measure, and the goal is to minimize the total expected discounted cost, composed of positive running and boundary costs, while satisfying some constraints also in this form. The basic idea is, by using the special features of the PDMPs, to re-write the problem via an embedded discrete-time Markov chain associated to the PDMP and re-formulate the problem as an infinite dimensional linear programming (LP) problem, via the occupation measures associated to the discrete-time process. It is important to stress however that our new discrete-time problem is not in the same framework of a general constrained discrete-time Markov Decision Process and, due to that, some conditions are required to get the equivalence between the continuous-time problem and the LP formulation. We provide in the sequel sufficient conditions for the solvability of the associated LP problem, based on a generalization of Theorem 4.1 in [8]. In the Appendix we present the proof of this generalization which, we believe, is of interest on its own. The paper is concluded with some examples to illustrate the obtained results.

We consider a network of agents whose objective is for the aggregate of their states to converge to a solution of a linear program in standard form. Each agent has limited information about the problem data and can communicate with other agents at discrete time instants of their choosing. Our main contribution is the synthesis of a distributed dynamics and a set of state-based rules, termed triggers, that individual agents use to determine when to opportunistically broadcast their state to neighboring agents to ensure asymptotic convergence to a solution of the linear program. Our technical approach to the algorithm design and analysis overcomes a number of challenges, including establishing convergence in the absence of a common smooth Lyapunov function, ensuring that the triggers are detectable by agents using only local information, accounting for asynchronism in the state broadcasts, and ruling out various causes of arbitrarily fast state broadcasting. Various simulations illustrate our results.

We consider a network of agents whose objective is for the aggregate of their states to converge to a solution of a linear program in standard form. Each agent has limited information about the problem data and can communicate with other agents at discrete time instants of their choosing. Our main contribution is the synthesis of a distributed dynamics and a set of state-based rules, termed triggers, that individual agents use to determine when to opportunistically broadcast their state to neighboring agents to ensure asymptotic convergence to a solution of the linear program. Our technical approach to the algorithm design and analysis overcomes a number of challenges, including establishing convergence in the absence of a common smooth Lyapunov function, ensuring that the triggers are detectable by agents using only local information, accounting for asynchronism in the state broadcasts, and ruling out various causes of arbitrarily fast state broadcasting. Various simulations illustrate our results.

A linear-programming decoder for \emph{nonbinary} expander codes is presented. It is shown that the proposed decoder has the maximum-likelihood certificate properties. It is also shown that this decoder corrects any pattern of errors of a relative weight up to approximately 1/4 \delta_A \delta_B (where \delta_A and \delta_B are the relative minimum distances of the constituent codes).

A class of valued constraint satisfaction problems (VCSPs) is characterised by a valued constraint language, a fixed set of cost functions on a finite domain. An instance of the problem is specified by a sum of cost functions from the language with the goal to minimise the sum. We study which classes of finite-valued languages can be solved exactly by the basic linear programming relaxation (BLP). Thapper and Zivny showed [20] that if BLP solves the language then the language admits a binary commutative fractional polymorphism. We prove that the converse is also true. This leads to a necessary and a sufficient condition which can be checked in polynomial time for a given language. In contrast, the previous necessary and sufficient condition due to [20] involved infinitely many inequalities. More recently, Thapper and Zivny [21] showed (using, in particular, a technique introduced in this paper) that core languages that do not satisfy our condition are NP-hard. Taken together, these results imply that a finite-valued language can either be solved using Linear Programming or is NP-hard.

We propose a novel Linear Program (LP) based formula- tion for solving jigsaw puzzles. We formulate jigsaw solving as a set of successive global convex relaxations of the stan- dard NP-hard formulation, that can describe both jigsaws with pieces of unknown position and puzzles of unknown po- sition and orientation. The main contribution and strength of our approach comes from the LP assembly strategy. In contrast to existing greedy methods, our LP solver exploits all the pairwise matches simultaneously, and computes the position of each piece/component globally. The main ad- vantages of our LP approach include: (i) a reduced sensi- tivity to local minima compared to greedy approaches, since our successive approximations are global and convex and (ii) an increased robustness to the presence of mismatches in the pairwise matches due to the use of a weighted L1 penalty. To demonstrate the effectiveness of our approach, we test our algorithm on public jigsaw datasets and show that it outperforms state-of-the-art methods.

Since the beginning of the development of interior-point methods, there exists a puzzling gap between the results in theory and the observations in numerical experience, i.e., algorithms with good polynomial bound are not computationally efficient and algorithms demonstrated efficiency in computation do not have a good or any polynomial bound. Todd raised a question in 2002: "Can we find a theoretically and practically efficient way to reoptimize?" This paper is an effort to close the gap. We propose two arc-search infeasible interior-point algorithms with infeasible central path neighborhood wider than all existing infeasible interior-point algorithms that are proved to be convergent. We show that the first algorithm is polynomial and its simplified version, if it terminates in finite iterations, has a complexity bound equal to the best known complexity bound for all (feasible or infeasible) interior-point algorithms. We demonstrate the computational efficiency of the proposed algorithms by testing all Netlib linear programming problems in standard form and comparing the numerical results to those obtained by Mehrotra's predictor-corrector algorithm and a recently developed more efficient arc-search algorithm (the convergence of these two algorithms is unknown). We conclude that the newly proposed algorithms are not only polynomial but also computationally competitive comparing to both Mehrotra's predictor-corrector algorithm and the efficient arc-search algorithm.

Since the beginning of the development of interior-point methods, there exists a puzzling gap between the results in theory and the observations in numerical experience, i.e., algorithms with good polynomial bound are not computationally efficient and algorithms demonstrated efficiency in computation do not have a good or any polynomial bound. Todd raised a question in 2002: "Can we find a theoretically and practically efficient way to reoptimize?" This paper is an effort to close the gap. We propose two arc-search infeasible interior-point algorithms with infeasible central path neighborhood wider than all existing infeasible interior-point algorithms that are proved to be convergent. We show that the first algorithm is polynomial and its simplified version, if it terminates in finite iterations, has a complexity bound equal to the best known complexity bound for all (feasible or infeasible) interior-point algorithms. We demonstrate the computational efficiency of the proposed algorithms by testing all Netlib linear programming problems in standard form and comparing the numerical results to those obtained by Mehrotra's predictor-corrector algorithm and a recently developed more efficient arc-search algorithm (the convergence of these two algorithms is unknown). We conclude that the newly proposed algorithms are not only polynomial but also computationally competitive comparing to both Mehrotra's predictor-corrector algorithm and the efficient arc-search algorithm.

Since the beginning of the development of interior-point methods, there exists a puzzling gap between the results in theory and the observations in numerical experience, i.e., algorithms with good polynomial bound are not computationally efficient and algorithms demonstrated efficiency in computation do not have a good or any polynomial bound. Todd raised a question in 2002: "Can we find a theoretically and practically efficient way to reoptimize?" This paper is an effort to close the gap. We propose two arc-search infeasible interior-point algorithms with infeasible central path neighborhood wider than all existing infeasible interior-point algorithms that are proved to be convergent. We show that the first algorithm is polynomial and its simplified version, if it terminates in finite iterations, has a complexity bound equal to the best known complexity bound for all (feasible or infeasible) interior-point algorithms. We demonstrate the computational efficiency of the proposed algorithms by testing all Netlib linear programming problems in standard form and comparing the numerical results to those obtained by Mehrotra's predictor-corrector algorithm and a recently developed more efficient arc-search algorithm (the convergence of these two algorithms is unknown). We conclude that the newly proposed algorithms are not only polynomial but also computationally competitive comparing to both Mehrotra's predictor-corrector algorithm and the efficient arc-search algorithm.

In future conflicts, the projected expenditure rates of artillery ammunition greatly exceed the ability of the Marine direct support artillery battery's ammunition transportation assets. It is therefore vital that the artillery battery commander be able to select the most effective mix of ammunition to carry on his organic transportation in a given tactical situation. Linear programing is a tool which the batten' commander can use to help solve this important problem. This thesis provides a linear program to assist him in this solution. In addition, with slight modification, this linear program can be of use to commanders at all levels of the Marine artillerv organization.

In future conflicts, the projected expenditure rates of artillery ammunition greatly exceed the ability of the Marine direct support artillery battery's ammunition transportation assets. It is therefore vital that the artillery battery commander be able to select the most effective mix of ammunition to carry on his organic transportation in a given tactical situation. Linear programing is a tool which the batten' commander can use to help solve this important problem. This thesis provides a linear program to assist him in this solution. In addition, with slight modification, this linear program can be of use to commanders at all levels of the Marine artillerv organization.

The handling of storage assignment and subroutine interconnections for the 704 linear programming codes are discussed. The available storage of the machine is divided into two main parts -- code and data. Their layouts re fairly independent of one another and the programs include routines for automatic 'housekeeping' during loading and running of a job. Flexibility is provided for systematic modifications in the programs or for machines with various storage facilities.

The decomposition principle of Dantzig and Wolfe (Operations Research, 8:101-111 (1960)) is a method for breaking large linear programs with a block diagonal structure into a set of smaller subprograms. As alternative decomposition scheme derived from a dynamic programming approach is proposed here. This results in a series of parametric linear subprograms whose recursive solution yields the solution to the original linear program.

Multi-stage stochastic linear programs (MSLPs) are notoriously hard to solve in general. Linear decision rules (LDRs) yield an approximation of an MSLP by restricting the decisions at each stage to be an affine function of the observed uncertain parameters. Finding an optimal LDR is a static optimization problem that provides an upper bound on the optimal value of the MSLP, and, under certain assumptions, can be formulated as an explicit linear program. Similarly, as proposed by Kuhn, Wiesemann, and Georghiou (Math. Program., 130, 177-209, 2011) a lower bound for an MSLP can be obtained by restricting decisions in the dual of the MSLP to follow an LDR. We propose a new approximation approach for MSLPs, two-stage LDRs. The idea is to require only the state variables in an MSLP to follow an LDR, which is sufficient to obtain an approximation of an MSLP that is a two-stage stochastic linear program (2SLP). We similarly propose to apply LDR only to a subset of the variables in the dual of the MSLP, which yields a 2SLP approximation of the dual that provides a lower bound on the optimal value of the MSLP. Although solving the corresponding 2SLP approximations exactly is intractable in general, we investigate how approximate solution approaches that have been developed for solving 2SLP can be applied to solve these approximation problems, and derive statistical upper and lower bounds on the optimal value of the MSLP. In addition to potentially yielding better policies and bounds, this approach requires many fewer assumptions than are required to obtain an explicit reformulation when using the standard static LDR approach. As an illustrative example we apply our approach to a capacity expansion model, and find that the two-stage LDR policy has expected cost between 20% and 34% lower than the static LDR policy, and in the dual yields lower bounds that are between 0.1% and 3.3% better.

From health surveys for public, the overweight condition of people happens frequently due to three causes eating too much, exercising too little, and eating incorrectly. For the normal person, it is difficult to know the situation means that acquiring and maintaining a healthy body weight. Therefore, this system tries to help the users to achieve a balanced diet in body weight management. For the body weight management system, the appropriate recipe will be calculated by linear programming for the users. To achieve a balanced diet with low calorie, the system can generate a recommended recipe for dinner, which intends to meet users' nutritional requirement according to their intake of breakfast and lunch. For the body weight management system, the appropriate recipe will be calculated by integer linear programming for the users. To achieve a balanced diet with low calorie, the system can generate a recommended recipe for dinner, which intends to meet users' nutritional requirement according to their intake of breakfast and lunch. By Phyu Phyu | Thin Thin Swe "Food Intake for Body Weight Management System using Linear Programming Model" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-5 , August 2019, URL: https://www.ijtsrd.com/papers/ijtsrd26735.pdf Paper URL https://www.ijtsrd.com/computer-science/cognitive-science/26735/food-intake-for-body-weight-management-system-using-linear-programming-model/phyu-phyu

The problem of optimizing a linear objective function,given a number of linear constraints has been a long standing problem ever since the times of Kantorovich, Dantzig and von Neuman. These developments have been followed by a different approach pioneered by Khachiyan and Karmarkar. In this paper we present an entirely new method for solving an old optimization problem in a novel manner, a technique that reduces the dimension of the problem step by step and interestingly is recursive. A theorem which proves the correctness of the approach is given. The method can be extended to other types of optimization problems in convex space, e.g. for solving a linear optimization problem subject to nonlinear constraints in a convex region.

By leveraging advanced techniques and models, Operations Research (OR) provides critical insights and strategic interventions across multiple domains, including transportation, communication, project management, and supply chain optimization. The field's multifaceted approach continues to drive efficiency and innovation in numerous industries. This paper thoroughly evaluates numerous approaches and methods researchers employ to model and investigate problems. Our objective is to bridge gaps in existing literature by examining recent advancements in this field. Network flow problems encompass the shortest path, maximal cost flow, and minimal cost flow problems. These critical elements are essential for understanding transportation dynamics, communication, and resource allocation networks. Furthermore, we explore real-life scenarios where these network flow problems arise, shedding light on their practical significance.

In this paper, we present an improved union bound on the Linear Programming (LP) decoding performance of the binary linear codes transmitted over an additive white Gaussian noise channels. The bounding technique is based on the second-order of Bonferroni-type inequality in probability theory, and it is minimized by Prim's minimum spanning tree algorithm. The bound calculation needs the fundamental cone generators of a given parity-check matrix rather than only their weight spectrum, but involves relatively low computational complexity. It is targeted to high-density parity-check codes, where the number of their generators is extremely large and these generators are spread densely in the Euclidean space. We explore the generator density and make a comparison between different parity-check matrix representations. That density effects on the improvement of the proposed bound over the conventional LP union bound. The paper also presents a complete pseudo-weight distribution of the fundamental cone generators for the BCH[31,21,5] code.

In recent work, semantic games of independence-friendly logic were studied in strategic form in terms of (mixed strategy) Nash equilibria. The class of strategic games of independence-friendly logic is contained in the class of win-loss, zero-sum two-player games. In this note we draw on the theory of linear programming to develop tools to analyze the value of such games. We give two applications of these tools to independence-friendly logic under the so-called equilibrium semantics.

We propose new optimal estimators for the Lipschitz frontier of a set of points. They are defined as kernel estimators being sufficiently regular, covering all the points and whose associated support is of smallest surface. The estimators are written as linear combinations of kernel functions applied to the points of the sample. The coefficients of the linear combination are then computed by solving related linear programming problem. The L_1 error between the estimated and the true frontier function with a known Lipschitz constant is shown to be almost surely converging to zero, and the rate of convergence is proved to be optimal.

We propose new estimates for the frontier of a set of points. They are defined as kernel estimates covering all the points and whose associated support is of smallest surface. The estimates are written as linear combinatio- ns of kernel functions applied to the points of the sample. The coefficients of the linear combination are then computed by solving a linear programming problem. In the general case, the solution of the optimizat- ion problem is sparse, that is, only a few coefficients are non zero. The corresponding points play the role of support vectors in the statistical learning theory. The L_1 error between the estimated and the true frontiers is shown to be almost surely converging to zero, and the rate of convergence is provided. The behaviour of the estimates on finite sample situations is illustrated on some simulations.

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.

We prove an elementary yet useful inequality bounding the maximal value of certain linear programs. This leads directly to a bound on the martingale difference for arbitrarily dependent random variables, providing a generalization of some recent concentration of measure results. The linear programming inequality may be of independent interest.

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The purpose of this study was to formulate a linear programming model to simulate a foundation type support program and to apply this model to a state support program for the public elementary and secondary school districts in the State of Iowa. The model was successful in producing optimal solutions to five objective functions proposed for testing it, and thus it is concluded that the use of a linear programming model to simulate a foundation type state program is indeed feasible. Tables present the study data, and an appendix provides the algebraic matrix used for the solution to each of the five problems. (Author/DB)

From the bitsavers.org collection, a scanned-in computer-related document. ibm :: 1620 :: general program library :: 10.1.008 Linear Programming II

This paper studies the linear optimization problem subject to a system of bipolar fuzzy relation equations with the max-product composition operator. Its feasible domain is briefly characterized by its lower and upper bound, and its consistency is considered. Also, some sufficient conditions are proposed to reduce the size of the search domain of the optimal solution to the problem. Under these conditions, some equations can be deleted to compute the minimum objective value. Some sufficient conditions are then proposed which under them, one of the optimal solutions of the problem is explicitly determined and the uniqueness conditions of the optimal solution are expressed. Moreover, a modified branch-and-bound method based on a value matrix is proposed to solve the reduced problem. A new algorithm is finally designed to solve the problem based on the conditions and modified branch-and-bound method. The algorithm is compared to the methods in other papers to show its efficiency.

Khachian's polynomial bound for finding a feasible solution to a relaxed linear program is an important theoretical result. Unfortunately, a polynomial bound does not imply a good algorithm because such a bound could be too large for problems of practical interest. For example, using the formulae in the original paper, practical problems with 3000 non-negative variables and 1000 equations (which are solved under one-half hour on IBM 370-168 using the simplex method) would involve over 10 to 15th power iterations and would take 50,000,000 years to solve using Khachian's method.

Through the use of linear programming techniques, the optimal number and positioning of air and ground ambulances within a theater of operations may be determined to ensure the orderly transport of casualties from the front lines to third echelon medical treatment facilities. The Probabilistic Location Set Covering Problem has been chosen as the core module for a linear programming model to assist in these determinations. The Optimal Placement of Casualty Evacuation Assets (OPTEVAC) model prompts the user to enter the dimensions of the theater, troop deployment nodes, types of evacuation assets available, and preferred locations of medical treatment facilities. The OPTEVAC model then provides output as to the required numbers of air and ground ambulances as well as the optimal positioning of those evacuation assets and ambulance exchange points.

An algorithm is described for determining the optimal solution of parametric linear and quadratic programming problems as an explicit piecewise linear function of the parameter. Each linear function is uniquely determined by an appropriate subset of active constraints. For every critical value of the parameter a new subset has to be determined. A simple rule is given for adding and deleting constraints from this subset. (Author)

An algorithm is described for determining the optimal solution of parametric linear and quadratic programming problems as an explicit piecewise linear function of the parameter. Each linear function is uniquely determined by an appropriate subset of active constraints. For every critical value of the parameter a new subset has to be determined. A simple rule is given for adding and deleting constraints from this subset. (Author)

A linear programing model and procedures for optimal assignment of students to attendance centers are presented. An example of the use of linear programing for the assignment of students to attendance centers in a particular school district is given. (CK)

A linear programing model and procedures for optimal assignment of students to attendance centers are presented. An example of the use of linear programing for the assignment of students to attendance centers in a particular school district is given. (CK)

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.

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.

We propose `Dracula', a new framework for unsupervised feature selection from sequential data such as text. Dracula learns a dictionary of $n$-grams that efficiently compresses a given corpus and recursively compresses its own dictionary; in effect, Dracula is a `deep' extension of Compressive Feature Learning. It requires solving a binary linear program that may be relaxed to a linear program. Both problems exhibit considerable structure, their solution paths are well behaved, and we identify parameters which control the depth and diversity of the dictionary. We also discuss how to derive features from the compressed documents and show that while certain unregularized linear models are invariant to the structure of the compressed dictionary, this structure may be used to regularize learning. Experiments are presented that demonstrate the efficacy of Dracula's features.

We consider the infinite dimensional linear programming (inf-LP) approach for solving stochastic control problems. The inf-LP corresponding to problems with uncountable state and input spaces is in general computationally intractable. By focusing on linear systems with quadratic cost (LQG), we establish a connection between this approach and the well-known Riccati LMIs. In particular, we show that the semidefinite programs known for the LQG problem can be derived from the pair of primal and dual inf-LPs. Furthermore, we establish a connection between multi-objective and chance constraint criteria and the inf-LP formulation.

Typical behavior of the linear programming (LP) problem is studied as a relaxation of the minimum vertex cover, a type of integer programming (IP) problem. A lattice-gas model on the Erd\"os-R\'enyi random graphs of $\alpha$-uniform hyperedges is proposed to express both the LP and IP problems of the min-VC in the common statistical-mechanical model with a one-parameter family. Statistical-mechanical analyses reveal for $\alpha=2$ that the LP optimal solution is typically equal to that given by the IP below the critical average degree $c=e$ in the thermodynamic limit. The critical threshold for good accuracy of the relaxation extends the mathematical result $c=1$, and coincides with the replica symmetry-breaking threshold of the IP. The LP relaxation for the minimum hitting sets with $\alpha\geq 3$, minimum vertex covers on $\alpha$-uniform random graphs, is also studied. Analytic and numerical results strongly suggest that the LP relaxation fails to estimate optimal values above the critical average degree $c=e/(\alpha-1)$ where the replica symmetry is broken.

We extend a relaxation technique due to Bertsimas and Nino-Mora for the restless bandit problem to the case where arbitrary costs penalize switching between the bandits. We also construct a one-step lookahead policy using the solution of the relaxation. Computational experiments and a bound for approximate dynamic programming provide some empirical support for the heuristic.

In this paper, we study linear programming based approaches to the maximum matching problem in the semi-streaming model. The semi-streaming model has gained attention as a model for processing massive graphs as the importance of such graphs has increased. This is a model where edges are streamed-in in an adversarial order and we are allowed a space proportional to the number of vertices in a graph. In recent years, there has been several new results in this semi-streaming model. However broad techniques such as linear programming have not been adapted to this model. We present several techniques to adapt and optimize linear programming based approaches in the semi-streaming model with an application to the maximum matching problem. As a consequence, we improve (almost) all previous results on this problem, and also prove new results on interesting variants.

We introduce a linear programming method to obtain bounds on the cardinality of codes in Grassmannian spaces for the chordal distance. We obtain explicit bounds, and an asymptotic bound that improves on the Hamming bound. Our approach generalizes the approach originally developed by P. Delsarte and Kabatianski-Levenshtein for compact two-point homogeneous spaces.

A multi-objective linear program (MOLP) using goal programming is developed as a decision support aid in determining optimal levels of those areas of Navy enlisted strength planning which are subject to centralized management control. Over a multi-year period these decisions include monthly inventories in each paygrade, monthly total inventories, monthly advancements in the top six paygrades, and monthly recruiting goals. The model incorporates the various budgetary, Congressional, and internal Navy force structure constraints inherent in the strength planning process while minimizing deviations from desired inventory goals, ensuring inventory stability, and determining optimal recruiting goals.

A multi-objective linear program (MOLP) using goal programming is developed as a decision support aid in determining optimal levels of those areas of Navy enlisted strength planning which are subject to centralized management control. Over a multi-year period these decisions include monthly inventories in each paygrade, monthly total inventories, monthly advancements in the top six paygrades, and monthly recruiting goals. The model incorporates the various budgetary, Congressional, and internal Navy force structure constraints inherent in the strength planning process while minimizing deviations from desired inventory goals, ensuring inventory stability, and determining optimal recruiting goals.

We present methods to synthesize cooperative strategies for multi-vehicle control problems using mixed integer linear programming. Complex multi-vehicle control problems are expressed as mixed logical dynamical systems. Optimal strategies for these systems are then solved for using mixed integer linear programming. We motivate the methods on problems derived from an adversarial game between two teams of robots called RoboFlag. We assume the strategy for one team is fixed and governed by state machines. The strategy for the other team is generated using our methods. Finally, we perform an average case computational complexity study on our approach.

linear programming ( production planning )

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A fuzzy set assigns to each element of a universe a membership degree within the interval [0, 1], thereby modeling imprecision and vagueness. A hyperfuzzy set extends this concept by associating each element with a nonempty subset of [0, 1], capturing both uncertainty and variability through a range of possible membership degrees. Building on this, a superhyperfuzzy set generalizes the framework further by assigning to each nonempty element in the  n th power-set hierarchy a nonempty subset of [0, 1], thus enabling the representation of recursively structured and hierarchical uncertainty. Linear programming is an optimization technique that aims to maximize or minimize a linear objective function subject to a set of linear equality and inequality constraints. Fuzzy linear programming generalizes this framework by incorporating fuzzy numbers into the objective coefficients and constraints, allowing for uncertainty in both parameters and feasible regions. In this paper, we propose mathematical models for Hyperfuzzy Linear Programming and Superhyperfuzzy Linear Programming, and briefly examine their theoretical properties. We hope that these models will provide a foundation for further validation, development, and refinement in future research.

In Grade 11 you were introduced to linear programming and solved problems by looking at points on the edges of the feasible region. In Grade 12 you will look at how to solve linear programming problems in a more general manner.