294 results
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2. Self-Scaling Variable Metric (SSVM) Algorithms. Part II: Implementation and Experiments
- Author
-
Oren, Shmuel S.
- Published
- 1974
3. Elements of Large Scale Mathematical Programming: Part II: Synthesis of Algorithms and Bibliography
- Author
-
Geoffrion, Arthur M.
- Published
- 1970
4. The Sequential Unconstrained Minimization Technique for Nonlinear Programing, a Primal-Dual Method
- Author
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Fiacco, Anthony V. and McCormick, Garth P.
- Published
- 1964
5. A COMMENT ON A PAPER OF MAXWELL.
- Author
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Sidney, Jeffrey B.
- Subjects
JOB shops management ,INTEGER programming ,MATHEMATICAL programming ,PRODUCTION management (Manufacturing) ,ALGORITHMS ,LINEAR programming ,MATHEMATICAL optimization ,MATHEMATICAL models ,MANAGEMENT science research ,OPERATIONS research - Abstract
The article presents comments on the management science paper "On Sequencing n Jobs on One Machine to Minimize the Number of Late Jobs," by William L. Maxwell. The paper focused on an integer programming formulation of a one-machine job shop problem. The author contends that Maxwell made in error in proving the validity of an optimal algorithm by applying cutting plane constraints to the problem. The author explains that the problem cannot be solved through traditional cutting plane procedures. The author provides a mathematical model in an attempt to correct the proof.
- Published
- 1972
- Full Text
- View/download PDF
6. A PRIMAL ALGORITHM TO SOLVE NETWORK FLOW PROBLEMS WITH CONVEX COSTS.
- Author
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Weintraub, Andres
- Subjects
CASH flow ,DIRECT costing ,MANAGEMENT science ,CONVEX functions ,CASH management ,MANAGEMENT ,ALGORITHMS ,STOCHASTIC convergence ,MANAGERIAL economics - Abstract
The problem of determining continuous flows of minimum cost in a network with convex cost functions is considered. The approach used is that of finding, for any given feasible flow, circuit flows of negative incremental costs. In the main theoretical result of this paper, it is proved that if at each stage, given a feasible nonoptimal flow X, the circuit flow with most negative incremental cost is added to X, linear convergence to the optimal solution will be obtained. In addition, this most negative incremental cost determines an upper bound on the difference in cost between the given feasible solution and the optimal. Based on these concepts, an algorithm, which preserves linear convergence, is presented to determine minimum cost flows in networks with convex costs in the arcs. Results of computer runs made for this algorithm are given. The special case of networks with linear costs is also considered. [ABSTRACT FROM AUTHOR]
- Published
- 1974
- Full Text
- View/download PDF
7. ON NONLINEAR FRACTIONAL PROGRAMMING.
- Author
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Dinkelbach, Werner
- Subjects
FRACTIONAL integrals ,NONLINEAR programming ,LINEAR programming ,MATHEMATICAL programming ,ALGORITHMS ,PARAMETER estimation ,QUADRATIC programming ,CONCAVE functions ,CONVEX functions ,POLYHEDRAL functions ,NUMERICAL solutions to Lagrange equations - Abstract
The main purpose of this paper is to delineate an algorithm for fractional programming with nonlinear as well as linear terms in the numerator and denominator. The algorithm presented is based on a theorem by Jagannathan [7] concerning the relationship between fractional and parametric programming. This theorem is restated and proved in a somewhat simpler way. Finally, it is shown how the given algorithm can be related to the method of Isbell and Marlow [6] for linear fractional programming and to the quadratic parametric approach by Ritter [10]. The Appendix contains a numerical example. [ABSTRACT FROM AUTHOR]
- Published
- 1967
- Full Text
- View/download PDF
8. ON SOME WORKS OF KANTOROVICH, KOOPMANS AND OTHERS.
- Author
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Charnes, A. and Cooper, W. W.
- Subjects
INDUSTRIAL management ,MANAGEMENT science ,LINEAR programming ,GAME theory ,MANAGEMENT games ,DECISION making ,DECISION theory ,ALGORITHMS ,RANDOM variables - Abstract
Commentary is presented for an article published in the July 1960 issue of "Management Science," written by L. V. Kantorovich with an introductory note by T. C. Koopmans. According to the author, in his introduction Koopmans addresses in particular a linear programming problem and a matrix game presented by Kantorovich. Also presented is an interpretation of the article's treatment of decision variables and mathematical optimization. The author notes that the article presents algorithms that have been improved from their contemporary presentations.
- Published
- 1962
- Full Text
- View/download PDF
9. ON THE MAXIMIZATION OF THE GEOMETRIC MEAN WITH LOGNORMAL RETURN DISTRIBUTION.
- Author
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Elton, Edwin J. and Gruber, Martin J.
- Subjects
LOGNORMAL distribution ,DISTRIBUTION (Probability theory) ,ALGORITHMS ,VARIANCES ,GEOMETRIC programming ,PORTFOLIO management (Investments) ,ECONOMIC equilibrium ,METHODOLOGY ,RATE of return - Abstract
In this paper we discuss the relevancy of the geometric mean as a portfolio selection criteria. A procedure for finding that portfolio with the highest geometric mean when returns on portfolios are lognormally distributed is presented. The development of this algorithm involves a proof that the portfolio with maximum geometric mean lies on the efficient frontier in arithmetic mean variance space. This finding has major implications for the relevancy of much of portfolio and general equilibrium theory. These implications are explored. [ABSTRACT FROM AUTHOR]
- Published
- 1974
- Full Text
- View/download PDF
10. APPLICATION OF QUASI-INTEGER PROGRAMMING TO THE SOLUTION OF MENU PLANNING PROBLEMS WITH VARIABLE PORTION SIZE.
- Author
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Armstrong, Ronald D. and Sinha, Prabhakant
- Subjects
MENU design (Printed ephemera) ,MATHEMATICAL programming ,ALGORITHMS ,INTEGER programming ,CAPITAL budget ,MENUS ,BRANCH & bound algorithms ,FOOD service research ,PLANNING - Abstract
This paper presents the application of a modified mixed-integer programming algorithm to plan menus in which the portion size of the menu items can vary over a specified positive range. Previous mathematical programming formulations of menu planning problems have either required the variables representing menu items to be bivalent, or have formulated the problem with food groups as decision variables and no integer requirements at all. The former gives rise to a zero-one programming problem and the latter to a "feed-mix" problem. In many instances, a more realistic formulation would require that if a menu item is offered, its portion size must be between a specified upper and lower bound. Although this paper addresses itself chiefly to menu planning, it is readily seen that problems in capital budgeting may be tractable with a similar formulation. It is shown how a branch-and-bound algorithm for mixed-integer programming of the type proposed by Beale and Tomlin can be modified to solve the quasi-integer programming problem resulting from a variable portion size formulation of menu planning problems. [ABSTRACT FROM AUTHOR]
- Published
- 1974
- Full Text
- View/download PDF
11. DETERMINATION OF OPTIMUM PACKAGING FREQUENCY OF ITEMS JOINTLY REPLENISHED.
- Author
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Goyal, S. K.
- Subjects
MANUFACTURING processes ,PACKAGING research ,OVERHEAD costs ,VARIABLE costs ,INVENTORY management systems ,MATHEMATICAL models ,ALGORITHMS ,ECONOMIC lot size - Abstract
This paper presents a systematic procedure for obtaining the optimum packaging frequencies for a number of items which are manufactured jointly but packaged individually immediately after manufacture. The method described in the paper is equally applicable to those problems where the optimum purchasing policy is to be obtained for a number of items procured from a single supplier. An example involving twenty jointly replenished items is solved to illustrate the procedure given in the paper. [ABSTRACT FROM AUTHOR]
- Published
- 1974
- Full Text
- View/download PDF
12. SOLVING THE "MARKETING MIX" PROBLEM USING GEOMETRIC PROGRAMMING.
- Author
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Balachandran, V. and Gensch, Dennis H.
- Subjects
MARKETING mix ,GEOMETRIC programming ,BUSINESS planning ,MATHEMATICAL programming ,LINEAR programming ,MARKETING research ,ALGORITHMS ,MATHEMATICAL optimization ,REGRESSION analysis - Abstract
This paper investigates the optimal allocation of the marketing budget within the marketing-mix decision variables so that sales (or profit) is maximized in a planning horizon. Since the influence of marketing mix variables upon sales are, in reality, nonlinear and interactive, a geometric programming algorithm is used that solves this problem. A procedure to estimate a functional of sales on the marketing mix and environmental variables utilizing the experienced judgments of the firm's executives and the raw data is provided. The derived functional is later optimized by the Geometric Programming algorithm under a constraint set consisting of budget and strategy restrictions imposed by a firm's marketing environment, and conditions under which the optimal solution is either local or global are identified. An empirical application for a large midwestern brewery is provided which utilizes and illustrates both the estimation an optimization procedures. [ABSTRACT FROM AUTHOR]
- Published
- 1974
- Full Text
- View/download PDF
13. IMPROVED COMBINATORIAL PROGRAMMING ALGORITHMS FOR A CLASS OF ALL-ZERO-ONE INTEGER PROGRAMMING PROBLEMS.
- Author
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Pierce, John F. and Lasky, Jeffery S.
- Subjects
MATHEMATICAL programming ,COMBINATORIAL optimization ,ALGORITHMS ,MODIFICATIONS ,DYNAMIC programming ,NONLINEAR programming ,INTEGER programming ,MANAGEMENT science ,MATHEMATICAL optimization ,PROBLEM solving - Abstract
In an earlier paper [20] combinatorial programming procedures were presented for solving a class of integer programming problems in which all elements are zero or one. By representing the problem elements in a binary computer as bits in a word and employing logical "and" and "or" operations in the problem-solving process, a number of problems involving several hundred integer variables were solved in a matter of seconds. In the present paper a number of improvements in these earlier algorithms are presented, including a new search strategy, methods for reducing the original problem, and mechanisms for feasibility filtering in multi-word problems. With these improvements problem-solving efficiency has been increased in many instances by an order of magnitude. In addition, the present paper contains computational experience obtained in solving problems for the k-best solutions. [ABSTRACT FROM AUTHOR]
- Published
- 1973
- Full Text
- View/download PDF
14. A DYNAMIC MODEL FOR BOND PORTFOLIO MANAGEMENT.
- Author
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Bradley, Stephen P. and Crane, Dwight B.
- Subjects
BOND funds ,DECISION theory ,DECISION making ,INVESTMENTS ,ECONOMIC indicators ,PORTFOLIO management (Investments) ,ALGORITHMS ,MATHEMATICAL decomposition ,MATHEMATICAL programming ,INVESTMENT analysis ,FINANCIAL management ,MATHEMATICAL models - Abstract
The bond portfolio problem is viewed as a multistage decision problem in which buy, sell, and hold decisions are made at successive (discrete) points in time. Normative models of this decision problem tend to become very large, particularly when its dynamic structure and the uncertainty of future interest rates and cash flows are incorporated in the model. In this paper we present a multiple period bond portfolio model and suggest a new approach for efficiently solving problems which are large enough to make use of as much information as portfolio managers can reasonably provide. The procedure utilizes the decomposition algorithm of mathematical programming and an efficient technique developed for solving subproblems of the overall portfolio model. The key to the procedure is the definition of subproblems which are easily solved via a simple recursive relationship. [ABSTRACT FROM AUTHOR]
- Published
- 1972
- Full Text
- View/download PDF
15. FINDING THE K SHORTEST LOOPLESS PATHS IN A NETWORK.
- Author
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Yen, Jin Y.
- Subjects
ALGORITHMS ,NETWORK analysis (Planning) ,OPERATIONS research ,BRANCH & bound algorithms ,MATHEMATICAL optimization ,MATHEMATICAL models ,LINEAR statistical models ,SYSTEMS engineering ,INDUSTRIAL management - Abstract
This paper presents an algorithm for finding the K loopless paths that have the shortest lengths from one node to another node in a network. The significance of the new algorithm is that its computational upper bound increases only linearly with the value of K. Consequently, in general, the new algorithm is extremely efficient as compared with the algorithms proposed by Bock, Kantner, and Haynes [2], Pollack [7], [8], Clarke, Krikorian, and Rausan [3], Sakarovitch [9] and others. This paper first reviews the algorithms presently available for finding the K shortest loopless paths in terms of the computational effort and memory addresses they require. This is followed by the presentation of the new algorithm and its justification. Finally, the efficiency of the new algorithm is examined and compared with that of other algorithms. [ABSTRACT FROM AUTHOR]
- Published
- 1971
- Full Text
- View/download PDF
16. AN ALGORITHM FOR THE QUADRATIC ASSIGNMENT PROBLEM.
- Author
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Graves, G. W. and Whinston, A. B.
- Subjects
RESOURCE allocation ,QUADRATIC assignment problem ,ASSIGNMENT problems (Programming) ,ECONOMIC models ,COMBINATORIAL optimization ,COMBINATORICS ,ECONOMIC statistics ,MATHEMATICAL statistics ,MATHEMATICAL models ,ALGORITHMS ,MATHEMATICAL analysis ,PROBLEM solving - Abstract
The paper presents a new approach to solving a class of combinatorial economic allocation problems. One member of this class is known as the quadratic assignment problem. Besides presenting an algorithm to solve this problem, we will discuss in general terms the techniques for treating combinatorial problems. A novel feature of the paper is the development of the statistical properties of the criterion function. These statistical properties are used in conjunction with a general enumerative procedure to form the main parts of the algorithm. Using the idea of confidence level enumeration, an extension of the algorithm is proposed which should allow for the effective treatment of large scale combinatorial problems. Finally we present some computational results in order to illustrate the effectiveness of the general approach. [ABSTRACT FROM AUTHOR]
- Published
- 1970
- Full Text
- View/download PDF
17. SOLVING PRODUCTION SMOOTHING PROBLEMS.
- Author
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Galbraith, Jay R.
- Subjects
INVENTORY control ,MATHEMATICAL optimization ,ALGORITHMS ,RESOURCE management ,FOUNDATIONS of arithmetic ,COST analysis ,PRODUCT management ,COST accounting ,ORGANIZATIONAL research ,RESOURCE allocation ,INDUSTRIAL costs ,ORGANIZATIONAL behavior - Abstract
This paper analyzes the problem of balancing resource capacity utilization against the costs of user delay or inventory investment. Instead of formulating algorithms to solve more classical versions of the problem, this paper describes ways that organizations can drop the cost curve prior to applying an optimizing algorithm. This is done by classifying the techniques and giving examples from organizations which have faced serious smoothing problems. Other organizations can improve their coat performance by adopting some of these techniques. [ABSTRACT FROM AUTHOR]
- Published
- 1969
- Full Text
- View/download PDF
18. OPTIMAL PRODUCTION SCHEDULING AND EMPLOYMENT SMOOTHING WITH DETERMINISTIC DEMANDS.
- Author
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Lippman, Steven A., Rolfe, Alan J., Wagner, Harvey M., and Yuan, John S.C.
- Subjects
PRODUCTION scheduling ,MATHEMATICAL variables ,LABOR supply ,LABOR costs ,INVENTORY control ,PRODUCTION control ,STATISTICAL smoothing ,INVENTORY accounting ,ALGORITHMS ,MATHEMATICAL models ,EDUCATION ,MANAGEMENT - Abstract
In this paper we study a model that minimizes the sum of production, employment smoothing, and inventory costs subject to a schedule of known demand requirements over a finite time horizon. The three instrumental variables are work force producing at regular-time, work force producing on overtime, and the total work force. Overtime is limited to be not more than a fixed multiple of regular time. The idle portion of the work force and the levels of inventory are resultant variables. We postulate the following shape characteristics for the cost functions production costs are convexlike, smoothing costs are V-shaped, and holding costs are increasing, both the production and holding cost functions need not be stationary. In this paper, we provide upper and lower bounds on the cumulative regular-time plus overtime work force for any sequence of demand requirements. We also give the form of an optimal policy when demands are monotone (either increasing or decreasing). Finally, we derive the asymptotic behavior of optimal policies when demands are monotone and the planning horizon becomes arbitrarily long. All of these results, which convey information about the numerical values of optimal policies, given specific demands and an initial level of inventory, depend only on the shape characteristics of the cost functions. Algorithmic techniques are discussed elsewhere. [ABSTRACT FROM AUTHOR]
- Published
- 1967
- Full Text
- View/download PDF
19. DECOMPOSITION OF PROJECT NETWORKS.
- Author
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Parikh, Shailendra C. and Jewell, William S.
- Subjects
PRODUCTION scheduling ,OPERATIONS research ,CRITICAL path analysis ,MANAGEMENT software ,PROJECT management ,ALGORITHMS ,COMPUTER storage devices ,MATHEMATICAL models ,COMPUTER networks ,SIMULATION methods & models ,MATHEMATICAL models in business ,MANAGEMENT science - Abstract
This paper considers "critical path" networks which are used for the planning and scheduling of projects that consist of well defined sequences of individual activities. When the number of activities is large, it becomes difficult to prepare a network for the project as one unit, and it may also be difficult to store the network in the high speed memory of a digital computer. Also, in the cases where there are two or more independent projects, which are weakly inter-related by common activities, the problem of efficient scheduling of all the projects becomes quite difficult. This paper presents a method to "tear" or "decompose" a project network into several subnetworks, schedule the subnetworks and then to put the subnetworks back together. A computational algorithm is first given for time-only networks; then two computational formulations are given for cost-time network of project subnetworks. The latter essentially generalize the algorithm due to Fulkerson, in order to handle piecewise linear, convex, cost-time curves for some or all of the activities. [ABSTRACT FROM AUTHOR]
- Published
- 1965
- Full Text
- View/download PDF
20. THE GENERALIZED STEPPING STONE METHOD FOR THE MACHINE LOADING MODEL.
- Author
-
Eisemann, Kurt
- Subjects
LINEAR programming ,ALGORITHMS ,MATHEMATICAL programming ,SIMPLEXES (Mathematics) ,MATHEMATICAL models ,MATHEMATICAL models in business ,TOPOLOGY ,NUMERICAL analysis ,BUSINESS mathematics ,MANAGEMENT science ,SIMULATION methods & models - Abstract
This paper gives a detailed description of an algorithm for the solution of a specialized Linear Programming model, to be called the Machine Loading model. It is a generalization of the Transportation Problem, in that weighting factors are applied to the individual elements which form the row and column sums. For the Machine Loading model, the simplex method reduces to a specialized algorithm which generalizes the stepping stone method of the Transportation Problem [2], [3]. With the resulting generalized stepping stone method, it becomes practicable to solve many large-scale problems for which the direct application of the simplex method would be impracticable. The present paper is restricted to a discussion of the following topics: (I) the general characteristics of the model and its topological features; (ii) a detailed solution algorithm, including a consideration of degenerate cases and the use of a computer; (iii) a more restrictive capacitated model and the corresponding modifications to the solution algorithm; and (iv) the complete illustrative solution of a numerical example. The purpose is to aid interested readers in gaining familiarity with the algorithm and facility in the solution of numerical problems. No derivations or proofs will be included. [ABSTRACT FROM AUTHOR]
- Published
- 1964
- Full Text
- View/download PDF
21. ON THE GENERALIZED TRANSPORTATION PROBLEM.
- Author
-
Balas, E. and Ivanescu, P. L.
- Subjects
TRANSPORTATION ,ALGORITHMS ,MATHEMATICAL models of industrial management ,MATHEMATICAL models in business ,MANAGEMENT science ,BUSINESS models ,MATHEMATICAL variables ,BUSINESS mathematics ,EQUATIONS ,ITERATIVE methods (Mathematics) ,PROBLEM solving - Abstract
The purpose of the present paper is to extend the loop-technique of the stepping-stone algorithm to the generalized transportation problem. The main result (Theorem and Corollary of §6) is, that passing from a basic feasible solution to another one may always be carried out by constructing a simple symmetrical or a double loop (as defined in §2) and computing new values for the variables only along this path. The amount of computations needed for this turns out to be substantially reduced as compared to the usual way of solving the system of equations relating the new basis to the old one. [ABSTRACT FROM AUTHOR]
- Published
- 1964
- Full Text
- View/download PDF
22. A HEURISTIC PROGRAM FOR LOCATING WAREHOUSES.
- Author
-
Kuehn, Alfred A. and Hamburger, Michael J.
- Subjects
WAREHOUSES -- Location ,ALGORITHMS ,LINEAR programming ,HEURISTIC programming ,PHYSICAL distribution of goods ,SHIPMENT of goods ,LOCATION analysis ,METHODOLOGY ,INDUSTRIAL applications ,PROBLEM solving research ,FLOW charts ,COMPUTER software - Abstract
The linear programing algorithms available for optimizing the routing of shipments in multi-plant, multi-destination systems cannot, in the current state of knowledge, be applied directly to the more general problem of determining the number and location of regional warehouses in large-scale distribution networks. This paper outlines a heuristic computer program for locating warehouses and compares it with recently published efforts at solving the problem either by means of simulation or as a variant of linear programing. The heuristic approach outlined in this paper appears to offer significant advantages in the solution of this class of problems in that it (1) provides considerable flexibility in the specification (modeling) of the problem to be solved, (2) can be used to study large-scale problems, that is, complexes with several hundred potential warehouse sites and several thousand shipment destinations, and (3) is economical of computer time. The results obtained in applying the program to small scale problems have been equal to or better than those provided by the alternative methods considered. [ABSTRACT FROM AUTHOR]
- Published
- 1963
- Full Text
- View/download PDF
23. A NOTE ON PARAMETRIC NETWORK FLOWS.
- Author
-
Minieka, Edward
- Subjects
ALGORITHMS ,COMPUTER algorithms ,COMPUTER networks ,NETWORK PC (Computer) ,LINEAR programming ,DATA flow computing ,MODULES (Algebra) ,MATHEMATICAL programming ,COMPUTER programming ,SPANNING trees - Abstract
In their paper [1], Doulliez and Rao present algorithms that solve two flow problems for a single source, multi-terminal network. The first problem that they solve is the construction of a flow that maximizes the value of t, where the demand at each sink is a nondecreasing, linear function of t. Given such a flow, the second problem that they solve is the construction of a flow that maximizes the value of t when the capacity of an arc is reduced. This paper supplies a finiteness proof for the first algorithm and sketches a finiteness proof for the second algorithm. The proofs are based on the well-known fact that a network possesses only a finite number of different spanning trees. [ABSTRACT FROM AUTHOR]
- Published
- 1973
- Full Text
- View/download PDF
24. A PARAMETRIC DECOMPOSITION APPROACH FOR THE SOLUTION OF UNCAPACITATED LOCATION PROBLEMS.
- Author
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Swain, Ralph W.
- Subjects
INDUSTRIAL location ,DECOMPOSITION method ,MANAGEMENT education ,STUDY & teaching of operations research ,FACILITY management ,MATHEMATICAL programming ,MATHEMATICAL models ,LOCATION analysis ,ALGORITHMS ,EDUCATION - Abstract
This paper presents a new approach to determining the optimal facility locations for uncapacitated location problems in two stages. First, it is shown that a subset of all solutions to the uncapacitated public facility location problem can be obtained by considering a closely related private location problem. The exact nature of the problem is established using the Generalised Lagrange Multiplier results given by Everett. In the second section of the paper a particularly efficient parametric decomposition algorithm, which is based on the results of the first section, is presented. Computational results are given which summarize the current level of experience with the algorithm. [ABSTRACT FROM AUTHOR]
- Published
- 1974
- Full Text
- View/download PDF
25. A LINEAR PROGRAMMING ANALOGUE, A DUALITY THEOREM, AND A DYNAMIC ALGORITHM.
- Author
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White, D.J.
- Subjects
DUALITY theory (Mathematics) ,LINEAR programming ,NONLINEAR programming ,ORGANIZATIONAL behavior ,MANAGEMENT science ,SIMPLEXES (Mathematics) ,ALGORITHMS ,COMPUTER simulation ,MANAGEMENT - Abstract
This paper considers fluid analogues for the standard linear programming problem and for a separable nonlinear programming problem. In the former case the usual duality results are demonstrated using the principle of minimum potential energy. In addition by examining the dynamics of the system a new method, referred to as the R-method, is derived for solving linear programmes, although it is not demonstrated that this has any computational advantages over the standard simplex method, except that degeneracy causes no problems. In the nonlinear case the weak Lagrangian principle is derived. The purpose of the paper is to demonstrate that analogue methods, while being impracticable as a physical method of solving optimisation problems, may give some insight into computational algorithms via dual energy concepts and/or dynamic behaviour. [ABSTRACT FROM AUTHOR]
- Published
- 1974
- Full Text
- View/download PDF
26. A SEQUENTIAL ALGORITHM FOR A CLASS OF PROGRAMMING PROBLEMS WITH NONLINEAR CONSTRAINTS.
- Author
-
Jagannathan, R.
- Subjects
THEORY of constraints ,SEQUENTIAL analysis ,NONLINEAR programming ,MANAGEMENT science ,ALGORITHMS ,CONJUGATE gradient methods ,QUADRATIC programming ,COMPUTER programming ,ECONOMIC convergence - Abstract
The purpose of this paper is to furnish a computational scheme for a class of programming problems with nonlinear constraints. The algorithm is sequential in nature, producing a sequence of feasible solutions whose limit points are optimal solutions of the original problem. Further, with a view to accelerating convergence we suggest two variants of the solution procedure. [ABSTRACT FROM AUTHOR]
- Published
- 1974
- Full Text
- View/download PDF
27. AN INTEGER PROGRAMMING ALGORITHM FOR PORTFOLIO SELECTION .
- Author
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Faaland, Bruce
- Subjects
INTEGER programming ,ALGORITHMS ,QUADRATIC programming ,INDIVIDUAL investors ,STOCKHOLDERS ,SIMULATION methods & models ,MANAGEMENT science ,COMPUTATIONAL complexity ,MATHEMATICAL models in business - Abstract
A mean-variance portfolio selection model suitable for the small investor is formulated as a sequence of quadratic integer programming problems. The special structure of these quadratic problems is exploited in a partial enumeration algorithm which uses cutting planes to accelerate convergence. Computational experience is reported on problems ranging in size from fifteen to fifty variables. [ABSTRACT FROM AUTHOR]
- Published
- 1974
- Full Text
- View/download PDF
28. SELF-SCALING VARIABLE METRIC (SSVM) ALGORITHMS. Part 2.
- Author
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Oren, Shmuel S.
- Subjects
SCALING laws (Statistical physics) ,ALGORITHMS ,FOUNDATIONS of arithmetic ,METRIC system ,MANAGEMENT science ,MATHEMATICAL variables ,APPROXIMATION theory ,QUALITATIVE research ,MATHEMATICAL models ,EXPERIMENTS - Abstract
This part of the paper introduces some possible implementations of Self-Scaling Variable Metric algorithms based on the theory presented in Part I. These implementations are analyzed theoretically and discussed qualitatively. A special class of SSVM algorithms is introduced, which has the additional property of being invariant under scaling of the objective function or of the variables. Experimental results are provided for a particular case of this class. This case has been tested in comparison to the DFP algorithm on a variety of functions with up to 50 variables. The results indicate that the new method has substantial advantage for functions with a large number of variables. [ABSTRACT FROM AUTHOR]
- Published
- 1974
- Full Text
- View/download PDF
29. MINIMAL COST CUT EQUIVALENT NETWORKS.
- Author
-
Picard, Jean-Claude and Ratlife, H. Donald
- Subjects
NETWORK analysis (Planning) ,FINANCIAL management ,OPERATIONS research ,PROBLEM solving ,MATHEMATICAL models ,GROUP extensions (Mathematics) ,COST control ,COST ,ALGORITHMS ,EDUCATION - Abstract
This paper is concerned with the following problem in network synthesis. Suppose that we are given a network with real-valued capacities on each arc. There is a cost associated with each arc which is proportional to the magnitude of the arc capacity. We wish to determine new arc capacities such that the capacity of each cut in the new network differs from the capacity of the corresponding cut in the original network by a specified constant and such that the total cost is minimized. This problem is shown to be equivalent to solving a minimum cost flow problem on a related network. [ABSTRACT FROM AUTHOR]
- Published
- 1973
- Full Text
- View/download PDF
30. SOME EMPIRICAL TESTS OF THE CRISS-CROSS METHOD.
- Author
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Zionts, Stanley
- Subjects
LINEAR programming ,MATHEMATICAL programming ,METHODOLOGY ,DYNAMIC programming ,MATHEMATICS ,ALGORITHMS ,SIMPLEXES (Mathematics) ,MATHEMATICAL optimization ,OPERATIONS research - Abstract
Randomly-generated linear programming problems of three different types and five different sizes were solved by the criss-cross method and by the simplex method. One hundred problems of each type and size were solved, and the results are overwhelmingly favorable to the criss-cross method. An improvement to the criss-cross method used in these tests is given, and the extension of the results of the paper to variations of the criss-cross and simplex methods is discussed. [ABSTRACT FROM AUTHOR]
- Published
- 1972
- Full Text
- View/download PDF
31. EXTREME POINT MATHEMATICAL PROGRAMMING.
- Author
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Kirby, M. J. L., Love, H. R., and Swarup, Kanti
- Subjects
MATHEMATICAL optimization ,LINEAR programming ,MATHEMATICAL programming ,DYNAMIC programming ,PRODUCTION scheduling ,PRODUCTION management (Manufacturing) ,MANUFACTURING process management ,INTEGER programming ,ALGORITHMS ,MATHEMATICAL models of industrial management ,NUMERICAL analysis ,OPERATIONS research - Abstract
The paper considers a class of optimization problems. The problems are linear programming problems: maximize ex subject to Ax = b with the additional constraint that x must also be an extreme point of a second convex polyhedron Dx = d. X ≥ 0. A cutting-plane algorithm for solving such problems is presented. Two numerical examples are also included. [ABSTRACT FROM AUTHOR]
- Published
- 1972
- Full Text
- View/download PDF
32. ON THE LOADING PROBLEM--A COMMENT.
- Author
-
Lev, Benjamin
- Subjects
ALGORITHMS ,HEURISTIC programming ,MATHEMATICAL programming ,MATHEMATICAL optimization ,RESOURCE allocation ,RESOURCE management ,MATHEMATICAL models ,MATHEMATICAL analysis ,NUMERICAL analysis ,OPERATIONS research ,PROBLEM solving - Abstract
The author responds to the paper "The Loading Problem," by Eilon and Christofides. He focuses on suggesting a solution to the problem. The recommended solution involves a heuristic algorithm that allegedly requires fewer computations than the one proposed by Eilon and Christofides. It is suggested that the simpler algorithm offered by the author reaches mathematical optimization as often as the aforementioned algorithm. It is further proposed that the algorithm presented by Eilon and Christofides cannot be used to solve multi-dimensional programming problems. The author goes on to discuss a numerical analysis that can be used for the allocation of industrial resources. Related mathematical models are discussed in detail. A table comparing the results of the two algorithms is also presented.
- Published
- 1972
- Full Text
- View/download PDF
33. EFFICIENT DISTRIBUTION OF RESOURCES THROUGH THREE LEVELS OF GOVERNMENT.
- Author
-
Cassidy, R. G., Kirby, M. J. L., and Raike, W. M.
- Subjects
RESOURCE allocation ,ASSET allocation ,OPERATIONS research ,GOVERNMENT policy ,ALGORITHMS ,DECISION making ,MATHEMATICAL models ,FEDERAL budgets ,PLANNING - Abstract
This paper presents a model for determining how a central government can most efficiently allocate resources among other levels of government. The model explicitly includes the fact that lower levels of government can make independent decisions once they have been given resources by the central government. A key feature of the model is the mathematical formulation of the central government's objective of distributing resources efficiently, while at the same time being as fair as possible to all those receiving allocations. An algorithm for solving the model is presented along with a numerical example. [ABSTRACT FROM AUTHOR]
- Published
- 1971
- Full Text
- View/download PDF
34. A SIMPLEX METHOD FOR A CLASS OF NONCONVEX SEPARABLE PROBLEMS.
- Author
-
Alloin, Guy
- Subjects
PROBLEM solving ,MATHEMATICAL statistics ,MATHEMATICAL programming ,CONVEX functions ,SEPARABLE algebras ,REAL variables ,NONLINEAR programming ,ALGORITHMS ,MATHEMATICAL models - Abstract
In this paper we present an extension of the simplex procedure to deal with separable programming problems where the objective is composed of functions which are either convex or concave and where the feasible solution set is convex. [ABSTRACT FROM AUTHOR]
- Published
- 1970
- Full Text
- View/download PDF
35. ELEMENTS OF LARGE-SCALE MATHEMATICAL PROGRAMMING PART I: CONCEPTS.
- Author
-
Geoffrion, Arthur M.
- Subjects
MATHEMATICAL programming ,MANAGEMENT science ,ALGORITHMS ,COMPUTER programming ,MATHEMATICAL optimization ,OPERATIONS research ,LINEAR programming ,LARGE scale systems ,BUSINESS mathematics ,MATHEMATICAL analysis ,MATHEMATICAL models - Abstract
A framework of concepts is developed which helps to unify a substantial portion of the literature on large-scale mathematical programming. These concepts fall into two categories. The first category consists of problem manipulations that can be used to derive what are often referred to as "master" problems; the principal manipulations discussed are Projection, Inner Linearization, and Outer Linearization. The second category consists of solution strategies that can be used to solve the master problems, often with the result that "subproblems" arise which can then be solved by specialized algorithms. The Piecewise, Restriction, and Relaxation strategies are the principal ones discussed. Numerous algorithms found in the literature are classified according to the manipulation/strategy pattern they can be viewed as using, and the usefulness of the framework is demonstrated by using it (see Part II of this paper) to rederive a representative selection of algorithms. The material presented is listed in the following order: The first section is introductory in nature, and discusses types of large-scale problems, the scope of discussion and the literature, and the notation used. The second section, entitled "Problem Manipulation: Source of 'Master' Problems" covers the subjects of projection, inner linearization and outer linearization. The third section, "Solution Strategies: Source of 'Subproblems'," discusses piecewise strategy, restriction and relaxation. The fourth section is entitled "Synthesizing Known Algorithms from Manipulations and Strategies," and is followed by a concluding section and an extensive bibliography. [ABSTRACT FROM AUTHOR]
- Published
- 1970
- Full Text
- View/download PDF
36. ADDENDUM TO STANKARD AND GUPTA'S NOTE ON LOT SIZE SCHEDULING.
- Author
-
Hodgson, Thom J.
- Subjects
ECONOMIC lot size ,INVENTORY control ,PRODUCTION scheduling ,DYNAMIC programming ,MATHEMATICAL programming ,INVENTORIES ,COST allocation ,INDUSTRIAL costs ,SYSTEMS engineering ,MATHEMATICAL models ,ALGORITHMS ,PROBLEM solving - Abstract
This paper presents a grouping procedure which yields improved solutions to Bomberger's Lot Size Scheduling Problem [1]. The group definition is a generalization of the group definition proposed by Stankard and Gupta [3]. Moreover, the generalization leads to a pseudo dynamic programming procedure which is readily programmable. Computational experience has shown the procedure to be extremely efficient. [ABSTRACT FROM AUTHOR]
- Published
- 1970
- Full Text
- View/download PDF
37. A CLASS OF INSIDE-OUT ALGORITHMS FOR GENERAL PROGRAMS.
- Author
-
Gould, F. J.
- Subjects
ALGORITHMS ,STOCHASTIC convergence ,NONLINEAR programming ,MATHEMATICAL programming ,MATHEMATICAL optimization ,GEOMETRICAL constructions ,MANAGEMENT science ,MATHEMATICAL models ,MATHEMATICAL analysis ,OPERATIONS research ,MATHEMATICAL functions ,PROBLEM solving - Abstract
In this paper the Fiacco-McCormick SUMT technique is embedded in a class of inside-out algorithms. Convergence is demonstrated for the nonlinear programming problem under fairly general conditions and the algorithms are interpreted in a geometric structure. [ABSTRACT FROM AUTHOR]
- Published
- 1970
- Full Text
- View/download PDF
38. DUALITY IN DISCRETE PROGRAMMING: II. THE QUADRATIC CASE.
- Author
-
Balas, Egon
- Subjects
QUADRATIC equations ,QUADRATIC programming ,MATHEMATICAL programming ,COMPUTER programming ,FUNCTIONAL equations ,ALGORITHMS ,NONLINEAR programming ,DYNAMIC programming ,INTEGER programming ,CONSTRAINT satisfaction ,MATHEMATICAL variables - Abstract
This paper extends the results of "Duality in Discrete Programming" [1] to the case of quadratic objective functions. The paper is, however, self-contained. A pair of symmetric dual quadratic programs is generalized by constraining some of the variables to belong to arbitrary sets of real numbers. Quadratic all-integer and mixed-integer programs are special cases of these problems. The resulting primal problem is shown, subject to a qualification, to have an optimal solution if and only if the dual has one, and in this case the values of their respective objective functions are equal. The dual of a mixed-integer quadratic program can be formulated as a minimax problem whose quadratic objective function is linear in the integer-constrained variables, and whose linear constraint set does not contain the latter. Based on this approach an algorithm is developed for solving integer and mixed-integer quadratic programs. [ABSTRACT FROM AUTHOR]
- Published
- 1969
- Full Text
- View/download PDF
39. A BACKLOGGING MODEL AND MULTI-ECHELON MODEL OF A DYNAMIC ECONOMIC LOT SIZE PRODUCTION SYSTEM--A NETWORK APPROACH.
- Author
-
Zangwill, Willard I.
- Subjects
PRODUCTION (Economic theory) ,ALGORITHMS ,DYNAMIC programming ,PRODUCTION management (Manufacturing) ,MATHEMATICAL optimization ,MATHEMATICAL models in business ,INVENTORY control ,MATHEMATICAL programming ,PRODUCT management ,INDUSTRIAL costs ,MATHEMATICAL models ,SYSTEMS engineering - Abstract
Two dynamic economic lot size production systems are analyzed in this paper, the first being a single product model with backlogging and the second a multi-echelon model. In each model the objective is to find a production schedule that minimizes the total production and inventory costs. A key conceptual difficulty is that the mathematically perplexing problem of minimizing a concave function is being considered, it is shown that both models are naturally represented via single source networks. The network formulations reveal the underlying structure of the models, and facilitate development of efficient dynamic programming algorithms for calculating the optimal production schedules. [ABSTRACT FROM AUTHOR]
- Published
- 1969
- Full Text
- View/download PDF
40. A NEW APPROACH TO DISCRETE MATHEMATICAL PROGRAMMING.
- Author
-
Graves, G. and Whinston, A.
- Subjects
INTEGER programming ,LINEAR programming ,MANAGEMENT science ,PRODUCTION management (Manufacturing) ,DISCRETE mathematics ,POPULATION statistics ,MATHEMATICAL functions ,MATHEMATICAL programming ,INFINITE groups ,COMBINATORIAL enumeration problems ,OPERATIONS research ,ALGORITHMS ,OPTIMAL designs (Statistics) ,MATHEMATICAL mappings - Abstract
The article presents an algorithm for solving a linear integer programming problem and describes the theoretical foundations for a new approach to integer programming. According to the authors, the approach may be described as an extension of the methods using enumeration, but by using a new, more powerful approach based on population statistics. The optimal solution to the integer programming problem in this paper is viewed as one of selecting the optimal function among a certain class of functions which map elements of one set into another. With this method in mind, the authors intend to view the different problems by characterizing the statistical structure of the associated finite class of maps.
- Published
- 1968
- Full Text
- View/download PDF
41. AN EXTENSION OF LAWLER AND BELL'S METHOD OF DISCRETE OPTIMIZATION WITH EXAMPLES FROM CAPITAL BUDGETING.
- Author
-
Mao, James C. T. and Wallingford, B. A.
- Subjects
CAPITAL budget ,ALGORITHMS ,MATHEMATICAL optimization ,LINEAR programming ,MANAGEMENT science ,CAPITAL investments ,INTEGER programming ,MATHEMATICAL programming ,OPERATIONS research - Abstract
The usefulness of integer programming as a tool of capital budgeting hinges on the development of an efficient solution technique. An algorithm based on partial enumeration has been developed by E. L. Lawler and M. D. Bell for solving integer linear programs with 0-1 decision variables; however their algorithm is not general enough to deal with all problems in which the objective function is quadratic. This paper extends Lawler and Bell's method so that it can be generally applied to integer quadratic programs. The new algorithm is illustrated by examples from capital budgeting. [ABSTRACT FROM AUTHOR]
- Published
- 1968
- Full Text
- View/download PDF
42. MINIMUM CONCAVE COST FLOWS IN CERTAIN NETWORKS.
- Author
-
Zangwill, Willard I.
- Subjects
PRODUCTION planning ,NETWORK analysis (Planning) ,PRODUCT management ,MATERIALS management ,INVENTORY control ,ALGORITHMS ,PRODUCTION (Economic theory) ,LINEAR programming ,MATHEMATICAL programming ,CONCAVE functions ,CONVEX functions ,INDUSTRIAL costs ,MANAGEMENT - Abstract
The literature is replete with analyses of minimum cost flows in networks for which the cost of shipping from node to node is a linear function. However, the linear cost assumption is often not realistic. Situations in which there is a set-up charge, discounting, or efficiencies of scale give rise to concave functions. Although concave functions can be minimized by an exhaustive search of all the extreme points of the convex feasible region, such an approach is impractical for all but the simplest of problems. In this paper some theorems are developed which explicitly characterize the extreme points for certain single commodity networks. By exploiting this characterization algorithms are developed that determine the minimum concave cost solution for networks with a single source and a single destination, for acyclic single source multiple destination networks, and for acyclic single destination multiple source networks. An interesting theorem then establishes that for either single source or single destination networks the multi-commodity case can be reduced to the single commodity case. [ABSTRACT FROM AUTHOR]
- Published
- 1968
- Full Text
- View/download PDF
43. ALGORITHMIC EQUIVALENCE IN LINEAR FRACTIONAL PROGRAMMING.
- Author
-
Wagner, Harvey M. and Yuan, John S.C.
- Subjects
ALGORITHMS ,LINEAR programming ,MATHEMATICAL variables ,HYPOTHESIS ,EQUIVALENCE relations (Set theory) ,MATHEMATICAL programming ,PROBLEM solving research ,MATHEMATICAL models ,EDUCATION - Abstract
This paper demonstrates the equivalence of several published algorithms for solving the so-called linear fractional programming problem. [ABSTRACT FROM AUTHOR]
- Published
- 1968
- Full Text
- View/download PDF
44. ON SOME PROPERTIES OF PROGRAMMING PROBLEMS IN PARAMETRIC FORM PERTAINING TO FRACTIONAL PROGRAMMING.
- Author
-
Jagannathan, R.
- Subjects
CONVEX programming ,MATHEMATICAL programming ,PARAMETER estimation ,ALGORITHMS ,NONLINEAR programming ,MANAGEMENT science ,FRACTIONAL calculus ,POLYHEDRAL functions ,MAXIMA & minima ,DYNAMIC programming - Abstract
This paper presents results which apply to convex programming problem in parametric form. The results secured are also related to the problem of fractional programming in a way which indicates computational possibilities for the latter class of problems. The results are extended to general non-linear programming problems with special reference to continuous criterion functions. As a particular case, the linear fractional programming problem is considered and, in conclusion, the results secured here are pointed up by reference to existing algorithms for the latter class of problems. [ABSTRACT FROM AUTHOR]
- Published
- 1966
- Full Text
- View/download PDF
45. INTEGER PROGRAMMING: METHODS, USES, COMPUTATION.
- Author
-
Balinski, M.L.
- Subjects
INTEGER programming ,PROBLEM solving ,METHODOLOGY ,MATHEMATICAL programming ,LINEAR programming ,ALGORITHMS ,DISCRETE groups ,KNAPSACK problems ,COMBINATORIAL enumeration problems ,BRANCH & bound algorithms ,TRANSPORTATION ,OPERATIONS research - Abstract
This paper attempts to present the major methods, successful or interesting uses, and computational experience relating to integer or discrete programming problems. Included are descriptions of general algorithms for solving linear programs in integers, as well as some special purpose algorithms for use on highly structured problems. This reflects a belief, on the author's part, that various clever methods of enumeration and other specialized approaches are the most efficacious means existent by which to obtain solutions to practical problems. A serious try at gathering computational experience has been made—but facts are difficult to uncover. The paper is written with intent to enable readers to read selected sections without having to read the whole. [ABSTRACT FROM AUTHOR]
- Published
- 1965
- Full Text
- View/download PDF
46. TOPOLOGY AND COMPUTATION OF THE GENERALIZED TRANSPORTATION PROBLEM.
- Author
-
Lourie, Janice R.
- Subjects
TRANSPORTATION ,TOPOLOGY ,ITERATIVE methods (Mathematics) ,ALGORITHMS ,MATHEMATICAL models of industrial management ,MATHEMATICAL models in business ,NUMERICAL analysis ,MANAGEMENT science ,COMPUTER software ,MATHEMATICAL models ,BUSINESS mathematics ,MANAGEMENT - Abstract
The topology of the Generalized Transportation Problem, at the end of each iteration of the stepping-stone method, is characterized by a variable number of loops to which are attached multiple branched side chains or trees. An efficient computer representation of this structure and methods for updating it are described in this paper. [ABSTRACT FROM AUTHOR]
- Published
- 1964
- Full Text
- View/download PDF
47. A DYNAMIC PROGRAMMING ALGORITHM FOR EMBEDDED MARKOV CHAINS WHEN THE PLANNING HORIZON IS AT INFINITY.
- Author
-
de Cani, John S.
- Subjects
DYNAMIC programming ,ALGORITHMS ,MARKOV processes ,MATHEMATICAL models of industrial management ,MATHEMATICAL models in business ,MATHEMATICAL programming ,LINEAR programming ,STOCHASTIC processes ,BUSINESS planning ,DISTRIBUTION (Probability theory) ,MATHEMATICAL optimization ,BUSINESS mathematics - Abstract
This paper presents an algorithm for the solution of dynamic programming problems requiring the determination of optimal policies for the control of a special class of stochastic processes when the time horizon of the planning period is at infinity. These processes can be mathematically described as discrete time parameter Markov chains with a finite number of states which have been "embedded" in continuous time in the sense that the time between transitions is a random variable whose probability distribution depends only on the states between which the transition takes place. Such processes are called Markov-renewal processes. The Markov processes considered by R. A. Howard in [1] are really two special cases of this somewhat wider class of stochastic processes. In these two special cases, the algorithm of this paper is identical with Howard's. In fact, with only slight modification, Howard's algorithm can be extended to this wider class of stochastic processes. [ABSTRACT FROM AUTHOR]
- Published
- 1964
- Full Text
- View/download PDF
48. A STRUCTURAL METHOD OF COMPUTING PROJECT COST POLYGONS.
- Author
-
Prager, William
- Subjects
LINEAR programming ,PRODUCTION scheduling ,NETWORK analysis (Planning) ,CONSTRUCTION industry planning ,ALGORITHMS ,MATHEMATICAL programming ,MATRICES (Mathematics) ,CRITICAL path analysis ,CIVIL engineering ,EDUCATION - Abstract
For a project that consists of numerous jobs subject to technological ordering restrictions the polygon representing project cost versus completion time is to be determined when the normal and crash completion times are known for each job and the cost of doing the job varies linearly between these times. A linear programming formulation of this problem was given by Keliey [1] and a network flow formulation by Fulkerson [2]. Since the traditional mathematical background of civil engineers includes neither linear programming nor network flow theory, these methods are not as widely used in the building industry as they deserve. This paper shows that Fulkerson's algorithm can be given a structural interpretation using concepts that are familiar to civil engineers. [ABSTRACT FROM AUTHOR]
- Published
- 1963
- Full Text
- View/download PDF
49. A NON-LINEAR EXTENSION OF THE SIMPLEX METHOD.
- Author
-
Wegner, Peter
- Subjects
NONLINEAR statistical models ,MATHEMATICAL programming ,ALGORITHMS ,NONLINEAR theories ,SIMPLEXES (Mathematics) ,SET theory ,ALGEBRA ,MATHEMATICS ,NONLINEAR functional analysis ,NONLINEAR integral equations ,NONNEGATIVE matrices ,DECISION making ,MATHEMATICAL models - Abstract
This paper describes an algorithm for the solution of mathematical programming problems having a linear objective function and non-linear constraints. The algorithm is basically an adaptation of the simplex method to the case of non-linear constraints. Certain complications are, however, introduced through non-linearity in the constraints, and it is shown how these can be overcome. [ABSTRACT FROM AUTHOR]
- Published
- 1960
- Full Text
- View/download PDF
50. A LINEAR PROGRAMMING APPROACH TO THE CHEMICAL EQUILIBRIUM PROBLEM.
- Author
-
Dantzig, George, Johnson, Selmer, and White, Wayne
- Subjects
CHEMICAL equilibrium ,LINEAR programming ,DYNAMIC programming ,PRODUCTION scheduling ,MATHEMATICAL programming ,ALGORITHMS ,LINEAR free energy relationship ,APPROXIMATION theory ,CONVEX functions ,MANAGEMENT science ,SEPARABLE algebras ,CHEMICAL engineers - Abstract
The well known chemical equilibrium problem is expressed in the form of minimizing the free energy of a mixture in order to compute the chemical composition at equilibrium. By piece-wise linear approximations to the free energy function, the problem becomes a linear program which can be solved by a standard code on a computing machine. Successive approximations give any degree of accuracy. [ABSTRACT FROM AUTHOR]
- Published
- 1958
- Full Text
- View/download PDF
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