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Opti-Money is a decision-support system for allocating assets that was developed at Bank Hapoalim, Israel's largest bank. Based on customer- and market-specific inputs, we solve a modified Markowitz-type nonlinear programming problem to produce optimal tailor-made investment portfolios in terms of asset classes. In its five years of operation, this mature system has provided excellent quantifiable results for the bank and its customers. In 2002, we held some 133,000 consultation sessions with 63,000 customers in which we used Opti-Money. Apart from the considerable prestige the system gained through its track record and uniqueness in the Israeli banking sector, it obtained net income 88 percent higher in customer accounts that used Opti-Money than in those that did not. The annual earnings over and above benchmarks to customers who follow the investment advice provided by the system total US$244 million. In 2002, the annual income for the bank directly attributed to Opti-Money exceeded US$31 million.

Quantitative policy analysis problems with hierarchical decision-making can be modeled as bilevel mathematical programming problems. In general, the solution of these models is very difficult; however, special cases exist in which an optimal solution can be obtained by ordinary mathematical programming techniques. In this paper, a two-stage approach for the formulation, construction, solution, and usage of bilevel policy problem is presented. An outline of an example for analyzing Israel's public expenditure policy is also given.

This paper deals with a stowage plan for containers in a container ship. Containers on board a container ship are placed in vertical stacks, located in many bays. Since the access to the containers is only from the top of the stack, a common situation is that containers designated for port J must be unloaded and reloaded at port I (before J) in order to access containers below them, designated for port I. This operation is called "shifting". A container ship calling at many ports may encounter a large number of shifting operations, some of which can be avoided by efficient stowage planning. In general, the stowage plan must also take into account stability and strength requirements, as well as several other constraints on the placement of containers. In this paper, we only deal with stowage planning in order to minimize the number of shiftings, without considering stability and several other constraints. First, we briefly present a 0-1 binary linear programming formulation that can find the optimal solution for stowage planning. However, finding the optimal solution using this model is quite limited because of the large number of binary variables and constraints needed for the formulation. Moreover, in [3] the stowage planning problem is shown to be NP-complete. For these reasons, the Suspensory Heuristic Procedure was developed.

In this paper, we deal with the proportional knapsack problem that is a variation on the ordinary knapsack problem. In the proportional knapsack problem, we look at filling an urn with objects having two characteristics: color and weight. The colors of the objects in the urn should be proportional to the distribution of the colors in the object universe, and the total weight of the objects in the urn should be as close as possible to the capacity of the urn. The formulation of the problem was motivated by a real-life application from the area of finance, called a dollar roll. We show that the proportional knapsack problem is NP-hard, and then, using sampling, develop a heuristic procedure for solving the problem.

Quantitative policy analysis may be defined as a process in which policy alternatives are generated and evaluated with respect to their achievement of policy goals. This paper develops a methodology for the modeling and solution of certain hierarchical decision-making problems. Weconsider the case of an upper-level policy-maker and a lower-level system that responds to the policy set by the upper level. The system's reaction to the policy alternative chosen (the behavioral problem) is formulated as a mathematical programming problem. The complete model is formulated as a special class of bi-level program for which a solution can be derived using ordinary mathematical programming techniques. In this way, a major difficulty of previous attempts at developing useful computational procedures for quantitative policy analysis, based on bi-level programming, has been removed. Particular attention is given to development of a planning model with respect to the uncertain nature of policy analysis problems and to the implementation of the model within the policy analysis process. Special cases and examples arc considered.

Linear programming integer programming graph theory and networks dynamic programming nonlinear programming multiobjective programming stochastic programming heuristic methods.

In this paper, a characterization of functions whose stationary points are global minima is studied. By considering the level sets of a real function as a point-to-set mapping, and by examining its semicontinuity properties, we obtain the result that a real function, defined on a subset ofR n and satisfying some mild regularity conditions, belongs to the above family iff the point-to-set mapping of its level sets is strictly lower semicontinuous. Mathematical programming applications are also mentioned.

Twenty-six consultant-assisted development projects were studied within the pharmaceutical industry in Israel. They were selected at random; some were successful and some were not. The work of the academic scientific consultants has been closely followed. It was found that their task is more to respond to the problems of management than to introduce new ideas. The academic consultant's main contribution was the transfer of existing published knowledge in a format applicable to practice. In addition, however, the consultant had to supply know-how on and supervision of work carried out within the academic institutions, utilizing the resources of the latter which are much more sophisticated than those of the industry of a developing country. The economic value of consultancy in coupling university and industry is also estimated. It was shown that in a developing country - where the size of the pharmaceutical industry is relatively small - the task of the consultant is more than that of an expert specialist: he is a valuable instrument of communication, information and implementation as well.

An augmented Lagrangian algorithm is used to find local solutions of geometric programming problems with equality constraints (GPE). The algorithm is Newton's method for unconstrained minimization. The complexity of the algorithm is defined by the number of multiplications and divisions. By analyzing the algorithm we obtain results about the influence of each parameter in the GPE problem on the complexity of an iteration. An attempt is made to estimate the number of iterations needed for convergence. In practice, certain hypotheses are tested, such as the influence of the penalty coefficient update formula, the distance of the starting point from the optimum, and the stopping criterion. For these tests, a random problem generator was constructed, many problems were run, and the results were analyzed by statistical methods.

The collection, collation and transformation of data into the form required by a large energy-economic model constitutes a significant part of the overall modelling activity. The correct basis for setting up a model data base rests on the separation of the model specification into the four areas of external data transformation, internal data representation, internal data transformation and the mathematical relationships. From this specification, the model may be generated. The internal data base is comprised of a series of tables, covering intersectoral transactions, capital stock transactions, energy sector operating characteristics, consumption patterns and other data. These tables may be compiled direct from source material, or via some research requiring prior data processing. In both cases the process should be fully documented, so that data accuracy is ensured and the data may be updated reliably when new material becomes available. It is recommended that a data coordinating function be established within the modelling group, in order to plan and control data-related activities.

We present a class of algebraic programs, illustrated by examples in optimal location-allocation problems, discrete approximations, and multiplier methods. This class can be characterized by objective and constraint functions consisting of absolute values of rational functions of posynomials raised to positive powers. Next, we derive an algorithm consisting of monomial condensations and cutting planes, extending the Avriel-Williams complementary geometric programming method.

The sequential gradient-restoration algorithm (SGRA) was developed in the late 1960s for the solution of equality-constrained nonlinear programs and has been successfully implemented by Miele and coworkers (Refs. 2 and 3) on many large-scale problems. The algorithm consists of two major sequentially applied phases. The first is a gradient-type minimization in a subspace tangent to the constraint surface, and the second is a feasibility restoration procedure. In Part 2, the convergence properties of the SGRA for the general case of nonlinear constraints are analyzed. It is shown that, for analytical convergence purposes, the feasibility restoration phase plays a crucial role. A slight modification of the original restoration algorithm is proposed, and global convergence of the modified version is proven. Finally, a slightly modified version of the complete algorithm is presented and global convergence is proven. The asymptotic rate of convergence of the SGRA is also analyzed. The reader is assumed to be familiar with the problem statement and the description of the SGRA, presented in Part 1 (Ref. 1).

In this paper, new classes of generalized convex functions are introduced, extending the concepts of quasi-convexity, pseudoconvexity, and their associate subclasses. Functions belonging to these classes satisfy certain local-global minimum properties. Conversely, it is shown that, under some mild regularity conditions, functions for which the local-global minimum properties hold must belong to one of the classes of functions introduced.

The sequential gradient-restoration algorithm (SGRA) was developed in the late 1960s for the solution of equality-constrained nonlinear programs and has been successfully implemented by Miele and coworkers on many large-scale problems. The algorithm consists of two major sequentially applied phases. The first is a gradient-type minimization in a subspace tangent to the constraint surface, and the second is a feasibility restoration procedure. In Part 1, the original SGRA algorithm is described and is compared with two other related methods: the gradient projection and the generalized reduced gradient methods. Next, the special case of linear equalities is analyzed. It is shown that, in this case, only the gradient-type minimization phase is needed, and the SGRA becomes identical to the steepest-descent method. Convergence proofs for the nonlinearly constrained case are given in Part 2.

A combined multisector model of economy-wide development planning and a process-wide planning of a major sector of the economy are considered. Static and dynamic versions of the model are derived, and various controlling mechanisms, objective functions and model refinements are presented. Application of the methodology to the development planning of the energy sector in Israel is also discussed.

In this note, a simple proof of a theorem concerning functions whose local minima are global is presented and some closedness properties of this class of functions are discussed.

In this paper, necessary and sufficient conditions for a local minimum to be global are derived. The main result is that a real function, defined on a subset ofRn, has the property that every local minimum is global if, and only if, its level sets are lower-semicontinuous point-to-set mappings.

Optimization of physical systems consisting of interrelated subsystems which can be formulated as geometric programming problems is considered. Necessary and sufficient conditions are derived for decomposing the optimization of a special class of such systems into a sequence of subsystem optimizations. An example is presented.

The solution of nonconvex nonlinear programs with sums ofr-convex functions is considered. An algorithm consisting of a sequence of approximating convex programs which converges to a Kuhn-Tucker point is described.

The problem of planning under uncertainty has many aspects; in this paper we consider the aspect that has to do with evaluating the state of information. We address ourselves to the question of how much better (i.e., how much more profitable) we could expect our plans to be if somehow we could know at planning time what the outcomes of the uncertain events will turn out to be. This expected increase in profitability is the “expected value of perfect information” and represents an upper bound to the amount of money that it would be worthwhile to spend in any survey or other investigation designed to provide that information beforehand. In many cases, the amount of calculation to compute an exact value is prohibitive. However, we derive bounds (estimates) for the value. Moreover, in the case of operations planning by linear or convex programming, we show how to evaluate these bounds as part of a post-optimal analysis.

A minimax search plan is developed for locating the maximum of a unimodal function of one variable with a sequence of blocks of simultaneous function evaluations. The proposed “block search strategy” is described for any number of simultaneous function evaluations (experiments) and for any number of blocks in the sequence. For one experiment per block it reduces to the Fibonacci strategy; in the case of only one block of experiments, to the simultaneous search plan. The “block search strategy” is optimal in the sense that for a required final interval of uncertainty, known to contain the point where the function attains its maximum, and for any given number of simultaneous experiments and blocks, it has the largest possible starting interval. A discrete variable version of the proposed search strategy is also described.

In a recent article in this Journal, R. Wilson developed duality relations and certainty equivalent formulas for a linear programming problem with normally distributed random demands. The purpose of the present letter is to establish such relations and formulas for quite general continuous probability distributions. In addition, our analysis reveals more clearly the role of the means and variances as they affect the certainty equivalents.

In a previous paper [Avriel, M., D. J. Wilde. 1966. Optimal search for a maximum with sequences of simultaneous function evaluations. Management Sci. 12 722.], the authors presented the minimax block search strategy for locating the maximum of unimodal functions of one variable by a sequence of simultaneous function evaluations. In the present work the nearly optimal minimax golden block search method is developed which has the advantage that the number of function evaluations need not be specified in advance. For one experiment per block it reduces to the well known golden section method. It is shown that this method is an excellent approximation of the block search strategy and that for a certain marginal resolution the two methods become identical.