Mordecai Avriel

Container ship stowage problem: complexity and connection to the coloring of circle graphs

This paper deals with a stowage plan for containers in a container ship. Since the approach to the containers on board the ship is only from above, it is often the case that containers have to be shifted. Shifting is defined as the temporary removal from and placement back of containers onto a stack of containers. Our aim is to find a stowage plan that minimizes the shifting cost. We show that the shift problem is NP-complete. We also show a relation between the stowage problem and the coloring of circle graphs problem. Using this relation we slightly improve Ungers upper bound on the coloring number of circle graphs.

The Value of Information and Stochastic Programming

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.

r-convex functions

A family of real functions, calledr-convex functions, which represents a generalization of the notion of convexity is introduced. This family properly includes the family of convex functions and is included in the family of quasiconvex functions. Some properties ofr-convex functions are derived and relations with other generalizations of convex functions are discussed.

Exact and approximate solutions of the container ship stowage problem

Abstract This paper deals with a stowage plan for containers in a container ship. Containers on board a container ship are placed in stacks, located in many bays. Since the access to the containers is only from the top of the stack, a common situation is that contianers 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 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 deal with stowage planning in order to minimize the number of shiftings, without considering stability constraints. First, a 0–1 binary linear programming formulating is presented that can find the optimal solution for stowage in a single rectangular bay of a vessel calling a given number of ports, assuming that the number of constainers to ship is known in advance. This model was successfully implemented using the GAMS software system. It was found, however, that finding the optimal solution using this model is quite limited, because of the large number of binary variables needed for the formulation. For this reason, several alternative heuristic algorithms were developed. The one presented here is based on a “reduced” transportation matrix. Containers with the same source and destination ports are stowed in full stacks as much as possible, and only the remaining containers are allocated by the binary linear programming model. This approach often allows the stowage planning of a much larger number of containers than using the exact formulation.

Generalized arcwise-connected functions and characterizations of local-global minimum properties

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.

Second order characterizations of pseudoconvex functions

Second order characterizations for (strictly) pseudoconvex functions are derived in terms of extended Hessians and bordered determinants. Additional results are presented for quadratic functions.

Calcium carbonate scale deposition on heat transfer surfaces

Abstract A Study is made of the mechanism of CaCO3 scale deposition on heat exchanger surfaces, occurring in the turbulent flow of water containing dissolved scale constituents under non-boiling conditions. A general rate model is presented which takes into account physical and chemical steps involved in scale deposition. Controlling mechanisms are elucidated by examining the dependence of scale growth rate on the pertinent basic parameters: flow velocity, scale surface temperature and water composition. Scale growth-rate varies with (Reynolds number)0.68and is only slightly dependent on surface temperature. This result signifies that CaCO3 scale deposition is diffusion controlled within the range of surface temperatures (67 to 85°C) and Reynolds numbers (13,000 to 42,000) covered by this study. The experimental data conform to the proposed model and suggest that in certain cases CO2 formed during the reaction remains in solution, while in others, it escapes as a gas on the reaction surface.

On Functions Whose Stationary Points Are Global Minima

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 ofRn 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.

Properties of the sequential gradient-restoration algorithm (SGRA), part 2: Convergence analysis

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).

Properties of the sequential gradient-restoration algorithm (SGRA), part 1: Introduction and comparison with related methods

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.