Stability criteria as constraints in a fleet of ships optimisation problem

Stability criteria as constraints in a fleet of ships optimisation problem Bogus³aw Oleksiewicz, D.Sc.,Eng. ABSTRACT The paper has been written within the European EUREKA Project E!2772, initiated and completed at the Faculty of Ocean Engineering & Ship Technology, Gdañsk University of Technology in the years 2001-2003. A problem has been solved concerning mathematical optimisation of a fleet of multipurpose sea-river vessels for European short-shipping regular lines, in the area of The North and Baltic Seas, on the level of marine transportation task, by the non-linear programming methods with constraints. A method is proposed which enables existing criteria of stability to be included as constraints in the optimisation model of a fleet. In the numerical examples, three typical criteria of intact stability: by IMO, PRS, and HSMB have been selected to demonstrate a post-optimisation feasibility analysis of principal parameters of ships. Keywords : maritime transportation, computer-aided ship design, optimisation, intact stability criteria 1. INTRODUCTION Computer aided ship design methods used at present, while offering automation of the design process, require its rationalisation and formalisation. In consequence, adequate mathematical models of the design object must be created which affect the design process by introducing a structure and terminology which unavoidably bounds reasoning to the terms of the model. In this case a fleet of ships at the stage of owner’s study is assumed to be an object and the task of optimising its main parameters is an objective of the adequate mathematical model. In consequence, the global structure of the model (further called an “optimisation model”) corresponds to that proposed by operational research methods in general and non-linear programming methods (NPM) in particular [3]. Within this structure, optimisation models consist of a set of sub-models of particular properties of the object which have been recognised as significant to the predictive features of the model. Optimisation models applied to fleet/ship design are definitely synthetic in nature. This feature requires the analytical representation of particular sub-models to be relatively simple. In consequence, sub-models usually neither become isomorphic with, nor conform to the physical structure of that part of object to which they are related. Such type of models is sometimes referred to as “non-structural” [19]. NPM require for all the sub-models concerned to be formulated as constraints. Among them there are always those concerning safety of an object. In ship design, a special interest in this group is focused on the stability of ships. In naval architecture today, the stability requirements are imposed in the form of legal regulations by such institutions as IMO, classification societies, governmental organisations and other bodies. An essential part of stability regulations are stability criteria. The paper deals with the problem of incorporating stability criteria as constraints in the optimisation model of fleet/ship design. At the initial stages of the design the principal difficulty is that the full geometry of a hull, necessary for the stability criteria to be applied, is usually unknown. A standard solution was to take into account the initial stability only, represented by the initial metacentric height GM0 [2], [4], [7], [12], [14]. The paper proposes an alternative approach, based on an idea introduced by Wiœniewski [20] and developed by Kupras [10], [11]. In this concept the full stability of ship can be accounted for by using systematic standard series of hull forms, following the methodology developed in ship resistance and power prediction. In order to accomplish the task, an attempt has been made to define all the stability-related geometrical characteristics of a ship analytically, based on the Series 60 body forms [19]. In consequence, an arbitrary criterion of intact stability can also be defined in an analytical way and so incorporated into optimisation model as a constraint. Stability aspects in the computer-aided modelling of ship design have been addressed on the background of the optimisation problem concerning a fleet of multipurpose sea-river vessels for European short-shipping regular lines, in the area of The North and Baltic Seas, on the level of marine transportation task, by non-linear programming methods with constraints. The problem has been undertaken within the European EUREKA Project [13] based on predictions that a significant increase of cargo transportation in Europe over the next 10 years (or probably after this period) will take place between Western Europe and the Central and East European countries. In the numerical examples, three criteria of intact stability: IMO [6], HSMB [5], and PRS [17] have been selected, as typical of contemporary stability regulations, to demonstrate the method in a post-optimisation, feasibility analysis of principal parameters of ships. 2. PROBLEM STATEMENT A fleet of ships consists of a number of homogeneous ships operating as a maritime transportation system in a certain environment. The transportation task for a fleet of ships is to carry goods between ports during a prescribed period of time. An optimum fleet to perform this task, given the particular (owner’s) data, is a general problem under discussion. A solution to this problem needs adequate functional and mathematical models. POLISH MARITIME RESEARCH, Special issue, 2004 39 Stability criteria as constraints in a fleet of ships optimisation problem 2.1. Functional model of a fleet In the particular case (Tab.2.1), a (potential) shipping line connects the furthest Western and Eastern regions of Europe (a). A corresponding model of shipping (b) is called a multi-port route model linking two areas of operation A and B with the two groups of clustered sea and hinterland river ports. There are two streams of goods transportation in the model: from A to B (called OUT) and back, from B to A (called IN). Ports A-0 and B-0 are the home and destination ports. For more details about the functional model of a fleet - see [13]. Tab. 2.1. An example of a shipping line and its graphical model ) ) 2.2. Optimisation model of a fleet The mathematical model chosen for the fleet optimisation problem can be described as deterministic, static, continuous, single level and single objective model, imposed and solved by non-linear programming methods. A standard formulation of the model, within NPM, is as follows: given a vector c (or a set C) of constants, find such a vector of decision variables x that minimises a single valued objective function Q(x,c) subject to a set of inequality constraints. An adequate mathematical form of the problem is : (2.1) It is generally assumed that Q(·) and gj(·) are all non-linear functions. The conditions for existence and uniqueness of the (optimum) solution to (2.1) can be found in [3]. Tab. 2.2. Elements of an op (...truncated)