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Optimization in design

Most design problems may be formulated as follows: determine a set of design variables (e.g. number of ships, individual ship size and speed in fleet optimization; main dimensions and interior subdivision of ship; scantlings of a construction; characteristic values of pipes and pumps in a pipe net) subject to certain relations between and restrictions of these variables (e.g. by physical, technical, legal, economical laws). If more than one combination of design variables satisfies all these conditions, we would like to determine that combination of design variables which optimizes some measure of merit (e.g. weight, cost, or yield).

3.1 Introduction to methodology of optimization

Optimization means finding the best solution from a limited or unlimited number of choices. Even if the number of choices is finite, it is often so large that it is impossible to evaluate each possible solution and then determine the best choice. There are, in principle, two methods of approaching optimization problems:

1.Direct search approach

Solutions are generated by varying parameters either systematically in certain steps or randomly. The best of these solutions is then taken as the estimated optimum. Systematic variation soon becomes prohibitively time consuming as the number of varied variables increases. Random searches are then employed, but these are still inefficient for problems with many design variables.

2.Steepness approach

The solutions are generated using some information on the local steepness (in various directions) of the function to be optimized. When the steepness in all directions is (nearly) zero, the estimate for the optimum is found. This approach is more efficient in many cases. However, if several local optima exist, the method will `get stuck' at the nearest local optimum instead of finding the global optimum, i.e. the best of all possible solutions. Discontinuities (steps) are problematic; even functions that vary steeply in one direction, but very little in another direction make this approach slow and often unreliable.

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86 Ship Design for Efficiency and Economy

Most optimization methods in ship design are based on steepness approaches because they are so efficient for smooth functions. As an example consider the cost function varied over length L and block coefficient CB (Fig. 3.1). A steepness approach method will find quickly the lowest point on the cost function, if the function K D f.CB; L/ has only one minimum. This is often the case.

Figure 3.1 Example of overall costs dependent on length and block coefficient

Repeating the optimization with various starting points may circumvent the problem of `getting stuck' at local optima. One option is to combine both approaches with a quick direct search using a few points to determine the starting point of the steepness approach. Also repeatedly alternating both methodsÐwith the direct approach using a smaller grid scale and range of variation each timeÐhas been proposed.

A pragmatic approach to treating discontinuities (steps) assumes first a continuous function, then repeats the optimization with lower and upper next values as fixed constraints and taking the better of the two optima thus obtained. Although, in theory, cases can be constructed where such a procedure will not give the overall optimum, in practice this procedure apparently works well.

The target of optimization is the objective function or criterion of the optimization. It is subject to boundary conditions or constraints. Constraints may be formulated as equations or inequalities. All technical and economical relationships to be considered in the optimization model must be known and expressed as functions. Some relationships will be exact, e.g. r D CB L B T; others

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will only be approximate, such as all empirical formulae, e.g. regarding resistance or weight estimates. Procedures must be sufficiently precise, yet may not consume too much time or require highly detailed inputs. Ideally all variants should be evaluated with the same procedures. If a change of procedure is necessary, for example, because the area of validity is exceeded, the results of the two procedures must be correlated or blended if the approximated quantity is continuous in reality.

A problem often encountered in optimization is having to use unknown or uncertain values, e.g. future prices. Here plausible assumptions must be made. Where these assumptions are highly uncertain, it is common to optimize for several assumptions (`sensitivity study'). If a variation in certain input values only slightly affects the result, these may be assumed rather arbitrarily.

The main difficulty in most optimization problems does not lie in the mathematics or methods involved, i.e. whether a certain algorithm is more efficient or robust than others. The main difficulty lies in formulating the objective and all the constraints. If the human is not clear about his objective, the computer cannot perform the optimization. The designer has to decide first what he really wants. This is not easy for complex problems. Often the designer will list many objectives which a design shall achieve. This is then referred to in the literature as `multi-criteria optimization', e.g. Sen (1992), Ray and Sha (1994). The expression is nonsense if taken literally. Optimization is only possible for one criterion, e.g. it is nonsense to ask for the best and cheapest solution. The best solution will not come cheaply, the cheapest solution will not be so good. There are two principle ways to handle `multi-criteria' problems, both leading to one-criterion optimization:

1.One criterion is selected and the other criteria are formulated as constraints.

2.A weighted sum of all criteria forms the optimization objective. This abstract criterion can be interpreted as an `optimum compromise'. However, the rather arbitrary choice of weight factors makes the optimization model obscure and we prefer the first option.

Throughout optimization, design requirements (constraints), e.g. cargo weight, deadweight, speed and hold capacity, must be satisfied. The starting point is called the `basis design' or `zero variant'. The optimization process generates alternatives or variants differing, for example, in main dimensions, form parameters, displacement, main propulsion power, tonnage, fuel consumption and initial costs. The constraints influence, usually, the result of the optimization. Figure 3.2 demonstrates, as an example, the effects of different optimization constraints on the sectional area curve.

Optimized main dimensions often differ from the values found in built ships. There are several reasons for these discrepancies:

1.Some built ships are suboptimal

The usual design process relies on statistics and comparisons with existing ships, rather than analytical approaches and formal optimization. Designs found this way satisfy the owner's requirements, but better solutions, both for the shipyard and the owner, may exist. Technological advances, changes in legislation and in economical factors (e.g. the price of fuel) are reflected immediately in an appropriate optimization model, but not when relying on partially outdated experience. Modern design approaches increasingly

88 Ship Design for Efficiency and Economy

Figure 3.2 Changes produced in sectional area curve by various optimization constraints: a is the basis form;

b is a fuller form with more displacement; optimization of carrying capacity with maximum main dimensions and variable displacement;

c is a finer form with the displacement of the basis form a, with variable main dimensions

Figure 3.3 Division of costs into length-dependent and length-independent

incorporate analyses in the design and compare more variants generated with the help of the computer. This should decrease the differences between optimization and built ships.

2.The optimization model is insufficient

The optimization model may have neglected factors that are important in practice, but difficult to quantify in an optimization procedure, e.g. seakeeping behaviour, manoeuvrability, vibrational characteristics, easy cargo-handling. Even for directly incorporated quantities, often important relationships are overlooked, leading to wrong optima, e.g.:

(a)A faster ship usually attracts more cargo, or can charge higher freight rates, but often income is assumed as speed independent.

(b)A larger ship will generally have lower quay-to-quay transport costs per cargo unit, but time for cargo-handling in port may increase. Often, the time in port is assumed to be size independent.

(c)In reefers the design of the refrigerated hold with regard to insulation and temperature requirements affects the optimum main dimensions. The additional investment and annual costs have to be included in the model to obtain realistic results.