Chapter 7 Multiple Constraints and Conflicting Objectives. Materials Selection in Mechanical Design, 4th Edition, 2010 Michael Ashby

Transcription

1 Chapter 7 Multiple Constraints and Conflicting Objectives

2 Multiple Constraints and Conflicting Objectives The selection of a material or process must satisfy several often conflicting constraints; a second class of problem involves more than one objective, and here the conflict is more severe

3 Strategies for tackling selection with multiple constraints and conflicting objectives Figure 7.1

4 Selection With Multiple Constraints Nearly all material selection problems are overconstrained, meaning there are more constraints than free variables Selection involves identifying the constraints and the objective and applying the following steps:

5 Figure 7.2 The screening stage imposes constraints on properties, on requirements such as corrosion resistance, or on the ability to be processed in a certain way The candidate materials that survive the screening stage are ranked using property charts

6 The simple process of screening and ranking becomes more complex for the special case of a single objective that can be limited by more than one constraint Example: The requirements of a tie-rod of minimum mass might specify both stiffness and strength, leading to two independent equations for the mass If stiffness is the dominant constraint, the mass of the rod is m 1 ; if it is strength, the mass is m 2 If the tie is to meet the requirements on both, its mass has to be the greater of m 1 and m 2

7 We seek the smallest value of a metric that is the larger of two or more alternatives Figure 7.3 One objective (here, minimizing mass) with two constraints leads to two performance equations, each with its own value of M

8 Analytical Method

9 Graphical Method Figure 7.4 Coupled selection can be done using performance metrics as in (a), or using material indices M and a coupling constant Cc as in (b)

10 Conflicting Objectives Real-life materials selection almost always requires that a compromise be reached between conflicting objectives

11 Trade-Off Strategies Strategy 1 A shortlist of materials can be identified by plotting the performance metrics against one another; solutions on or near the trade-off surface offer the best compromise, the rest can be rejected Figure 7.5 Strategy 2 One objective can be reformulated as a constraint; in this example, an upper limit is set on cost; however, this is not a true optimization Figure 7.6

12 The trade-off surface identifies the subset of solutions that offer the best compromises between objectives. To obtain a single solution, we must aggregate the various objectives into a single objective function, formulated such that its minimum defines the most preferable solution. To do this we define a locally linear penalty function Z Figure 7.7

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14 Relative Penalty Functions When we seek a better material for an existing application, it is more helpful to compare the new material choice with the existing one; To do this we define a relative penalty function Figure 7.8

15 An exchange constant is a measure of the penalty of unit increase in a performance metric, or it is the value or utility of a unit decrease in the metric

16 Figure 7.9 A cost-mass trade-off plot for bicycles. The tangent to the tradeoff surface at any point gives an estimate of the exchange constant. It depends on the application. To a consumer seeking a cheap bike for shopping, the value of weight savings is low ($20/kg). To an enthusiast who wants performance, it can be high ($2000/kg).

17 Figure 7.10 It is often the case that a single material (or subset of materials) is optimal over a wide range of values of the exchange constant. Then approximate values for exchange constants are sufficient to reach precise conclusions about the choice of materials.