Discrete Event Dynamic Systems: Theory and Applications, 10, 307–324, 2000.
c 2000 Kluwer Academic Publishers, Boston. Manufactured in The Netherlands. °
Ordinal Hill Climbing Algorithms for Discrete Manufacturing Process Design Optimization Problems KELLY A. SULLIVAN United Airlines, Chicago, Illinois SHELDON H. JACOBSON
[email protected] Department of Mechanical and Industrial Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois, USA 61801
Abstract. This paper introduces ordinal hill climbing algorithms for addressing discrete manufacturing process design optimization problems using computer simulation models. Ordinal hill climbing algorithms combine the search space reduction feature of ordinal optimization with the global search feature of generalized hill climbing algorithms. By iteratively applying the ordinal optimization strategy within the generalized hill climbing algorithm framework, the resulting hybrid algorithm can be applied to intractable discrete optimization problems. Computational results on an integrated blade rotor manufacturing process design problem are presented to illustrate the application of the ordinal hill climbing algorithm. The relationship between ordinal hill climbing algorithms and genetic algorithms is also discussed. This discussion provides a framework for how the ordinal hill climbing algorithm fits into currently applied algorithms, as well as to introduce a bridge between the two algorithms. Keywords: stochastic algorithms, ordinal optimization, optimization, hill climbing, genetic algorithms, manufacturing process design
1.
Introduction and Motivation
The Materials Process Design Branch of the Air Force Research Laboratory (Wright Patterson Air Force Base, Dayton, USA) is faced with the challenge of identifying optimal manufacturing process designs, where the finished unit meets certain geometric and microstructural specifications, while being produced at minimum cost. To date, the very expensive and time intensive approach of trial and error (on the shop floor) has been used to identify feasible manufacturing process designs. Each manufacturing process can affect the geometry and/or microstructure of the manufactured unit. Associated with each manufacturing process are input and output parameters. The input parameters can be classified as controllable or uncontrollable. An exhaustive enumerative search through all possible manufacturing process design sequences and controllable input parameter combinations can be undertaken. The difficulty with such an exhaustive search is the prohibitive amount of time it will take, even using computer simulation models executed on state-of-the art computers. The Materials Process Design Branch, in conjunction with researchers at Ohio University (Athens, Ohio, USA) (Fischer et al., 1997; Gunasekera et al., 1996), have developed computer simulation models of manufacturing processes (such as forging and machining). These deterministic computer simulation models move the search for optimal manufactur-
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ing process designs from the shop floor to a computer platform. This move encourages the development of optimization algorithms to identify optimal/near-optimal manufacturing process designs. The purpose of this paper is to introduce ordinal hill climbing algorithms as an optimization tool for discrete manufacturing process design optimization problems. Ordinal hill climbing algorithms incorporate concepts from ordinal optimization (namely, the idea of using ordinal rankings to search for good designs over a large set of possible designs) into a hill climbing algorithm framework, by iteratively applying features of the ordinal optimization procedure in the generalized hill climbing algorithm framework. The paper is organized as follows: Section 2 provides background information on the discrete manufacturing process design optimization problem, definitions needed to describe the problem, and specific details related to the application of ordinal hill climbing algorithms to the discrete manufacturing process design optimization problem for an integrated blade and rotor component. Section 3 briefly describes ordinal optimization and generalized hill climbing algorithms. The ordinal hill climbing algorithm framework is also formally introduced. Section 4 reports computational results with various ordinal hill climbing algorithm formulations. The computational results reported here are able to obtain similar (and in some cases, identical) quality designs (as measured by the cost function values) compared to results reported in Jacobson et al., 1997, 1998 with generalized hill climbing algorithms. This suggests that optimization using ordinal information (through the ordinal hill climbing algorithm formulation) can be effective in finding optimal designs. This illustrates that ordinal information, when properly used, can be as powerful as using precise objective function values. Section 5 contains concluding comments. An appendix describes the relationship between ordinal hill climbing algorithms and genetic algorithms. This discussion establishes a bridge between the ordinal hill climbing algorithms and genetic algorithms. 2.
Background
The following definitions and notation are needed to describe algorithms for discrete manufacturing process design optimization problems. 2.1.
Definitions
Let the manufacturing processes be denoted by P1 , P2 , . . . , Pn (e.g., machining, forging). Associated with each manufacturing process Pi , i = 1, 2, . . . , n, are a set of controllable input parameters (e.g., die dimension, die speed, post-process workpiece geometry), a set of uncontrollable input parameters (e.g., pre-process workpiece geometry), and a set of output parameters (e.g., strain rate, workpiece geometry). Note that the output parameters for a particular process may serve as the uncontrollable input parameters for a subsequent process. Moreover, the controllable input parameters can be continuous or discrete. A sequence of processes, together with a particular set of controllable input parameters defines a manufacturing process design; label such designs D1 , D2 , . . . , D N . Note that if any of the controllable input parameters are continuous, then N = +∞; otherwise, N
0 close to zero) designs. Stopping criterion that can be used for the ordinal hill climbing algorithm include looking at the top α M designs, with α > 0 close to zero, and stopping the algorithm if this top α M set of designs does not change over some specified number of iterations. Each of these modifications results in a large selection of ordinal hill climbing algorithm variations that can be used to address the IBR discrete manufacturing process design optimization problem. 4.
Computational Results
The utility of ordinal hill climbing algorithms can be empirically assessed by determining its strengths and limitations in identifying optimal discrete manufacturing process design sequences and controllable input parameters using computer simulation models. To this end, computational results for the IBR manufacturing process designs are reported. Yang et al. (1997) present results on using ordinal optimization to address this IBR manufacturing process design optimization problem. They also show how linear regression can be used to capture dependencies in the data, hence improve the effectiveness of ordinal optimization for this and other discrete optimization problems with data imprecision and model bias. Three different ordinal hill climbing algorithm formulations (defined by three different Rk representations), together with three different neighborhood functions (used to update the selected design set at each iteration) are applied to the three computer simulation IBR discrete manufacturing process design sequences. The algorithms are applied to each of the three sequences individually, with the algorithm used to find the optimal set of input parameter values for each sequence. The following sections describe how these ordinal hill climbing algorithm formulations were implemented. 4.1.
Ordinal Hill Climbing Algorithm Formulations
Three ordinal hill climbing algorithm formulations (i.e., three different Rk representations) are used to identify optimal controllable input parameter values for the three discrete manufacturing process design sequences for the IBR part discussed in Section 2.2. The first ordinal hill climbing algorithm formulation (i.e., R1,k ) consists of setting the hill climbing variable Rk = R1,k = M/2 to a constant value over all iterations. The second ordinal hill climbing algorithm formulation (i.e., R2,k ) initially sets the hill climbing variable to one (i.e., R0 = R2,0 = 1) and deterministically increments Rk (by one) every I /M iterations, where I = total number of iterations executed. For example, if the stopping criterion is I = 1, 000
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iterations and M = 10, then I /M = 100, hence R100∗ n+ j = n + 1, j = 0, 1, 2, . . . , 99, n = 0, 1, 2, . . . , 9, R1000 = 10, or Rk = [floor (k/100) + 1] for k = 0, 1, . . . , 999, and R1000 = 10, where floor (x) = the greatest integer less than or equal to x. The third ordinal hill climbing algorithm formulation (i.e., R3,k ) sets Rk = R3,k according to a truncated geometric probability distribution with probability qk = [floor (k/100) + 1]−1 for three formulations for Rk k = 0, 1, . . . , I , where Rk = Min {Geometric (qk ), M}. All PM satisfy the two conditions for the Rk selection criteria (i.e., m=1 P{Rk = m} = 1 and R0 = 1 with probability one). Moreover, the second and third formulations also satisfy the third condition described in Section 3 (i.e., Rk → M with probability one as k → +∞). Note also that the second formulation is a deterministic version of the third formulation, similar to how threshold accepting can be described as a deterministic version of simulated annealing (see Johnson, 1996). After the Rk value is generated, the ordinal hill climbing algorithm keeps the best Rk = rk designs (i.e., set E 1 ) and completes the new selected design set by generating M − Rk new design(s) (i.e., set E 2 ) using the neighborhood function according to the following rule: If Rk ≥ (M − Rk ), generate one neighbor of each of the best (M − Rk ) designs If Rk < (M − Rk ), generate floor[(M − Rk )/(Rk )] neighbors of each of the best Rk designs and one additional neighbor of each of the best (M − Rk ) −Rk∗ floor[(M − Rk )/(Rk )] designs. To illustrate this rule, if M = 20 and Rk = 12, then the algorithm will generate one neighbor of each of the best eight designs. On the other hand, if M = 10 and Rk = 3, then the algorithm will generate two neighbors of each of the best three designs and one additional neighbor of the best design. 4.2.
Neighborhood Functions
For a fixed manufacturing process design sequence, the neighborhood function of the ordinal hill climbing algorithm generates design neighbors by moving between controllable input parameters. Three neighborhood functions were implemented for each of the three (fixed) manufacturing process design sequences. The first neighborhood function (i.e., η1 ) randomly selects a single process in a given manufacturing process design sequence, and randomly changes a single controllable input parameter value associated with the selected process (using the discrete uniform over the set of possible values, as listed in the Appendix). The second neighborhood function (i.e., η2 ) is similar to the first neighborhood function, except that it randomly changes a single controllable input parameter in each of the processes of the manufacturing process design sequence (rather than in just one particular process). Similar to the second neighborhood function, the third neighborhood function (i.e., η3 ) considers every process and every controllable input parameter of the manufacturing process design sequence. However, the third neighborhood function considers changing each controllable input parameter of all the processes using independent and identically distributed Bernoulli random variables, with a specified probability p (for the results presented,
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p = .3). For example, the design sequence P1 P2 P5 P6 consists of four processes, but only the first three have controllable input parameters (the final workpiece geometry is fixed). The neighborhood function then considers each controllable input parameter associated with each of these three processes, and individually decides (with probability p) whether each should be changed. Once a controllable input parameter is (probabilistically) picked to be changed, then its value is changed based on a discrete uniform random variable over its set of possible values (as described for neighborhood function η1 ). 4.3.
Cost Function Components
The cost function (Jacobson et al., 1997,1998) evaluates the cost, in US dollars, associated with a discrete manufacturing process design for the IBR component. The initial cost is the cost of the initial billet, which depends on the dimensions of the billet and the specific metal being processed. The costs for the forging processes include: i) set up costs, ii) post-inspection costs, iii) die wear costs, iv) press run costs, v) cost of possible strain-induced-porosity damage in the workpiece. Penalties are incurred with the forging processes when i) the press capacity is exceeded, ii) the aspect ratio of the workpiece is too large, iii) the geometry of the workpiece conflicts with the die geometry. The cost to machine the workpiece is the cost of the material removed from the workpiece, where a penalty cost is incurred when the geometry of the workpiece conflicts with the desired final geometry of the workpiece after machining. After the workpiece is processed, a mandatory ultrasonic non-destructive evaluation cost and, if necessary, a cost of heat treatment is accrued. In addition, the final microstructure of the workpiece is evaluated; if the microstructure violates predetermined specifications, a penalty cost is incurred. All penalties are translated into US dollars in the cost function. For example, the cast ingot costs include the materials costs, which are based on the volume of the titanium-aluminum alloy of the initial billet, at a rate of $0.188/cm3 . The upset forging costs and the blocker forge process costs include the cost of initially reducing the initial billet size, the cost of the flat die, the cost of each flat die forging operation, the press run cost, as well as penalties for unacceptable aspect ratios, press capacities, the probability of cracking. The machine preform costs, the rough machining costs, and the finished machining costs include the cost of setting up the machine, as well as a cost of
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Table 1. Ordinal hill climbing computational results for manufacturing process design sequence P1 P2 P5 P6 . J for R j,k
η Rule
M
ρ
µ
σ
Minimum
Maximum
1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 3 3
1 1 2 2 3 3 1 1 2 2 3 3 1 1 2 2 3 3
10 20 10 20 10 20 10 20 10 20 10 20 10 20 10 20 10 20
.83 .83 1.0 1.0 .87 .87 .87 .83 1.0 1.0 .87 .87 .83 .87 .97 1.0 .83 .83
4512.563 4512.254 1955.522 1919.473 4214.342 4311.817 4412.822 4512.563 1955.254 1955.216 4086.021 4313.064 4592.390 4372.351 3666.678 2706.336 4573.888 4532.990
1005.500 1006.329 133.652 0.590 1294.794 1210.380 1108.714 1005.500 135.073 136.762 1346.786 1207.827 860.432 1103.932 959.038 579.562 880.180 954.692
1927.010 1919.280 1919.280 1919.280 1919.280 1919.280 1919.280 1927.010 1919.280 1919.280 1921.351 1927.010 2753.819 1927.010 2380.772 1927.010 2753.819 2437.648
5303.75 5303.75 2447.172 1921.213 5305.75 5305.75 5305.75 5305.75 2457.076 2462.287 5303.75 5305.75 5303.75 5303.75 5303.75 4027.159 5303.75 5303.75
performing the operation, based on the volume of material removed (set at $0.043/cm3 , $0.043/cm3 and $0.17/cm3 , respectively.) Lastly, the costs used for the penalties were several magnitude larger than all other costs (i.e., $108 US), to ensure that the best manufacturing process designs obtained by the algorithm were feasible (note that this value was obtained through trial-and-error experimentation). For additional information on the cost function, see Medina (1998).
4.4.
Results
Computational results with the three ordinal hill climbing algorithm formulations and the three neighborhood functions are reported for the three manufacturing process design sequences for the IBR component (Tables 1–3). All ordinal hill climbing algorithm runs used I = 1000 iterations as the stopping criterion. Thirty replications of each ordinal hill climbing algorithm application were executed, each initialized with a different initial design and a different seed for the random number generator. Common initial designs were used across the three ordinal hill climbing algorithm formulations (with the manufacturing process design sequences and neighborhood functions fixed), for all thirty replications. For the first replication, a feasible controllable input parameters setting was used, set by the user. The initial input parameters setting for the remaining twenty-nine replications were obtained by randomly selecting a neighbor of the first replication’s feasible controllable input parameters setting using the second neighborhood function discussed in Section 4.2. Results are reported for two different selected design set sizes, M = 10 and M = 20. The proportion of replications (ρ) that obtained a feasible design for the best design found over
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Table 2. Ordinal hill climbing computational results for manufacturing process design sequence P1 P2 P4 P6 . J for R j,k
η Rule
M
ρ
µ
σ
Minimum
Maximum
1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 3 3
1 1 2 2 3 3 1 1 2 2 3 3 1 1 2 2 3 3
10 20 10 20 10 20 10 20 10 20 10 20 10 20 10 20 10 20
0.67 0.9 1.0 1.0 0.67 1.0 0.67 0.9 1.0 1.0 0.7 0.93 0.67 0.9 0.93 1.0 0.67 0.97
4657.349 3858.260 2239.848 2239.077 4657.297 2332.819 4656.627 3671.857 2242.043 2238.361 3684.270 3486.050 4819.780 3861.257 3632.069 2531.316 4732.828 3632.148
2533.922 953.446 4.147 2.261 2533.973 502.430 2534.644 1046.361 8.657 0.0 1464.901 1135.639 2474.592 948.306 1610.978 377.455 3480.205 971.153
2245.279 2238.361 2238.361 2238.361 2244.247 2238.361 2238.361 2238.361 2238.361 2238.361 2238.361 2248.077 2255.596 2244.247 2255.596 2238.361 2255.596 2255.596
10948.30 4997.86 2255.596 2248.077 10948.30 4992.923 10948.30 4997.86 2280.072 2238.361 6343.91 4997.86 10948.30 4997.86 10902.04 3801.400 10948.30 4997.86
Table 3. Ordinal hill climbing computational results for manufacturing process design sequence P1 P2 P3 P4 P6 . J for R j,k
η Rule
M
ρ
µ
σ
Minimum
Maximum
1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 3 3
1 1 2 2 3 3 1 1 2 2 3 3 1 1 2 2 3 3
10 20 10 20 10 20 10 20 10 20 10 20 10 20 10 20 10 20
0.57 0.57 1.0 1.0 0.63 0.63 0.6 0.57 1.0 1.0 0.73 0.63 0.57 0.57 .90 .97 0.53 0.67
6791.049 6788.178 2296.602 2282.006 6265.132 5569.833 6170.021 6435.287 2335.318 2309.241 4747.889 6161.381 6904.323 6855.574 5097.388 3585.545 6803.058 6220.348
2921.927 2926.662 27.546 26.542 3162.821 3259.155 3140.613 3029.347 101.667 46.920 3204.358 3258.135 2527.700 2821.112 2130.477 1164.320 2805.881 2936.288
2325.632 2245.469 2245.469 2245.469 2245.469 2245.469 2245.469 2245.469 2245.469 2245.469 2245.469 2245.469 3139.817 2843.926 2848.651 2304.012 2913.452 2879.562
11614.15 11614.15 2360.016 2340.901 11614.15 11614.15 11614.15 11614.15 2827.837 2382.882 11614.15 11614.15 11614.15 11614.15 11108.40 8466.28 11614.15 11614.15
all I iterations is reported. The mean (µ) and standard deviation (σ ), as well as the minimum and maximum cost function values, were computed using the best design obtained over all I iterations among the 30ρ replications that yielded feasible designs. All computational experiments were executed on a SUN ULTRA-10 workstation (128 Mb RAM). Each
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set of thirty replications took approximately 60 CPU minutes for M = 10 and 120 CPU minutes for M = 20. The results in Tables 1–3 illustrate the performance of the three ordinal hill climbing algorithm formulations and the three neighborhood functions on the three manufacturing process design sequences. All three of the Rk formulations yielded comparable results. The results from a larger design set (M = 20), were better than those from a smaller design set (M = 10), since a larger design set allows the algorithm to visit, evaluate, and retain a larger number of designs from the design space. Comparing the three neighborhood functions, the results with the second function were superior to the results with the first and third functions. The second function consistently allowed the algorithm to find a feasible design for all thirty replications. The first function is very myopic (conservative) in how it traverses the design space, while the third function is more aggressive; the second function provides an effective balance between the other two neighborhood functions. Comparing the three manufacturing process design sequences, the first design sequence (P1 P2 P5 P6 ) yielded the minimum cost to produce the IBR component. This is consistent with manufacturing process practices, since machining is more cost effective if the material being worked is sufficiently inexpensive and easy to work (i.e., soft versus hard). This is also consistent with the results reported in Jacobson et al. (1998). Therefore, the ordinal hill climbing algorithm obtained results that are comparable with the generalized hill climbing resulted reported in Jacobson et al. (1998). This suggests that using ordinal rankings may be as effective as using precise values when attempting to find an optimal design over a large design space. However since ρ was less than one for some implementations, this suggests that it may be necessary to replicate the experiments to ensure that the best design obtained is indeed feasible, hence the best design cost function value is not contaminated with penalty costs. The results presented both here and in Jacobson et al. (1998) suggest that there is one globally optimal design (namely, sequence P1 P2 P5 P6 , with cost 1919.28). The range and diversity of possible designs is quite large, due in part to the penalty costs. This is also suggested by the typically high variances among the designs found by the algorithms. However, there are several designs clustered around the globally optimal design. For example, from Table 1, all (but one of) the best input parameter values resulted in a minimum cost design within 1% of the globally optimal design, as measured by the cost function. Therefore, the initial input parameter values and the algorithm applied are important in actually finding the globally optimal design, but are not critical in being able to move to within a reasonable proximity of the optimal design. Note that the initial designs were not chosen to be close to the global optimal design, hence did not impact the results. Overall, the computational results are consistent with what would have been obtained using trial and error on the shop floor. The advantage of using ordinal hill climbing algorithms and computer simulation manufacturing processes is the speed and efficiency at which these results can be obtained, at a fraction of the cost that would be spent if trial and error on the shop floor were required.
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Summary
Ordinal hill climbing algorithms have been proposed to address an IBR discrete manufacturing process design optimization problem using computer simulation models. The ordinal hill climbing algorithm incorporates the design space reduction feature of ordinal optimization with the global optimization hill climbing feature of generalized hill climbing algorithms to identify optimal/near-optimal manufacturing process designs. Computational results with the IBR component problem suggest that using ordinal information within a generalized hill climbing algorithm framework is sufficient to obtain similar (and in some cases identical) optimal solutions. The flexibility offered by the ordinal hill climbing algorithm framework, through the choice of the hill climbing variable and the neighborhood function used in updated the selected subsets, provides a rich avenue for future research directions in determining how to fine-tuning the algorithm’s performance for different classes of problems. Research is in progress to better understand and appreciate the interplay between ordinal optimization, generalized hill climbing algorithms, and GA, as well as how to best exploit their complementary properties in a single algorithmic formulation. The discussion in Appendix 2 (on how two particular GA formulations can be modeled as ordinal hill climbing algorithms) is an important step towards understanding the behavior and performance of GA. The convergence results for GHC algorithms may be particularly useful in addressing this issue (see Johnson, 1996; Johnson and Jacobson, 2001a,b). Work is in progress to obtain a more complete understanding of these issues, and their impact on how these algorithms can be designed to produce effective algorithm implementations for hard problems.
Acknowledgments The authors would like to thank Dr. Neal Glassman, Dr. W. Garth Frazier, Mr. Enrique Medina, Dr. James Malas, Dr. William Mullins, Dr. Jay Gunasekera, Dr. Yu-Chi (Larry) Ho, and Dr. Michael Yang, for their support on this research project. The authors would like to thank the four anonymous referees for their thoughtful insights and comments, resulting in a significantly improved manuscript. The authors also want to thank Mr. Zafar Ansari for his insights into the relationship between GA and ordinal hill climbing algorithms. This research is supported by the Air Force Office of Scientific Research (F49620-98-10111, F49620-98-1-0432). The second author is supported in part by the National Science Foundation (DMI-9907980).
Appendix 1 The following are the input parameters for the seven processes. Note that all the length measurements (i.e., radius, height) are in inches, and all the temperatures are in fahrenheit.
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All speeds are in inches per second. The die friction factors are unitless. Process
Input Parameters
Possible Values
P0
Billet Radius [2, 2.5, 3.0, 3.5, 4.0, 4.5, 5.0, 5.5, 6.0] Billet Height [1.0, 2.0, 3.0] ∗ Billet Radius P1 Die Geometry: Height 1 [1.5, 2.0, 2.5, 3.0, 3.5] Height 2 [1.5, 2.0, 2.5, 3.0, 3.5] Height 3 [1.5, 2.0, 2.5, 3.0, 3.5] Die Speed [.4, .6, .8, 1.0, 1.2] Die Friction Factor [.2, .4, .6, .8] Die Temperature/Ambient Temperature [1562, 1607, 1652, 1697, 1742, 1787, 1832] P2 Radius [2.0, 2.5, . . . , 5.0] Height [1.5, 2.0, 2.5, 3.0, 3.5] P3 Die Geometry: Radius 1 [1.5, 1.75, . . . , 3.5] Height 1 [0.5, 0.625, 0.75, 0.875] Radius 2 9.0 (fixed) Height 2 [1.0, 1.25, 1.5, 1.75, 2.0] Die Speed [0.4, 0.6, 0.8, 1.0, 1.2] Die Friction Factor [0.2, 0.4, 0.6, 0.8] Die Temperature/Ambient Temperature [1562, 1607, 1652, 1697, 1742, 1787, 1832] P4 Radius 1 [1.25, 1.50, . . . , 3.25] Height 1 [0.5, 0.625, 0.75, 0.875] Radius 2 [1.5, 1.75, . . . , 3.5] Height 2 [0.5, 0.75, 1.0, 1.25, 1.5, 1.75, 2.0] Radius 3 [3.5, 3.75, . . . , 5.0] Height 3 [1.0, 1.25, 1.5, 1.75, 2.0] P5 Radius 1 3.0 (fixed) Height 1 0.5 (fixed) Radius 2 3.0 (fixed) Height 2 1.5 (fixed) Radius 3 4.0 (fixed) Height 3 1.5 (fixed) P6 Radius [1.5, 2.0, . . . , 4.0] Velocity [0.1, 0.5, 1.0, 1.5] Die Friction Factor [0.2, 0.4, 0.6, 0.8] Die Length [4.0, 6.0, 8.0, 10.0] Temperature [1562, 1652, . . . . . . , 2372] (∗ Reminder: there is a option for Die Geometry, shape = 1 for conical shape, this value is fixed
Appendix 2 Bridges Between Genetic Algorithms and Ordinal Hill Climbing Algorithms This section looks at the relationship between genetic algorithms and ordinal hill climbing algorithms. Genetic algorithms (GA) are an optimization strategy that has been successfully applied to numerous discrete optimization problems. The foundation of GA is derived from the Darwinian notion of “survival of the fittest” (Reeves, 1993); this notion suggests that as time evolves, parents produce new offspring that are more acceptable than members of previous populations. In other words, over a period of time, parents are continually
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mating to produce successive generations of children that are closer to optimality than their predecessors. The GA ordinal component of comparing cost function values over a set of children (i.e., designs) suggest that there may be a natural bridge between GA and ordinal hill climbing algorithms. Though GA have proven to be effective for addressing intractable discrete optimization problems, and can be classified as a type of hill climbing approach, its link with generalized hill climbing algorithms (through the ordinal hill climbing formulation) provides a well-defined relationship with other generalized hill climbing algorithms (like simulated annealing and threshold accepting). Therefore, such an analysis provides useful insights and observations that may fuel further research into both ordinal hill climbing and genetic algorithms, and how they fit together on a broader scale. The bridge between GA and the ordinal hill climbing algorithm framework is defined through the hill climbing variable Rk and the method by which each successive selected subset is updated (based on a neighborhood function definitions). In particular, the hill climbing variable Rk for ordinal hill climbing algorithms determines the number of parents carried over from the selected set (population) D(k) to the set E 1 . Once the set E 1 has been determined, its members can be mated (e.g., according to a crossover rule) to produce M − Rk offspring. Therefore, the GA concept of mating serves as the neighborhood function in the ordinal hill climbing algorithm framework. This set of M − Rk offspring, E 2 , together with the set E 1 becomes the next selected set or population (i.e., D(k + 1) = E 1 ∪ E 2 ). Basic GA consist of three components: i) Reproduction ii) Crossover iii) Mutation Reproduction is the process by which individual parents are evaluated for mating and inclusion in future populations. Variants of GA can be described by employing different hill climbing variables Rk (i.e., the selection of parents to be contained in E 1 varies with the choice of Rk ). Each parent in a population is evaluated according to its cost function value. In general, parents with good (lower) cost function values are more likely to be among the Rk selected parents for the set E 1 . However, for diversification, the set E 1 can be a mix of both good and bad parents. Crossover is a process by which offspring are generated based upon the cost function values of the parents. A particular crossover method needs to be defined before the GA is executed. Crossover may consist of generating a neighbor of a parent in E 1 to produce offspring or combining different parts of two parents in E 1 to produce offspring. The following is the ordinal hill climbing algorithm formulation of simple GA, which uses reproduction and crossover components: Select a set of M initial designs D(0) ⊂ Ä Set the iteration number k = 0 Repeat Order the designs in D(k) from smallest to largest cost function values
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Generate Rk (= rk ) Keep the best (smallest cost function value) rk designs from the set D(k) as parents for creating offspring. Call this set E 1 . Generate offspring of E 1 according to the crossover rule to obtain M − rk new designs. Call this set E 2 . Set D(k + 1) ← E 1 ∪ E 2 . k ←k+1 Until stopping criterion is met Mutation involves the random alteration of a parameter value in the offspring. Mutation can play either a primary or secondary role in GA, depending on the desired aggressiveness of GA. If the entire population has cost function values that are relatively close to each other, the search may easily get caught at a local optimum. In this situation, it may be desirable to increase the probability of mutation to ensure the inclusion of designs in the set E 1 sufficiently different from those already in the population. The following is the ordinal hill climbing formulated simple GA with a mutation component. Select a set of M initial designs D(0) ⊂ Ä Set the mutation probability Set the iteration number k = 0 Repeat Order the designs in D(k) from smallest to largest cost function values Generate Rk (= rk ) Keep the best (smallest cost function value) rk designs from the set D(k) as parents for creating offspring. If the largest and smallest cost function values of parents among these rk values are relatively close to each other, increase the mutation probability, hence the diversity of the parents set. Call this set E 1 . Generate offspring of E 1 according to the crossover rule to obtain M − rk new designs. Call this set E 2 . Set D(k + 1) ← E 1 ∪ E 2 . k ←k+1 Until stopping criterion is met At the end of each iteration, the new population becomes the population of parents that will be used to perform reproduction and crossover to generate yet another new population. This process is repeated until the algorithm terminates; at termination, the final population will (hopefully) contain an optimal parent (i.e., an optimal design). These ordinal hill climbing algorithm formulations for GA illustrate the power and flexibility of the ordinal hill climbing algorithm framework. It should be noted that by defining the procedure by which the selected subset in ordinal hill climbing algorithms is updated without a neighborhood function, it is possible to obtain an ordinal hill climbing algorithm formulation that is not a GA (such as by simply randomly generating designs to obtain set E 2 ). Moreover, problem-specific implementations for GA may not fit into the ordinal hill climbing framework. Further research is needed to fully assess the relationship between
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ordinal hill climbing algorithms and GA. However, the two formulations presented provide a first step towards developing a bridge between GA and other search strategies like simulated annealing, threshold accepting, and tabu search (Johnson, 1996) using the generalized hill climbing paradigm (Johnson and Jacobson, 2001a,b). Moreover, these formulations serve to illustrate the power of the ordinal optimization strategy in addressing deterministic discrete optimization problems. References Chen, C. H., and Kumar, V. 1996. Motion planning of walking robots in environments with uncertainty. Proceedings of IEEE Conference on Robotics and Automation. Dai, L. 1996. Convergence properties of ordinal comparison in the simulation of discrete event dynamic systems. Journal of Optimization Theory and Application 91(2): 363–388. Fischer, C. E., Gunasakera, J. S., and Malas, J. C. 1997. Process model development for optimization of forged disk manufacturing processes. Steel Forgings, Second Volume, ASTM STP 1257. E. G. Nisbett and A. S. Melilli, eds. American Society for Testing and Materials. Gunasakera, J. S., Fischer, C. E., Malas, J. C., Mullins, W. M., and Yang, M. S. 1996. Development of process models for use with global optimization of a manufacturing system. Proceedings of the ASME Symposium on Modeling, Simulation, and Control of Metal Processing, ASME-International Mechanical Engineering Congress, Atlanta, GA, November. Ho, Y. C., Sreenivas, R., and Vakili, P. 1992. Ordinal optimization of discrete event dynamic systems. Journal of Discrete Event Dynamical Systems 2(2): 61–88. Jacobson, S. H., Sullivan, K. A., and Johnson, A. W. 1997. Generalized hill climbing algorithms for discrete manufacturing process design problems using computer simulation models. Proceedings of the 11th European Simulation Multiconference, Istanbul, Turkey, June 1–4, pp. 473–478. Jacobson, S. H., Sullivan, K. A., and Johnson, A. W. 1998. Discrete manufacturing process design optimization using computer simulation and generalized hill climbing algorithms. Engineering Optimization 31: 247–260. Johnson, A. W. 1996. Generalized hill climbing algorithms. Ph.D. Dissertation, Department of Industrial and Systems Engineering, Virginia Tech, Blacksburg, Virginia. Johnson, A. W., and Jacobson, S. H. 2001a. A class of convergent generalized hill climbing algorithms. Applied Mathematics and Computation (to appear). Johnson, A. W., and Jacobson, S. H. 2001b. A general convergence result for hill climbing algorithms. Discrete Applied Mathematics (to appear). Lee, L. H., Lau, T. W. E., and Ho, Y. C. 1999. Explanation of goal softening in ordinal optimization. IEEE Transactions on Automatic Control 44(1): 94–99. Medina, E. A. 1998. Cost function for discrete event dynamical system optimization program. Technical Report, Austral Engineering and Software, Inc., Athens, Ohio. Reeves, C. R. 1993. Modern Heuristic Techniques for Combinatorial Problems. New York: John Wiley & Sons, Inc. Yang, M. S., Lee, L. H., and Ho, Y. C. 1997. On stochastic optimization and its applications to manufacturing. Proceedings of AMS Conference on Stochastic Problems in Manufacturing, Lecture Notes in Applied Mathematics 33, pp. 317–331.
