A numerical example along with sensitivity analysis is presented and compared with previous work. Keywords. Strength, Stress, Reliability, Optimization, Engineering design ... influence the failure of a product are often probabilistic in nature, it is ... first step in optimizing reliability-cost in system design is developing statistics ...
A Cost-Effective Strength-Stress Reliability Modeling and Optimization in Engineering Design Mohammad T. Khasawneh, Shannon R. Bowling, Sittichai Kaewkuekool, and Byung Rae Cho Department of Industrial Engineering Clemson University Clemson, South Carolina 29634-0920 Abstract Because of manufacturing variability, the strength of a component may vary significantly and unpredictably. When a component is put into use, it is subjected to a stress, which is also unpredictable. If the strength of the component is sufficient to withstand stress, then the component is functional. Otherwise, the component fails immediately. Achieving higher reliability requires higher costs. This paper first shows two reliability optimization models using normally distributed strength and stress: one is to maximize reliability subject to maximally allowable cost associated with reliability, while the other is to minimize the cost subject to a required minimum reliability level. The paper then proceeds by showing how these two conflicting decision criteria are comprised in a joint manner by presenting a proposed cost-reliability compromise model. A numerical example along with sensitivity analysis is presented and compared with previous work.
Keywords Strength, Stress, Reliability, Optimization, Engineering design
1. Introduction Quality, a desirable characteristic that a product or service should possess, is a key factor leading to business success, growth, and an enhanced competitive position. Because of today's competitive market, it is often not only desirable but also necessary to maximize the reliability of a product to ensure customer satisfaction and product success. The challenge that business faces is not only to develop products that are reliable but also take into consideration the cost factors. Achieving high levels of reliability while minimizing cost often poses problems and limitations for engineers during the design stage. Therefore, a cost-reliability compromise will always exist in the context of system design. A primary goal of reliability and design engineers is to choose the best structural and mechanical designs, considering factors such as cost, reliability, weight and volume (see Kapur and Lamberson, 1977). The question then arises of how to incorporate these factors into a model that will achieve optimum results. Because the factors that influence the failure of a product are often probabilistic in nature, it is important to incorporate the randomness of the design variables into a model that optimizes the final product. However, this randomness motivates some designers to believe that component failure may be entirely eliminated by using a preconceived margin as a safety factor. The reliability of a component is an important factor that needs to be considered at earlier stages of design. It has been proven that conventional design methodologies may not be adequate from a reliability point of view. A new probabilistic design methodology has been introduced and it explicitly identifies all the important design parameters and variables. The two important random variables that have been considered are stress and strength. Hence, determining the probability distributions for these variables is a key step in calculating component reliability. For a certain mode of failure, the reliability of a component with respect to the particular mode of failure is the probability that the strength of the component is greater than the stress acting on the component. Therefore, the costreliability compromise involves incorporating nonlinear optimization models with two random variables stress and strength as decision variables.
2. Research Motivation The first step in optimizing reliability-cost in system design is developing statistics about stress and strength distributions. Consider the case in which strength and stress are independent and normally distributed. Information about a strength distribution can be obtained from material properties while those about a stress distribution can be obtained from load statistics history.
Two models have been introduced in the literature to address such kind of problems (Kapur, 1975). In essence, the objective of the first model is to find optimal process parameters (i.e. mean and standard deviation) for the stress and strength parameters by minimizing total cost while maintaining a desired minimum reliability level. The objective of the second model is to find optimal statistics of stress and strength by maximizing reliability while maintaining minimum cost budget available. The mathematical formulations of these models are stated below: Nonlinear Optimization Model I min TC = C1 ( µS ) + C 2 (σ S ) + C3 ( µs ) + C 4 (σ s ) µS − µs
s.t.
σ 2S + σ s2
≥Z
(1) (2)
Nonlinear Optimization Model II max Z = ( µS − µs ) / σ S2 + σ s2
(3)
s.t. C1 ( µS ) + C 2 (σ S ) + C 3 ( µs ) + C 4 (σs ) ≤ r
(4)
where µS = mean value of the strength; σ S = standard deviation of the strength; µs = mean value of the stress; σ s = standard deviation of the stress; C1 ( µS ) = cost function for mean strength; C 2 (σ S ) = cost function for standard deviation of strength; C 3 ( µs ) = cost function for mean stress; and C 4 (σs ) = cost function for standard deviation of stress. By examining models I and II, we can observe that the objective function and the constraint are switched and share the same decision variables. Both models address the same problem in a separate manner. Therefore, the motivation of this research is to minimize the total cost and maximize the reliability in a simultaneous manner. Proof. The proof for the total cost and reliability functions is presented. The probability density functions for both the stress (s) and strength (S), are given below, respectively: f s (s ) =
1 σs
1s −µ s exp − 2 σs 2π
2
,
−∞ ≤ s ≤ ∞
(5)
1 S − µ 2 S , −∞ ≤ S ≤ ∞ f S (S ) = exp − (6) 2 σ S σ S 2π The reliability, R, is equal to P(S-s>0). By defining a random variable y = S-s, y is normally distributed random variable since S and s are both normally distributed. Hence, the mean and standard deviations of y are given in the following form, respectively: 1
µ y = µ S − µ s , and σ y = σS2 + σ s2
(7)
Therefore, the Reliability function is given below: ∞
∫σ
R= P(y>0) =
0
Using the transformation z =
1 y
y − µy
R=
σy
1 y − µy exp − 2 σy 2π
2
dy
(8)
, we can rewrite the reliability equation in the following form: ∞
1 2π
∫e
−
−
z2 2
dy
(9)
µS −µs σ 2S +σ s2
Based on the above equation, we can clearly observe that the reliability can be found using normal tables, where it should be noted that the new variable z is the standard normal variable. We can also see that the reliability depends
on the lower limit of the integral. That is, lowering the lower limit will result in higher values of reliability. Thus, we can identify the coupling equation in the following form: Z* =
µS − µs σ S2 + σ s2
(10)
The value of Z* is used to find the reliability from the cumulative standard normal tables. The higher the value of Z* , the higher the reliability. The total cost is then given by the summation of the individual cost functions for the mean and standard deviation for both stress and strength. This can be written in the following form: TC = C1 ( µS ) + C 2 (σ S ) + C 3 ( µs ) + C 4 (σs )
(11)
3. The Proposed Model The main objective of this research is to develop a strength-stress nonlinear optimization model that addresses cost and reliability in a simultaneous manner, which is a multiple response optimization problem. Because the desirability function approach is one of the most useful methods in dealing with such types of problems, we attempt to use this approach for finding the optimum solution. For each response Yi (x) , a desirability function d i (Yi ) assigns numbers between 0 and 1 to the possible values of Yi , with d i (Yi ) = 0 representing a completely undesirable value and d i (Yi ) = 1 representing a completely desirable or ideal response value. The individual desirability functions are then combined using the geometric mean, which gives the overall desirability D :
D = (d1 (Y1 ) * d 2 (Y2 ) * ...* d k (Yk ))1 / k
(12)
where k denotes the number of responses. Notice that if any response Yi is completely undesirable, then the overall desirability is zero (i.e. d i (Yi ) = 0 ). In our model we have two response functions (i.e. total cost, TC, and reliability, Z). The desirability function for each of these will be d 1 (TC ) and d 2 ( Z ) . The value of k in our case is 2. The method will attempt to find optimum operating conditions that will provide the "most desirable" response values. The desirability approach consists of developing individual desirability functions for each response. To this end, an S-type desirability function for minimizing the total cost is combined with an L-type desirability function for maximizing system reliability. The individual desirability functions for both the cost and reliability, respectively, are shown below: 1. 0 , TC < T r1 TC − U d 1 (TC ) = , T ≤ TC ≤ U (13) T − U 0 , TC > U 0 ,Z < L r2 Z − L d 2 ( Z ) = ,L ≤ Z ≤ t (14) t − L 1. 0 ,Z > t where L = lower value for the desirable response; U = upper value for the desirable response; T = a possible maximum value for the response; t = a possible minimum value for the response; r1 = a parameter to determine how strictly the target value of the total cost is desired; and r2 = a parameter to determine how strictly the target value of the reliability is desired. Consequently, the following optimization model is developed:
r2 µS − µs −L C ( µ ) + C (σ ) + C ( µ ) + C (σ ) − U r1 σ S2 + σ s2 2 S 3 s 4 s max 1 S . T −U t−L
1/ 2
(15)
4. Numerical Example Information on the cost functions for the statistical parameters of strength and stress is required in order to develop the final cost-reliability optimization model. As the strength of the material increases, the cost associated with that value increases. Maintaining higher average strength values require controlled manufacturing processes environment and better heat treatment processes, and hence results in higher cost values. Similarly, ensuring less variability results in the strength values, since it requires higher machinery precision results in higher costs. On the other hand, reducing the values of the mean and standard deviation for the stress of material generally provides higher reliability values, which require additional costs. The American Society of Metals (ASM, 1969) has developed various tables showing statistical data for costs and properties of selected materials. These data have been also used by Kapur (1974) and fitted polynomial functions were developed for the discrete data with a range from 30,000 to 75,000 psi. The fitted functions for the different cost functions are given below: c1 (µ S ) = 0.0002µ 1S .135 , 30 , 000 ≤ µS ≤ 75 ,000 psi (16)
c 2 (σ S ) = 800σ −S 0 .476 ,
1, 000 ≤ σ S ≤ 10 ,000 psi
(17)
c3 (µ s ) = 8997µ s−0 .513 , c 4 (µ S ) = 366µ S−0. 358 ,
10 , 000 ≤ µs ≤ 68 , 000 psi
(18)
500 ≤ σs ≤ 7, 500 psi
(19)
The above results give the following response function for the total cost:
TC = 0. 0002µ 1S .135 + 800σ S−0 .476 + 8997µ −s 0. 513 + 366σ s−0 .358
(20)
Establishing Limits and Target Values for Response Functions. The use of the desirability function in the present case requires knowledge about the lower, upper, and target values for both response functions (i.e. TC and Z). The target and upper values for the total cost were set by finding the minimum and maximu m possible values of the TCfunction based on the variation in the parameter limits (i.e. T = $78.97, U = $217.51). The target and lower values for the Z-function were found in a similar manner (i.e. t = 58.14, L = 2.33). Based on the results obtained above, therefore, the final optimization model can be written in the following form: 217.51 − 0 .0002µ1.135 + 800σ− 0.476 + 8997 µ− 0.513 + 366σ −0.358 S S s s max 138.54
(
)
µS − µs − 2 .33 r1 σS2 + σs2 . 55.81
r 2 1 / 2
(21)
s.t. 30 , 000 ≤ µS ≤ 75 ,000
(22.a)
1, 000 ≤ σ S ≤ 10 ,000
(22.b)
10 ,000 ≤ µs ≤ 68 ,000
(22.c)
500 ≤ σs ≤ 7, 500
(22.d)
5. Results The proposed model is a nonlinear optimization model with four decision variables, namely means and standard deviations of both strength and stress. In order to solve the optimization model, a numerical algorithm that
enumerates all possible values of the decision variables and their corresponding optimum desirability-function values was developed and used. As stated earlier, r1 and r2 are exponents that determine how strictly the target value of its related response is desired, and the effect of varying these values on the optimum solution was investigated while solving the model. For that purpose, a new parameter that represents that relative importance of each response compared to the other was introduced. There are certain expectations related to changes in the relative-importancer ratio (i.e. RIR = 1 ), and those are clearly stated in the discussion section of this paper. The algorithm was used r2 to find optimum responses at different values of RIR. The sample results of the optimum responses at different values of the desirability exponents are shown in Table 1. The effects of changing the RIR on the optimum total cost and optimum reliability are shown in Figures 1 and 2. From the figures, it can be clearly seen that increasing the RIR value decreases both optimum values of the total cost and the reliability. The obtained table and figures are shown below: Table 1: Sample optimum results for different values of the desirability exponents. r1 r2 r1 µ* σ* µ* σ* TC * Z* R* S
r2 1 2 3 4 5 6 7 8 9 10
0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1
10 20 30 40 50 60 70 80 90 100
S
(psi) 43794 42091 41552 41290 41136 41035 40961 40909 40866 40833
s
(psi) 2601 3380 3694 3862 3966 4037 4088 4127 4157 4182
(psi) 26069 26160 26191 26205 26215 26220 26224 26227 26230 26231
s
(psi) 2452 3229 3545 3714 3819 3891 3942 3982 4012 4037
($) 127.228 121.186 119.283 118.359 117.814 117.456 117.202 117.012 116.866 116.748
4.959 3.408 3.000 2.815 2.710 2.642 2.595 2.560 2.533 2.512
1.000 1.000 0.999 0.998 0.997 0.996 0.995 0.995 0.994 0.994
* Indicates optimum values.
Total Cost .VS. Relative Importance Ratio (r1/r2)
Reliability .VS. Relative Importance Ratio (r1/r2) 1
235
0.999 0.998
195
Reliability
Total Cost ($)
215
175 155
0.997 0.996 0.995
135
0.994 0.993
115 0
20
40
60
80
100
RIR (r1/r2)
Figure 1: The variation of the optimum total cost as a function of relative importance ratio.
0
20
40
60
80
100
r1/r2
Figure 2: The variation of the optimum reliability as a function of relative importance ratio.
Figure 3 shows the variation of the optimum values of the decision variables as a function of the relative importance ratio, RIR. It can be clearly seen an increase in RIR will result in a decrease in the mean strength value, while the mean stress and the standard deviation of both strength and stress decrease.
Optimum Decision Variables
OPtimum Decision Variables .VS. RIR 80000 70000 60000 50000 40000 30000 20000 10000 0
Mean(S) StD(S) Mean(s) StD(s) 0
20
40
60
80
100
RIR (r1/r2)
Figure 3: The variation of the optimum decision variables as a function of relative importance ratio.
6. Discussion and Conclusions This paper has provided a cost-effective strength-stress reliability modeling approach in engineering design using the concept of desirability function. The proposed model enables designers to optimize both reliability and total cost in a simultaneous manner using normally distributed functions for both stress and strength, which addressed the disadvantages of previous models discussed in the literature. As we can see in Figure 1, the total cost decreases as the RIR increases, where the curve asymptotes to a certain value at fairly large values of RIR. Large values of RIR means we are giving more importance to the total cost function and in a way ignoring the effect of the reliability function. As RIR approaches a large enough value, our model becomes identical to that of Kapur (1975) and gives almost the same optimum results (i.e. TC= $ 115.71, R=0.99, µ *S = 40520, σ *S = 4405, µ s* = 26250, σ *s = 4255 psi). In this case, the model provides optimum total cost at the lowest possible reliability level. On the other hand, having a zero value for RIR represents the case where our main concern is the reliability function and the importance of the total cost value is neglected. This will result in optimum reliability at a maximum total cost (i.e. TC= $217.51, R=1.0), which is never the case in a real-world application. The same argument applies to Figures 2, where it can be seen that increasing RIR decreases the z-value and hence the reliability. Several extensions could be investigated in the future by considering different distribution to overcome the limitation of normal distributions, in addition to incorporating time dependent strength-stress models.
Biographical Sketch Mohammad T Khasawneh is a PhD student in the Department of Industrial Engineering at Clemson University. He received his B.S. and M.S. in Mechanical Engineering from Jordan University of Science and Technology, Jordan. His research interests are in the development of advanced technology to solve interesting human-machine systems design problems, and modeling humans in process and quality control systems. Shannon R Bowling is a PhD student in the Department of Industrial Engineering at Clemson University. He received his B.S. in Electrical Engineering Technology from Bluefield State College, WV, and M.S. in Quality Management from East Tennessee State University. His research focus is in the use of advanced technology systems for improving human performance in the aircraft maintenance industry. Sittichai Kaewkuekool is a PhD student in the Department of Industrial Engineering at Clemson University. He received his B.S. in Production Technology Education from King Mongkut’s Institute of Technology, Thonbui, Thailand, and M.S. in Industrial Engineering from the University of Miami, FL. His research focus is in the use of advanced technology systems for improving human performance in quality control systems. Byung Rae Cho is an associate professor in the Department of Industrial Engineering at Clemson University. He received his M.S. in Industrial and Systems Engineering from Ohio State University and his Ph.D. in Industrial Engineering from the University of Oklahoma. His specialization is quality engineering with an emphasis in robust design and tolerance synthesis.
References 1234-
Kapur, K. C., 1975, "Optimization in Design by Reliability", AIIE transactions, pp. 185-192. Kapur, K. C., and Lamberson, L. R., 1977, “Reliability in Engineering Design”, John Wiley & Sons, Inc. Cho, B. R., 2001, “Lecture notes on Industrial Testing and Quality (IE 871), Clemson University, SC. American Society for Metals (ASM), 1969, I, Properties and Selection of Materials, II, Heat treatment, Cleaning and Finishing, III, Machining; all volumes published by American Society for Metals, Metals Park, Ohio.
