bsc thesis: scheduling policies for a repair shop problem · bsc thesis: scheduling policies for a...
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BSc Thesis: Scheduling policies for a repair shopproblem
Dopheide, J.J.Student Number: 371175
July 3, 2016
Econometrics and operations researchSupervisor: Oosterom, C.D. van
First Reader: Dekker, R.
Abstract
In this paper we take another look at the Myopic(R) scheduling policy proposedby Liang et al. (2013). The Myopic(R) policy is much more time efficient, comingwith a small loss of optimality. This policy is useful when one repair shop for everymachine type is combined to one central repair shop. The other case Liang et al.(2013) look at is the base case, where one repair shop per fleet is used.
In this paper multiple search methods are proposed to find the optimal costs forboth the base case as the central repair shop case under the Myopic(R) policy. Thesesearch methods are based on the convexity of the cost functions of the base caseand the CRS case. which is proven for the first and conjectured for the second. Theproposed methods show great improvement in calculation time.
In Liang et al. (2013) it is shown that the CRS is preferred over the base case forcertain a certain factor α. We show for which values the cost minima of the base caseand the CRS case are in equilibrium., which means that we know when it is benificialto use which repair shop strategy.
Contents
1 Introduction 1
2 Notation 2
3 Methodology 33.1 Base Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
3.1.1 Convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.1.2 Search methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.2 Central Repair Shop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.2.1 Myopic(R) policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2.2 Global balance equations . . . . . . . . . . . . . . . . . . . . . . . . 103.2.3 Search methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.3 BC/CRS-equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4 Results 124.1 Base Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.2 Central Repair Shop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.3 BC/CRS-equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
5 Conclusion 16
6 Discussion and further research 16
7 Appendix 19
1 INTRODUCTION 1
1 Introduction
In a production line, the manufacturing of products is often partitioned into severalstages where in every stage a different kind of machine is used. Within each stage thesame type of machine is used. All the machines together in one stage are then called afleet. Machines break down from time to time and have to be repaired. When a machineis down the operation targets cannot be met, so until the machine is repaired, the man-ufacturing plant experiences downtime-costs. To decrease the downtime-costs, it can beuseful to have spare machines in stock, so the broken ones can be replaced immediately,and the production process is not interrupted. These spares are accompanied by holdingcosts per type of machine. Therefore, it is of interest to determine a good balance betweenthe number of spares and the long-term expected downtime, so the costs for productionare at its minimum. Also, the repair shop structure is of importance. A manufacturercould use a different repair shop per fleet, or a general repair shop for all the differentfleets together (with a higher service (repair) rate). Sahba et al. (2013) shows that a centralrepair shop can be more efficient.
Sahba et al. (2013) looks at the option where there are spares kept per fleet, but also thecase where there is a general stock where all fleets can draw resources from. This last caseis beyond the scope of this research. When using a central repair shop, the scheduling ofthe repairs is of importance as well. This is caused by the fact that the downtime costsfor every machine is different. The machine that the decreases the downtime costs themost when repaired relatively to the repair time, has to be repaired first to save as muchas possible.
In this paper the Myopic(R) policy proposed by Liang et al. (2013) will be re-examined.The Myopic(R) policy is very useful, since, according to the results in Liang et al. (2013),the computational time decreases drastically compared to time required to find the opti-mal solution, while outcomes do not differ a lot from the optimal solution. The minimumcosts are determined by searching over different number of spares in stock per machinetype, and calculating the system costs for those numbers. The costs of the central repairshop while using the Myopic(R) will be compared to the one repair shop per fleet case.
Since the Myopic(R) policy is introduced to decrease computational time, it is of in-terest to use a clever way of finding the optimal number of spares as well, to decrease thecomputational time even more. In Liang et al. (2013) the way to find the optimal numberof spares is undefined and therefore we assume that they used complete enumeration,which is slow. Very few literature could be found regarding keeping spare inventory forbreaking down machines. This is said in Liang et al. (2013) as well. Therefore the purposeof this research is to make sure we find the optimal value of stock and find a method todecrease the computational time of finding the optimal number of spares kept in stockper fleet.
Intuitively, the system cost function for the number of spares is first non-increasingand then increasing. This means that there is a global minimum for the system costs. Thisintuitive assumption is supported by the fact that adding spares has a constant increase incosts, since the holding costs per machine type are constant. Furthermore, increasing thenumber of spares, causes a decrease in probability of having less machines operationalthan required, and therefore the downtime costs are non-increasing. Taylor and Jackson(1954) results show that probability of machines is the number of spare machines in theone repair shop per fleet case is strictly decreasing descending in their graphs, which maybe the case in the central repair shop case as well. This assumption has to be checked andmay be the basis for a clever search algorithm to find the optimal number of spares per
2 NOTATION 2
fleet. We will prove that for the multiple repair shop case the cost function is indeed con-vex and use this result to propose two fast ways of finding the optimal number of sparesper fleet. We will show by graphs that the (marginal) cost functions in the central repairshop case are also convex for the cases we examine and will propose search methods forfinding the optimal number of spares.
Also, Liang et al. (2013) show that a central repair shop is more cost efficient thanseparate repair shops under the assumption of higher repair rates per fleet. In their paperthey assumed when two fleets are repaired in the same repair shop, their repair timesincrease with factor two. We assume that the central repair shop case is indeed beneficialover the base case when all the repair rates are multiplied by factor r when r repair shopsare combined in one repair shop. Another purpose of this research is to find out for whatfactor α the central repair shop is not preferred over the base case any more. In otherwords, we want to find out for which factor α the optimal system costs for the centralrepair shop and the separate shops are in equilibrium.
It is of interest to find this equilibrium to know for which production lines a centralrepair shop should be used, and for which lines a separate repair shop is beneficial. Alsoa hybrid method could be used, such that some machine types are repaired in the samerepair shop and others have its own repair shop or are combined in another separaterepair shop.
2 Notation
To reproduce the case where we use a different repair shop per fleet of machines and thecentral repair shop case while using the Myopic(R) scheduling policy, we first introducenotation. We use similar notation asLiang et al. (2013). From now on we will refer to therepair shop per fleet as the base case (BC), and to the combined repair shop as the centralrepair shop (CRS).
We consider r different fleets with identical machines types i, i = 1, 2, .., r within eachfleet. The number of required machines within a fleet to operate at full strength is denotedas Ni. Since we allow to keep spares per fleet as well, the number of spares kept instock per fleet is defined as Si. The total number of machines of type i owned by themanufacturer equals Ni + Si.
The number of machines that is operational and active at time t is defined as Wi(t),where 0 ≤ Wi(t) ≤ Ni. The number of functional machines that are kept in stock attime t is defined as Ii(t), where 0 ≤ Ii(t) ≤ Si. Since we will always operate at thehighest operation level, such that all functional will be used if possible. This means thatIi(t) = 0 when Wi(t) < Ni. The total number of functional machines is defined as Ai(t) =Wi(t) + Ii(t). When Ai is equal or larger than Ni, fleet i operates at full strength. WhenAi is smaller than Ni the fleet has Ni − Ai down machines.
When there are less machines operational than required to operate at full strength(Wi(t) < Ni), the manufacturer experiences less production than is optimal. This re-sults in less turnover/profit, which means that the manufacturer experiences extra costs.Therefore we define bi as the costs per time unit per machine that is down. The totalholding costs per time unit is hi × Si, where hi are the holding costs per time unit permachine. The downtime costs and holding costs are assumed to be constant per machinetype.
We say that a system i is in state ni, when there are ni machines functional. A machinefrom fleet i is repaired with a rate µi and a machine breaks down with rate λi. Thebreak down rate is state dependent, since the expected time till the first break down is
3 METHODOLOGY 3
dependent on the number of machines that are in use. The break down rate in state niequals λi ·min{ni, Ni}.
3 Methodology
3.1 Base Case
We will first look into the BC. The BC can be modeled as a birth-and-death process. Abirth takes place when a machine is repaired and a death takes place when a machinebreaks down. Therefore a birth takes place with rate µi and a death takes place with λi.
In the BC every fleet has its own repair shop. This means that the probability ofhaving n functional machines in fleet i, is independent of the number of machines func-tional in fleet j, where i 6= j. Therefore, we can consider this model as r different queu-ing systems. Every repair shop has its own state space S that consists of the numbers1, 2, ..., Ni + Si, corresponding to the number of functional machines in every state. Sincethe fleets are independent in the BC we will solve the different systems one by one, andfrom now on we drop the subscript i for notational convenience.
Now we can define the probability that the system is in state n and we define theprobability of being in state n while having S spares in stock as follows. Because the statespace is dependent on the number of spares Si, the state probabilities are dependent onSi as well. Therefore, we use the following notation:
p(n|S) =µn
λn·∏ni=1 min{i,N}
∑N+Sn=0
µn
λn·∏ni=1 min{i,N}
(1)
We need to construct the cost function, because we want to minimize the total pro-duction costs, such that:
C∗BC =r
∑i=1
Ci(S∗i )
is minimal.The state probabilities in equation 1 are an important part here. The cost function is
defined as follows:
C(S) = h · S + bN
∑n=0
(N − n)p(n|S) (2)
The first term are the total holding costs for the machines kept in stock, and the secondterm are the costs for the down machines. We need to minimize this function for everyfleet i to determine the minimum costs.
To find the optimal value of S, for which the function C(S) is at its minimum, wecompute the costs for different values of S. In Liang et al. (2013) the search method isundefined and therefore we assumed that they used complete enumeration with a pre-defined maximum value of S. The choice of this maximum S is also undefined. Thechoice of a maximum Smax is hard, because taking Smax large, means that a lot of differentoptions have to be considered and computed, when complete enumeration is used. Thistakes a lot of computation time. Taking Smax small, might cause that the global minimumof the cost function is not included in the interval, such that S∗ /∈ {0, 1, ..., Smax}. There-fore the optimal S cannot be found when Smax is too small. This means that it is of greatinterest to find out if there are other possibilities to find S∗.
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3.1.1 Convexity
As mentioned in Section 1, we want to show that the cost function is convex in the BC.Convexity in general for an integer function y : Z→ R, ∀x ∈ Z is defined as follows:
y(x + 1)− y(x) ≤ y(x + 2)− y(x + 1)
We will prove that the function given in 2 is convex.
Theorem 1: The cost function of the base case is convex, or equivalently:
C(S + 1)− C(S) ≤ C(S + 2)− C(S + 1) (3)
To prove this theorem we will first prove the following lemma.
Lemma 1: The probability function of the system being in state n given a stock Sis convex in S:
p(n|S + 1)− p(n|S) ≤ p(n|S + 2)− p(n|S + 1) ∀n (4)
Proof of Lemma 1: For notational convenience we will first introduce φn = µn
λn·(
n!∏n
i=N+1 i ·∏n−Ni=1 (N)
)such that Equation 1 becomes p(n|S) = φn
∑N+Sn=0 φn
. First we will work out the left hand side
of the equation and next the right hand side.
LHS:
p(n|S + 1)− p(n|S) = φn
∑N+S+1n=0 φn
− φn
∑N+Sn=0 φn
=φn
∑N+S+1n=0 φn
−∑N+S+1
n=0 φn
∑N+Sn=0 φn
∑N+S+1n=0 φn
∑N+Sn=0 φn
· φn
∑N+Sn=0 φn
=φn
∑N+S+1n=0 φn
−∑N+S+1
n=0 φn
∑N+Sn=0 φn
· φn
∑N+S+1n=0 φn
=
(1− ∑N+S+1
n=0 φn
∑N+Sn=0 φn
)· φn
∑N+S+1n=0 φn
=
(∑N+S
n=0 φn
∑N+Sn=0 φn
− ∑N+S+1n=0 φn
∑N+Sn=0 φn
)· φn
∑N+S+1n=0 φn
=
(− φN+S+1
∑N+Sn=0 φn
)· φn
∑N+S+1n=0 φn
RHS:
p(n|S + 2)− p(n|S + 1) = · · · =
(− φN+S+2
∑N+S+1n=0 φn
)· φn
∑N+S+2n=0 φn
=
(− φN+S+2
∑N+S+2n=0 φn
)· φn
∑N+S+1n=0 φn
3 METHODOLOGY 5
We now have to show that:(− φN+S+1
∑N+Sn=0 φn
)· φn
∑N+S+1n=0 φn
≤
(− φN+S+2
∑N+S+2n=0 φn
)· φn
∑N+S+1n=0 φn
Or equivalently:
− φN+S+1
∑N+Sn=0 φn
≤ − φN+S+2
∑N+S+2n=0 φn
(5)
To show this inequality we need to substitute φn back into the equation:
− φN+S+1
∑N+Sn=0 φn
= −µN+S+1
λN+S+1·∏N+S+1i=1 min{i,N}
∑N+Sn=0
µn
λn·∏ni=1 min{i,N}
= −( µ
λ )N+S+1
∑N+Sn=0 (
µλ )
n ·[
∏N+S+1i=1 min{i,N}∏n
i=1 min{i,N}
] (6)
≤ −( µ
λ )N+S+1
∑N+Sn=0 (
µλ )
n ·[
∏N+S+2i=1 min{i,N}
∏n+1i=1 min{i,N}
] (7)
≤ −( µ
λ )N+S+1
λµ ∏N+S+2
i=1 min{i, N}+ ∑N+Sn=0 (
µλ )
n ·[
∏N+S+2i=1 min{i,N}
∏n+1i=1 min{i,N}
] (8)
= −( µ
λ ) · (µλ )
N+S+1
∏N+S+2i=1 min{i, N}+ ∑N+S
n=0 (µλ )
n+1 ·[
∏N+S+2i=1 min{i,N}
∏n+1i=1 min{i,N}
]= −
( µλ )
N+S+2
∑N+S+1n=0 ( µ
λ )n ·[
∏N+S+2i=1 min{i,N}∏n
i=1 min{i,N}
]
= −µN+S+2
λN+S+2·∏N+S+2i=1 min{i,N}
∑N+S+1n=0
µn
λn·∏ni=1 min{i,N}
= − φN+S+2
∑N+S+2n=0 φn
The inequality between 6 and 7 is explained by the fact that:
∏N+S+1i=1 min{i, N}∏n
i=1 min{i, N} =
∏N+S+2i=1 min{i,N}
∏n+1i=1 min{i,N}
∏N+S+2i=1 min{i,N}
∏n+1i=1 min{i,N}
· ∏N+S+1i=1 min{i, N}∏n
i=1 min{i, N}
=∏N+S+2
i=1 min{i, N}∏n+1
i=1 min{i, N}· min{n + 1, N}
min{N + S + 2, N}
≤ ∏N+S+2i=1 min{i, N}
∏n+1i=1 min{i, N}
The inequality between 7 and 8 is explained by the fact that the term λµ ∏N+S+2
i=1 min{i, N}is always positive, since λ, µ, N > 0. Therefore inequality 5 holds and Lemma 1 (4) isproven.
3 METHODOLOGY 6
Lemma 1 will be useful for proving Theorem 1.
Proof of Theorem 1:
C(S + 1)− C(S) =
(h(S + 1) + b
N
∑n=0
(N − n)p(n|S + 1)
)−(
hS + bN
∑n=0
(N − n)p(n|S))
= h + bN
∑n=0
(N − n)[
p(n|S + 1)− p(n|S)]
(9)
≤ h + bN
∑n=0
(N − n)[
p(n|S + 2)− p(n|S + 1)]
(10)
=
(h(S + 2) + b
N
∑n=0
(N − n)p(n|S + 2)
)−(
h(S + 1) + bN
∑n=0
(N − n)p(n|S + 1)
)= C(S + 2)− C(S + 1)
The inequality between 9 and 10 holds, because of Lemma 1. This proofs Theorem 1(3) that the cost function of the BC is convex.
The convexity of the BC cost function will be very useful in proposing a method thatreduces the computation time for finding the optimal value of S, and still guarantees thatwe find S∗, the number of spares in stock for which the operation costs are at its globalminimum.
Figure 1: Cost functions for two fleets with its own repair shops (BC)
0 2 4 6 8 10 12 14 16 18 2010
15
20
25
30
35
40
Number of spares in stock
Tota
l costs
Cost function of BC
N1 = 10, λ
1 = 0.09, µ
1 = 1.0, h
1 = 1.0, b
1 = 20
N2 = 5, λ
2 = 0.09, µ
2 = 0.5, h
2 = 0.9, b
2 = 18
In Figure 1 an example of the cost function is shown. The convexity is very clear forthese two functions.
3.1.2 Search methods
The first method we propose to find S∗ uses the property of a convex function, that whenit starts increasing, it will never decrease anymore. The algorithm computes the costsfor every S, until the cost function value starts increasing. This means that the previousiteration corresponds to the number of spares S at which the production costs are at itsminimum. The pseudo-code for this algorithm is as follows:
3 METHODOLOGY 7
Pseudo Code Base Case 1
Set C(0)→ infSet S→ 0repeat
k→ S + 1Compute C(S)
until C(S) > C(S− 1);C∗ → C(S− 1)S∗ → S− 1Return S∗, C∗
In Figure 1 we show an example of the cost functions for two fleets (correspondingto the first row of results in Appendix 7). If we use complete enumeration, and we setSmax for both fleets at 20, 40 function evaluations have to be computed. Since the minimaare at S1 = 9 and S2 = 6, the algorithm we propose only uses 19 function evaluations.Another advantage is that it costs more time to solve a system of equations when thenumber of equations and variables increase. When S is large, there are more states inwhich the system can be and therefore there are more variables and equations. Thereforeit is beneficial to compute the cost function only for small values of S, which is done inthe proposed method.
Another search method that is proposed is the Fibonacci search method. The Fi-bonacci is similar to the well-known golden search algorithm, but is suitable for an inte-ger function such as the cost function 2 in Section 3.1. The Fibonacci search algorithm isa four point search method that shrinks a start interval iteratively wherein the minimumof the cost function will be. In the end the interval consists of four function values andthe minimum will be searched over these four values, and S∗ will be the correspondingstock to this minimum.
The advantage of this method is that function evaluations get reused, just as in thegolden search method, which saves computation time. The downside of this searchmethod is that it needs an upper bound. We did not look into how to find a upper boundthat guarantees that the minimum will indeed lie within the interval.
The pseudo-code of the algorithm is shown on the next page. We define Fi as theith Fibonacci number and Fn as the smallest Fibonacci number greater than the presetupper bound. The new upper bound of the interval will be Fn, such that the minimum issearched in the interval [0, Fn].
3 METHODOLOGY 8
Pseudo Code Base Case 2
Set L→ 0Set U → FnSet CL, CU → infSet S1 → Fn−2Set S2 → Fn−1Compute C1 → C(S1)Compute C2 → C(S2)for j = 3, ..., n− 1 do
if C1 < C2 thenSet U → S2Set CU → C2Set S2 → S1Set C2 → C1Set S1 → L + Fn−jCompute C1 → C(S1)
elseSet L→ S1Set CL → C1Set S1 → S2Set C1 → C2Set S2 = U − Fn−jCompute C2 = C(S2)
endendif CL equals inf then
Compute CL → C(L)endif CL equals inf then
Compute CU → C(U)endSet C∗ → min({CL, C1, C2, CU})Set S∗ to corresponding SReturn C∗, S∗
3.2 Central Repair Shop
In the Central Repair Shop case (CRS), all machines are repaired at the same shop, butwith a higher repair rate. For now we assume that µi = r · µBC
i . Since every machineis repaired at the same shop, the state probabilities for every fleet are not independentanymore. This means that the problem becomes much larger and costs more time tosolve.
Let us first define the vector n = (n1, n2, ..., nr), which states the number of functionalmachines of every fleet. We also define vector S = (S1, S2, ..., Sr), which states the num-ber of spares kept in stock per fleet. Our goal is to find the optimal vector S for whichthe operation costs for the CRS are at its minimum. Therefore we need to minimize thecost function, which is slightly different in the CRS compared to the BC, due to the de-pendence between the fleets:
3 METHODOLOGY 9
C∗ = minS
{r
∑i=1
Ci(S)
}where,
Ci(S) = hi · Si + bi
Ni
∑n=0
(Ni − n)pi(n)
The minimum costs depend on the scheduling policy that is used in the CRS. Thescheduling policy states which machine type is going to be repaired when not all ma-chines are functional. Liang et al. (2013) propose the Myopic(R) policy, which is fasterthan the optimal scheduling, and has a low cost difference with the optimal schedulingpolicy.
3.2.1 Myopic(R) policy
The Myopic(R) policy chooses to repair the machine type first for which the cost differ-ence relatively to its repair and breakdown rate, between repairing and not repairing thismachine is the largest. This cost rate difference is dependent on the number of type imachines in stock in a certain state. Therefore we define xi as the inventory position ofmachine type i. The cost difference rate is defined as follows for xi > 0:
∆cRi (xi) = −bi
Ni+xi
∑m=xi+1
pRi (m) (11)
The probability pRi (m) is the probability that m machines will break during the look
ahead time. The look ahead time in the Myopic(R) policy is the repair time. Liang et al.(2013) show that for 0 ≤ xi ≤ m:
pRi (m) =
Ni !(Ni−m+xi)!
µiλi
(Niλi)xi
(Niλi+µi)xi
∏m−xij=0 (Ni +
µiλi− j)
(12)
When xi = 0, there can be precisely Ni functional machines, or less. When xi = 0 andthere are exactly ni = Ni machines functional , the costs are defined as:›
cRi (xi) = −bi ·
(Niλi)
Niλi + µi(13)
When xi = 0 and the system is not fully operational, such that ni < Ni, the expecteddowntime rate costs are defined as:
cRi (xi) = −bi (14)
For every state n we can determine which machine has the lowest expected downtime
cost rate difference. The machine type that corresponds to this minimum for µi ·∆cRi (xi)
λiis
the machine that will be repaired in this state. As can be seen, we compensate for therepair and break down rate in this fraction, such that we look at the relative downtimecosts. Since we now know which machine will be repaired in every state we can computethe global balance equations:
3 METHODOLOGY 10
3.2.2 Global balance equations
To find the long-term state probabilities, we compose a system of equations.
The left hand side of the equation (first line) is the rate that the process leaves staten = (n1, n2, ..., nr).
The first term of the right hand side (second line) is the rate that the process enters staten = (n1, n2, ..., nr) by a break down of any of the machines.
The second term of the right hand side (third line) is the rate that the process enters staten = (n1, n2, ..., nr) by a repair.
[r
∑i=1
λi(ni) + µ(n)
]P(n) =
r
∑i=1
[λi(ni + 1) · P(n + ei)] +
r
∑i=1
[µ(n− ei) · Ii · P(n− ei)]
where ei is a unit-vector.
Where
λi(ni) = min{ni, Ni} · λi
is the failure rate of a type i machine given that there are min{ni, Ni} functional and activemachines.
Where
µ(n) =
{µi, if not all machines are functional0, otherwise
where i is the to be repaired machine based on the Myopic(R) scheduling policy in Sec-tion 3.2.1.
Where
Ii =
{1, if type i machine in function µ(.) is scheduled for repair0, otherwise
At last we add the normalizing equation, such that the total probability equals one:
∑n
P(n) = 1
We will solve this system by using the MATLAB function for solving a system of equa-tions.
3 METHODOLOGY 11
3.2.3 Search methods
Intuitively the CRS cost function is convex, just as the BC cost function. When the multi-ple stocks Si increase, the downtime costs are expected to be decreasing descending, andtherefore expected to be convex in S. The holding costs increase linear, so are convex inS as well. Since the sum of two convex function is also convex, the the CRS cost functionhas to be convex as well if the assumption on the downtime costs holds. We could notprove this assumption for the CRS, since it is too complex for the time span we have.
Figure 2: Cost functions for two fleets sharing the same repair shop (CRS)
Therefore we checked this assumption by numerical experiments. In Figure 2 thecost function is shown for the experiment corresponding to the first row of the results inAppendix 7. As we expected we can see in both graphs clear convex behavior. Althoughwe do not know for sure if convexity holds in every case, we propose an algorithm whichwill decrease the computational time drastically if this assumption indeed holds.
The algorithm is the same as the first proposed method in the BC for a given stock forall the fleets except for fleet 1, so for a given S2, ..., Sr. This means the stop at first increase-method is used in one dimension. In the other dimensions we use complete enumerationto guarantee convergence to the Myopic(R) optimum.
If the cost function of the Myopic(R) is convex, this method will converge. Since wehave no proof of the convexity of this function, there is no guarantee that the optimalvector S will be found. Therefore this algorithm is a heuristic for the Myopic(R) policy inthe CRS.
Another method we propose, is also based on the assumption of convexity in the costfunction. It is almost the same as the previous, but were we first only used the stop at firstincrease-method in one direction, we will now use it in all directions. This method is alsoknown as the coordinate descent-method.
The algorithm will search for an optimal value for Si given Sj, ∀j 6= i ∈ R, whereR = {1, 2, ..., r}. We will use the line search method from Section 3.1.2 to find S∗i . Whenthe optimal value for Si is found, another direction i will be chosen, and another optimumwill be searched for along the new line. The process will repeat itself until no moreimprovement is found. The algorithm will search in the following order of directions:i = 1, 2, ..., r, 1, 2, ..., r....
A slight adjustment has to be notified according to the line search method, becausewe used to start at the search at S0
i = 0, such that we only have to increase Si to find the
4 RESULTS 12
minimum costs. Now that S0i ≥ 0 in each line search, the optimum can be on either sides
of the current Si. We propose to first look if a decrease is found when Si is increased, andif not so, we search in the other direction along the line.
The downside of this method that it does not guarantee convergence to the mini-mum, even for convex functions. The upside is that this method will be faster since theinformation from the previous line-search will be used in the next.
3.3 BC/CRS-equilibrium
In the BC there is one repair shop per machine type, and in the CRS there is only onerepair shop. This means that the CRS has more repairs to handle than the BC, since thebreak down rate per shop in the BC is ni ·λi, ∀i, and in the CRS this rate equals ∑r
i=1 ni ·λi.The CRS can only handle one machine at the time and the BC at most r different machinesat the same time.
For trivial reasons, the BC will therefore be lower in operational costs when µCRSi =
µBCi . Keeping the repair rates at the same in the BC and the CRS, does not make sense,
since you will just lose workforce. In Liang et al. (2013), they use µi = 2 · µBCi , when
r = 2. Therefore we assume that they used α = r in µCRSi = α · µBC
i . This makes sensewhen for example in every shop there is an equal number of repairmen, and the types ofmachines do not require a specific skill set. The total workforce is then combined in thecentral repair shop resulting in α = r.
There are also a lot of reasons to argue the α = r assumption. For example, if amachine type does need a specific skill to be repaired, combining workforce from otherrepair shops in one central repair shop will result not in µi = r · µBC
i , since the work-force from a repair shop j, where j 6= i, cannot repair machine i at the same rate as theworkforce from repair shop i.
Because the α = r assumption may be argued, we are going to look for which valueof α the operation costs in the CRS are equal to the costs in the BC. We will do this tosearch over different values of α using the golden search method. For r = 2, we use theinitial interval of α = [1, 2], because at α = 1 the BC is preferred over the CRS and asshown by Liang et al. (2013) the CRS is preferred when α = 2, with r = 2. Furthermorethe function we evaluate is the absolute value of the difference between the CRS and BCoperation costs ( f (α) = |CBC − CCRS(α)|, where CCRS(α) is the cost function of the CRS,with µi = α · µBC
i ). Since intuitively we know the minimum of this function equals 0, thestopping criterium we use is f (α) < 104. We also set the maximum number of iterationson 25.
4 Results
Before we take a look at the results of the several , we have to introduce numerical exam-ples. We will use the same instances as Liang et al. (2013) use. We consider a productionline with r = 2 fleets, with the following parameters:
• Setting h1 = 1, we consider the following holding cost rates for fleet 2: h2 ={0.9, 0.7, 0.5};
• We consider the following down time cost rate to holding cost rations: b1h1
= b2h2
=
{20, 30};
• The fleet sizes are: (N1, N2) ∈ {(10, 5), (10, 10, (10, 15), (50, 25), (50, 50), (100, 50)};
4 RESULTS 13
• When N1 = 10, 100 (N1 = 50), we set µ1 = 2 (µ1 = 1), and we consider µ1µ2∈
{2, 1, 23};
• As an approximate measure of the repair shop utilization, we set u = λ1 N1µ1
= λ2 N2µ2∈
{0.45, 0.35, 0.25} corresponding to high, medium and low levels of repair shop uti-lization.
This results in a total of 3× 2× 6× 3× 3 = 324 instances which have to be computedfor all methods. The results can be found in Appendix 7.
The results of these numerical experiments did agree with the results in Liang et al.(2013), except for the coordinate descent-method. This is caused by the fact that it is aheuristic that does not always converge to the Myopic(R) optimum. For that reason thecoordinate descent-method has its own cost and optimal stock columns in the tables withresults.
4.1 Base Case
When we use complete enumeration we have to set an Smax to find the optimal numberof spare machines. For the BC we used an Smax = 50 for both fleets, which we assumedto be an arbitrarily large enough number.
In Section 3.1 we propose two search methods for finding the optimal stock S for theBC making use of the convexity of its cost function. In Appendix 7 the extensive resultscan be found.
To find out the performance of the method we look at the relative computation timedifference as a measure, where TCE is the computation time for using complete enumer-ation, where TS is the computation time of the proposed stop at first increase-method, andwhere TF is the computation time of the Fibonacci-method:
∆CES =
TCE − TS
TCE
∆CEF =
TCE − TF
TCE
∆FS =
TF − TS
TF
Table 1: Summary of BC results
min. (%) mean (%) median (%) max. (%)
∆CS 8.9 86.2 91.5 93.9
∆CF 61.11 83.1 84.3 99.4
∆FS -1192.5 -11.2 4.6 29.6
Table 1 summarizes the relative performance of the search methods in the BC. Theresults indicate that the proof of convexity makes it possible to decrease the calculationtime drastically.
4 RESULTS 14
4.2 Central Repair Shop
The computation time for the instances with N1, N2 being large, will be large, since thenumber of variables and equations in the system of equations of Section 3.2.2 increases.When we use complete enumeration we have to set an Smax to find the optimal numberof spare machines. Since we have limited time, we chose to look at the results of Lianget al. (2013) and chose an Smax based on their findings. We decided that Smax
i should beconstant for a given Ni. The values for Smax
i can be found below.
(N1, N2)
(10, 5) (10, 10) (10, 15) (50, 25) (50, 50) (100, 50)
Smax1 20 20 20 25 25 25
Smax2 16 20 22 25 25 31
To determine the performance of the several search methods in the CRS while usingthe Myopic(R) policy, we propose the following measures:
∆CES1 =
TCE − TS1
TCE
∆CECD =
TCE − TF
TCE
∆S1CD =
TS1 − TCD
TS1
Here is TCE the computation time of finding the minimum while using complete enu-meration again, TS1 the stop at first increase-method in one dimension and TCD the coordi-nate descent method where the stop at first increase-method is used in every dimension.
Due to the time constraints not all results for the complete enumeration could beobtained. The last 40 instances are not created. In 44 out of 324 cases the same minimumas with the methods that did converge. These are therefore not included in the measures.For the other two methods all proposed instances are obtained and therefore included in∆S1
CD.
Table 2: Summary of CRS - Myopic(R) policy results
min. (%) mean (%) median (%) max. (%)
∆CES1 44.5 82.2 85.8 91.8
∆CECD 84.4 96.8 98.3 99.7
∆S1CD 50.2 85.5 88.1 96.9
Table 2 summarizes the relative performance of the search methods in the BC. Theresults indicate that the two heuristics save a lot of computation time.
The stop at first increase-method in one direction converged in for all instances, whichsuggests that cost function for the CRS is also convex.
The coordinate descent method does not converge to the optimal stock S∗ for all in-stances. In 44 out of 324 the method did not converge and higher costs were found. Thisis caused by the fact that in some cases no improvement of costs could be found in anydirection i, but only in a direction where you adjust Si and Sj with i 6= j at the same time.
4 RESULTS 15
We only summarize the relative increase for the 44 instances that did not converge,and not include the instances for which the coordinate descent method did converge. Therelative positive deviations from the minimum are summarized in Table 3.
Table 3: Deviation from the minimum costs
min. (%) mean (%) median (%) max. (%)
1.7 0.4 0.2 1.7
The results indicate that the coordinate descent method does not deviate much fromthe minimum costs for the Myopic(R) policy and save a lot of computation time.
4.3 BC/CRS-equilibrium
To find the equilibria we used the stop at first increase-method, since it converges thefastest to the Myopic(R) optimum. Due to time constraints, and the long computationtimes, even for the stop at first increase-method, we did not compute all values of α. Wedid compute alpha for (N1, N2) ∈ {(10, 5), (10, 10), (10, 15), (50, 25)}. The obtained val-ues for α can be found in Appendix 7.
The average value of α equals 1.601. The results showed that their was a high cor-relation between the utilization factor u and the equilibrium factor α. The correlationbetween α and u equals 0.972. This is also shown in the scatter plot in Figure 3 and inTable 4 we give the mean values of α for different utilization factors:
Figure 3: Scatter plot showing the relation between utilization rate and equilibrium factor
α
0.25 0.3 0.35 0.4 0.45
u
1.3
1.4
1.5
1.6
1.7
1.8
1.9Repair shop utilization vs. equilibrium factor α plot
Table 4: Mean alpha for different utilization rates
u 0.25 0.35 0.45
α 1.399 1.601 1.803
5 CONCLUSION 16
5 Conclusion
We can conclude that due to the proof of convexity of the BC cost function smart searchmethods can be used to decrease the calculation time significantly for finding the min-imum production line costs. The stop at first increase-method performs better than theFibonacci search method in most of the cases, with a computation time decrease of re-spectively 86.2% and 83.1%. The better performance of the stop at first increase-methodis mostly faster due to the fact that the optimal stock for the Myopic(R) policy is smallfor most instances and therefore this method does not use many iterations to find theminimum costs. Also, this method always converges to minimum costs. When a largestock is needed for to assure minimum costs, the Fibonacci method is preferred. The dis-advantage of the Fibonacci method is that a maximum possible stock has to be set. Thismaximum number of spares is unknown, and therefore we do not know if the minimumwill lie in the preset interval [0, Smax]. The stop at first increase-method is therefore pre-ferred. If there would be a method to estimate the optimal stock, a hybrid method couldbe beneficial. Then the Fibonacci method will be used when the expected optimal stockis large.
For the Myopic(R) policy we proposed two heuristics to find the minimum costs. Thefirst method will converge to the minimum if the cost function is convex, which seemsvery likely according the results, because in all the proposed instances the method con-verged. The average decrease in time is 82.2%. The second method we proposed has aneven smaller average computation time (a decrease of 96.8 %), but does not converge insome cases (13.5%). The cost difference with the optimal value is on average 0.4% for thenot converged cases. This means that the loss is really small and therefore the coordinatedescent method has a great performance, since it saves lot of computation time. In Lianget al. (2013) an average loss of 0.66% was considered as close enough to the optimum.Due to this fast search method another 0.0005% is added, which can be considered as aninsignificant increase.
Finding the equilibria where the BC has the same costs as the CRS with the Myopic(R)scheduling policy, has shown that the utilization rate is highly correlated with the factorα. We think this is caused by the fact that a high utilization makes combining repairshops less beneficial, and the assumption that a high factor α corresponds to the CRSbeing a less profitable substitute for the BC. Combining shops with a high utilization isless profitable since combining makes sense when the workforce in one shop can be usedfor repairing other machine types. If the workforce is already occupied all the time withrepairing the machine type of its shop, it cannot be used for repairing other machinestypes.
6 Discussion and further research
Although improvements are made, there are several limitations to this research. Due totime constraints there was a lack code optimizing. This means that the computation timescould have been lower, influencing the result. On the other hand, it is expected that thiswill reduce the computation times for all methods and that the relative difference will beapproximately the same.
Also, not all instances were considered in this research due to the same time con-straint as mentioned before. This means that we cannot use our recommendations for all
REFERENCES 17
instances.For the complete enumeration, Fibonacci- and the stop at first increase-method in one
direction for the CRS case, a maximum value for S had to be set before solving the prob-lem. We did not find a way to find an upper bound for S such that convergence to theminimum is guaranteed. In further research it will be useful to find a way to set an Smax
that assures convergence.For the CRS we set Smax by looking at the results from Liang et al. (2013). This means
that if we had to choose Smax without this knowledge, and we would have chosen an-other value, it could have been lower, meaning we would not find the minimum, or itcould have been higher and the computation time difference in Section 4.1 and 4.2 wouldbe even greater and the proposed stop at first increase-method in the BC and the coordi-nate descent method in the CRS are even more preferred.
Another thing that has to be looked in, is what happens when the utilization increases.The results suggest that the CRS will not be an improvement anymore when the singlerepair shops have a higher utilization, since the factor α might be greater than 2 when theutilization factor for the single repair shops increase.
References
Liang, W. K., Balcıoglu, B., and Svaluto, R. (2013). Scheduling policies for a repair shopproblem. Annals of Operations Research, 211(1):273–288.
Sahba, P., Balcıoglu, B., and Banjevic, D. (2013). Spare parts provisioning for multiplek-out-of-n: G systems. IIE transactions, 45(9):953–963.
Taylor, J. and Jackson, R. (1954). An application of the birth and death process to theprovision of spare machines. OR, 5(4):95–108.
REFERENCES 18
7 APPENDIX 19
7 Appendix
Tabl
e5:
The
para
met
ers
and
corr
espo
ndin
gm
inim
alco
sts
for
the
BCan
dth
eM
yopi
c(R
)pol
icy
wit
hth
eco
mpu
tati
onti
mes
ofth
ese
vera
lsea
rch
met
hods
and
the
equi
libri
umfa
ctor
α.
PAR
AM
ETER
SBa
seC
ase
Myo
pic(
R)
Myo
pic(
R)
Equi
libri
um
N1
N2
λ1
λ2
µ1
µ2
h 1h 2
b 1b 2
Cos
tsS 1
S 2T C
ET S
T FC
osts
S 1S 2
T CE
T SC
osts
S 1S 2
T CD
α
105
0.09
0.09
21
10.
920
1827
.314
96
0.03
30.
008
0.00
515
.317
37
21.9
73.
564
15.3
173
70.
962
1.73
410
50.
070.
072
11
0.9
2018
14.9
296
50.
025
0.00
50.
003
8.26
24
21.0
372.
323
8.26
24
0.59
11.
568
105
0.05
0.05
21
10.
920
188.
774
33
0.02
40.
002
0.00
25.
071
22
20.7
392.
218
5.07
12
20.
197
1.38
410
50.
090.
182
21
0.9
2018
27.3
149
60.
022
0.00
40.
003
16.2
64
620
.674
3.62
316
.305
55
0.65
51.
751
105
0.07
0.14
22
10.
920
1814
.929
65
0.02
40.
001
0.00
28.
528
33
20.6
622.
519
8.52
83
30.
378
1.59
710
50.
050.
12
21
0.9
2018
8.77
43
30.
022
0.00
10.
002
5.14
22
220
.75
2.33
85.
142
22
0.18
81.
409
105
0.09
0.27
23
10.
920
1827
.314
96
0.02
30.
002
0.00
216
.144
73
20.6
934.
144
16.1
447
30.
398
1.74
510
50.
070.
212
31
0.9
2018
14.9
296
50.
016
0.00
10.
003
8.66
34
220
.71
2.63
68.
663
42
0.26
31.
593
105
0.05
0.15
23
10.
920
188.
774
33
0.02
70.
001
0.00
35.
242
220
.803
2.32
25.
242
20.
203
1.41
910
50.
090.
092
11
0.9
8072
47.3
1718
140.
021
0.00
40.
003
24.4
685
1420
.674
8.15
324
.468
514
2.90
51.
813
105
0.07
0.07
21
10.
980
7222
.122
98
0.01
80.
002
0.00
311
.743
46
20.9
884.
066
11.7
434
60.
979
1.59
210
50.
050.
052
11
0.9
8072
12.5
485
50.
021
0.00
20.
003
7.19
43
320
.757
3.03
57.
194
33
0.36
71.
385
105
0.09
0.18
22
10.
980
7247
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1814
0.01
50.
004
0.00
426
.807
713
20.7
239.
174
26.8
668
122.
793
1.82
710
50.
070.
142
21
0.9
8072
22.1
229
80.
016
0.00
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003
12.2
675
520
.696
4.01
612
.44
64
0.62
91.
625
105
0.05
0.1
22
10.
980
7212
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55
0.01
60.
001
0.00
37.
264
33
20.7
582.
948
7.26
43
30.
354
1.41
410
50.
090.
272
31
0.9
8072
47.3
1718
140.
016
0.00
30.
004
26.2
7914
520
.729
10.4
26.3
3113
60.
957
1.82
105
0.07
0.21
23
10.
980
7222
.122
98
0.01
60.
001
0.00
212
.292
64
20.7
724.
1812
.292
64
0.55
91.
619
105
0.05
0.15
23
10.
980
7212
.548
55
0.02
30.
001
0.00
27.
404
33
20.8
052.
946
7.40
43
30.
366
1.41
510
50.
090.
092
11
0.7
2014
24.8
119
60.
019
0.00
10.
003
13.1
513
721
.267
3.11
413
.151
37
0.87
1.71
105
0.07
0.07
21
10.
720
1413
.463
65
0.01
70.
001
0.00
27.
199
24
20.7
22.
433
7.19
92
40.
468
1.54
510
50.
050.
052
11
0.7
2014
7.88
63
30.
016
0.00
10.
002
4.51
22
20.6
172.
224
4.51
22
0.19
41.
364
105
0.09
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22
10.
720
1424
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96
0.01
60.
001
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214
.025
37
20.6
463.
305
14.0
664
60.
674
1.72
510
50.
070.
142
21
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2014
13.4
636
50.
015
0.00
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003
7.52
53
320
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2.40
47.
525
33
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61.
575
105
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22
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720
147.
886
33
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001
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563
22
20.6
52.
209
4.56
32
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198
1.39
310
50.
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272
31
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2014
24.8
119
60.
023
0.00
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14.4
15
420
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1413
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001
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272.
439
7.79
84
20.
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1.58
810
50.
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152
31
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2014
7.88
63
30.
017
0.00
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003
4.64
22
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982.
205
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22
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11
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017
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003
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780
5619
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98
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001
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027
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11.2
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50.
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43
320
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2.93
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33
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416
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1022
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001
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37
20.7
132.
942
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70.
693
1.63
310
50.
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072
11
0.5
2010
11.9
966
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0.00
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002
6.01
32
420
.756
2.24
86.
013
24
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489
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10.
520
106.
999
33
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33.
949
22
20.7
092.
208
3.94
92
20.
198
1.31
910
50.
090.
182
21
0.5
2010
22.3
099
60.
015
0.00
10.
003
11.1
983
720
.675
3.10
311
.198
37
0.69
31.
645
105
0.07
0.14
22
10.
520
1011
.996
65
0.01
70.
001
0.00
36.
434
24
20.8
052.
344
6.43
42
40.
512
1.52
210
50.
050.
12
21
0.5
2010
6.99
93
30.
016
00.
003
3.98
42
220
.651
2.20
53.
984
22
0.17
21.
363
105
0.09
0.27
23
10.
520
1022
.309
96
0.01
60.
001
0.00
411
.854
37
20.6
173.
242
11.8
554
60.
629
1.65
610
50.
070.
212
31
0.5
2010
11.9
966
50.
017
0.00
10.
003
6.76
63
421
.303
2.37
86.
766
34
0.43
1.54
910
50.
050.
152
31
0.5
2010
6.99
93
30.
021
0.00
10.
003
4.04
22
21.0
022.
233
4.04
22
0.2
1.39
910
50.
090.
092
11
0.5
8040
38.3
5418
140.
017
0.00
30.
003
16.3
895
1420
.608
5.88
16.3
895
142.
309
1.75
810
50.
070.
072
11
0.5
8040
17.6
859
80.
018
0.00
20.
002
8.61
84
620
.847
3.83
58.
618
46
0.78
1.52
910
50.
050.
052
11
0.5
8040
9.97
95
50.
020.
001
0.00
25.
616
33
20.9
22.
952
5.61
63
30.
365
1.31
910
50.
090.
182
21
0.5
8040
38.3
5418
140.
016
0.00
30.
003
18.0
465
1521
.152
6.73
418
.039
516
2.35
91.
771
105
0.07
0.14
22
10.
580
4017
.685
98
0.01
60.
001
0.00
39.
157
47
20.8
133.
898
9.15
74
70.
758
1.56
310
50.
050.
12
21
0.5
8040
9.97
95
50.
020.
001
0.00
25.
644
33
20.6
652.
929
5.64
43
30.
379
1.36
610
50.
090.
272
31
0.5
8040
38.3
5418
140.
017
0.00
30.
003
19.4
676
1520
.818
7.70
319
.467
615
2.44
11.
781
105
0.07
0.21
23
10.
580
4017
.685
98
0.01
80.
002
0.00
59.
688
47
20.6
883.
959
9.68
84
70.
903
1.59
310
50.
050.
152
31
0.5
8040
9.97
95
50.
016
0.00
10.
003
5.72
53
320
.707
2.92
15.
725
33
0.36
21.
408
1010
0.09
0.04
52
11
0.9
2018
30.5
99
0.01
80.
002
0.00
317
.296
39
46.5
427.
111
17.2
963
91.
708
1.78
710
100.
070.
035
21
10.
920
1815
.826
66
0.01
90.
001
0.00
38.
628
24
47.5
814.
878.
628
24
0.88
81.
591
1010
0.05
0.02
52
11
0.9
2018
9.08
23
30.
023
0.00
10.
003
5.16
72
246
.642
5.77
45.
167
22
0.46
1.39
610
100.
090.
092
21
0.9
2018
30.5
99
0.01
90.
002
0.00
218
.832
48
46.4
098.
059
19.0
557
51.
223
1.81
1010
0.07
0.07
22
10.
920
1815
.826
66
0.01
60.
001
0.00
29.
033
446
.462
5.13
9.03
34
0.76
91.
627
1010
0.05
0.05
22
10.
920
189.
082
33
0.01
90.
001
0.00
35.
252
246
.438
4.64
85.
252
20.
455
1.42
410
100.
090.
135
23
10.
920
1830
.59
90.
018
0.00
20.
003
18.7
888
446
.503
8.98
18.8
129
30.
762
1.80
810
100.
070.
105
23
10.
920
1815
.826
66
0.01
80.
001
0.00
29.
156
43
46.3
625.
218
9.15
64
30.
693
1.62
810
100.
050.
075
23
10.
920
189.
082
33
0.02
40.
001
0.00
25.
368
22
46.9
744.
725.
368
22
0.45
11.
435
7 APPENDIX 20Ta
ble
6:Th
epa
ram
eter
san
dco
rres
pond
ing
min
imal
cost
sfo
rth
eBC
and
the
Myo
pic(
R)p
olic
yw
ith
the
com
puta
tion
tim
esof
the
seve
rals
earc
hm
etho
dsan
dth
eeq
uilib
rium
fact
orα
.
PAR
AM
ETER
SBa
seC
ase
Myo
pic(
R)
Myo
pic(
R)
Equi
libri
um
N1
N2
λ1
λ2
µ1
µ2
h 1h 2
b 1b 2
Cos
tsS 1
S 2T C
ET S
T FC
osts
S 1S 2
T CE
T SC
osts
S 1S 2
T CD
α
1010
0.09
0.04
52
11
0.9
8072
51.5
8518
180.
034
0.00
40.
004
26.7
295
1646
.577
14.8
0326
.729
516
5.91
91.
836
1010
0.07
0.03
52
11
0.9
8072
23.0
659
90.
017
0.00
20.
003
12.1
294
646
.593
8.41
112
.129
46
1.40
41.
604
1010
0.05
0.02
52
11
0.9
8072
12.8
595
50.
018
0.00
10.
002
7.30
53
346
.477
6.14
67.
305
33
0.69
31.
388
1010
0.09
0.09
22
10.
980
7251
.585
1818
0.01
70.
005
0.00
329
.889
716
46.3
4620
.668
30.0
269
146.
271.
853
1010
0.07
0.07
22
10.
980
7223
.065
99
0.01
60.
002
0.00
312
.764
56
46.5
398.
605
12.7
645
61.
386
1.64
310
100.
050.
052
21
0.9
8072
12.8
595
50.
023
0.00
10.
002
7.38
63
347
.187
6.17
67.
386
33
0.60
11.
423
1010
0.09
0.13
52
31
0.9
8072
51.5
8518
180.
018
0.00
40.
003
29.3
4916
646
.801
25.1
3329
.349
166
1.70
51.
8510
100.
070.
105
23
10.
980
7223
.065
99
0.01
80.
002
0.00
312
.884
74
46.6
098.
5412
.884
74
0.69
21.
638
1010
0.05
0.07
52
31
0.9
8072
12.8
595
50.
018
0.00
10.
002
7.54
93
346
.746
6.13
17.
549
33
0.58
11.
4310
100.
090.
045
21
10.
720
1427
.289
99
0.01
70.
002
0.00
214
.641
39
46.5
596.
776
14.6
413
91.
652
1.76
810
100.
070.
035
21
10.
720
1414
.16
66
0.02
0.00
10.
002
7.49
32
446
.537
4.87
17.
493
24
0.68
1.56
910
100.
050.
025
21
10.
720
148.
126
33
0.02
60.
001
0.00
34.
586
22
46.4
084.
713
4.58
62
20.
306
1.37
410
100.
090.
092
21
0.7
2014
27.2
899
90.
017
0.00
20.
002
16.0
043
946
.813
7.38
216
.004
39
1.89
91.
788
1010
0.07
0.07
22
10.
720
1414
.16
66
0.01
70.
001
0.00
27.
883
446
.467
4.90
87.
883
40.
699
1.60
610
100.
050.
052
21
0.7
2014
8.12
63
30.
016
0.00
10.
002
4.64
92
246
.492
4.60
44.
649
22
0.29
1.40
710
100.
090.
135
23
10.
720
1427
.289
99
0.01
80.
002
0.00
316
.715
74
46.3
978.
095
16.7
157
40.
963
1.80
110
100.
070.
105
23
10.
720
1414
.16
66
0.02
0.00
10.
003
8.15
94
346
.446
5.66
78.
159
43
0.45
31.
626
1010
0.05
0.07
52
31
0.7
2014
8.12
63
30.
024
0.00
10.
003
4.74
12
246
.493
4.64
24.
741
22
0.30
31.
432
1010
0.09
0.04
52
11
0.7
8056
46.1
5518
180.
018
0.00
40.
005
22.4
575
1747
.375
14.6
4122
.457
517
5.70
21.
824
1010
0.07
0.03
52
11
0.7
8056
20.6
379
90.
018
0.00
20.
003
10.5
754
746
.608
8.47
610
.575
47
1.52
1.58
310
100.
050.
025
21
10.
780
5611
.506
55
0.01
70.
001
0.00
26.
493
33
46.3
916.
113
6.49
33
30.
581.
369
1010
0.09
0.09
22
10.
780
5646
.155
1818
0.01
70.
005
0.00
425
.137
618
46.5
3419
.395
25.1
376
185.
804
1.84
210
100.
070.
072
21
0.7
8056
20.6
379
90.
020.
004
0.00
311
.125
47
47.1
38.
397
11.1
254
71.
584
1.62
210
100.
050.
052
21
0.7
8056
11.5
065
50.
024
0.00
20.
002
6.55
13
346
.415
6.08
96.
551
33
0.58
41.
407
1010
0.09
0.13
52
31
0.7
8056
46.1
5518
180.
018
0.00
50.
003
26.6
8214
746
.361
21.9
7126
.682
147
3.71
1.84
910
100.
070.
105
23
10.
780
5620
.637
99
0.01
70.
004
0.00
311
.524
65
46.6
689.
839
11.5
246
50.
994
1.64
310
100.
050.
075
23
10.
780
5611
.506
55
0.01
80.
002
0.00
26.
683
346
.795
6.33
36.
683
30.
595
1.43
1010
0.09
0.04
52
11
0.5
2010
24.0
799
90.
018
0.00
20.
003
11.9
83
1046
.456
6.45
211
.98
310
1.60
91.
7410
100.
070.
035
21
10.
520
1012
.494
66
0.01
80.
002
0.00
36.
358
24
46.4
184.
767
6.35
82
40.
668
1.53
810
100.
050.
025
21
10.
520
107.
173
30.
024
0.00
10.
002
4.00
62
246
.645
4.64
4.00
62
20.
297
1.34
410
100.
090.
092
21
0.5
2010
24.0
799
90.
017
0.00
20.
002
12.9
953
1051
.212
6.89
712
.995
310
1.56
1.75
710
100.
070.
072
21
0.5
2010
12.4
946
60.
016
0.00
10.
002
6.72
25
46.5
114.
898
6.72
93
40.
672
1.57
210
100.
050.
052
21
0.5
2010
7.17
33
0.01
70.
001
0.00
34.
048
22
46.4
954.
622
4.04
82
20.
365
1.38
210
100.
090.
135
23
10.
520
1024
.079
99
0.01
80.
002
0.00
313
.859
49
46.9
087.
235
13.8
594
91.
618
1.77
110
100.
070.
105
23
10.
520
1012
.494
66
0.01
70.
001
0.00
26.
946
34
46.3
085.
086
6.94
63
40.
715
1.60
110
100.
050.
075
23
10.
520
107.
173
30.
024
0.00
10.
002
4.11
42
246
.369
4.62
94.
114
22
0.31
11.
406
1010
0.09
0.04
52
11
0.5
8040
40.7
2518
180.
026
0.00
40.
004
18.1
545
1846
.479
14.7
1818
.154
518
5.26
11.
809
1010
0.07
0.03
52
11
0.5
8040
18.2
19
90.
018
0.00
20.
003
8.96
24
746
.508
8.38
48.
962
47
1.37
41.
555
1010
0.05
0.02
52
11
0.5
8040
10.1
525
50.
018
0.00
10.
002
5.68
23
346
.438
6.10
35.
682
33
0.59
21.
338
1010
0.09
0.09
22
10.
580
4040
.725
1818
0.01
60.
004
0.00
320
.184
520
46.7
4116
.596
20.1
845
205.
668
1.82
610
100.
070.
072
21
0.5
8040
18.2
19
90.
017
0.00
20.
002
9.39
14
746
.501
8.46
69.
391
47
1.36
31.
5910
100.
050.
052
21
0.5
8040
10.1
525
50.
018
0.00
10.
002
5.71
63
346
.521
6.10
45.
716
33
0.55
51.
376
1010
0.09
0.13
52
31
0.5
8040
40.7
2518
180.
025
0.00
40.
003
21.9
916
2046
.412
18.7
0621
.991
620
6.04
41.
836
1010
0.07
0.10
52
31
0.5
8040
18.2
19
90.
018
0.00
20.
004
9.88
84
756
.175
8.28
49.
925
61.
284
1.62
110
100.
050.
075
23
10.
580
4010
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55
0.01
90.
001
0.00
25.
813
346
.807
6.08
85.
813
30.
588
1.41
210
150.
090.
032
11
0.9
2018
32.5
159
110.
019
0.00
20.
003
18.5
833
1071
.409
11.5
5718
.583
310
3.42
21.
796
1015
0.07
0.02
32
11
0.9
2018
16.2
426
60.
018
0.00
20.
003
9.24
92
591
.299
7.53
29.
249
25
1.45
1.61
110
150.
050.
017
21
10.
920
189.
235
33
0.01
90.
001
0.00
35.
593
22
82.6
336.
832
5.59
32
20.
631.
4310
150.
090.
062
21
0.9
2018
32.5
159
110.
026
0.00
20.
003
19.6
74
971
.362
12.2
5119
.691
58
3.17
81.
813
1015
0.07
0.04
72
21
0.9
2018
16.2
426
60.
022
0.00
10.
002
9.21
83
471
.234
7.41
99.
314
43
0.89
31.
632
1015
0.05
0.03
32
21
0.9
2018
9.23
53
30.
018
0.00
10.
004
5.32
22
71.3
896.
793
5.32
22
0.63
21.
427
1015
0.09
0.09
23
10.
920
1832
.515
911
0.02
10.
002
0.00
419
.144
84
70.9
8712
.973
19.1
448
41.
464
1.80
610
150.
070.
072
31
0.9
2018
16.2
426
60.
028
0.00
10.
004
9.04
24
371
.718
7.41
59.
042
43
0.63
11.
621
1015
0.05
0.05
23
10.
920
189.
235
33
0.02
40.
001
0.00
45.
247
22
71.0
176.
814
5.24
72
20.
424
1.41
410
150.
090.
032
11
0.9
8072
54.0
6218
210.
025
0.00
50.
005
28.0
85
1871
.562
22.3
6828
.08
518
11.1
431.
839
1015
0.07
0.02
32
11
0.9
8072
23.5
149
100.
019
0.00
20.
004
12.7
224
771
.475
13.1
4312
.722
47
2.76
21.
615
1015
0.05
0.01
72
11
0.9
8072
13.0
155
50.
019
0.00
10.
005
7.66
93
471
.767
9.09
37.
669
34
1.02
91.
413
1015
0.09
0.06
22
10.
980
7254
.062
1821
0.01
80.
005
0.00
530
.839
717
71.3
4629
.868
30.9
759
159.
831
1.85
410
150.
070.
047
22
10.
980
7223
.514
910
0.01
90.
002
0.00
712
.952
56
71.0
8812
.49
13.0
56
51.
594
1.64
510
150.
050.
033
22
10.
980
7213
.015
55
0.02
30.
001
0.00
37.
468
33
77.9
768.
986
7.46
83
30.
824
1.42
510
150.
090.
092
31
0.9
8072
54.0
6218
210.
026
0.00
50.
003
29.8
1516
610
3.10
336
.873
29.8
1516
62.
586
1.85
110
150.
070.
072
31
0.9
8072
23.5
149
100.
019
0.00
20.
003
12.7
376
484
.735
12.2
8712
.737
64
1.20
41.
633
1015
0.05
0.05
23
10.
980
7213
.015
55
0.01
90.
001
0.00
37.
383
33
70.9
348.
966
7.38
33
30.
803
1.41
6
7 APPENDIX 21Ta
ble
7:Th
epa
ram
eter
san
dco
rres
pond
ing
min
imal
cost
sfo
rth
eBC
and
the
Myo
pic(
R)p
olic
yw
ith
the
com
puta
tion
tim
esof
the
seve
rals
earc
hm
etho
dsan
dth
eeq
uilib
rium
fact
orα
.
PAR
AM
ETER
SBa
seC
ase
Myo
pic(
R)
Myo
pic(
R)
Equi
libri
um
N1
N2
λ1
λ2
µ1
µ2
h 1h 2
b 1b 2
Cos
tsS 1
S 2T C
ET S
T FC
osts
S 1S 2
T CE
T SC
osts
S 1S 2
T CD
α
1015
0.09
0.03
21
10.
720
1428
.857
911
0.01
90.
002
0.00
316
.237
311
71.5
2311
.332
16.2
373
113.
376
1.79
810
150.
070.
023
21
10.
720
1414
.484
66
0.01
90.
001
0.00
38.
151
25
71.2
197.
393
8.15
12
51.
525
1.60
210
150.
050.
017
21
10.
720
148.
245
33
0.02
70.
001
0.00
34.
982
23
71.1
136.
839
4.98
22
30.
735
1.42
110
150.
090.
062
21
0.7
2014
28.8
579
110.
021
0.00
30.
003
17.4
043
1170
.98
12.0
4317
.408
410
3.01
61.
815
1015
0.07
0.04
72
21
0.7
2014
14.4
846
60.
018
0.00
20.
002
8.26
73
471
.052
7.42
68.
267
34
1.21
61.
629
1015
0.05
0.03
32
21
0.7
2014
8.24
53
30.
017
0.00
10.
002
4.80
82
272
.317
6.77
74.
808
22
0.62
31.
4310
150.
090.
092
31
0.7
2014
28.8
579
110.
019
0.00
20.
004
17.8
717
570
.986
12.6
8417
.871
75
1.90
51.
825
1015
0.07
0.07
23
10.
720
1414
.484
66
0.01
90.
001
0.00
38.
267
43
70.8
877.
409
8.26
74
30.
655
1.63
810
150.
050.
052
31
0.7
2014
8.24
53
30.
029
0.00
10.
003
4.73
32
270
.906
6.69
94.
733
22
0.43
81.
4310
150.
090.
032
11
0.7
8056
48.0
8118
210.
021
0.00
50.
004
24.1
235
1971
.57
23.2
7224
.123
519
11.0
251.
8410
150.
070.
023
21
10.
780
5620
.986
910
0.01
90.
002
0.00
311
.219
47
71.5
4413
.103
11.2
194
72.
71.
602
1015
0.05
0.01
72
11
0.7
8056
11.6
275
50.
019
0.00
10.
003
6.81
83
471
.208
9.07
86.
818
34
1.08
91.
401
1015
0.09
0.06
22
10.
780
5648
.081
1821
0.01
80.
006
0.00
326
.676
620
70.9
6129
.379
26.6
766
209.
885
1.85
610
150.
070.
047
22
10.
780
5620
.986
910
0.01
90.
003
0.00
311
.498
47
71.9
3912
.395
11.6
015
61.
856
1.63
510
150.
050.
033
22
10.
780
5611
.627
55
0.02
20.
001
0.00
26.
738
33
71.0
888.
952
6.73
83
30.
833
1.42
210
150.
090.
092
31
0.7
8056
48.0
8118
210.
026
0.00
70.
003
28.0
415
870
.933
35.5
6228
.04
158
3.46
41.
8610
150.
070.
072
31
0.7
8056
20.9
869
100.
026
0.00
20.
003
11.6
116
571
.592
13.8
8711
.611
65
1.36
1.65
1015
0.05
0.05
23
10.
780
5611
.627
55
0.02
20.
001
0.00
36.
652
33
70.8
779.
016
6.65
23
30.
832
1.42
810
150.
090.
032
11
0.5
2010
25.1
989
110.
020.
002
0.00
312
.558
311
71.4
329.
762
12.5
583
112.
558
1.76
710
150.
070.
023
21
10.
520
1012
.725
66
0.01
90.
002
0.00
36.
442
25
71.4
127.
014
6.44
22
51.
038
1.55
1015
0.05
0.01
72
11
0.5
2010
7.25
53
30.
019
0.00
10.
003
4.03
22
71.1
946.
732
4.03
22
0.55
81.
349
1015
0.09
0.06
22
10.
520
1025
.198
911
0.01
70.
002
0.00
313
.786
311
71.1
1110
.664
13.7
863
112.
627
1.78
610
150.
070.
047
22
10.
520
1012
.725
66
0.02
80.
001
0.00
36.
835
25
78.6
927.
16.
856
34
0.91
31.
587
1015
0.05
0.03
32
21
0.5
2010
7.25
53
30.
018
0.00
10.
003
4.07
52
271
.14
6.71
64.
075
22
0.42
41.
387
1015
0.09
0.09
23
10.
520
1025
.198
911
0.01
90.
002
0.00
314
.861
411
71.3
2711
.447
14.8
614
112.
921.
803
1015
0.07
0.07
23
10.
520
1012
.725
66
0.01
90.
001
0.00
37.
107
34
71.2
47.
246
7.10
73
40.
948
1.61
810
150.
050.
052
31
0.5
2010
7.25
53
30.
019
0.00
10.
003
4.14
72
271
.344
6.73
64.
147
22
0.71
41.
415
1015
0.09
0.03
21
10.
580
4042
.101
1821
0.01
90.
005
0.00
318
.777
519
71.0
1520
.551
18.7
775
198.
493
1.82
210
150.
070.
023
21
10.
580
4018
.459
910
0.02
70.
002
0.00
39.
037
47
70.9
4212
.086
9.03
74
72.
074
1.56
110
150.
050.
017
21
10.
580
4010
.239
55
0.02
0.00
10.
003
5.70
13
471
.326
8.96
75.
701
34
0.97
31.
341
1015
0.09
0.06
22
10.
580
4042
.101
1821
0.01
80.
005
0.00
321
.065
521
72.2
0527
.626
21.0
655
218.
681.
841
1015
0.07
0.04
72
21
0.5
8040
18.4
599
100.
018
0.00
20.
003
9.51
44
771
.344
12.0
999.
514
47
1.89
31.
598
1015
0.05
0.03
32
21
0.5
8040
10.2
395
50.
018
0.00
10.
003
5.74
63
372
.843
9.21
75.
746
33
0.83
21.
3810
150.
090.
092
31
0.5
8040
42.1
0118
210.
021
0.00
60.
003
23.1
476
2270
.918
30.6
2723
.147
622
9.26
61.
851
1015
0.07
0.07
23
10.
580
4018
.459
910
0.02
50.
002
0.00
310
.063
48
71.0
1812
.139
10.0
855
61.
791
1.63
110
150.
050.
052
31
0.5
8040
10.2
395
50.
019
0.00
10.
003
5.84
73
371
.178
9.01
95.
847
33
0.81
91.
417
5025
0.00
90.
009
10.
51
0.9
2018
43.9
7316
130.
048
0.00
80.
006
21.3
534
1214
53.1
0423
4.25
21.3
534
1251
.142
1.81
450
250.
007
0.00
71
0.5
10.
920
1818
.27
77
0.04
50.
004
0.00
59.
274
34
1451
.782
145.
621
9.27
43
418
.71
1.60
550
250.
005
0.00
51
0.5
10.
920
189.
697
44
0.05
40.
002
0.00
55.
304
22
1459
.374
137.
172
5.30
42
28.
348
1.39
850
250.
009
0.01
81
11
0.9
2018
43.9
7316
130.
037
0.00
80.
006
23.8
795
1214
57.4
0428
0.98
523
.907
611
62.8
681.
834
5025
0.00
70.
014
11
10.
920
1818
.27
77
0.03
30.
004
0.00
59.
83
414
53.6
1814
8.78
69.
83
417
.843
1.64
350
250.
005
0.01
11
10.
920
189.
697
44
0.03
50.
004
0.00
55.
408
22
1516
.766
137.
988
5.40
82
28.
321.
434
5025
0.00
90.
027
11.
51
0.9
2018
43.9
7316
130.
044
0.00
80.
006
23.6
6212
414
60.2
5732
6.78
323
.662
124
22.0
541.
8350
250.
007
0.02
11
1.5
10.
920
1818
.27
77
0.03
50.
004
0.00
69.
941
53
1456
.937
154.
573
9.94
15
39.
119
1.64
250
250.
005
0.01
51
1.5
10.
920
189.
697
44
0.03
50.
002
0.00
65.
556
22
1431
.981
138.
098
5.55
62
28.
297
1.44
550
250.
009
0.00
91
0.5
10.
980
7267
.511
2824
0.03
70.
022
0.00
631
.08
521
1433
.031
443.
566
31.0
986
2014
5.90
51.
855
5025
0.00
70.
007
10.
51
0.9
8072
25.6
111
100.
042
0.00
60.
006
12.6
854
714
32.6
5423
6.73
212
.685
47
33.6
781.
6150
250.
005
0.00
51
0.5
10.
980
7213
.495
66
0.03
40.
004
0.00
57.
463
33
1436
.603
176.
888
7.46
33
312
.694
1.39
350
250.
009
0.01
81
11
0.9
8072
67.5
1128
240.
034
0.01
60.
009
35.5
37
2214
40.8
7858
7.46
435
.602
920
186.
174
1.86
450
250.
007
0.01
41
11
0.9
8072
25.6
111
100.
044
0.00
60.
008
13.5
185
614
84.5
3923
4.55
13.5
185
628
.734
1.65
150
250.
005
0.01
11
10.
980
7213
.495
66
0.03
40.
003
0.00
97.
562
33
1431
.683
176.
895
7.56
23
310
.892
1.42
950
250.
009
0.02
71
1.5
10.
980
7267
.511
2824
0.03
40.
015
0.00
734
.727
216
1437
.15
797.
076
34.7
2721
639
.777
1.86
450
250.
007
0.02
11
1.5
10.
980
7225
.61
1110
0.03
90.
006
0.00
513
.61
74
1430
.218
238.
872
13.6
17
416
.122
1.64
650
250.
005
0.01
51
1.5
10.
980
7213
.495
66
0.04
20.
003
0.00
67.
764
314
40.0
9417
8.85
17.
764
38.
258
1.43
350
250.
009
0.00
91
0.5
10.
720
1439
.744
1613
0.03
60.
009
0.00
617
.866
313
1430
.713
223.
365
17.8
663
1352
.834
1.79
750
250.
007
0.00
71
0.5
10.
720
1416
.47
70.
034
0.00
40.
005
8.05
52
514
38.7
8814
3.31
68.
055
25
18.1
991.
583
5025
0.00
50.
005
10.
51
0.7
2014
8.68
54
40.
045
0.00
30.
005
4.70
42
214
36.6
1613
7.49
24.
704
22
8.29
11.
377
5025
0.00
90.
018
11
10.
720
1439
.744
1613
0.04
10.
008
0.00
620
413
1433
.632
255.
476
204
1357
.264
1.81
450
250.
007
0.01
41
11
0.7
2014
16.4
77
0.03
60.
004
0.00
58.
505
35
1454
.179
147.
309
8.50
53
518
.089
1.62
150
250.
005
0.01
11
10.
720
148.
685
44
0.03
40.
002
0.00
64.
782
22
1454
.054
137.
578
4.78
22
28.
381
1.41
450
250.
009
0.02
71
1.5
10.
720
1439
.744
1613
0.04
60.
009
0.00
621
.267
105
1458
.475
298.
332
21.2
6710
526
.851
1.82
550
250.
007
0.02
11
1.5
10.
720
1416
.47
70.
066
0.00
50.
006
8.84
43
1514
.793
151.
639
8.84
43
11.6
551.
642
5025
0.00
50.
015
11.
51
0.7
2014
8.68
54
40.
034
0.00
40.
006
4.89
82
215
04.9
0613
6.92
84.
898
22
8.24
61.
439
7 APPENDIX 22Ta
ble
8:Th
epa
ram
eter
san
dco
rres
pond
ing
min
imal
cost
sfo
rth
eBC
and
the
Myo
pic(
R)p
olic
yw
ith
the
com
puta
tion
tim
esof
the
seve
rals
earc
hm
etho
dsan
dth
eeq
uilib
rium
fact
orα
.
PAR
AM
ETER
SBa
seC
ase
Myo
pic(
R)
Myo
pic(
R)
Equi
libri
um
N1
N2
λ1
λ2
µ1
µ2
h 1h 2
b 1b 2
Cos
tsS 1
S 2T C
ET S
T FC
osts
S 1S 2
T CE
T SC
osts
S 1S 2
T CD
α
5025
0.00
90.
009
10.
51
0.7
8056
60.8
5428
240.
048
0.01
90.
007
25.8
45
2114
56.3
4942
3.09
525
.84
521
142.
71.
845
5025
0.00
70.
007
10.
51
0.7
8056
22.9
6911
100.
095
0.00
60.
006
10.9
994
714
59.6
6523
2.71
710
.999
47
30.4
861.
589
5025
0.00
50.
005
10.
51
0.7
8056
12.0
836
60.
061
0.00
40.
005
6.62
93
314
57.0
9317
6.10
76.
629
33
12.3
121.
3750
250.
009
0.01
81
11
0.7
8056
60.8
5428
240.
033
0.01
70.
009
29.5
496
2414
52.6
2261
3.52
629
.549
624
153.
571.
857
5025
0.00
70.
014
11
10.
780
5622
.969
1110
0.03
50.
009
0.00
811
.785
47
1452
.176
231.
964
11.7
854
734
.537
1.62
950
250.
005
0.01
11
10.
780
5612
.083
66
0.04
0.00
50.
009
6.7
33
1459
.618
177.
934
6.7
33
10.6
151.
413
5025
0.00
90.
027
11.
51
0.7
8056
60.8
5428
240.
033
0.01
60.
0131
.833
197
1453
.468
682.
569
31.8
3319
738
.698
1.85
950
250.
007
0.02
11
1.5
10.
780
5622
.969
1110
0.03
60.
006
0.00
512
.201
65
1452
.33
234.
554
12.2
016
521
.374
1.65
150
250.
005
0.01
51
1.5
10.
780
5612
.083
66
0.04
0.00
30.
007
6.85
83
314
81.3
3517
6.51
36.
858
33
10.5
361.
434
5025
0.00
90.
009
10.
51
0.5
2010
35.5
1616
130.
041
0.00
80.
006
13.9
283
1214
51.7
3619
8.01
813
.928
312
40.6
651.
761
5025
0.00
70.
007
10.
51
0.5
2010
14.5
37
70.
033
0.00
40.
005
6.68
92
514
59.1
5814
1.21
26.
689
25
16.3
231.
542
5025
0.00
50.
005
10.
51
0.5
2010
7.67
34
40.
034
0.00
30.
005
4.10
42
214
53.8
213
8.07
34.
104
22
5.44
11.
338
5025
0.00
90.
018
11
10.
520
1035
.516
1613
0.04
20.
010.
006
15.7
464
1314
59.3
6323
0.17
615
.746
413
42.9
161.
776
5025
0.00
70.
014
11
10.
520
1014
.53
77
0.04
50.
006
0.00
57.
233
514
50.8
216
6.33
37.
233
515
.353
1.58
350
250.
005
0.01
11
10.
520
107.
673
44
0.03
30.
004
0.00
64.
157
22
1457
.846
156.
306
4.15
72
25.
446
1.38
950
250.
009
0.02
71
1.5
10.
520
1035
.516
1613
0.03
30.
013
0.00
617
.29
414
1508
.839
271.
811
17.3
345
1347
.306
1.78
950
250.
007
0.02
11
1.5
10.
520
1014
.53
77
0.04
60.
004
0.00
57.
755
35
1452
.977
153.
147.
755
35
18.1
041.
617
5025
0.00
50.
015
11.
51
0.5
2010
7.67
34
40.
034
0.00
30.
005
4.24
12
214
53.5
8413
7.08
14.
241
22
5.44
41.
435
5025
0.00
90.
009
10.
51
0.5
8040
54.1
9628
240.
038
0.01
50.
007
20.1
465
2114
52.2
2141
4.99
120
.146
521
122.
768
1.82
450
250.
007
0.00
71
0.5
10.
580
4020
.329
1110
0.03
90.
006
0.00
69.
185
47
1461
.705
227.
969
9.18
54
730
.936
1.55
350
250.
005
0.00
51
0.5
10.
580
4010
.671
66
0.03
40.
004
0.00
55.
752
34
1459
.107
175.
424
5.75
23
412
.632
1.33
150
250.
009
0.01
81
11
0.5
8040
54.1
9628
240.
040.
016
0.00
723
.116
624
1451
.773
519.
428
23.1
166
2412
2.75
91.
839
5025
0.00
70.
014
11
10.
580
4020
.329
1110
0.05
0.00
60.
008
9.92
84
814
76.3
8522
9.93
9.92
84
828
.347
1.59
350
250.
005
0.01
11
10.
580
4010
.671
66
0.06
50.
004
0.00
85.
837
34
1452
.765
175.
887
5.83
73
412
.548
1.38
450
250.
009
0.02
71
1.5
10.
580
4054
.196
2824
0.03
80.
018
0.00
925
.83
725
1455
.059
586.
312
25.8
37
2514
4.19
21.
845
5025
0.00
70.
021
11.
51
0.5
8040
20.3
2911
100.
033
0.01
60.
006
10.7
485
814
58.2
0723
4.75
410
.91
66
23.8
141.
629
5025
0.00
50.
015
11.
51
0.5
8040
10.6
716
60.
045
0.00
40.
006
5.95
43
314
50.7
6117
5.87
35.
954
33
10.5
511.
433
5050
0.00
90.
005
10.
51
0.9
2018
47.3
9316
160.
040.
017
0.00
723
.176
414
7484
.042
1288
.705
23.1
764
1429
2.17
7N
A50
500.
007
0.00
41
0.5
10.
920
1818
.723
77
0.04
10.
005
0.00
79.
436
35
7437
.687
685.
794
9.43
63
561
.413
NA
5050
0.00
50.
003
10.
51
0.9
2018
9.77
24
40.
050.
003
0.00
65.
335
22
7408
.392
661.
706
5.33
52
231
.786
NA
5050
0.00
90.
009
11
10.
920
1847
.393
1616
0.04
0.01
10.
007
26.4
855
1474
07.7
1511
.692
26.4
855
1440
8.32
5N
A50
500.
007
0.00
71
11
0.9
2018
18.7
237
70.
044
0.00
90.
007
10.0
294
474
07.1
0271
9.24
210
.029
44
56.8
09N
A50
500.
005
0.00
51
11
0.9
2018
9.77
24
40.
046
0.00
40.
006
5.44
62
274
10.2
6665
3.53
35.
446
22
31.9
66N
A50
500.
009
0.01
41
1.5
10.
920
1847
.393
1616
0.04
0.01
40.
007
26.3
0815
474
78.1
9217
90.0
5226
.308
154
116.
756
NA
5050
0.00
70.
011
11.
51
0.9
2018
18.7
237
70.
049
0.00
50.
007
10.1
735
373
95.9
7173
0.22
910
.173
53
48.3
69N
A50
500.
005
0.00
81
1.5
10.
920
189.
772
44
0.04
10.
003
0.00
75.
604
22
7474
.047
656.
406
5.60
42
232
.081
NA
5050
0.00
90.
005
10.
51
0.9
8072
71.3
4928
280.
042
0.02
10.
012
32.9
925
2374
53.7
8821
85.0
6233
.007
622
845.
849
NA
5050
0.00
70.
004
10.
51
0.9
8072
26.0
8211
110.
048
0.00
80.
007
12.8
394
771
03.6
9911
33.6
3412
.839
47
152.
886
NA
5050
0.00
50.
003
10.
51
0.9
8072
13.5
76
60.
039
0.00
40.
006
7.49
93
371
59.2
7584
6.44
7.49
93
344
.434
NA
5050
0.00
90.
009
11
10.
980
7271
.349
2828
0.04
60.
024
0.01
138
.299
725
7133
.128
3003
.015
38.3
649
2310
12.5
68N
A50
500.
007
0.00
71
11
0.9
8072
26.0
8211
110.
047
0.01
70.
011
13.7
295
771
98.3
111
30.4
8513
.778
66
117.
487
NA
5050
0.00
50.
005
11
10.
980
7213
.57
66
0.04
10.
008
0.00
87.
604
33
7127
.738
842.
757
7.60
43
338
.929
NA
5050
0.00
90.
014
11.
51
0.9
8072
71.3
4928
280.
051
0.04
80.
011
37.4
9624
672
18.5
236
65.4
1237
.496
246
226.
236
NA
5050
0.00
70.
011
11.
51
0.9
8072
26.0
8211
110.
040.
010.
007
13.8
767
571
63.1
4111
44.7
1813
.876
75
97.0
67N
A50
500.
005
0.00
81
1.5
10.
980
7213
.57
66
0.04
10.
004
0.00
77.
787
43
7123
.998
846.
887
7.78
74
336
.177
NA
5050
0.00
90.
005
10.
51
0.7
2014
42.4
0516
160.
062
0.01
10.
007
19.2
613
1471
96.8
7311
71.8
9619
.261
314
267.
105
NA
5050
0.00
70.
004
10.
51
0.7
2014
16.7
527
70.
052
0.00
50.
006
8.17
32
571
26.7
8669
7.16
88.
173
25
72.1
43N
A50
500.
005
0.00
31
0.5
10.
720
148.
743
44
0.04
0.00
30.
007
4.72
92
271
50.8
3767
0.58
44.
729
22
32.4
41N
A50
500.
009
0.00
91
11
0.7
2014
42.4
0516
160.
047
0.01
50.
009
21.9
854
1671
80.5
4513
96.2
2221
.985
416
293.
744
NA
5050
0.00
70.
007
11
10.
720
1416
.752
77
0.04
30.
009
0.00
78.
663
35
7235
.986
714.
165
8.66
33
571
.253
NA
5050
0.00
50.
005
11
10.
720
148.
743
44
0.04
30.
005
0.00
64.
812
22
7151
.04
669.
145
4.81
22
232
.054
NA
5050
0.00
90.
014
11.
51
0.7
2014
42.4
0516
160.
051
0.01
80.
007
23.7
1412
671
63.7
1116
59.9
823
.739
135
110.
147
NA
5050
0.00
70.
011
11.
51
0.7
2014
16.7
527
70.
040.
007
0.00
79.
076
44
7151
.416
728.
684
9.07
64
448
.279
NA
5050
0.00
50.
008
11.
51
0.7
2014
8.74
34
40.
043
0.00
50.
006
4.93
62
271
47.8
2467
2.65
74.
936
22
32.0
3N
A50
500.
009
0.00
51
0.5
10.
520
1037
.416
1616
0.04
0.01
30.
007
15.2
753
1573
94.8
7310
42.3
4415
.275
315
228.
07N
A50
500.
007
0.00
41
0.5
10.
520
1014
.781
77
0.04
50.
005
0.00
76.
865
25
7432
.689
691.
068
6.86
52
564
.997
NA
5050
0.00
50.
003
10.
51
0.5
2010
7.71
54
40.
040.
004
0.00
74.
123
22
7435
.399
692.
175
4.12
32
221
.595
NA
5050
0.00
90.
009
11
10.
520
1037
.416
1616
0.05
20.
012
0.00
717
.364
416
7412
.284
1217
.044
17.3
644
1623
5.59
1N
A50
500.
007
0.00
71
11
0.5
2010
14.7
817
70.
061
0.00
50.
006
7.28
93
574
42.2
370
1.55
87.
289
35
57.6
58N
A50
500.
005
0.00
51
11
0.5
2010
7.71
54
40.
042
0.00
40.
007
4.17
92
274
03.9
6966
6.06
74.
179
22
21.5
9N
A50
500.
009
0.01
41
1.5
10.
520
1037
.416
1616
0.06
10.
013
0.00
719
.213
418
7229
.082
1396
.555
19.2
445
1627
2.76
2N
A50
500.
007
0.01
11
1.5
10.
520
1014
.781
77
0.05
10.
010.
006
7.69
53
573
45.3
571
6.75
47.
695
35
63.0
33N
A50
500.
005
0.00
81
1.5
10.
520
107.
715
44
0.04
70.
005
0.00
94.
269
22
7587
.133
686.
645
4.26
92
221
.596
NA
7 APPENDIX 23Ta
ble
9:Th
epa
ram
eter
san
dco
rres
pond
ing
min
imal
cost
sfo
rth
eBC
and
the
Myo
pic(
R)p
olic
yw
ith
the
com
puta
tion
tim
esof
the
seve
rals
earc
hm
etho
dsan
dth
eeq
uilib
rium
fact
orα
.
PAR
AM
ETER
SBa
seC
ase
Myo
pic(
R)
Myo
pic(
R)
Equi
libri
um
N1
N2
λ1
λ2
µ1
µ2
h 1h 2
b 1b 2
Cos
tsS 1
S 2T C
ET S
T FC
osts
S 1S 2
T CE
T SC
osts
S 1S 2
T CD
α
5050
0.00
90.
005
10.
51
0.5
8040
56.3
2928
280.
044
0.03
0.01
121
.547
524
7543
.511
2121
.188
21.5
475
2465
7.39
5N
A50
500.
007
0.00
41
0.5
10.
580
4020
.591
1111
0.04
0.01
10.
019.
404
47
7542
.011
1124
.063
9.40
44
711
3.56
9N
A50
500.
005
0.00
31
0.5
10.
580
4010
.713
66
0.04
80.
005
0.01
5.76
34
7563
.225
871.
006
5.76
34
50.9
62N
A50
500.
009
0.00
91
11
0.5
8040
56.3
2928
280.
042
0.02
20.
012
24.8
356
2775
63.0
5426
72.5
7824
.835
627
665.
137
NA
5050
0.00
70.
007
11
10.
580
4020
.591
1111
0.04
0.00
90.
007
9.98
34
875
74.3
9711
25.4
089.
983
48
139.
936
NA
5050
0.00
50.
005
11
10.
580
4010
.713
66
0.05
10.
005
0.00
75.
848
34
7543
.598
860.
553
5.84
83
448
.977
NA
5050
0.00
90.
014
11.
51
0.5
8040
56.3
2928
280.
040.
024
0.00
827
.885
631
7626
.718
3280
.902
27.9
017
2974
8.54
NA
5050
0.00
70.
011
11.
51
0.5
8040
20.5
9111
110.
041
0.00
90.
007
10.6
455
775
55.3
4311
34.4
2310
.645
57
118.
534
NA
5050
0.00
50.
008
11.
51
0.5
8040
10.7
136
60.
050.
004
0.00
65.
983
475
63.5
6886
8.14
55.
983
444
.84
NA
100
500.
009
0.00
92
11
0.9
2018
50.8
519
160.
050.
021
0.00
923
.463
414
3413
9.73
672
89.7
5823
.463
414
1406
.957
NA
100
500.
007
0.00
72
11
0.9
2018
19.1
117
70.
062
0.01
0.00
99.
467
35
3413
1.01
242
85.6
799.
467
35
384.
294
NA
100
500.
005
0.00
52
11
0.9
2018
9.82
24
40.
050.
006
0.00
85.
344
22
3431
8.43
740
45.3
475.
344
22
206.
195
NA
100
500.
009
0.01
82
21
0.9
2018
50.8
519
160.
056
0.01
80.
008
26.8
255
1534
655.
968
8650
.188
26.8
296
1419
05.3
23N
A10
050
0.00
70.
014
22
10.
920
1819
.111
77
0.05
20.
008
0.00
910
.068
44
3344
0.68
443
22.5
4510
.068
44
346.
776
NA
100
500.
005
0.01
22
10.
920
189.
822
44
0.04
80.
005
0.01
5.45
62
233
427.
998
4001
.083
5.45
62
220
6.64
8N
A10
050
0.00
90.
027
23
10.
920
1850
.85
1916
0.08
60.
021
0.01
226
.597
154
3387
6.96
410
024.
321
26.5
9715
456
0.75
4N
A10
050
0.00
70.
021
23
10.
920
1819
.111
77
0.05
0.00
90.
0110
.212
53
3377
7.68
344
29.7
8210
.212
53
293.
285
NA
100
500.
005
0.01
52
31
0.9
2018
9.82
24
40.
065
0.00
50.
012
5.61
62
233
861.
641
4049
.498
5.61
62
220
5.68
2N
A10
050
0.00
90.
009
21
10.
980
7275
.062
3228
0.05
10.
039
0.01
133
.291
623
3389
4.04
211
625.
762
33.2
916
2336
11.3
45N
A10
050
0.00
70.
007
21
10.
980
7226
.46
1111
0.05
90.
011
0.01
12.8
814
734
509.
803
6792
.306
12.8
814
776
9.06
7N
A10
050
0.00
50.
005
21
10.
980
7213
.62
66
0.05
30.
007
0.00
87.
513
333
926.
595
5170
.887
7.51
33
284.
681
NA
100
500.
009
0.01
82
21
0.9
8072
75.0
6232
280.
048
0.03
50.
011
38.6
678
2533
807.
278
1584
8.56
438
.667
825
5117
.674
NA
100
500.
007
0.01
42
21
0.9
8072
26.4
611
110.
060.
013
0.00
913
.77
57
3396
0.57
467
04.6
9613
.816
66
600.
709
NA
100
500.
005
0.01
22
10.
980
7213
.62
66
0.04
80.
010.
009
7.61
53
3N
A51
29.1
437.
615
33
250.
007
NA
100
500.
009
0.02
72
31
0.9
8072
75.0
6232
280.
060.
037
0.01
37.8
124
6N
A18
458.
255
37.8
124
692
3.97
9N
A10
050
0.00
70.
021
23
10.
980
7226
.46
1111
0.06
0.01
10.
0113
.915
75
NA
6473
.239
13.9
157
547
4.94
8N
A10
050
0.00
50.
015
23
10.
980
7213
.62
66
0.05
20.
006
0.01
7.79
34
3N
A48
94.4
657.
793
43
201.
787
NA
100
500.
009
0.00
92
11
0.7
2014
45.8
6119
160.
069
0.01
90.
014
19.5
33
15N
A63
35.8
9819
.533
414
1151
.379
NA
100
500.
007
0.00
72
11
0.7
2014
17.1
47
70.
054
0.00
80.
012
8.20
83
5N
A40
02.0
478.
208
35
383.
28N
A10
050
0.00
50.
005
21
10.
720
148.
793
44
0.05
60.
005
0.01
34.
737
22
NA
3830
.791
4.73
72
219
6.61
NA
100
500.
009
0.01
82
21
0.7
2014
45.8
6119
160.
052
0.01
60.
009
22.2
834
16N
A74
56.5
3622
.283
416
1515
.413
NA
100
500.
007
0.01
42
21
0.7
2014
17.1
47
70.
056
0.00
80.
009
8.70
23
5N
A40
92.1
098.
702
35
423.
573
NA
100
500.
005
0.01
22
10.
720
148.
793
44
0.06
80.
004
0.00
84.
822
22
NA
3912
.275
4.82
22
219
8.11
1N
A10
050
0.00
90.
027
23
10.
720
1445
.861
1916
0.06
60.
027
0.00
924
.01
126
NA
8744
.257
24.0
2113
551
5.74
9N
A10
050
0.00
70.
021
23
10.
720
1417
.14
77
0.05
50.
022
0.00
99.
116
44
NA
4207
.436
9.11
64
429
9.55
5N
A10
050
0.00
50.
015
23
10.
720
148.
793
44
0.05
30.
005
0.00
94.
947
22
NA
3816
.172
4.94
72
219
6.48
3N
A10
050
0.00
90.
009
21
10.
780
5667
.552
3228
0.06
50.
030.
011
27.5
515
24N
A11
000.
213
27.5
515
2433
28.3
69N
A10
050
0.00
70.
007
21
10.
780
5623
.714
1111
0.05
0.01
10.
011
11.1
594
7N
A63
26.5
1911
.159
47
664.
514
NA
100
500.
005
0.00
52
11
0.7
8056
12.1
926
60.
058
0.00
60.
009
6.65
93
4N
A48
79.2
846.
659
34
309.
986
NA
100
500.
009
0.01
82
21
0.7
8056
67.5
5232
280.
047
0.03
20.
012
31.9
436
27N
A14
181.
968
31.9
436
2736
26.5
NA
100
500.
007
0.01
42
21
0.7
8056
23.7
1411
110.
051
0.01
40.
012
11.9
885
7N
A63
30.8
9611
.988
57
635.
589
NA
100
500.
005
0.01
22
10.
780
5612
.192
66
0.06
20.
006
0.01
26.
744
33
NA
4865
.622
6.74
43
323
8.86
1N
A10
050
0.00
90.
027
23
10.
780
5667
.552
3228
0.05
0.03
30.
012
34.7
7621
8N
A17
243.
773
34.7
922
710
16.4
54N
A10
050
0.00
70.
021
23
10.
780
5623
.714
1111
0.06
30.
011
0.01
12.4
976
6N
A63
81.6
812
.501
75
389.
848
NA
100
500.
005
0.01
52
31
0.7
8056
12.1
926
60.
052
0.00
60.
009
6.91
23
3N
A49
14.7
576.
912
33
236.
097
NA
100
500.
009
0.00
92
11
0.5
2010
40.8
7219
160.
052
0.01
80.
009
15.1
633
15N
A57
90.1
9815
.163
315
1109
.568
NA
100
500.
007
0.00
72
11
0.5
2010
15.1
77
70.
059
0.00
70.
012
6.82
25
NA
3934
.552
6.82
25
388.
14N
A10
050
0.00
50.
005
21
10.
520
107.
764
44
0.05
0.00
40.
009
4.13
12
2N
A38
10.1
014.
131
22
134.
207
NA
100
500.
009
0.01
82
21
0.5
2010
40.8
7219
160.
065
0.02
40.
0117
.419
416
NA
6644
.105
17.4
194
1611
91.9
78N
A10
050
0.00
70.
014
22
10.
520
1015
.17
77
0.04
90.
008
0.01
7.40
43
5N
A40
60.7
367.
404
35
339.
79N
A10
050
0.00
50.
012
21
0.5
2010
7.76
44
40.
082
0.00
40.
014
4.18
82
2N
A38
06.0
694.
188
22
134.
052
NA
100
500.
009
0.02
72
31
0.5
2010
40.8
7219
160.
068
0.01
70.
013
19.4
824
18N
A77
53.1
7619
.486
517
1389
.161
NA
100
500.
007
0.02
12
31
0.5
2010
15.1
77
70.
068
0.00
70.
012
8.00
13
6N
A42
04.1
138.
001
36
410.
312
NA
100
500.
005
0.01
52
31
0.5
2010
7.76
44
40.
051
0.00
50.
012
4.27
82
2N
A38
25.5
214.
278
22
134.
171
NA
100
500.
009
0.00
92
11
0.5
8040
60.0
4132
280.
062
0.02
90.
012
21.4
075
23N
A10
130.
069
21.4
075
2327
71.8
42N
A10
050
0.00
70.
007
21
10.
580
4020
.969
1111
0.05
0.01
0.01
49.
314
7N
A62
11.0
389.
314
757
7.06
7N
A10
050
0.00
50.
005
21
10.
580
4010
.763
66
0.06
0.00
60.
012
5.76
63
4N
A49
31.1
835.
766
34
309.
913
NA
100
500.
009
0.01
82
21
0.5
8040
60.0
4132
280.
048
0.03
80.
0124
.874
627
NA
1310
9.55
124
.874
627
2898
.722
NA
100
500.
007
0.01
42
21
0.5
8040
20.9
6911
110.
049
0.01
0.00
810
.105
48
NA
6261
.44
10.1
054
861
7.32
4N
A10
050
0.00
50.
012
21
0.5
8040
10.7
636
60.
058
0.00
60.
008
5.85
53
4N
A48
90.5
085.
855
34
274.
42N
A10
050
0.00
90.
027
23
10.
580
4060
.041
3228
0.04
90.
028
0.01
128
.14
730
NA
1633
0.89
828
.14
730
3547
.839
NA
100
500.
007
0.02
12
31
0.5
8040
20.9
6911
110.
063
0.01
10.
009
10.9
645
8N
A63
39.3
2110
.964
58
624.
86N
A10
050
0.00
50.
015
23
10.
580
4010
.763
66
0.05
50.
006
0.00
95.
988
34
NA
4928
.95
5.98
83
427
4.35
1N
A