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OPTIMALIT Y OF ZERO -IN VENTORYPOLIC IE S FOR UNRELIA BL EMANUFAC TURING SY STEM S

T. Bielecki*and P. R. Kumar!October 15, 1999

AbstractWe showthat there are ranges of parameter valuesdescribing an unre-

liable manufacturing system for which zero-inventory policies are exactlyoptimal even when there is uncertainty in manufacturing capacity. Thisresult may be initially surprising since it runs counter to the argumentthat inventories are buffers against uncertainty and that therefore onemust strive to maintain a strictly positive inventory as long as there isany uncertainty. However, there is a deeper reason why this argumentdoes not hold, and why a zero-inventory policy can be optimal even in thepresence of uncertainty. This provable optimality reinforces the case forzero-inventory policies, which is currently made on the separate groundsthat it enforces a healthy discipline on the entire manufacturing process.

In recent years, the goals of "zero-inventory" and "stockless production" haveattracted much attention (Hall 1983). Such policies enforce a strict disciplineon the entire manufacturing process that has several beneficial consequences forthe overall production system.

In this paper, we examine the question of whether there are any conditionsunder which a zero-inventory policy is actually optimal. In this connection itshould be noted that it is sometimes argued that a zero-inventory policy onlypasses the costs from, say, the assembly line to the parts manufacturer, andthat, therefore, any advantages accruing from enforcing such a policy are onlyside benefits resulting from the greater discipline governing the system, and notdirectly from the policy itself. After all, the reasoning goes, positive inventoriesare used as a buffer against uncertainties in supply or demand, and so as long asthere is any uncertainty whatsoever in the system, one should strive to maintain

* (Please address all correspondence to the second author below). Main College ofPlanningand Statistics, Warsaw, Poland.

tDepartment ofElectrical and Computer Engineering and Coordinated Science Laboratory,University of Illinois, 1101W. Springfield Avenue, Urbana, Illinois 61801/USA.

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a strictly positive buffer. Thus, zero inventory levels can only be optimal whenthere is no uncertainty at all, and that is never possible!

In contrast to this conclusion, we show in this paper that there are con-ditions under which a zero-inventory policy is actually p ro va bl y o pt ima l, evenwhen there is uncertainty, and that furthermore such optimality of the zero-inventory policy results whenever the system is designed to be sufficiently, butnot necessarily perfectly, efficient.

The model with respect to which we exhibit these conclusions is that of asimple failure prone manufacturing system producing a commodity. The timebetween failures is random and modeled as an exponentially distributed randomvariable with mean l/Ql, while the repair time is exponentially distributed withmean l/Qo. When functioning, the system can produce at any rate up to amaximum of r units/time; while broken down it cannot produce at all. At alltimes the commodity is being depleted at a demand rate of d units/time. Thetotal inventory can be negative, which just corresponds to a backlog. Positiveinventories are assessed a cost at a rate of c+ dollars per unit commodity perunit time, while negative inventories are assessed a similar cost of c". We seekthe optimal production policy for this system which minimizes the long runaverage expected cost incurred per unit time,

(1 )where x(t) = inventory level at time t (see below), x+ = max(O, x) is its positivepart, and x- = max(O, -x ) its negative part. Let u(t) be the production rateat time t, then clearly,

x(t) f a t (u(s) - d)dsis the inventory at time t, and let I(t) = 1 or 0, respectively, depending onwhether the system is functioning or not. Clearly, u(t) = whenever I(t) =0, and so we only need to determine the optimal production rate when themanufacturing system is up, i.e., in state I(t) = 1.

We show that there is a number z* , called the o ptim al in ve nto ry le ve l, towardwhich the production should be aimed, i.e.,

u(t) if x (t)z* , I(t) = 1difx(t) = z *, I (t) = 1rifx(t)z*,I(t) = 1.

(2 )(3 )(4 )

Thus, if current inventory x(t) exceeds z* , the system should produce nothing; ifx(t) is less than the optimal level z* , the system should produce at the maximumrate r; if the inventory level exactly equals z* , then the system should produceexactly enough to meet demand and thereby keep the inventory level at z* .

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The quantity z*, the optimal inventory level, is therefore key to the wholeproblem, and it is explicitly given by:

z* (l if rql(c++C-) 1 dr-dd1 < an ---c+(r - d)(qo + qd - ql q or-dd+ 00 if ---ql qo

1 1 [ rq l (c + + c-) ]((qo/dd) - (qI/(r - d))) og c+(r - d)(qo + qdotherwise.

(5 )(6 )

(7 )Note that l/ql is the mean up-time and so r - d f q, is the maximum total

production in a mean up-time. Similarly l/qo is the mean down-time and sod/ qo is the total depletion in a mean down-time. So if r - d] ql < d/ q o, then thesystem does not have the capacity to meet demand even if it produces at fullcapacity when up. Thus, z* = + 00 under these conditions.

So consider only the case r - d f q, > d/qo. Then we have the interestingsituation that the optimal inventory level z* is exactly 0 whenever


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The average inventory cost (1) corresponding to this policy, which we denoteby J(z), can then be computed as the expected cost with respect to the steadystate distribution, i.e., it can be written as,

J(z) (13)We will shortly minimize J(z) over z to obtain z *.

We claim first thatPZ((z, 00), i) = 0 for i = 0, 1

PZ({z} , l ) =: 0: > O .(14a)(14b)

Note from (11 and 12) that if the inventory level XZ (0) starts with a value largerthan z, then it is depleted at rate d until it hits z. Thereafter (10 and 12) ensurethat it never again rises above Z ; see Figure 1. Hence, XZ(t) ~ z for all t large

Figure 1: Comparison of inventory behavior under policies with z and z + ,,(,when xz+,y (0) = XZ (0) + "(.enough, and so (14a) follows. To see (14b) which asserts that there is a strictlypositive pro ba bility m ass at z, note that whenever x(t) hits z with I(t) = 1, itstays at z until Iswitches to O . So x(t) does spend a positive fraction of timeat exactly the level z, which implies that the point z has positive probabilitymass, giving (14b).Lastly, let us make the reasonable assumption that except for the massP" ({z}, 1), the distribution P" (dx, i) has a probability density function pi(x).Hence QZ(dx) also has a probability density function qZ(x), except for the massQ Z ( {z }) at z.

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The property (14b) is key to the optimality of a zero-inventory policy. Bychoosing z = 0 one can ensure that with probability the system has zero inven-tory, and therefore w ith probability one is paying no inventory cost. Of course,we must balance against this the possibly extra cost incurred by choosing z = O .If is la rg e e nough, then the advantages will outweigh the disadvantages and soa zero inventory policy with z = 0 is optimal. This is at heart the main reasonfor optimality of zero inventory policy.

To decide when is large enough to make it advantageous to choose z = 0, weneed to compare costs with z = 0 and z - I - O . For doing this it is helpful to notethat translating z merely translates the probability distribution P"; that is

(15)where by the notation A + "(we mean the set A translated by ,,(, or

A+,,(:= {x:(x-"()EA}.To see (15), suppose that {Td are the switching times of I(t), i.e.,

I(t) 0 for T2n ~ tT 2n+l1 for T2nH ~ tT2n+2.

If {XZ ( t) , It)} and {xz+,y ( t) , It)} are the processes corresponding to the choicesof z and z + ,,(,with in itia l c ho ic e of

then from (9-12) we will have

xZ+'Y(t) = XZ(t) +"( for all t 2 o .Again, see Figure 1. Since the paths are translates of each other, so are thestationary probability distributions.

We can now optimize J(z) quite readily. From (15) it follows that movingfrom a choice of 0 to a negative value of z merely shifts the distribution pOleftwards by z units (thereby resulting in PZ). From (14a) it follows that

J(z) - J(O ) = c-Izl for z ~ o .Hence, the optimal z will never be negative.

Now compare the advantage in going from a choice of 0 to a positive valueof z. There are three components; translating the probability distribution on(-00, -z ) to (-00,0) yields an advantage in cost of c-zQO( -00, -z), the move-ment of the probability mass from 0 to z costs ac+ z, and translating the distri-bution over (-z,O) to (O,z) yields a difference of

j_z [c-Ixl- c+(x + z)]QO(dx).6


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So,J(z) - J(O) [ac+ z - c: zQO(( -00, -z))]

+ f _ : (c+(x + z) + c-x)QO(dx)],forzO.

(16)

The derivative at 0 isddz (J(z) - J(O ))]z=o = ac+ - c-QO(( -00,0)).

For a zero-inventory policy to be optimal, it is necessary that this be positive,which happens when

(17)i.e., the cost of the probability mass at 0 is large enough. Moreover, for z > 0,(djdz)[J(z) - J(O)] can be computed to be,

ddz [J(z) - J(O)]zo= ac+ - (1- a)c- + (c+ + c-)QO (( -z, 0)),

which is also positive when (17) holds. Hence (17) is both necessary and suffi-cient for z = 0 to yield a minimum of J(z).

The basic reason for the optimality of an exactly zero inventory policy shouldnow be intuitively clear.

Using QO(( -00,0)) = 1 - a, we can rewrite (17) asa(c++c-) 2 '. (18)

Next, we need to compute a. This is done by determining p i (-) and a insteady state. (Recall that p i (-) is the density function of P" (dx, i) in (-00, z)).Let P:'\) and at denote the corresponding quantities at time t. Consideringthe probabilities at time (t + h), for small h > 0, we have:!~,t+h(x)dx = (1- qOh)! p~ ,t(y)dyA A+hd

+q1h! p~,t(y)dy+o(h)A-(r-d)hip~ ,t+h(X)dX qOh! p~ ,t(y)dyA+hd+(1- q1h)! p~ ,t(y)dy +o(h),A-(r-d)h

for z f j_ Aat+h (1_ qlh)at + j Z p~,t(y)dy + o(h).z-(r-d)h

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of an infinitesimal operator. Finally, (29d) denotes a stability property that wewant 7f to possess. Here and in what follows, E", denotes the expected valuewhen a policy 7f is used.

We now have the following theorem which gives sufficient conditions of theDynamic Programming type for a policy to be optimal with respect to theaverage cost criterion. In what follows we use the notation, c(x) :=c+x+ +c"z ".Theorem. Suppose 7f* is an admissible stable policy, and there exist continu-ously differentiable functions W(x, i) and a constant J* such that

[ *( )_dJdW(x,i)7f x, Z dx-qdW(x, i) - W(x, 1 - i)] + c(x) - J* = 0

for i= 0, 1 and all x[ *( 1) d J dW(x, 1) _ . [ d J dW(x, 1)7f x, - - mm u-dx O : : ; u : : ;r dx

IW(x, i)1 ~ k1x2 + k2 for some i..:(30a)(30b)(30c)

Then1 i T 1 i Tim -E",* c(x(t))dt = J* ~ limooT--+oo-E", c(x(t))dtT--+ooToT 0

for all admissible stable 7f and all initial conditions x(O).

Proof: The proof uses the (main) Dynkin formula, see Karlin and Taylor(1981). To compute the infinitesimal operator, let f(x, i) be continuously dif-ferentiable in x and vanishing outside a compact set. Then for an admissible 7fwe have

E",[f(x(h),J(h))lx(O) = x,J(O) = i]Prob(l jump of Iin (0, h))E",[f(x(h),J(h))1x(O) = x,1(O) = i, 1 jump of Iin (0, h)]+Prob(O jumps of Iin (0, h))E",[f(x(h),J(h))1

x(O) = x,1(O) = i, 0 jumps of Iin (0, h)] + o(h)qih I o h f (x + 1 0 8 if(y, i)dy+ l h if(y, 1- i)dy) ~+(1 - qih)f(x + I o h if(y, i)dy) + o(h),where if :=7f - d

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= qd(x, 1- i)h + (1- qih) [f(X , i) + df~~ i) f a h 7f(Yi , i )dy 1+o(h) for somex ~ ~ ~ x + h.Using (29c), we get

1. E7r[f(x(h),J(h))lx(O) = x,J(O) = i]-(x,i)IIfl _.::____:____:___:__c____:____:__:_:___:__:__---'----__:___:__--=---_:__'------'-----'-h t o h= df~~ i) [7r(x, i) - d ] + qdf(x, 1- i) - f(x, i)].

From Dynkin's Formula we have,

E7r [ f a T {df(X(i; I(t)) [7r(x(t),J(t)) - d ]+qI( t) [ f(x( t) , 1-(t)) - f(x(t),J(t))]} dt

= E7r[f(x(T),J(T )]- f(x(O ),J(O )).Now we apply the above formula to W noting that since Ix(t)1 is bounded byIx(O)I+ (r + d)T for 0 ~ t ~ T we can modify W outside this range so that itvanishes outside some compact set. So we have

E7r [ f a T {dW (X~;, I(t)) [7r(x(t), I(t)) - d ]+qI( t) [W(x( t) , 1- I(t)) - W (x(t), I(t))]dt

= E7r[W (x(T),J(T))]- W (x(O ),J(O )).From (30a and 30b) it follows that,

E7r f a T c(x(t))dt 2 T J* + W(x(O),J(O))-E7r[W(x(T),I(T))].

Dividing by T, taking the limit and using (30c) and (29d) to see thatE7r [W (x(T), I(T) )]

T -+ 0,yields

. E7rJoT c(x(t))dt *lim OOT--+oo > J .T -A similar computation with 7r* yields equality.

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We need to find W( , ) and J * which satisfy the theorem. For this wetake advantage of the fact that the average cost can often be obtained as thelimit of the discounted cost problem (8), which has been solved in Akella andKumar. We exclude the trivial case (6 ) where (r - d)/ql < d/qo and computethe limits J* := limy.j.o)'W')'(x, i); W ( x , i) := l im')' .j.o[W')'(x, i) - W')'(z*, 1)] and7r*(x,i):= lim')'.j.07r')'(x,i),where W')' is the optimal discounted cost obtained inAkella and Kumar and 7r)' is the corresponding optimal policy. The results ofthese computations are summarized in Tables I and II.

J*

W ( x )

7r*(x , i)= d if x = 0,i= 1

= 0 otherwise.

Table 1: Computation Summary for Case I:

r if x O , i= 1

In both Cases I and IIit can be verified (the latter being much more tedious)that except possibly for the admissibility and stability of 7r*,the given J *, W ,and 7r*satisfy (30a, 30b, and 30c). Turning to the verification of admissibilityof 7r*, the conditions (29a) and (29c) pose no problem, while condition (29b)is proved to hold in Akella and Kumar. Therefore, we need only to prove thestability of 7r*,that (29d) holds; then 7r*will be proved to be both admissible andstable, and consequently optimal through the use of the Verification Theorem.

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3 Stability of Optimal PolicyWe intend to prove the stability of 7[*, i.e., to show that

lim -T 1E",* (x 2(T)) = O.T--+oo (31)This condition essentially states that the inventory grows more slowly than thesquare-root of time, and it is clear that any reasonable policy that we wishto implement will have to possess this property. Here, we need to prove thisstability property in order to invoke the sufficient conditions for optimality givenin the Theorem of the preceding section.

Just for clarity, we focus on the z* = 0 case; the details are similar in thecase of z* > O.

Let {r-} be the sequence of stopping times at which the process (x(t),J(t))hits (0,0), that is

TO inf{t 2 0 : x(t) = O,J(t) = O},Tn+! inf{tTn : X(t) = O,J(t) = O},

(with inf > :=+00). These form renew al tim es for the process (x , I). LetF(s) := PrOb(Tn+! - Tn ~ s)

be the common probability distribution ofthe time between successive renewals,and let

N(t) := sup{n: Tn ~ t}be the number of renewals which have taken place prior to time t. The quantity

8(t) := t - TN(t)is the current life which denotes the elapsed time since the last renewal. Sincex( TN(t)) = 0 and the inventory cannot deplete at a rate faster than d, we have

o 2 x(t) 2 -8(t)dand so

Hence to show (31), it suffices to show that

(32)We therefore need to examine the second moment of the current life randomvariable 8(T).

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Fix l and consider a typical renewal period, which is the time interval[ T l, T I + 1 ] . Let

am+ 1T I( note that I(ao) = 0)inf{ t > am : I (t) - I - I(am)

(33)(34)

be the successive times at which Iswitches, and let(35)

be, respectively, the lengths of times which Ispends in states 0 and 1 beforea switch occurs. Note that {Ym} is a sequence of independent and identicallyexponentially distributed random variables with mean l/qo, while the {Zm} areindependent and exponentially distributed with mean 1/q1. When I= 0 theinventory depletes at rate d; when 1= 1 the inventory builds up at rate (r - d)(until the inventory returns to 0, whereupon it stays at 0). So

Hence if

then

min { a 2m : ~ [ ( r - d)Z i - dY i] > o } + Zm*H.Since

r-d d---- 20,q1 qothe Law of Large Numbers shows that TIH < +00 a.s. Hence F(s) is a properdistribution.

In what follows, we will assume without loss of generality that TO = O. LetGT(s) := Prob(8(T ) ~ s)

be the probability distribution ofthe current life at time T. From renewal theory(see Karlin and Tyalor, p. 193, 1975) we know that

GT(s) F(T) _foT-s[l_ F (T - y)]dm (y)sT1 s 2 T,

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wherem(t) := E[N(t)]

is the mean number of renewals. Hence we may compute the second moment of8(T) as follows,

E [8 2 (T)]1 0 00 s2dGT(s)T2[1- F (T)] + I o T- s2dGT(s)T2[1- F (T)] + I o T- 1 0 8 2ydydG T(s)T2[1- F (T)] + I o T- 2y I o T- dG T(s)dyT2[1- F (T)] + loT 2y[GT(T-) - GT(y)]dyT2[1 - F (T)]+ loT 2y I o T-Y [1- F (T - z)]dm(z)dyT2[1 - F (T)]+ loT (T - z)2[1 - F (T - z)]dm(z). (36)

We now intend to apply the Key Renewal Theorem (see Feller 1968) to obtainthe limit of (36) as T -+ 00. To do so, we need to show that

F is not lattice. (37a)lim T2[1 - F(T)] = O. (37b)T--+oo

The distribution function F has finite mean. (37c)The function t2[1 - F (t)] is directly Riemannintegrable. (37d)

In the lemma which follows we will prove all four properties. Then, uponapplying the Key Renewal Theorem to (36), we will have1 1 00lim E7r* [82(T)] = - t2[1- F(t)]dt < +00.T--+oo M 0So (32), and the stability property (31) will follow readily. To complete thisargument, we thus need the following lemma.Lemma. The properties (37a-37d) hold.

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Proof: We will prove two facts which are sufficient for the lemma.F(t) is continuous.

There exists k and 8 > 0 such that(38a)

1- F(t) < ktHO" for all large t. (38b)Clearly (37a) follows from (38a); (37b) follows from (38b); and (37c) follows from(38b) through the fact that the mean of F can be computed as 1 0 00[1- F(t)]dt.Finally, (37a) shows that t2 [1 - F(t)] is Riemann integrable over every finiteinterval, and (38b) is then a sufficient condition for direct Riemann integrability(see Feller).

To prove both (38a) and (38b), let Tl := 0 and define {am,Ym,Zm} as in(35). Let x(t) = -d for a2m ~ ta2m+1 and x(t) = r - d for a2m-1 ~ ta2m,which are, respectively, the intervals in which I(t) = 0 and I(t) = 1.

For a 2 0, defineF(t,a) := Prob(x(s) 2 a for some 0 < s ~ t)

be the probability distribution of the stopping time when x first rises to a levela. Clearly,

F(t, a) a= 0 for t < --,-r-dsince x cannot rise more rapidly than at rate (r - d ). By conditioning on aland a2 we get the renewal equationc:oe-qoO"ldalox 10 qIe-q1 (0"2-0"1)d(a2 - ada+O"ld/r-dF(t, a ) c:qoe-qOO"ldalola + O " l d)/(r-d)X qIe-q1 (0"2-0"1)o

xF (t - a2, a + a2 d + aIr - a2r)d(a2 - ad. (39)Let the right hand side above be defined as (HF)(t,a), i.e., the action of anoperator H acting on functions F(x,y) with x ~ t. Clearly,

sup I(H f)(x,w) - (H g)(x,w )1x"S,tw

~ [1 - e-qo t] sup If(x, w) - g(x, w)l,m::;tw

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and is therefore a contraction with respect to the sup norm. Moreover, H pre-serves continuity and so the unique solution F of (39) is continuous. Noting thatF(t) is the convolution of F(t,O), with a mean l/ql exponential distribution,we have proved (38a).

To prove (38b), note that by the Principle of Large Deviations (Varadhan1982) the quantity

1-p ( I ~t ( l i - : 0 ) I ~ Eand I ~ t(, - ~) I ~ 8)

decreases exponentially in n. This implies that

n

i=l

also decreases exponentially in n. Sincer-d d-->-,ql qo

there exist E, 8 > so thatWith such a choice we see that

1- p (t(li + Zi) ~ n ( 2 _ + 2 _ + E + 8 )i=l qo ql

and t ( ( r - d)Zi - dli) > 0 ) ,decreases exponentially in n. This implies that so does

and therefore also [1- F(n)], proving (38b).

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4 CONCLUDING REMARKSWe have shown that zero-inventory policies can be optimal even when there isuncertainty in manufacturing systems. This adds support to the use of suchpolicies, which are currently the focus of much attention due to the greaterdiscipline that they enforce on the manufacturing system.

The particular problem for which we have obtained the solution is somewhatrestrictive. It would be useful to relax several of the features. For example, thedemand rate, which is considered a constant here, may be time-varying andfeaturing linear growth as well as seasonal and cyclical components. It is alsoimportant to take into account the change-over cost when switching productionrates. Also, it would be useful to consider more general batch manufacturingproblems.'

In a different direction, it is of much interest to examine the complex modelof a multi-machine, multi-product flexible manufacturing system featured inKimemia and Gershwin (1983) to show that there are ranges of mean times be-tween failures and mean repair times under which zero-inventory policies remainoptimal.

ACKNOWLEDGEMENTSThe authors are grateful to the editor, associate editor and the two referees for anumber of constructive suggestions including those in Section ?? The researchof the second author has been supported by the U.S. Army Research Officeunder Contract No. DAAG-29-85-K0094, and by the Joint Services ElectronicsProgram under Contract No. NOOOI4-85-C0149. The work of the first authorwas supported by a Fulbright Scholarship while he was visiting the Universityof Illinois.

REFERENCESAkella, R, and P. R Kumar. 1986. Optimal Control of Production Rate

in a Failure Prone Manufacturing System. IEEE Trans. on AutomaticControl, AC-31, 2, 116-126.Feller, W. 1968. An Introduction to Probability Theory and Its Applications,

Vol. II, John Wiley & Sons, New York.Hall, R W. 1983. Zero Inventories, Dow Jones-Irwin Press. Homewood, Ill.Karlin, S., and H. M. Taylor. 1975. A First Course in Stochastic Processes,

Academic Press, New York.IThe authors are grateful to a reviewer for these suggestions.

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Karlin, S., and H. M. Taylor. 1981. A Second Course in Stochastic Processes,Academic Press, New York.

Kimemia, J. G., and S. B. Gershwin. 1983. An Algorithm for the ComputerControl of Production in Flexible Manufacturing Systems. lEE Transac-tions, 15, 353-362.

Varadhan, S. R. S. 1982. Large Deviations and Applications, SIAM-CBMSPress, Philadelphia.

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Table 2: Computation Summary for Case II : c + ( r - d ) ( q o + q d - q l r ( C + + C - ) ~ 0

W ( x )

where

J*

A = [ r ' 1 _ ! _ d ; ! _ ~ ] . A = [ - 1 " ~ ] . b = [ r - = ,_ ld ] . b = [~]1 -qo 'lQ ' 2 'lQ -qo ' 1 l' 2 r=r d d =r Ii dand

z* = 1 1 [ r q l ( C + + C - ) 1( ~ - r ' 1 _ ! _ d ) og c + ( r - d ) ( q o + q d .

7r*(x,i) = riJxz*,i = 1=diJx=z*,i=l= 0 otherwise.

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