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Economic Control Chart Models with Cycle Duration Constraints

187

W T-l Q .    1 f •

Ź(—-y TT J V¹ e dv

₌ i M + ^e _

^    X+0 ^^vX+0^{; i!}

-v

Substitute: Jv’e^vdv = -e^v (i!) X tt :

j=0^J!

_ i-

»Xt-v r-1 i

•y y<—)*—

^    ^ + 6 M £o X+0 j!

Substitute back to original variables: t-x = v/(X + 0):

1 Xx _e^xi'^(x+e)(,‘^x) ^ X, 0    ; (X + 0)^j(t-x)^j

- - « —z Z(^e>-«-

X+0

i=0 j=0

Switch order of summation:

I Xx e^to-^9<,"°« m e ₀(>.+e)⁾(t-x)⁾ e —r-r-^ L<ttx)---

X + 0 j=o j=j X+0

j!

i Xx e^ta-«^M>£(X + 0)^j(t-x)ⁱ.£, 9 „

x+e

j! _i=j X. +0

^e    â€”L-1;-Z(rTS-)

j=0

Apply geometrie series formula:

Xx-0(t-x) r-1

I Xx _ ę_____

X ⁶    X + 0

£ (X + 0)l(t - x)l j=0 ^j!

f—T-

X + 0

x + 0

Simplify to obtain the finał expression: