INVENTORY MODELS WITH TWO CONSECUTIVE DELIVERY MODES
3.5. The Nonstationary Infinite-Horizon Problem
which contradicts (3.77). As a result, the case z^ > y^ does not arise. D The theorem says that in case (i), there are two base-stock levels—fast and slow—with the fast base-stock level y^ being smaller than the slow base-stock level z^. Moreover, when the inventory position is too low (that is, smaller than 2/p, then we order up to y^ via the fast mode and order an additional amount from yl up to z^ via the slow mode. On the other hand, when the inventory position is too high (that is, larger than z^), then we order nothing. Finally, if the inventory position is neither too low nor too high (that is, when it is between the levels y^ and z p , then we simply order up to z^ via the slow mode. In case (ii), there is only one base-stock level zj, and if the inventory position is too low (that is, smaller than z p , then we order up to z^ via the fast mode and order nothing via the slow mode. On the other hand, if the inventory position is too high (that is, larger than z p , we order altogether nothing.
the value function of the infinite-horizon problem is a solution of (3.79) and the decision that attains the infimum in (3,79) is an optimal nonanticipative policy.
Our method is that of successive approximation of the infinite-horizon problem by longer and longer finite-horizon problems.
Let us, therefore, examine the finite-horizon approximation J„fc(x„,5„_i, il^) of (3.78), which is obtained by the first /c-period truncation of the infinite- horizon problem. The objective function for this problem is to minimize
'Jn,k\^ni ^n—l^'^ni \-^ ^ ^)) n+k
= Hn(xn) + Y.^^'"^ \cl{Fi) + CaSe) + aHi+iiXe+i) (3.80) Let Vn^ki^m Sn-i,in) be the value function of the truncated problem—that is,
Vn,k{^n,Sn-uin) = _ mf \ Jn,k{Xn, Sn-l,ili, {F, S)) \ . (3.81)
{F,S)eAn,k I J Since (3.80) is a finite-horizon problem on the interval (n, n -|- A:), we can apply
Theorem 3.2 to prove that Vn^k(xn,Sn-iiin) satisfies the dynamic program- ming equations
= Hn+e{xn+e) + mf [cl^^i^) + Q+^(a-)
+aE Un+C+l,k-e~l{Zn+e+l{Xn+e + (f>),cr,In+i+l)\ p
i = 0,...,k-l, (3.82)
where
- Hn+k{xn+k) + inf [Cl^ki^) + C'^+ki<^)
-j-aE Hn+k+l{Zn+k+l{Xn+k + '^))j | , (3.83)
Zn+£+l{t) = t + Sn+e-l - gn+e{in+e^ ^n+h '^n+^)-
To get the optimal policy for the infinite-horizon problem, we assume that there exist constants c > 0 and M > 0 such that for all /c > 1,
\clixi) - Cl{x2)\ <c-\xi- X2\, (3.84)
\C'kM - Clix2)\ < c • |xi - X2I, (3.85)
\Hk{xi) - Hk(x2)\ <c-\xi- X2\, (3.86)
E[gkillllvk)] <M. (3.87) Furthermore, we assume that
Cl(t) + E[Hk+i{t- gkillJlvk))] - - 0 0 as t ^ 00, (3.88) CI(t) + E[Hk+i{t-gk{llllvk))] -^cx) as t - > o o , (3.89) uniformly hold with respect to k.
It follows from (3.84)-(3.89) that for any {xn, s„_i) and (x„, s„_i),
\^n,k\^ni ^n—1: '^ni \-^ 1 ^)) 'Jn,k\^ni ^n—li '^n^ \-^ 1 ^))\
n+k
<C-\Xn-Xn\ + Yl <^^""^^ (c ' ^ n - Xn\ -\-C • \Sn-l - S n - l | )
C
< -. ( k n - £n\ + \Sn~l - Sn-~l\) • (3.90)
i — a Therefore, we have
I »^n,fci^n?'^n—15 ^n/ ~" ^n,fc v^n?'^n—1? ^ n / |
< Z ( k n - ^ n | + \Sn-l ^ «§n-l|) • (3.91)
1 — a
T H E O R E M 3.6 Assume that (3Al (3.4)-(3.5), and i3M)--(3,S9) hold Then the limit ofVn^ki^nj ^n-i,^^) exists as k —^ oo. Letting the limit be denoted by V^{xn, Sn-i,in)> we have
(i) V^{xn^ Sn-i^i]i) is convex and Lipschitz continuous in (x^^ ^n-i) on
(—00, +(X)) X [O5 + o c ) ;
(ii) V^{xn^ Sn-i^in) is a solution of (3,19);
(iii) there exist functions Fji{xn'> 5^-1, i^) and Sn{xn^ ^n-i^^n) \^hich pro- vide the infima in (3.79) with U^{xn^ 5^-1, z^) == V ^ ( x ^ , Sn-i^i^). and
( F , 5 ) = {(Fn{Xn,Sn~l,in),Sn{Xn,Sn~l,in)), Tl > l}
is an optimal nonanticipative policy—that is,
Vr{xuso,i\) = J r ( x i , 5 o , i l , ( F , 5 ) )
inf {j^{xuSo,il{F,S))\
,S)eA I J
iF,S)eA
Proof First we show that there exists a function V^{xn^ Sn-i^in) such that
l i m Vn^k{Xn,Sn~l,in) = ^ ^ ^ ( X n , 5 ^ - 1 , ^ n ) ' ( 3 - 9 2 ) fc—>oo
Let
(3.93) attain the infimum on the right-hand side of (3.82)-(3.83). Note that
"^n+kK^n+ki ^n+k-lj^n+k) ~ ^'
Thus,
^n,k{^ni ^n—15 ^ n / ^ ^n.kv^ni ^n—li '^n? v-^ ^,AJ? ^n,k))
^ ^n,k—l\^nj ^n—1^ ^n? v-^ n,fc—1? *^n,/c—1 j j
> _inf < Jn
s))\
iF,S)eA ^ ^
= Vn,k-l{Xn,Sn-l,ii), (3.94) which implies that for fixed a:n»Sn-i» and i^, V^n.fcC^^n; Sn-i,^n) is an increasing
sequence in k. On the other hand, for any k,
where O is a policy of ordering nothing at each period by both fast and slow modes. From (3.86) and (3.87), J^(x;i,Sn_i,iJj,0) < oo. Consequently, (3.92) follows from (3.94).
By Theorem 3.1, we know that for each k, Vn^ki^n^ s„_i, i^) is convex and Lipschitz continuous in {xn,Sn-i) on (—oo, +oo) x [0, +oo). Hence (i) follows from (3.92).
Next, we show that V^(xn, s^-i, i^) is a solution of (3.79). Using Theorem 3.2,
cr>0
+ a E [Vn+l,k-l{Xn + Sn-1 + 0 - ^ n ( 4 ' ^ n ' ^n), O", ^ n + l ) ] }
(3.96) Taking limits on both sides of (3.96) with respect to k, we get
<Hnixn)+M{cl{cl,) + C^{a)
<7>0
+aE [V^iiXn + S n - 1 + 0 - 9n{ili, ll, Vn), Cf, ^ n + l ) ] } •
(3.97) Using (3.88)-(3.89), there exists a Q > 0 such that for all n and k
F^+ii^n+i,sn+i-uin+e) < Q, 0 < i < k, (3.98) S'^+eixn+i, sn+e-uii+e) <Q. 0 < £ < /c - 1, (3.99) where Fl^^^{xn+e, Sn+e-iJl+e) ^^^ S^+e(^n+e, Sn+e-i^il^^^) are defined by (3.93). Furthermore, by (3.94), for any £ < k,
*'^n,k\Xn^ Sn—1, Iji)
+ Q E | y n + l , f c - l \Xn-\- Sn-l + F!^(Xn, Sn-\,i\) - gn{i\jl,Vn),
+ a E
(3.100) Fixed ^ and let /c —> oc. In view of (3.98) and (3.99), we can, for any given n, Xn, Sn-l, and z^, extract a converging subsequence
^ n v^n^ "^n—1? '^'nj? ^ n v ^ n ? ^^n—1? ^n/
Let
l i m ( r^ [Xn-i Sji—lf Z72J? ^nx-^n-} Sn—1^ "^nJ
~ V^n V'^n^'^n—1? ^n/5 ^ n v^n?'^n—1?'^n/j (3.101)
From the uniform integrability of
Vn^u{Xn H- S n - 1 + 0 " Qniii, ll, Vn), CT, I^^-^) (sCC ( 3 . 9 1 ) ) ,
we can pass to the limit on the right-hand side of (3.100). We obtain (noting that the left-hand side converges as well)
> HniXn) + CliF^iXn^ ^ n - b 4 ) ) + C^iS^{Xn^ 8^-1 A)) + a E \Vn+l^i {Xn + Sn~l + F^{Xn, ^ n - l , 4 ) "^ 9n{i]i, ll, Vn),
^n v^n?'^n-l?'^n/? ^n+1 j j •
(3.102)
Furthermore,
^n\Xn) I O^ [r^ [Xfij ^n—li '^n/i " ^n+i\^n \^n? ^n—l, ^nj) +aE [Vn+l^i (xn + Sn-l + F^(Xn, ^ ^ - l , 4 ) - gn{i\jl, Vn),
^n v^n? %—1? ^n/? ^ n + l j j
^^ J^n\Xn) '^ ^n\^n v^n?-^n—1?'^nJ/ ' ^nv*^n \^n, ^n—l^'^n)) + a E [V^i {Xn + Sn-l + F^{Xn, ^ n - l , 4 ) " 9n{^lt, ll, Vn),
^n v^n? •^n-l?'^n/? ^n+1 j j
<T>0 ^
+ Q ; E [ V ; , ° J I ( x n - h S n - l + 0 - P n ( 4 ^ ^ n ' ' ^ n ) , c r , / i + l ) ] L
(3.103) Therefore, by (3.102) and (3.103),
^n \Xni Sn—li'^n)
> / / , ( . : „ ) + i n f | c / ( 0 ) + C^(cr)
<7>0 ^
(3.104) which and (3.97) imply (ii) of the theorem, (iii) can be proved along the line of
the proof of Theorem 3.3. The detail is omitted here. D
REMARK 3.6 Theorem 3.6 does not imply that there is a unique solution of the dynamic programming equations (3.79). In addition, it is possible to show that the value function is the minimal positive solution of (3.79). Furthermore, it is also possible to obtain a uniqueness proof, provided that the cost functions Cl{-), C^() and Hni-) are subject to additional conditions.
To derive the optimality of a base-stock policy in the same way as in Section 3.4, we still make assumptions (3.67)-(3.68). Let
Gn{t) = cl- {t-Xn - S n - l ) - < i + E[Hn+l{t - gn{inJn^Vn))].
Let ?/* be a minimum of the function Gn{t)- Furthermore, let
Ln{t) = Ct,t + E[V^,{t - gniijlvn))] + lGn{t) - Gn{y*n)] ' Kvl " 0 , and let 2:* be a minimum of the function Ln{t), Similar to Theorem 3.5, we have the following result.
T H E O R E M 3.7 Assume that0A\ (3.4H3.5), (3.67H3.68), a^J(3.86H3.87) hold. Then the policy {f^i^V) 8^^^^ by the following is an optimal nonantici- pative policy:
(i) when yl < 4 ,
KJn'^^n) ~ \ K^^^n ^n "
[ (0,0), (ii) when y* > 4 ,
1 {^n ~ ^n •
(/„*,<) :=<^ (0,0), I (0,0),
- S n - l ) ,
- S n - 1 , 0 ) , if if if
if y^<Xn-\- Sn-i < ;
V ^n *^ "^n ' Sn—i',
^ n 1 S T ^ — I ^ Zj^y
^n "^ ^n ' ^n—1 S Vni yl < Xn + Sn-1.
"ni
REMARK 3.7 When Cl{u) = cl • u and C^iu) = c^ • u with cl > 0 and c^ > 0, (3.84)-<3.85), and (3.88)-(3.89) hold. Thus we do not need to specify that (3.84)-(3.85), and (3.88)-(3.89) hold in the theorem.
Proof of Theorem 3.7 The proof is similar to the proof of Theorem 3.5, and
is omitted. D