- Email: [email protected]
An inventory model for manufacturing systems with delivery time guarantees
Pergamon
Plh S0305-0548(97)00077-4
ComputersOpsRes.Vol.25, No. 5, pp. 367-377, 1998 © 1998ElsevierScienceLtd.All rights rc'~xved Printed in GreatBritain 0305-0548/98 $19.00+0.00
AN INVENTORY MODEL FOR MANUFACTURING SYSTEMS
WITH DELIVERY TIME GUARANTEES Wai Ki Chinglat:~ Department of Mathematics, The Hong Kong University of Science and Technology, Poeple's Republic of China Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Sbatin, Hung Kong, Poeple's Republic of China
(Received December 1996; in revisedform August 1997) Scope and Purpose---In this paper, we study manufacturing systems of one machine which produces one type of product. A delivery time guarantee is offered to customers for each unit of product. There is a growing trend of using delivery time guarantee in commercial companies (e.g. United Parcel Service) as a marketing strategy to attract customers and compete in the market-place. Numerous papers have studied the relationships among the delivery time, the pricing and the capacity, see for instance [2,8,9,11,14]. Here we derived an optimal bound for the maximum backlog when a service level is given by using techniques in queueing theory. In our study, the production process and arrival of orders are modeled by Markovian process. We employ (s,S) policy as the production control. The optimal values of s and S can be obtained by enumeration. Abstract--This paper studies optimal (s,S) policies for production planning in one-machine manufacturing systems. The machine produces one type of product with delivery time guarantees on the products offered to the customers. The inter-arrival time of the demand and the processing time for one unit of prodnct are assumed to be exponentially distributed. The total delivery time (total lead time) consists of two parts: the cycle time and the delivery time. The cycle time is the time between the arrival of an order and the time requested product leaves the manufacturing system. The delivery time is the time from the manufacturing system to the customer. We model the delivery time by a shifted exponential distribution. Unbiased and consistent estimators are derived for this distribution. The analytical form of the steady state probability distribution for the inventory levels is derived. The average profit of the system can be written in terms of the resulting probability distribution. Hence the optimal (s,S) policy can be obtained by varying different possible values ofs and $. © 1998 Elsevier Science Ltd. All rights reserved 1. INTRODUCTION In this paper, w e study optimal (s,S) policies for the production planning in manufacturing systems with
delivery time guarantees. There is a growing trend in using the delivery time guarantee in commercial companies as a marketing strategy to attract customers. For example, the United Parcel Services guarantees the next day delivery by 8:30 am. Pizza Hut in Hong Kong offers free pizza if the ordered pizza cannot be served within 20 min. Lucky a major supermarket chain in California, use a "3's a crowd" campaign, which guarantees a new checkout counter will be open if there are more than 3 people waiting in its checkout queue. Wells Fargo Bank offers a "5 min maximum wait policy" which offers five dollars to the customer if the customer waits for more than 5 min in line. Numerous papers have studied the relationships between the delivery time, the pricing and the capacity, see for instance [2,8,9,11,14]. However, to our best knowledge, the application of delivery time guarantee as a marketing strategy in manufacturing systems seems not yet addressed in literature. In our study, we assume that the delivery time guarantee T and the unit price P are the main factors affecting the demand rate A=A(P,T) of the
product. One standard formulation for the relation (Cobb-Douglas) is A = A ( P , T ) = t r P - ~' T - ~2,
(1)
where ~'~,z2 are the price and delivery time guarantee elasticity respectively, and tr is a positive constant; see [ 10,11 ] for instance. In the delivery time guarantee strategy, a delivery time guarantee of T on each unit of ordered product is offered to the costumers. In [7], Ching and Zhou considered optimal (s,S) policies for one-machine t Email: [email protected]. :~Wai Ki Ching received his B.Sci. and M.Phil. degrees in Mathematics from the University of Hong Kong. He is a teaching assistance in the department of Mathematics, the Hung Kong University of Science and Technology and a part time Ph.D. student in the department of Systems Engineering and Engineering Management, the Chinese University of Hong Kong. He has published numerous papers in Applied Mathematics and Engineering Journals and conference proceedings. His research interests include queueing systems, numerical analysis, simulation and operations research. 367
368
W.K. Ching
manufacturing system with delivery guarantee. In their model, an order can always be served within the delivery time T whenever there are products available in the inventory. However, the model ignored an important and stochastic factor, the delivery time. Here in our model, we assume that the orders are served in the first come first serve principle. The delivery time consists of two parts: the cycle time and the delivery time. The cycle time is the time between the arrival of an order and the time requested product leaves the manufacturing system. The delivery time is the time for the requested product from the manufacturing system to the customer. In our study, we model the delivery time by a shitted exponential distribution. Unbiased and consistent estimators are derived for the distribution. Here we consider a one-machine system. The machine produces one type of product, the inter-arrival time of the demand and the processing time for one unit of product are assumed to be exponentially distributed. Finite backlog of the demands is allowed in the system. A cost of cB is assigned to each unit of backlog and a cost of c~ is assigned to each unit of inventory. Using the inventory level as an indicator, our interest is to determine the production strategy such that the average profit is maximized. We employ the (s,S) policy as the production control. An (s,S) policy is characterized by two integers s and S. The machine keeps on producing until an inventory level of S is reached. Once this inventory level is reached, the production is stopped by shutting down the machine. We let the inventory level falls to s, (s
SYSTEM
In this section, w e present a one-machine manufacturing system with product delivery time guarantee. Let us define the following notations: /~ - ~---themean processing time for a demand for one unit of the product, A - ~---themean inter-arrivaltime for one unit of the product,
O=operating cost for one unit of the product, P (P> O)--'unit price of the product, T---the guarantee time of delivery, s
An inventory model for manufacturing systems with delivery time guarantees
369
unit of rejected demand. To determine the optimal values of s and S, we have to set an appropriate guaranteed delivery time. In the next section, we study the delivery time distribution. 3. THE DELIVERY TIME D I S T R I B U T I O N
In this section, we study the delivery time distribution of an order. The total lead time of an order equals to the sum of the cycle time and the delivery time of the product. The processing time of an product is assumed to be exponentially distributed with parameter/z. If an arrival order is not a backlog, then the cycle time (time between the arrival of order and the time requested product leaves the manufacturing system) is zero. But if the order is in thejth backlog then the probability density function of the cycle time of the order is the Erlangian function of j-phase: f~(t) =
#.(p.t)~e -~' J!
Usually the delivery time is greater than a minimum time To. The time To may due to the packing of the products or the waiting time for transportation schedule. In our study, we model the delivery time by a "shifted" exponential distribution of parameters To and fl: {0fle-~ T(t)=
if t-->To, ift
(2)
A similar formulation has been used by Karmarkar [13] in the fractile estimation. In the following, we derive unbiased and consistent estimators for TOand 1/fl respectively. The estimators here are not unique, other forms of unbiased and consistent estimators are available, see [12] for instance. We first let
{ TI,T2..... T~} be a sample of n delivery times, i" be the mean of the sample and Tc~)=min{T~}. Lemma 1. Let T~<--Tt2)~--...<--T{.) be n order statistics generated by the exponential distribution f(t) =/3e -~,
then the probability density function for T.) is given by
g(t)=n~-,#t, E(T~I))=I/(nfl), andVar(T,))= I/(n3)~.
ProofSince each random variable/'tois exponentiallydistributedwith parameter fl,we
have
Pr°b(X)03->Y)= I tic-~ dt, for i=I,2.....n. Thus, i f X , ) ~ Y w e must have all the X~o>-Y Therefore, we have
Prob(X,)>-Y)=[ I ~c-/~dt]"=[e-#rl"=e-"#r. Differentiateboth sideswith respect to Y,we have the probabilitydensity functionof T,) given by
g(t)=n~¢-"#'. For more detailabout order statisticsdistributions,see [12].The probabilitydensity functiong(t) is still an exponential distributionwith parameter n//.Thus, we have E(T
EfTo))= To+ ~-fl. Therefore To) is an biased estimator and so is (7"- To)). Here, according to the bias, we consider the
estimators ~'o and 1/]~ for To and 1/~ satisfying the equations:
370
W.K. Ching
{ ~o ~. T(l) - 1~(hiS),
(3)
1/[3=7"- ~'o. Solving the equations, we get
{ ~o=(nT,) - T)l(n - 1), 11[3=n(7"- To)l(n - 1).
(4)
In the following, we prove that ~ro and 1//~ are unbiased and consistent estimators for To and l/fl respectively. Proposition 1. The estimators l"o=(nT,)-T)/(n-1) consistent estimators for To and 1/fl respectively.
and ll/3=n(7"-To)l(n-1)
are unbiased and
Proof. Since 1
E(~o) = ~
1
(nE(T.))- E(~)= n - 1 (n(To+fl/n)- To-fl)=To
and E(1/~)= ~
n
(E(~ - E(T(t)))=
n
(1/fl+ To- TO- l/(n~)= 1/~,
the estimators 1/fl and ~'o are unbiased. Moreover, by Lamma 1, we have
Var(1/~)= ~
n2 (Var(i")+Var(To)))=~ n(n~
+
1)
and
n2
Var(~'o)= ~
1 l 1 Var(T(,)) + (n - l) 2 Var(~= ((n- l)fl)2 + (n(n- l)fl)"
Thus lim Var(I/]~)=it--*eo lim Var(~'o)=O,
W--,m
and therefore ll~md ~'oare consistent estimators. • 4. DETERMINATIONOF MAXIMUMALLOWABLEBACKLOG In manufacturing system of limited production capacity, to guarantee customers a delivery time of the product, there should be an upper limit for the maximum allowable backlog m. Here we propose a simple estimation of the maximum allowable backlog m. Suppose we have k backlogged orders, the probability that they can be offered within the guarantee time T should be greater than a pre-determined value T (0< ~,< 1). The reliability value (or the service level) ~ can be 90% or 95% for example. The following proposition gives an inequality relating the quantities % T and the maximum allowable backlog m. Proposition 2. Given the delivery guarantee time T and the reliability value % the maximum allowable backlog m is the largest positive integer k which satisfies the inequah'ties: (1 - 3,)e'~r-ro)>- ~.~ ( T - To)J(#j -/~k(/z- B) ~-k)) +/zke -~#-~xr-To
,.0 - - - - - - i T - - -
- - -
for
(5)
3')e~r-r°)--- k~l (p.(T- To))2 +/zk(T - To)te-~r-r° for/.t--// j=o j! kt Proof. The processing time for one product is exponentially distributed with mean 1//~, therefore the probability density function for the total processing time of k products is given by the Erlangian function f,(t)=p.(la)*-~ e -~/(k- 1)!. The delivery time is modeled by the shifted exponential distribution (1
fie #' if t_>To g(t)= O if t
(6)
An inventory model for manufacturing systems with delivery time guarantees
--
7.
E ] --
it-
k
371
D ' @
°..
~
p.
p.
®.
...:
®
Q
The machine is shut off and the inventory level is i.
D
The machine is turned on and the inventory level isj. Fig. 1.
The parameters fl and To are estimated by the unbiased and consistent estimators in Proposition 1. Hence we have
r~ro ~)1"-----/'t(~)k-l(k-~• e - ' q l
- e
-/~r-ro-,)] &>--7.
By making use of the formula
~)k-l -~ 1(k_ )----~ e dt=l-e-~rkj~o
j!
we get the inequalities (5). Therefore the maximum allowable backlog m is the largest integer k satisfying the above inequality. • 5. THE INVENTORY LEVELS When the desirable guaranteed delivery time T and the unit price P of the product are decided, the maximum allowable backlog m (Proposition 2) and the demand rate A (cf. (1)) can be determined. We then employ the (s,S) policy in the production planning of the manufacturing system. We note that under the (s,S) policy, the inventory levels can take integer values in [ - re,S]. When the inventory level reaches S, the machine is shut off. When the machine is shut off, the inventory levels can take integer values in [s,S] and once it falls to s, the production is re-started. Therefore there are ( m + 2 S - s ) possible states in the inventory system, see Fig. 1. We order the states as follows. The first ( S - s ) states correspond to the situation that the machine is mined off, and the inventory levels are ordered from (s+ 1) up to S. The next (re+S) states correspond to the situation that the machine is turned on, and we order the inventory levels from ( S - 1) down to - m. We the construct the generator matrix for the machine-inventory system. A generator matrix Z of a Markov process gives the transition rate [Z]# from statej to state i. Under our ordering, we obtain the following generator matrix A for the machine-inventory system: A O
-A A
O
.,.
0 -A ..°
0
A[ O -A O
.°.
A
-A
°..
0
A+/z -A
-/z A+/z
-/z
..°
°..
-A
S-s+l "S-s+2 ...
2(S-
A+/.t .°.
-A 0
1 2
s)
° . .
A+p -A
--ft
/z
:
m + 2S - s
372
W.K. Ching
The generator matrix A is irreducible, has zero column sum, positive diagonal entries and non-positive off-diagonal entries. Therefore A has one dimensional null space; see Varga [16]. The steady state probability distribution is the normalized positive null vector of A. 6. T H E S T E A D Y S T A T E D I S T R I B U T I O N
OF THE INVENTORY
LEVELS
In this section, we derive the analytic form of the steady state probability distribution of the inventory levels. To obtain the probability distribution vector p, we perform the following row operations to A. Beginning from the last row of A, we add the ith row of A to the ( i - 1 ) t h row of A, for i=(m + 2 S - s ) , ( m + 2 S - s - 1)..... 2. The purpose of these row operations is to eliminate the upper subdiagonal ofA. We get a lower triangular matrix B:
B=
0 -a -a -a -a -,a A
0 a 0
0 0 ...... a 0
0 a 0
0 /~ -a
i
0 I-*
1 2 :
] S-s+l "S-s+2
0
. . . . . . . . .
-a
-a
(7)
"
/z
0
2 ( S - s)
.........
:
;
-a
0
/.*
-A
0
i
I~ m + 2 S - s
By applying the forward substitution to (7), we have the following proposition concerning the steady state probability distribution of the machine-inventory system. Proposition 3. The steady state probability distribution p o f A can be written as follows:
a P=(Pt,P2 .... ,p,,+2s_s)=a-'(1,V,U)', llx(s_,)=(1,1 ..... 1), p= --. /x For p ~ 1, we have
v
=
u
=
a'
----
- - P( l - p , .... 1 - p S - ' ) , 1-p p(1 - p s - s) (p, p2 ..... pro+s), 1--p pm.e(pS _ p') + (1 -- p ) ( S - s) (1 _p)2
(8)
For p= 1, we have
v
=
(1,2,3 ..... S - s ) ,
u
=
(S-s,S-s
=
1 ~(S-s)(3+S+s+2m).
a
..... S - s ) ,
(9)
P r o o f We first note that i f p is a right null vector of A (i.e. A p = 0 ) then p is also a right null vector o f B (i.e. Bp=0). We discuss the case when p # 1, the case p = 1 being similar. Letpt = a - J>0, we observe from (7) that p ~ = a - t for i = 1 . . . . . ( S - s ) . By forward substitution, we obtain
Ps-s+i--
131
lP(1 _pi), i=1 ..... ( S - s ) 1-p -
-
and a - tp(1 -
P2(s- ~)+~=
1-p
pS-,) IJ, j = 1..... (m + s).
An inventory model for manufacturing systems with delivery time guarantees
373
The normalization constant a is obtained by the relation 1=
m+2S-s ~, p i = a i=l
pm+2(pS-p~)+(1-p)(S-s) -l
(1 - p ) ~
• "
We observed that the steady state probability of the first ( S - s ) are equal. In fact, the Markov chain formed by the (S - s) states is close to that of Erlangian distribution of ( S - s) phases with parameter A. The generator matrix of the Erlangian distribution is given the following ( S - s) x ( S - s) matrix E: A E
-A
0
~
.
.
.
.
.
.
A -A
-A A
and clearly the steady state probability of this generator matrix is given by (l/(S-s))(1,1 ..... 1)' which has equal probability for each state. From the distribution vector p, we get the marginal probability of the inventory levels. P r o p o s i t i o n 4. The marginal probability distribution f o r the inventory levels is
q = (q-
m .....
qo,ql ..... qs)"
Here q~ is the steady state probabiHty that the inventory level is i, and i = - m .... ,s ..... S. For p ~ l, we have or=
pm+2(pS _ p~)+ (1 -- p ) ( S - s) (1 _p)2
and _ P (l__pS-s)pr-i, Ot I i _ p
qi=
a -I(
1 -Pp ( 1 - - p S - t ) ) ,
i= - m ..... (s - 1),s, (1o)
i = ( s + l ) .....
For p= 1, we have a = ( 1 / 2 ) ( S - s)(3 + S+s+ 2m), and fa-'(S-s), qi= ~ a - I( l + S --
i = - m ..... 0 ..... ( s - 1),s, i=(s + l ) ..... S.
(ll)
Proof. From the ordering of the states (see Fig. 1), we have qs=Ps_s, qs_i=Ps_~_i+Ps_s+~, i= 1,2..... (S - s - 1)
and q_m+i=Pm+2S_s_i, i=0,1 ..... (re+s).
The proof is completed by using Proposition 3. • 7. THEAVERAGE PROFIT The average profit of the system can be written as follows: (P - O).S(s,S) - l(s,S) - B(s,S) - P(s,S),
where S(s,S), I(s,S), B(s,S) and P(scS) are the average sales, the average inventory holding cost, the average backlog cost and the average demand rejection penalty cost respectively. Here P and O are the unit price and unit operation cost of the product respectively. In terms of the inventory distribution q, the average sales S(s~g), the average inventory cost l(s,S) and the average demand rejection penalty cost P(s,S) can be written as
374
W.K. Ching S
S
S(s,S)= A i~=lqi, I(s,S)=cj i~=liqi and P(s,S)=cpAq_m, respectively. To obtain the average backlog penalty cost we need the following proposition. Proposition
5. The probability w~ that the jth backlog order cannot be offered within the delivery time
T is given by -i ] z j e - (~ - g ) ( r - r o) /
e
(/-~-/3)s
i!
e
i!
k!
j,
for/z~fl,
(12)
for pt=fl.
'
Proof. The result can be proved directly by Proposition 2. • Given k (k>- 1) backlog orders, the probability b *~that i orders out of the k are delayed can be written in terms ofwj. For examples, when k=3, the probabilities b3;, i=0,1,2,3 are given by b3o = (1-wl)(1-w2)(1-w3), bal = Wl(1- w2)(l - w3)+(1 - Wl)W2(l b32 = (1 - wt)w2w3+wt(1 - - W 2 ) W 3 + W I W 2 ( 1
b33
=
w3)+(1 --
-
wOO - w2)w3,
(13)
W3),
W l W 2 W 3.
The expected number of delay when there are k backlog orders is then given by • ~ffi~ib~. When an order is not in backlog, the probability that it will be delay is given by e-~r-r0) (of. (2)). Hence, the average backlog cost is given by
B(s'S)=cB{ ~ [ ( ~i.,i b ~ ) q - k ] + e - I ~ r - r ° ' ~if,q i } However, from the numerical examples we observed that when the service level T is close to 1 (e.g. T=0.95), the backlog cost is negligible when compare with the inventory cost and the profit. This can be explained by the fact that we have set a high service level (reliability) for the system. Therefore, in the following, we assume that the average profit of the manufacturing system is given by S
[A(P - O) -
i=l
cli]q
cpAq-m.
i -
(14)
8. S P E C I A L C A S E A N A L Y S I S
In this section, we consider the case of allowing infinite backlog with negligible backlog cost ca . In this situation, the average profit is given by S(s,S)- I(s,S). We prove that under some conditions, the optimal (s,S) policy is of hedging point type (i.e. S= (s + 1)) which is the interesting case in [4-7]. Proposition
6. Under the permission of infinite backlog and zero backlog cost, if 0
-
~/5 2 -
1
A(P - O)(1 -
and q->
1_p_p2
p)p2
,
then the optimal (s~.g)policy is of hedging point type, i.e. S=(s+ 1). Proof. Using (10) we have S(s,S) - / ( s , S ) =
(A(P - O) + (Cl/(1 - p)))( l - p2f(s,S)) - q(S + s) 1 - p'+~s,S)
where
pS -- pS Ks,S) =
(1 - p ) ( s - s)"
05)
An inventory model for manufacturing systems with delivery time guarantees
375
When infinity backlog is allowed (m--. oo), the average profit is given by
(
H(s,S)= a ( P - O ) + ~
cl)
(1-p2f(s,S))-cl(S+s).
(16)
To maximize H(s,S), it is equivalent to minimize
(
c,)
(
c,)
G(s,S)= A ( P - O ) + I~_
(pZf(s,S))+q(S+s).
Let t = ( S - s ) > - 1, we write
G(s,t)= A ( P - O ) + 1 - ~
(P2f(s's+t))+cI(t+2s)"
In the following, we prove that G(s,t) is an increasing function in t for any given s. Thus for any given s, G(s,t) attains its minimum at t= 1. Indeed,
OG(s,t)
(A(P-O)(1
0--~ =
1 -tptlogp)
- - p ) + C l ) p s+2 ( p t _
(1 - p)2
t2
+c~
(A(P- O)(I - p)+ci)p ~÷2 ( p t - 1) > (1 - p)2 t---T--- +ci - (A(P - O)(1 - p ) + C l ) p
s+2
-
(A(P- O)(1 - p) + Cl)p 2
q-Cl~
(1 -p)
(1 -p)
+C I
= - A(P - O)(1 - p)p2+ (1 - p - p2)cl(1 - p)-->O. The first inequality comes from fact that - tp' log p>0. The last inequality comes from the condition (15). The second inequality comes from the fact that the negative function (pt_ 1)/t 2 is an increasing function in t and we give the proof as follows: p'-I t~
p-1 = -~(l+p'+...+p
<'-'>)
_p-1
- t(t+l-----~(l+ _ p-1 -
t(t+
)(l+pl+..'+p <'-l))
{(l+p+...+p¢,_t))+ 1
1-----)
t (1 +p+...+p(t-1))}
< p-1
p-1
-- t(t+ 1) (1 +p+...+p~)--< ~
(1 +p+...+p')
p,+l _ 1 (t+ 1)2 • Thus, the proposition is proved. • As a corollary, when the optimal policy is of hedging point type, the average profit is given by (16) with S=(s+ 1), i.e. CI
H(s,s+ 1)=(A(P - O)+ 1 - ~ )(1 - p2f(s,s+ 1)) - ci(2s+ 1). We observe that
=-
A(P-O)+ 1-~
p2(I°gp)2p~<0'
and therefore H(s,~+ 1) is a convex function in s. Thus, the maximum point is given by the unique solution of
W. K. Ching
376
Table 1. The optimal (s',S') for different p p m s S"
C(s',S')
0.1 32 l 2 $773.1
dH(s,S+ds1) _ _
0.2 20 2 3 $766.1
0.3 12 3 4 $ 757.8
(
A(P- O)+ ~
9. N U M E R I C A L
c,)
0.4 9 4 5 $748.4
0.5 7 5 6 $737.3
p2(log p)pS_ 2c,=0.
EXAMPLES
In what follows, we give some numerical examples. Recall that the demand rate
A=oe -~,T -~'.(cf. (1)). We let the elasticity constants z,, r2, and o-to be 2, 1, and 1000, respectively. The parameters P, O, A, p and c, are set to satisfy the condition (15) in Proposition 6. The unit price P is $100 and the unit operating cost O is $20. The guaranteed delivery time T is 1 day. The parameters To and fl of the delivery distribution are to be 0.1 and 10 respectively. The maximum inventory capacity/max is 100. The inventory cost c, per unit of product is $10. Furthermore, we let the reliability factor to be 90%, the penalty cost cp to be $2000. For the delay cost cB, we set it to be zero. In Table 1 we give the optimal pair (s',S ") and their corresponding average profit C(s',S °) for different values of p= A//z. We found that although the backlog cost cB and penalty cost cp are included, the optimal policy is still of hedging point type. In Tables 2-6, we investigate the optimal profit of the above numerical examples when a constraint Table 2. The optimal profit for different d when p=0.1 d
C(s',s'+d) d
C(s',s'+d)
1 $773.1 7 $749.8
2 $771.7 8 $745.0
3 $768.1 9 $740.1
4 $763.9 10 $735.2
5 $759.3 20 $690.6
6 $754.6 40 $590.9
Table 3. The optimal profit for different d when p=0.2 d
C(s',s'+d) d
C(s'~*+d)
1 $766.1 7 $746.8
2 $763.6 8 $742.5
3 $759.8 9 $738.0
4 $757.5 10 $733.5
5 $754.5 20 $690.8
Table 4. The optimal profit for different dwhen d
C(s',s'+d) d
C(s',s'+d)
1 $757.8 7 $739.9
2 $755.1 8 $736.4
3 $754.2 9 $732.8
4 $751.6 10 $728.9
6 $750.8 40 $591.6
p=0.3
5 $748.1 20 $689.8
6 $744.1 40 $591.6
Table 5. The optimal profit for different d when p=0.4 d
C(s',s'+a') d
C(s ",s"+d)
1 $748.4 7 $734.4
2 $747.2 8 $730.9
3 $745.9 9 $727.1
4 $743.3 10 $723.1
5 $739.9 20 $685.3
6 $737.4 40 $591.2
Table 6. The optimal profit for different d when p=0.5 d
C(s',s'+d) d
C(s',s'+d)
1 $737.3 7 $725.7
2 $736.0 8 $722.4
3 $735.2 9 $718.7
4 $733.1 I0 $714.9
5 $730.4 20 $636.0
6 $728.4 40 $589.6
An inventory model for manufacturing systems with delivery time guarantees
377
S - s = d is imposed on the optimization. We found that the derivation o f the average profit from the optimal average profit grows in polynomial rate (more than linear rate) in d. Thus, the average profit is sensitive to the values o f s and S.
10. CONCLUDING REMARKS In this paper, we have discussed optimal (s,S) policies for production planning with delivery time guarantees. We have established a relation (Proposition 2) between the desirable maximum allowable backlog m and the guaranteed delivery time Z Analytic form o f the average profit has also been derived. By varying different possible values o f s and S, the optimal (s,S) policy can be obtained. A sufficient condition (Proposition 6) for the hedging point policy to be optimal has also been given. It is interesting to relax the condition for hedging point policy to be optimal and extend the model to the case when there are set-up time and set-up cost for the manufacturing system. These issues are studied in another paper
[3]. Many other interesting problems remain open. For example, what if the machines are unreliable? Extension to the ease o f multi-item is also interesting. Furthermore, other distributions (e.g. the Weibull distribution) can be used to model the delivery time. Another possible extension o f our model is the following non-linear programming problem. We do not pre-determine the unit price P and the guaranteed delivery time T (Recall that the demand rate is a function o f P and T, of. (1).) Rather, they are part o f the decisions. The generalized formulation is given as follows: maxe, r~,s
{ E [ ai ( P - O ) - c l i ] q ~ - c p a q _ . } i =
s.t. (5),
s < S < - I ~ , and 1 - m .
Acknowledgements---The author would like to thank the referees for their helpful comments and constructive suggestions in revising the paper. Research supported by RGC Earmarked Grant CUHK-489/95Eand Hong Kong CroucherFoundationGrant.
REFERENCES I. Akella, R. and Kumar, R, Optimal control of production rate in a failure prone manufacturingsystems. IEEE Transactions on Automatic Control, 1986, 31, 116-126. 2. Buss, A., Lawrence, S. and Kropp, D., Volume and capacity interaction in facility design, lie Transactions on Design and Manufacturing, 1994, 26, 36--49. 3. Ching, W., Optimal (S,s) policies with delivery time guarantees for manufacturing systems with early set-up, submitted, 1997. 4. Ching, W. and Zhou, X., Hedging point production planning for failure prone manufacturing systems. In: Proceedings of the 5th Conferenceof the Operational Research Society of Hong Kong, 1995, pp. 163-172. 5. Ching, W. and Zhou, X., Matrix method for productionplanning in failure prone manufacturingsystems. In: Lecture Notes in Control and Information Sciences, 214. Springer-Verlag,London, 1996, pp. 3-30. 6. Ching, W., Chart, R. and Zhou, X., Cireulant praconditioners for Markov modulated poisson processes and their applications to manufacturingsystems. S1AMJournal on Matrix Analysis and Its Applications, 1997, 18, 464-481. 7. Ching W. and Zhou, X., Optimal (S,s) policies for productionplanningwith deliverytime guarantees. To appear in: AMS-SIAM Proceedings of Applied Mathematics. AmericanMathematical Society, 1997, pp. 71-82. 8. Dewan, S. and Mendelson, H., User delay costs and internal pricing for a service facility. Management Science, 1990, 36, 1502-1517. 9. Donohue, K., The economies of capacity and marketing measure in a simple manufacturing environment. Production and Operations Management, 1994, 3, 78--99. 10. Freeland, J., Coordination strategies for productionand marketing in a functionallydecentralizedfirm. lie Transactions, 1980, 12, 126-132. 1I. Hill, A. and Khosla, I., Models for optimal leadtime reduction. Production and Operations Management, 1992, 1, 185-197. 12. Hogg R. and Craig, A., Introduction to Mathematical Statistics. Collier--MacMillan, 1978. 13. Karmarkar, U., A robust forecastingtechniques for inventoryand leadtime management.Journal of Operations Management, 1994, 12, 45-54. 14. Li, L., The role of inventory in delivery-timecompletion.Management Science, 1992,38, 182-197. 15. Monden,Y., Toyota Production System. Industrial Engineering ManufacturingPress, Atlanta, GA, 1983. 16. Varga, R., Matrix lterativeAnalysis. Prentice-Hall, New Jersey, 1963.
Plh S0305-0548(97)00077-4
ComputersOpsRes.Vol.25, No. 5, pp. 367-377, 1998 © 1998ElsevierScienceLtd.All rights rc'~xved Printed in GreatBritain 0305-0548/98 $19.00+0.00
AN INVENTORY MODEL FOR MANUFACTURING SYSTEMS
WITH DELIVERY TIME GUARANTEES Wai Ki Chinglat:~ Department of Mathematics, The Hong Kong University of Science and Technology, Poeple's Republic of China Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Sbatin, Hung Kong, Poeple's Republic of China
(Received December 1996; in revisedform August 1997) Scope and Purpose---In this paper, we study manufacturing systems of one machine which produces one type of product. A delivery time guarantee is offered to customers for each unit of product. There is a growing trend of using delivery time guarantee in commercial companies (e.g. United Parcel Service) as a marketing strategy to attract customers and compete in the market-place. Numerous papers have studied the relationships among the delivery time, the pricing and the capacity, see for instance [2,8,9,11,14]. Here we derived an optimal bound for the maximum backlog when a service level is given by using techniques in queueing theory. In our study, the production process and arrival of orders are modeled by Markovian process. We employ (s,S) policy as the production control. The optimal values of s and S can be obtained by enumeration. Abstract--This paper studies optimal (s,S) policies for production planning in one-machine manufacturing systems. The machine produces one type of product with delivery time guarantees on the products offered to the customers. The inter-arrival time of the demand and the processing time for one unit of prodnct are assumed to be exponentially distributed. The total delivery time (total lead time) consists of two parts: the cycle time and the delivery time. The cycle time is the time between the arrival of an order and the time requested product leaves the manufacturing system. The delivery time is the time from the manufacturing system to the customer. We model the delivery time by a shifted exponential distribution. Unbiased and consistent estimators are derived for this distribution. The analytical form of the steady state probability distribution for the inventory levels is derived. The average profit of the system can be written in terms of the resulting probability distribution. Hence the optimal (s,S) policy can be obtained by varying different possible values ofs and $. © 1998 Elsevier Science Ltd. All rights reserved 1. INTRODUCTION In this paper, w e study optimal (s,S) policies for the production planning in manufacturing systems with
delivery time guarantees. There is a growing trend in using the delivery time guarantee in commercial companies as a marketing strategy to attract customers. For example, the United Parcel Services guarantees the next day delivery by 8:30 am. Pizza Hut in Hong Kong offers free pizza if the ordered pizza cannot be served within 20 min. Lucky a major supermarket chain in California, use a "3's a crowd" campaign, which guarantees a new checkout counter will be open if there are more than 3 people waiting in its checkout queue. Wells Fargo Bank offers a "5 min maximum wait policy" which offers five dollars to the customer if the customer waits for more than 5 min in line. Numerous papers have studied the relationships between the delivery time, the pricing and the capacity, see for instance [2,8,9,11,14]. However, to our best knowledge, the application of delivery time guarantee as a marketing strategy in manufacturing systems seems not yet addressed in literature. In our study, we assume that the delivery time guarantee T and the unit price P are the main factors affecting the demand rate A=A(P,T) of the
product. One standard formulation for the relation (Cobb-Douglas) is A = A ( P , T ) = t r P - ~' T - ~2,
(1)
where ~'~,z2 are the price and delivery time guarantee elasticity respectively, and tr is a positive constant; see [ 10,11 ] for instance. In the delivery time guarantee strategy, a delivery time guarantee of T on each unit of ordered product is offered to the costumers. In [7], Ching and Zhou considered optimal (s,S) policies for one-machine t Email: [email protected]. :~Wai Ki Ching received his B.Sci. and M.Phil. degrees in Mathematics from the University of Hong Kong. He is a teaching assistance in the department of Mathematics, the Hung Kong University of Science and Technology and a part time Ph.D. student in the department of Systems Engineering and Engineering Management, the Chinese University of Hong Kong. He has published numerous papers in Applied Mathematics and Engineering Journals and conference proceedings. His research interests include queueing systems, numerical analysis, simulation and operations research. 367
368
W.K. Ching
manufacturing system with delivery guarantee. In their model, an order can always be served within the delivery time T whenever there are products available in the inventory. However, the model ignored an important and stochastic factor, the delivery time. Here in our model, we assume that the orders are served in the first come first serve principle. The delivery time consists of two parts: the cycle time and the delivery time. The cycle time is the time between the arrival of an order and the time requested product leaves the manufacturing system. The delivery time is the time for the requested product from the manufacturing system to the customer. In our study, we model the delivery time by a shitted exponential distribution. Unbiased and consistent estimators are derived for the distribution. Here we consider a one-machine system. The machine produces one type of product, the inter-arrival time of the demand and the processing time for one unit of product are assumed to be exponentially distributed. Finite backlog of the demands is allowed in the system. A cost of cB is assigned to each unit of backlog and a cost of c~ is assigned to each unit of inventory. Using the inventory level as an indicator, our interest is to determine the production strategy such that the average profit is maximized. We employ the (s,S) policy as the production control. An (s,S) policy is characterized by two integers s and S. The machine keeps on producing until an inventory level of S is reached. Once this inventory level is reached, the production is stopped by shutting down the machine. We let the inventory level falls to s, (s
SYSTEM
In this section, w e present a one-machine manufacturing system with product delivery time guarantee. Let us define the following notations: /~ - ~---themean processing time for a demand for one unit of the product, A - ~---themean inter-arrivaltime for one unit of the product,
O=operating cost for one unit of the product, P (P> O)--'unit price of the product, T---the guarantee time of delivery, s
An inventory model for manufacturing systems with delivery time guarantees
369
unit of rejected demand. To determine the optimal values of s and S, we have to set an appropriate guaranteed delivery time. In the next section, we study the delivery time distribution. 3. THE DELIVERY TIME D I S T R I B U T I O N
In this section, we study the delivery time distribution of an order. The total lead time of an order equals to the sum of the cycle time and the delivery time of the product. The processing time of an product is assumed to be exponentially distributed with parameter/z. If an arrival order is not a backlog, then the cycle time (time between the arrival of order and the time requested product leaves the manufacturing system) is zero. But if the order is in thejth backlog then the probability density function of the cycle time of the order is the Erlangian function of j-phase: f~(t) =
#.(p.t)~e -~' J!
Usually the delivery time is greater than a minimum time To. The time To may due to the packing of the products or the waiting time for transportation schedule. In our study, we model the delivery time by a "shifted" exponential distribution of parameters To and fl: {0fle-~ T(t)=
if t-->To, ift
(2)
A similar formulation has been used by Karmarkar [13] in the fractile estimation. In the following, we derive unbiased and consistent estimators for TOand 1/fl respectively. The estimators here are not unique, other forms of unbiased and consistent estimators are available, see [12] for instance. We first let
{ TI,T2..... T~} be a sample of n delivery times, i" be the mean of the sample and Tc~)=min{T~}. Lemma 1. Let T~<--Tt2)~--...<--T{.) be n order statistics generated by the exponential distribution f(t) =/3e -~,
then the probability density function for T.) is given by
g(t)=n~-,#t, E(T~I))=I/(nfl), andVar(T,))= I/(n3)~.
ProofSince each random variable/'tois exponentiallydistributedwith parameter fl,we
have
Pr°b(X)03->Y)= I tic-~ dt, for i=I,2.....n. Thus, i f X , ) ~ Y w e must have all the X~o>-Y Therefore, we have
Prob(X,)>-Y)=[ I ~c-/~dt]"=[e-#rl"=e-"#r. Differentiateboth sideswith respect to Y,we have the probabilitydensity functionof T,) given by
g(t)=n~¢-"#'. For more detailabout order statisticsdistributions,see [12].The probabilitydensity functiong(t) is still an exponential distributionwith parameter n//.Thus, we have E(T
EfTo))= To+ ~-fl. Therefore To) is an biased estimator and so is (7"- To)). Here, according to the bias, we consider the
estimators ~'o and 1/]~ for To and 1/~ satisfying the equations:
370
W.K. Ching
{ ~o ~. T(l) - 1~(hiS),
(3)
1/[3=7"- ~'o. Solving the equations, we get
{ ~o=(nT,) - T)l(n - 1), 11[3=n(7"- To)l(n - 1).
(4)
In the following, we prove that ~ro and 1//~ are unbiased and consistent estimators for To and l/fl respectively. Proposition 1. The estimators l"o=(nT,)-T)/(n-1) consistent estimators for To and 1/fl respectively.
and ll/3=n(7"-To)l(n-1)
are unbiased and
Proof. Since 1
E(~o) = ~
1
(nE(T.))- E(~)= n - 1 (n(To+fl/n)- To-fl)=To
and E(1/~)= ~
n
(E(~ - E(T(t)))=
n
(1/fl+ To- TO- l/(n~)= 1/~,
the estimators 1/fl and ~'o are unbiased. Moreover, by Lamma 1, we have
Var(1/~)= ~
n2 (Var(i")+Var(To)))=~ n(n~
+
1)
and
n2
Var(~'o)= ~
1 l 1 Var(T(,)) + (n - l) 2 Var(~= ((n- l)fl)2 + (n(n- l)fl)"
Thus lim Var(I/]~)=it--*eo lim Var(~'o)=O,
W--,m
and therefore ll~md ~'oare consistent estimators. • 4. DETERMINATIONOF MAXIMUMALLOWABLEBACKLOG In manufacturing system of limited production capacity, to guarantee customers a delivery time of the product, there should be an upper limit for the maximum allowable backlog m. Here we propose a simple estimation of the maximum allowable backlog m. Suppose we have k backlogged orders, the probability that they can be offered within the guarantee time T should be greater than a pre-determined value T (0< ~,< 1). The reliability value (or the service level) ~ can be 90% or 95% for example. The following proposition gives an inequality relating the quantities % T and the maximum allowable backlog m. Proposition 2. Given the delivery guarantee time T and the reliability value % the maximum allowable backlog m is the largest positive integer k which satisfies the inequah'ties: (1 - 3,)e'~r-ro)>- ~.~ ( T - To)J(#j -/~k(/z- B) ~-k)) +/zke -~#-~xr-To
,.0 - - - - - - i T - - -
- - -
for
(5)
3')e~r-r°)--- k~l (p.(T- To))2 +/zk(T - To)te-~r-r° for/.t--// j=o j! kt Proof. The processing time for one product is exponentially distributed with mean 1//~, therefore the probability density function for the total processing time of k products is given by the Erlangian function f,(t)=p.(la)*-~ e -~/(k- 1)!. The delivery time is modeled by the shifted exponential distribution (1
fie #' if t_>To g(t)= O if t
(6)
An inventory model for manufacturing systems with delivery time guarantees
--
7.
E ] --
it-
k
371
D ' @
°..
~
p.
p.
®.
...:
®
Q
The machine is shut off and the inventory level is i.
D
The machine is turned on and the inventory level isj. Fig. 1.
The parameters fl and To are estimated by the unbiased and consistent estimators in Proposition 1. Hence we have
r~ro ~)1"-----/'t(~)k-l(k-~• e - ' q l
- e
-/~r-ro-,)] &>--7.
By making use of the formula
~)k-l -~ 1(k_ )----~ e dt=l-e-~rkj~o
j!
we get the inequalities (5). Therefore the maximum allowable backlog m is the largest integer k satisfying the above inequality. • 5. THE INVENTORY LEVELS When the desirable guaranteed delivery time T and the unit price P of the product are decided, the maximum allowable backlog m (Proposition 2) and the demand rate A (cf. (1)) can be determined. We then employ the (s,S) policy in the production planning of the manufacturing system. We note that under the (s,S) policy, the inventory levels can take integer values in [ - re,S]. When the inventory level reaches S, the machine is shut off. When the machine is shut off, the inventory levels can take integer values in [s,S] and once it falls to s, the production is re-started. Therefore there are ( m + 2 S - s ) possible states in the inventory system, see Fig. 1. We order the states as follows. The first ( S - s ) states correspond to the situation that the machine is mined off, and the inventory levels are ordered from (s+ 1) up to S. The next (re+S) states correspond to the situation that the machine is turned on, and we order the inventory levels from ( S - 1) down to - m. We the construct the generator matrix for the machine-inventory system. A generator matrix Z of a Markov process gives the transition rate [Z]# from statej to state i. Under our ordering, we obtain the following generator matrix A for the machine-inventory system: A O
-A A
O
.,.
0 -A ..°
0
A[ O -A O
.°.
A
-A
°..
0
A+/z -A
-/z A+/z
-/z
..°
°..
-A
S-s+l "S-s+2 ...
2(S-
A+/.t .°.
-A 0
1 2
s)
° . .
A+p -A
--ft
/z
:
m + 2S - s
372
W.K. Ching
The generator matrix A is irreducible, has zero column sum, positive diagonal entries and non-positive off-diagonal entries. Therefore A has one dimensional null space; see Varga [16]. The steady state probability distribution is the normalized positive null vector of A. 6. T H E S T E A D Y S T A T E D I S T R I B U T I O N
OF THE INVENTORY
LEVELS
In this section, we derive the analytic form of the steady state probability distribution of the inventory levels. To obtain the probability distribution vector p, we perform the following row operations to A. Beginning from the last row of A, we add the ith row of A to the ( i - 1 ) t h row of A, for i=(m + 2 S - s ) , ( m + 2 S - s - 1)..... 2. The purpose of these row operations is to eliminate the upper subdiagonal ofA. We get a lower triangular matrix B:
B=
0 -a -a -a -a -,a A
0 a 0
0 0 ...... a 0
0 a 0
0 /~ -a
i
0 I-*
1 2 :
] S-s+l "S-s+2
0
. . . . . . . . .
-a
-a
(7)
"
/z
0
2 ( S - s)
.........
:
;
-a
0
/.*
-A
0
i
I~ m + 2 S - s
By applying the forward substitution to (7), we have the following proposition concerning the steady state probability distribution of the machine-inventory system. Proposition 3. The steady state probability distribution p o f A can be written as follows:
a P=(Pt,P2 .... ,p,,+2s_s)=a-'(1,V,U)', llx(s_,)=(1,1 ..... 1), p= --. /x For p ~ 1, we have
v
=
u
=
a'
----
- - P( l - p , .... 1 - p S - ' ) , 1-p p(1 - p s - s) (p, p2 ..... pro+s), 1--p pm.e(pS _ p') + (1 -- p ) ( S - s) (1 _p)2
(8)
For p= 1, we have
v
=
(1,2,3 ..... S - s ) ,
u
=
(S-s,S-s
=
1 ~(S-s)(3+S+s+2m).
a
..... S - s ) ,
(9)
P r o o f We first note that i f p is a right null vector of A (i.e. A p = 0 ) then p is also a right null vector o f B (i.e. Bp=0). We discuss the case when p # 1, the case p = 1 being similar. Letpt = a - J>0, we observe from (7) that p ~ = a - t for i = 1 . . . . . ( S - s ) . By forward substitution, we obtain
Ps-s+i--
131
lP(1 _pi), i=1 ..... ( S - s ) 1-p -
-
and a - tp(1 -
P2(s- ~)+~=
1-p
pS-,) IJ, j = 1..... (m + s).
An inventory model for manufacturing systems with delivery time guarantees
373
The normalization constant a is obtained by the relation 1=
m+2S-s ~, p i = a i=l
pm+2(pS-p~)+(1-p)(S-s) -l
(1 - p ) ~
• "
We observed that the steady state probability of the first ( S - s ) are equal. In fact, the Markov chain formed by the (S - s) states is close to that of Erlangian distribution of ( S - s) phases with parameter A. The generator matrix of the Erlangian distribution is given the following ( S - s) x ( S - s) matrix E: A E
-A
0
~
.
.
.
.
.
.
A -A
-A A
and clearly the steady state probability of this generator matrix is given by (l/(S-s))(1,1 ..... 1)' which has equal probability for each state. From the distribution vector p, we get the marginal probability of the inventory levels. P r o p o s i t i o n 4. The marginal probability distribution f o r the inventory levels is
q = (q-
m .....
qo,ql ..... qs)"
Here q~ is the steady state probabiHty that the inventory level is i, and i = - m .... ,s ..... S. For p ~ l, we have or=
pm+2(pS _ p~)+ (1 -- p ) ( S - s) (1 _p)2
and _ P (l__pS-s)pr-i, Ot I i _ p
qi=
a -I(
1 -Pp ( 1 - - p S - t ) ) ,
i= - m ..... (s - 1),s, (1o)
i = ( s + l ) .....
For p= 1, we have a = ( 1 / 2 ) ( S - s)(3 + S+s+ 2m), and fa-'(S-s), qi= ~ a - I( l + S --
i = - m ..... 0 ..... ( s - 1),s, i=(s + l ) ..... S.
(ll)
Proof. From the ordering of the states (see Fig. 1), we have qs=Ps_s, qs_i=Ps_~_i+Ps_s+~, i= 1,2..... (S - s - 1)
and q_m+i=Pm+2S_s_i, i=0,1 ..... (re+s).
The proof is completed by using Proposition 3. • 7. THEAVERAGE PROFIT The average profit of the system can be written as follows: (P - O).S(s,S) - l(s,S) - B(s,S) - P(s,S),
where S(s,S), I(s,S), B(s,S) and P(scS) are the average sales, the average inventory holding cost, the average backlog cost and the average demand rejection penalty cost respectively. Here P and O are the unit price and unit operation cost of the product respectively. In terms of the inventory distribution q, the average sales S(s~g), the average inventory cost l(s,S) and the average demand rejection penalty cost P(s,S) can be written as
374
W.K. Ching S
S
S(s,S)= A i~=lqi, I(s,S)=cj i~=liqi and P(s,S)=cpAq_m, respectively. To obtain the average backlog penalty cost we need the following proposition. Proposition
5. The probability w~ that the jth backlog order cannot be offered within the delivery time
T is given by -i ] z j e - (~ - g ) ( r - r o) /
e
(/-~-/3)s
i!
e
i!
k!
j,
for/z~fl,
(12)
for pt=fl.
'
Proof. The result can be proved directly by Proposition 2. • Given k (k>- 1) backlog orders, the probability b *~that i orders out of the k are delayed can be written in terms ofwj. For examples, when k=3, the probabilities b3;, i=0,1,2,3 are given by b3o = (1-wl)(1-w2)(1-w3), bal = Wl(1- w2)(l - w3)+(1 - Wl)W2(l b32 = (1 - wt)w2w3+wt(1 - - W 2 ) W 3 + W I W 2 ( 1
b33
=
w3)+(1 --
-
wOO - w2)w3,
(13)
W3),
W l W 2 W 3.
The expected number of delay when there are k backlog orders is then given by • ~ffi~ib~. When an order is not in backlog, the probability that it will be delay is given by e-~r-r0) (of. (2)). Hence, the average backlog cost is given by
B(s'S)=cB{ ~ [ ( ~i.,i b ~ ) q - k ] + e - I ~ r - r ° ' ~if,q i } However, from the numerical examples we observed that when the service level T is close to 1 (e.g. T=0.95), the backlog cost is negligible when compare with the inventory cost and the profit. This can be explained by the fact that we have set a high service level (reliability) for the system. Therefore, in the following, we assume that the average profit of the manufacturing system is given by S
[A(P - O) -
i=l
cli]q
cpAq-m.
i -
(14)
8. S P E C I A L C A S E A N A L Y S I S
In this section, we consider the case of allowing infinite backlog with negligible backlog cost ca . In this situation, the average profit is given by S(s,S)- I(s,S). We prove that under some conditions, the optimal (s,S) policy is of hedging point type (i.e. S= (s + 1)) which is the interesting case in [4-7]. Proposition
6. Under the permission of infinite backlog and zero backlog cost, if 0
-
~/5 2 -
1
A(P - O)(1 -
and q->
1_p_p2
p)p2
,
then the optimal (s~.g)policy is of hedging point type, i.e. S=(s+ 1). Proof. Using (10) we have S(s,S) - / ( s , S ) =
(A(P - O) + (Cl/(1 - p)))( l - p2f(s,S)) - q(S + s) 1 - p'+~s,S)
where
pS -- pS Ks,S) =
(1 - p ) ( s - s)"
05)
An inventory model for manufacturing systems with delivery time guarantees
375
When infinity backlog is allowed (m--. oo), the average profit is given by
(
H(s,S)= a ( P - O ) + ~
cl)
(1-p2f(s,S))-cl(S+s).
(16)
To maximize H(s,S), it is equivalent to minimize
(
c,)
(
c,)
G(s,S)= A ( P - O ) + I~_
(pZf(s,S))+q(S+s).
Let t = ( S - s ) > - 1, we write
G(s,t)= A ( P - O ) + 1 - ~
(P2f(s's+t))+cI(t+2s)"
In the following, we prove that G(s,t) is an increasing function in t for any given s. Thus for any given s, G(s,t) attains its minimum at t= 1. Indeed,
OG(s,t)
(A(P-O)(1
0--~ =
1 -tptlogp)
- - p ) + C l ) p s+2 ( p t _
(1 - p)2
t2
+c~
(A(P- O)(I - p)+ci)p ~÷2 ( p t - 1) > (1 - p)2 t---T--- +ci - (A(P - O)(1 - p ) + C l ) p
s+2
-
(A(P- O)(1 - p) + Cl)p 2
q-Cl~
(1 -p)
(1 -p)
+C I
= - A(P - O)(1 - p)p2+ (1 - p - p2)cl(1 - p)-->O. The first inequality comes from fact that - tp' log p>0. The last inequality comes from the condition (15). The second inequality comes from the fact that the negative function (pt_ 1)/t 2 is an increasing function in t and we give the proof as follows: p'-I t~
p-1 = -~(l+p'+...+p
<'-'>)
_p-1
- t(t+l-----~(l+ _ p-1 -
t(t+
)(l+pl+..'+p <'-l))
{(l+p+...+p¢,_t))+ 1
1-----)
t (1 +p+...+p(t-1))}
< p-1
p-1
-- t(t+ 1) (1 +p+...+p~)--< ~
(1 +p+...+p')
p,+l _ 1 (t+ 1)2 • Thus, the proposition is proved. • As a corollary, when the optimal policy is of hedging point type, the average profit is given by (16) with S=(s+ 1), i.e. CI
H(s,s+ 1)=(A(P - O)+ 1 - ~ )(1 - p2f(s,s+ 1)) - ci(2s+ 1). We observe that
=-
A(P-O)+ 1-~
p2(I°gp)2p~<0'
and therefore H(s,~+ 1) is a convex function in s. Thus, the maximum point is given by the unique solution of
W. K. Ching
376
Table 1. The optimal (s',S') for different p p m s S"
C(s',S')
0.1 32 l 2 $773.1
dH(s,S+ds1) _ _
0.2 20 2 3 $766.1
0.3 12 3 4 $ 757.8
(
A(P- O)+ ~
9. N U M E R I C A L
c,)
0.4 9 4 5 $748.4
0.5 7 5 6 $737.3
p2(log p)pS_ 2c,=0.
EXAMPLES
In what follows, we give some numerical examples. Recall that the demand rate
A=oe -~,T -~'.(cf. (1)). We let the elasticity constants z,, r2, and o-to be 2, 1, and 1000, respectively. The parameters P, O, A, p and c, are set to satisfy the condition (15) in Proposition 6. The unit price P is $100 and the unit operating cost O is $20. The guaranteed delivery time T is 1 day. The parameters To and fl of the delivery distribution are to be 0.1 and 10 respectively. The maximum inventory capacity/max is 100. The inventory cost c, per unit of product is $10. Furthermore, we let the reliability factor to be 90%, the penalty cost cp to be $2000. For the delay cost cB, we set it to be zero. In Table 1 we give the optimal pair (s',S ") and their corresponding average profit C(s',S °) for different values of p= A//z. We found that although the backlog cost cB and penalty cost cp are included, the optimal policy is still of hedging point type. In Tables 2-6, we investigate the optimal profit of the above numerical examples when a constraint Table 2. The optimal profit for different d when p=0.1 d
C(s',s'+d) d
C(s',s'+d)
1 $773.1 7 $749.8
2 $771.7 8 $745.0
3 $768.1 9 $740.1
4 $763.9 10 $735.2
5 $759.3 20 $690.6
6 $754.6 40 $590.9
Table 3. The optimal profit for different d when p=0.2 d
C(s',s'+d) d
C(s'~*+d)
1 $766.1 7 $746.8
2 $763.6 8 $742.5
3 $759.8 9 $738.0
4 $757.5 10 $733.5
5 $754.5 20 $690.8
Table 4. The optimal profit for different dwhen d
C(s',s'+d) d
C(s',s'+d)
1 $757.8 7 $739.9
2 $755.1 8 $736.4
3 $754.2 9 $732.8
4 $751.6 10 $728.9
6 $750.8 40 $591.6
p=0.3
5 $748.1 20 $689.8
6 $744.1 40 $591.6
Table 5. The optimal profit for different d when p=0.4 d
C(s',s'+a') d
C(s ",s"+d)
1 $748.4 7 $734.4
2 $747.2 8 $730.9
3 $745.9 9 $727.1
4 $743.3 10 $723.1
5 $739.9 20 $685.3
6 $737.4 40 $591.2
Table 6. The optimal profit for different d when p=0.5 d
C(s',s'+d) d
C(s',s'+d)
1 $737.3 7 $725.7
2 $736.0 8 $722.4
3 $735.2 9 $718.7
4 $733.1 I0 $714.9
5 $730.4 20 $636.0
6 $728.4 40 $589.6
An inventory model for manufacturing systems with delivery time guarantees
377
S - s = d is imposed on the optimization. We found that the derivation o f the average profit from the optimal average profit grows in polynomial rate (more than linear rate) in d. Thus, the average profit is sensitive to the values o f s and S.
10. CONCLUDING REMARKS In this paper, we have discussed optimal (s,S) policies for production planning with delivery time guarantees. We have established a relation (Proposition 2) between the desirable maximum allowable backlog m and the guaranteed delivery time Z Analytic form o f the average profit has also been derived. By varying different possible values o f s and S, the optimal (s,S) policy can be obtained. A sufficient condition (Proposition 6) for the hedging point policy to be optimal has also been given. It is interesting to relax the condition for hedging point policy to be optimal and extend the model to the case when there are set-up time and set-up cost for the manufacturing system. These issues are studied in another paper
[3]. Many other interesting problems remain open. For example, what if the machines are unreliable? Extension to the ease o f multi-item is also interesting. Furthermore, other distributions (e.g. the Weibull distribution) can be used to model the delivery time. Another possible extension o f our model is the following non-linear programming problem. We do not pre-determine the unit price P and the guaranteed delivery time T (Recall that the demand rate is a function o f P and T, of. (1).) Rather, they are part o f the decisions. The generalized formulation is given as follows: maxe, r~,s
{ E [ ai ( P - O ) - c l i ] q ~ - c p a q _ . } i =
s.t. (5),
s < S < - I ~ , and 1 - m .
Acknowledgements---The author would like to thank the referees for their helpful comments and constructive suggestions in revising the paper. Research supported by RGC Earmarked Grant CUHK-489/95Eand Hong Kong CroucherFoundationGrant.
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