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Optimal production policy for decaying items with decreasing demand
168
European Journal of Operational Research 43 (1989) 168-173 North-Holland
Theory and Methodology
Optimal production policy for decaying items with decreasing demand T.C.E. C H E N G *
Department of Actuarial and Management Sciences, Unioersity of Manitoba, Winnipeg, Manitoba, Canada R3T 2N2
Abstract: In this paper an infinite-horizon inventory problem is considered in which items are decaying at a constant rate and the demand rate follows an exponentially decreasing function. An analytical model based on dynamic programming is developed from which closed-form optimal resuls are derived. Earlier results for a similar problem without item deterioration can be obtained as a special case of our model. A special condition which establishes the general validity of the optimal results is also presented. Finally an illustrative example is provided to demonstrate how the theoretical results can be applied to solve a given problem. Keywords: Inventory, dynamic programming, EOQ
Introduction
This paper is an extension of the classical economic order quantity (EOQ) problem to the case where the inventory items are decaying over time with demand decreasing in some known fashion. This inventory problem is often faced by manufacturers in the chemical processing and pharmaceutical industries where products, such as organic solvents for making medicine, are highly volatile and fast becoming obsolete as a result of rapid success in research and development. Designing an optimal production policy for such inventory items requires the determination of an optimal sequence of replenishment intervals over an infinite planning horizon. * This research was undertaken while the author was an academic visitor at the Management Studies Group, Engineering Department, University of Cambridge. Received June 1988, revised November 1988
Early work on inventory problems with deteriorating items has been undertaken by Ghare and Schrader (1963). Their basic model has subsequently been extended by Covert and Philip (1973), Misra (1975), Shah (1977), Tadikamalla (1978), Elsayed and Teresi (1983), Hollier and Mak (1983), Kang and Kim (1983), and Cheng (1986), among others. For inventory problems with varying demand rates, Donaldson (1977), and Smith (1977) are among the pioneers who have presented analytical optimal results. While Donaldson uses calculus to study the finite-horizon problem with a linear trend in demand, Smith applies the dynamic programming technique to deal with the infinitehorizon problem with an exponentially decreasing demand rate. The work of Donaldson has inspired a wealth of subsequent research into problems with linearly varying demand rates by various researchers, who include Henery (1979), Silver (1979), Buchanan (1980), Phelps (1980), Ritchie
0377-2217/89/$3.50 © 1989, Elsevier Science Publishers B.V. (North-Holland)
T. C E. Cheng / Optimalproductionpolicy with decreasingdemand (1980, 1984), Mitra et al. (1984), Goyal et al. (1985), and Goyal (1987), to name a few. So far it seems that study of inventory problems with decaying items and variable demand rates is almost non-existent in the literature. The only exception is perhaps the work of Hollier and Mak (1983). While the paper of Hollier and M a k is concerned with the finite-horizon problem, we will focus our attention on the infinite-horizon problem in this paper. As with Hollier and Mak's model, we assume that the demand rate is exponentially decreasing. But we will follow the approach of Smith (1977) by employing dYnamic programming (DP) to derive closed-form optimal results for our infinite-horizon problem. We will show that all the results in Smith (1977) can be readily obtained as a special case of the more general results presented in this paper. In addition, we will show that the results for the optimal number of batches are valid only when the unit cost of production of the item satisfies a special condition.
Model and assumptions The inventory problem addressed in this paper is to minimize the combined costs of setups, production and inventory-holding associated with meeting the total future demand from stock for a decaying item which is obsolete. Unlike the classical EOQ problem which is only concerned with the optimal order size, see Hadley and Whitin (1963), our problem is to find the optimal number of batches, and the corresponding optimal batch quantities, to minimize the total cost over an infinite planning horizon. We define the following notations to be used to develop the total cost model for subsequent analysis.
D(t) I(t)
= Setup cost per batch (c > 0). = Holding cost per item per unit time (h > 0 ) . = Unit cost of production per item ( p > 0). = a e --~t= Exponentially decreasing dem a n d rate at time t (13>~0, D(t)>~O, t>~0). = Inventory level at time t ( I ( t ) >1O, t >1 0).
169
X
= Constant rate of decay per item in inventory per unit time (X >/0). Fn(a ) = Minimum total cost of setups, production and inventory-holding, consistent with supplying all future demand from stock; starting with no inventory and an initial demand rate of a; and making exactly n batches (n ~ I + U {0}). S(a, t ) = C o s t of inventory-holding associated with producing a batch at time 0 when the demand rate is a to meet the total demand in the period 0 to t at which the inventory is exhausted and a new batch is required. In addition, we make the following explicit assumptions about the model: (1) Rate of demand for the items follows an exponential decreasing function with parameters a and/3. (2) Items are decaying over time with a constant rate of decay X. (3) N o shortages are permitted. (4) /3 is greater than X. Although we fully appreciate the restrictive nature of Assumption 1, we require it to simplify the mathematical analysis to derive the optimal resuits. In addition, as argued by Smith (1977), Assumption 1 is not as restrictive as it first appears since, in the words of Smith, " b y varying the parameter, a very wide range of demand types can be approximated". Assumptions 2 and 3 are standard for treating EOQ problems with decaying items, see Ghare and Schrader (1963), so they are included in our model. Assumption 4 is required to ensure that sufficient items are available to meet demand after decay has been accounted for. With a constant rate of decay X, the change in inventory level of the items over time will be non-linear. Specifically, the inventory level I ( t ) (0 ~< t ~< to) is governed by the following first-order differential equation, where t o is the time instant at which inventory first drops to zero.
dI(t)/dt
+ X I ( t ) = - a e -Bt
(1)
which, upon solving, yields
I ( t ) = [ a e - X ' / ( / 3 - )k)] {e - ( ~ - x ) ' - e -~B-x)t° }.
(2)
T CE. Cheng / Optimalproduction poficy with decreasing demand
170
Evidently, the total number of items produced in the first batch is equal to the starting inventory I(0) which will last until time t 0. It follows that the total cost of production for the first batch is given by
horizon. Following Smith's argument, we see that the boundary conditions (6) requires that t~* must be infinite, i.e. with only one batch remaining, it must meet all future demand. Thus, letting n = 0 and q* ---, oo in (7), we obtain
pI(O) = p [ a / ( t i - )t)] {1 - e-(~-x)'° }.
F l ( a ) = c + (1 + t i p / h ) { h a / [ t i ( t i - ) t ) ] } .
(3)
It is also clear that the inventory-holding cost incUrred in the period 0 to t o is equal to the quantity hfo <, < toI(t) dt. Thus,
S(a, to)
(8)
Expression (8) prompts us to conjecture that, in general, when n + 1 batches are to be produced over the entire planning horizon, the following proposition would be true.
Proposition 1.
= hfo
F~+,(a) = (n + 1)c + a n + l h a / [ t i ( t t - X)]
= (ha/[ti(ti-X)]} x {(1 - e -too) + ,8 e-Pro(1 - eXto)/X }.
(4)
It is interesting to note that by letting 2, approach zero in (4) and applying the 1' Hospital's rule for limits, one can obtain the result given in Smith (1977).
Optimal replenishment instants
with al=(l+flp/h
and n ~ I + U ( O } .
Proof. We will prove the proposition by mathematical induction. It is obvious from (8) that the proposition is true for n = 0. Assuming that the proposition is true for n = k, we obtain from (7)
Fk+,(a)
In order to derive the optimal solution to our inventory problem, we follow the approach presented in Smith (1977) which formulates the problem as a dynamic program as follows:
=c+ Min{(ha/[fl(fl-)~)]) t>~0
X [(1 + tip~h)(1
t>~O
+ kc +
(5)
-
e -e')
+ f l e-/~'(1 _ eXt)/X]
F.+l(a ) = c + Man (pl(O) + S(a, t) + F , ( a e-Z')}
)
akha
- ~)1 }
e-B'/[ti(ti
= (k + 1)c + h a / [ t i ( t i - 2,)1 {(1 + tip~h)
subject to the following boundary conditions: -Min F0(a)={O
ifa=0,ifa>0.
e - ~ ' [ 1 - ak - t i ( 1 - eXt)/X ] }.
t>~O
(6)
It is easy to show that Substituting (3) and (4) into (5) yields
tt+,
F,+l(a)
=
in( [ti - (1 - ,,k)x]/(ti
- x)},
(9)
--c + Min { ( h a / [ f l ( t i - ) Q ] )
therefore
t>~0
X [(1 + tip~h)(1 - e -~,)
Fk+l( a ) = ( k + 1)c + ( h a / [ t i ( t i - )t )])
+ 13 e - " ( 1 - eX')/)~]
+F,(ae-#')}.
X [(1 + tip~h) (7)
Let t * l , t*, t*,-1 . . . . , q* denote the optimal replenishment instants, given that n + 1 batches are to be produced over the infinite planning
-
e -(e-x)tt+']
= (k + 1)c + a k + , h a / [ t i ( t t --•)], where ak+ 1 = (1 +
tip/h) - e -(B-x)tt+l
T.C.E. Cheng / Optimal production policy with decreasing demand
171
therefore
which, on substituting (9), becomes a,+ 1 = (1 + t i p ~ h ) - {1 - a k / ( 1 -- t i / h ) } ( ' - a / x )
2~
-(ti/h + r- 2)/r!]
Following the principle of induction, proof of the proposition is established immediately. Thus, given that the number of batches to be produced is n + 1, we can use (10) to solve for the sequence ( a l, a 2 , . . . , a , + a } recursively and stubstitute them into (9) to compute the set of optimal replenishment instants ( t*+ 1, t*, t*n - - a . . . . . tl* }. Again, it is interesting to note that by letting both p ---)0 and X ~ 0, one can obtain the results in Smith (1977, eqs. (6) and (7)).
Y', [ ( t i / x - 1 ) ( t i / x ) ( t i / X + l )
a k -- a k + 1 =
(lo)
X ( aJ(ti/X
- 1)} r - t i p / h .
Since fl > X from Assumption 4 and a~ > 0 for all k ~ I +, it follows that a , - a,+ 1 > 0 iff condition (11) holds. This completes the proof of the proposition. C o r o l l a r y . a 1 = a 2 --- a 3 . . . .
p = (h/ti)
E
iff
[(ti/X- 1)(ti/X)(ti/X + 1) •(ti/X + r - 2)/r!]
Optimal number of batches
)( { a n / (
We first present and prove a proposition, the results of which will facilitate the determination of the optimal number of batches for a given problem. Proposition 2. The sequence { a 1, a 2 .... } is strictly positive. In addition, it is strictly decreasing iff E
p < (h/ti)
[ ( t i / X - 1 ) ( t i / X ) ( t i / X + 2)
2 ~ < r { oo
•(ti/X + r - 2)/r!] ×{a,/(fl/X-1)}
~ forallnEI
+.
(11)
Proof. To prove the first part, we note that expression (9) requires that t.*+, = ( a / x )
ln{[ti
- (1 - a . ) X ] / ( t i
- X ) } > 0.
Thus the result that a . > 0 for all n ~ I + follows immediately. To prove the second part, we expand the second term on the right-hand side of expression (10) by the Binomial Theorem to obtain a , + 1 = (1 + t i p / h ) -{1 -a, +
E
[ ( t i / X - 1 ) ( t i / X ) ( t i / ) ~ + 1)
2~
-(ti/k + r- 2)/r!]
× {a/(ti/X-
1)} r },
ti/~
-- 1)} r
for all n ~ I +.
(12)
When condition (11) holds, we can follow Smith's approach to determine the optimal number of batches. Since under such a condition, the sequence { a 1, a 2. . . . } is strictly decreasing, it follows from Proposition 1 that the sequence of minimum total cost functions {F,+](ct ), F , ( a ) . . . . . Fl(a)} have increasing intercepts at the ordinate but decreasing slopes. In addition, the set of intersection points of adjacent minimum total cost functions, ( D , : F,( D , ) = F,+1( D , ) }, is strictly increasing. The { D, } are given by D, = cti(ti - X ) / [ h( a , - a , + l ) ] .
(13)
As observed by Smith, for all values of D,_ 1 ~< a ~< D, the least cost is given by F , ( a ) and the optimal number of batches is n. Clearly, if ~ ~< D~ only one batch need be made. We should bear in mind that this simple rule for determining the optimal number of batches is valid only when condition (11) holds. However, if the unit cost of prodution is negligible, i.e. p---, 0, then condition (11) always holds and the rule is always valid. Alternatively, condition (11) is always true when = 0, corresponding to the case of non-deteriorating inventory items. Thus, this rule is valid for the problem considered by Smith (1977) in which the inventory items do not decay. Furthermore, in the limiting case when condition (12) holds, the sequence of minimum total cost functions will have
T C.E. Cheng / Optimalproduction policy with decreasing demand
172
the same slopes. Thus the least cost in such a situation is given by F~(a). Hence only one batch should be made with a size equal to all future demand a/(fl - ~ ). Finally, we will briefly discuss the condition for the final batch. We see from (13) that the final batch should be made when the following condition holds:
(10) and noting that a 1 = (1 + tip~h) by definition, we obtain a I = 1,
(15)
a 2 = ½,
(16)
a, =
(17)
Substituting (15) and (16), and (16) and (17), into (13) respectively yields
a <~D 1 = c f l ( f l - X ) / [ h ( a I - a2) ] = [ c f l ( f l - X)/h] { f l / ( f l - X)) (#/x-l) which can be rewritten as
or
~ ) <~EOQ- { i l l ( r -
h)}/~/2x/V~-.
(18)
O z = 600.
(19)
Since a = 350 lies between D1 and D 2, two batches should be produced according to the optimal batching rule, i.e. n* = 2. It follows that the least cost is Fz(350), which can be determined using the result of Proposition 1 as follows:
{ a / ( fl - X) } 2 <~[2 ca/h ] { fl/( fl - X) } t]/x/2,
a/(fl-
D1 = 200,
(14)
F2 (350) = 2(100) + (½)(0.05)(350)/(1 - ½) = $217.5.
It is easy to see that the left-hand side of expression (14) is the total future demand while the fight-hand side is the product of a constant depending on the problem parameters and the standard EOQ formula. Thus the stopping rule can be stated as follows: Make the final batch with a size equal to the total future demand when it is less than the EOQ multiplied by a constant given by the second term on the right-hand side of
(14). When ?, ---, 0 the second term on the right-hand side of (14) becomes lim { fl/(fl
-
(20)
To find the optimal interval for the first batch, we use (9) to obtain t~' = 2 In 2 = 1.39 days.
(21)
Finally, we apply the standard EOQ formula to find EOQ = [2ca/hp/2 = 1183.
(22)
It follows from the stopping rule (14) that the final (second) batch should be produced with a size equal to total future demand when it is less than EOQ/(2v/2 -) = 419.
~ ) } / ] / 2 x / v ~ = ( e / 2 ) 1/2
A---,0
which, once again, is identical to the result presented in Smith (1977, eq. (10)).
An example For a given problem, the following data have been estimated: c = $100/batch, h = $ 0 . 5 / i t e m / day, p = $0.01/item, a = 350 items/day, fl = 1 item/day, 2~= 0.5 items/day. To solve this problem, we first note that p is small compared with the other costs. So we can safely assume that p--* 0 to simplify the procedure to find the optimal solution. Using equation
Conclusions We have considered an infinite-horizon inventory problem in which items are decaying at a constant rate and the demand rate follows an exponentially decreasing function. An analytical model based on dynamic programming has been developed from which closed form optimal results have been derived. We have noted that earlier results for a similar problem without item deterioration can be easily obtained from a special case of our model. We have also presented a special condition which establishes the general validity of the optimal batching rule. Finally we have provided a numerical example to demon-
T. C. E, Cheng / Optimal production policy with decreasing demand
s t r a t e h o w to a p p l y the theories to f i n d the optim a l s o l u t i o n to a g i v e n p r o b l e m .
Acknowledgements T h e a u t h o r wishes to a c k n o w l e d g e the N a t u r a l Sciences a n d E n g i n e e r i n g R e s e a r c h C o u n c i l of C a n a d a for p r o v i d i n g p a r t i a l f i n a n c i a l s u p p o r t for this research. H e is also g r a t e f u l to P r o f e s s o r S.R. W a t s o n of the M a n a g e m e n t S t u d i e s G r o u p , a n d P r o f e s s o r C. A n d r e w of the M a n u f a c t u r i n g E n gineering Group, Engineering Department, Univ e r s i t y of C a m b r i d g e for p r o v i d i n g the facilities to u n d e r t a k e this research.
References Buchanan, J.T. (1980), "Alternative solution methods for the inventory replacement problem under increasing demand", Journal of the Operational Research Society 31,615-620. Cheng, T.C.E. (1986), "EOQ with limited backorder delays", Computers and Operations Research 13, 477-480. Covert, R.P., and Philip, G.C. (1973), "An EOQ model for items with Weibull distribution deterioration", A I I E Transactions 5, 4-9. Donaldson, W.A. (1977), "Inventory replacement policy for a linear trend in demand--An analytical solution", Operational Research Quarterly 28, 663-670. Elsayed, E.A., and Teresi, C. (1983), "Analysis of inventory systems with deteriorating items", International Journal of Production Research 21,449-460. Ghare, P.M., and Schrader, G.F. (1963), "A model for exponentially decaying inventory", Journal of Industrial Engineering 14, 5-11. Goyal, S.K. (1987), "A simple heuristic method for determining economic order interval for linear demand", Engineering Costs and Production Economics 11, 53-57.
17 3
Goyal, S.K., Kusy, M., and Soni, R. (1985), "Economic replenishment interval for an item having positive liner trend in demand", Engineering Costs and Production Economics 10, 253-255. Hadley, G., and Whitin, T.M. (1963), Analysis of Inventory systems, Prentice-Hall, Englewood Cliffs, NJ. Henery, R.J. (1979), "Inventory replacement policy for increasing demands", Journal of the Operational Research Society 30, 611-617. Hollier, R.H., and Mak, K.L. (1983), "Inventory replenishment policies for deteriorating items in a declining market", International Journal of Production Research 21,813-826. Kang, S., and Kim, I.T. (1983), "A study on the price and production level of the deteriorating inventory system", International Journal of Production Research 21, 899-908. Misra, R.B. (1975), "Optimal production lot-size model for a system with deteriorating inventory", International Journal of Production Research 13, 5-12. Mitra, A., Cox, J.F., and Jesse, R.R. (1984), "A note on determining order quantities with a linear trend in demand", Journal of the Operational Research Society 35, 141-144. Phelps, R.I. (1980), "Optimal inventory rule for a linear trend in demand with a constant replacement period", Journal of the Operational Research Society 31,439-442. Ritchie, E. (1980), "Practical inventory replacement policies for a linear trend in demand followed by a period of steady demand", Journal of the Operational Research Socie(v 31, 605-613. Ritchie, E. (1984), "The EOQ for linear increasing demand: A simple optimal solution", Journal of the Operational Research Society 35, 949-952. Shah, Y.K. (1977), "An order-level lot-size inventory model for deteriorating items", A I l E Transactions 9, 1-6. Silver, E.A. (1979), "A simple inventory replacement decision rule for a linear trend in demand", Journal of the Operational Research Society 30, 71-75. Smith, P.H. (1977), "Optimal production policies for items with decreasing demand", European Journal of Operational Research 1, 365-367. Tadikamalla, P.R. (1978), "An EOQ inventory model for items with gamma distributed deterioration", AIIE Transactions 10, 1-5.
European Journal of Operational Research 43 (1989) 168-173 North-Holland
Theory and Methodology
Optimal production policy for decaying items with decreasing demand T.C.E. C H E N G *
Department of Actuarial and Management Sciences, Unioersity of Manitoba, Winnipeg, Manitoba, Canada R3T 2N2
Abstract: In this paper an infinite-horizon inventory problem is considered in which items are decaying at a constant rate and the demand rate follows an exponentially decreasing function. An analytical model based on dynamic programming is developed from which closed-form optimal resuls are derived. Earlier results for a similar problem without item deterioration can be obtained as a special case of our model. A special condition which establishes the general validity of the optimal results is also presented. Finally an illustrative example is provided to demonstrate how the theoretical results can be applied to solve a given problem. Keywords: Inventory, dynamic programming, EOQ
Introduction
This paper is an extension of the classical economic order quantity (EOQ) problem to the case where the inventory items are decaying over time with demand decreasing in some known fashion. This inventory problem is often faced by manufacturers in the chemical processing and pharmaceutical industries where products, such as organic solvents for making medicine, are highly volatile and fast becoming obsolete as a result of rapid success in research and development. Designing an optimal production policy for such inventory items requires the determination of an optimal sequence of replenishment intervals over an infinite planning horizon. * This research was undertaken while the author was an academic visitor at the Management Studies Group, Engineering Department, University of Cambridge. Received June 1988, revised November 1988
Early work on inventory problems with deteriorating items has been undertaken by Ghare and Schrader (1963). Their basic model has subsequently been extended by Covert and Philip (1973), Misra (1975), Shah (1977), Tadikamalla (1978), Elsayed and Teresi (1983), Hollier and Mak (1983), Kang and Kim (1983), and Cheng (1986), among others. For inventory problems with varying demand rates, Donaldson (1977), and Smith (1977) are among the pioneers who have presented analytical optimal results. While Donaldson uses calculus to study the finite-horizon problem with a linear trend in demand, Smith applies the dynamic programming technique to deal with the infinitehorizon problem with an exponentially decreasing demand rate. The work of Donaldson has inspired a wealth of subsequent research into problems with linearly varying demand rates by various researchers, who include Henery (1979), Silver (1979), Buchanan (1980), Phelps (1980), Ritchie
0377-2217/89/$3.50 © 1989, Elsevier Science Publishers B.V. (North-Holland)
T. C E. Cheng / Optimalproductionpolicy with decreasingdemand (1980, 1984), Mitra et al. (1984), Goyal et al. (1985), and Goyal (1987), to name a few. So far it seems that study of inventory problems with decaying items and variable demand rates is almost non-existent in the literature. The only exception is perhaps the work of Hollier and Mak (1983). While the paper of Hollier and M a k is concerned with the finite-horizon problem, we will focus our attention on the infinite-horizon problem in this paper. As with Hollier and Mak's model, we assume that the demand rate is exponentially decreasing. But we will follow the approach of Smith (1977) by employing dYnamic programming (DP) to derive closed-form optimal results for our infinite-horizon problem. We will show that all the results in Smith (1977) can be readily obtained as a special case of the more general results presented in this paper. In addition, we will show that the results for the optimal number of batches are valid only when the unit cost of production of the item satisfies a special condition.
Model and assumptions The inventory problem addressed in this paper is to minimize the combined costs of setups, production and inventory-holding associated with meeting the total future demand from stock for a decaying item which is obsolete. Unlike the classical EOQ problem which is only concerned with the optimal order size, see Hadley and Whitin (1963), our problem is to find the optimal number of batches, and the corresponding optimal batch quantities, to minimize the total cost over an infinite planning horizon. We define the following notations to be used to develop the total cost model for subsequent analysis.
D(t) I(t)
= Setup cost per batch (c > 0). = Holding cost per item per unit time (h > 0 ) . = Unit cost of production per item ( p > 0). = a e --~t= Exponentially decreasing dem a n d rate at time t (13>~0, D(t)>~O, t>~0). = Inventory level at time t ( I ( t ) >1O, t >1 0).
169
X
= Constant rate of decay per item in inventory per unit time (X >/0). Fn(a ) = Minimum total cost of setups, production and inventory-holding, consistent with supplying all future demand from stock; starting with no inventory and an initial demand rate of a; and making exactly n batches (n ~ I + U {0}). S(a, t ) = C o s t of inventory-holding associated with producing a batch at time 0 when the demand rate is a to meet the total demand in the period 0 to t at which the inventory is exhausted and a new batch is required. In addition, we make the following explicit assumptions about the model: (1) Rate of demand for the items follows an exponential decreasing function with parameters a and/3. (2) Items are decaying over time with a constant rate of decay X. (3) N o shortages are permitted. (4) /3 is greater than X. Although we fully appreciate the restrictive nature of Assumption 1, we require it to simplify the mathematical analysis to derive the optimal resuits. In addition, as argued by Smith (1977), Assumption 1 is not as restrictive as it first appears since, in the words of Smith, " b y varying the parameter, a very wide range of demand types can be approximated". Assumptions 2 and 3 are standard for treating EOQ problems with decaying items, see Ghare and Schrader (1963), so they are included in our model. Assumption 4 is required to ensure that sufficient items are available to meet demand after decay has been accounted for. With a constant rate of decay X, the change in inventory level of the items over time will be non-linear. Specifically, the inventory level I ( t ) (0 ~< t ~< to) is governed by the following first-order differential equation, where t o is the time instant at which inventory first drops to zero.
dI(t)/dt
+ X I ( t ) = - a e -Bt
(1)
which, upon solving, yields
I ( t ) = [ a e - X ' / ( / 3 - )k)] {e - ( ~ - x ) ' - e -~B-x)t° }.
(2)
T CE. Cheng / Optimalproduction poficy with decreasing demand
170
Evidently, the total number of items produced in the first batch is equal to the starting inventory I(0) which will last until time t 0. It follows that the total cost of production for the first batch is given by
horizon. Following Smith's argument, we see that the boundary conditions (6) requires that t~* must be infinite, i.e. with only one batch remaining, it must meet all future demand. Thus, letting n = 0 and q* ---, oo in (7), we obtain
pI(O) = p [ a / ( t i - )t)] {1 - e-(~-x)'° }.
F l ( a ) = c + (1 + t i p / h ) { h a / [ t i ( t i - ) t ) ] } .
(3)
It is also clear that the inventory-holding cost incUrred in the period 0 to t o is equal to the quantity hfo <, < toI(t) dt. Thus,
S(a, to)
(8)
Expression (8) prompts us to conjecture that, in general, when n + 1 batches are to be produced over the entire planning horizon, the following proposition would be true.
Proposition 1.
= hfo
F~+,(a) = (n + 1)c + a n + l h a / [ t i ( t t - X)]
= (ha/[ti(ti-X)]} x {(1 - e -too) + ,8 e-Pro(1 - eXto)/X }.
(4)
It is interesting to note that by letting 2, approach zero in (4) and applying the 1' Hospital's rule for limits, one can obtain the result given in Smith (1977).
Optimal replenishment instants
with al=(l+flp/h
and n ~ I + U ( O } .
Proof. We will prove the proposition by mathematical induction. It is obvious from (8) that the proposition is true for n = 0. Assuming that the proposition is true for n = k, we obtain from (7)
Fk+,(a)
In order to derive the optimal solution to our inventory problem, we follow the approach presented in Smith (1977) which formulates the problem as a dynamic program as follows:
=c+ Min{(ha/[fl(fl-)~)]) t>~0
X [(1 + tip~h)(1
t>~O
+ kc +
(5)
-
e -e')
+ f l e-/~'(1 _ eXt)/X]
F.+l(a ) = c + Man (pl(O) + S(a, t) + F , ( a e-Z')}
)
akha
- ~)1 }
e-B'/[ti(ti
= (k + 1)c + h a / [ t i ( t i - 2,)1 {(1 + tip~h)
subject to the following boundary conditions: -Min F0(a)={O
ifa=0,ifa>0.
e - ~ ' [ 1 - ak - t i ( 1 - eXt)/X ] }.
t>~O
(6)
It is easy to show that Substituting (3) and (4) into (5) yields
tt+,
F,+l(a)
=
in( [ti - (1 - ,,k)x]/(ti
- x)},
(9)
--c + Min { ( h a / [ f l ( t i - ) Q ] )
therefore
t>~0
X [(1 + tip~h)(1 - e -~,)
Fk+l( a ) = ( k + 1)c + ( h a / [ t i ( t i - )t )])
+ 13 e - " ( 1 - eX')/)~]
+F,(ae-#')}.
X [(1 + tip~h) (7)
Let t * l , t*, t*,-1 . . . . , q* denote the optimal replenishment instants, given that n + 1 batches are to be produced over the infinite planning
-
e -(e-x)tt+']
= (k + 1)c + a k + , h a / [ t i ( t t --•)], where ak+ 1 = (1 +
tip/h) - e -(B-x)tt+l
T.C.E. Cheng / Optimal production policy with decreasing demand
171
therefore
which, on substituting (9), becomes a,+ 1 = (1 + t i p ~ h ) - {1 - a k / ( 1 -- t i / h ) } ( ' - a / x )
2~
-(ti/h + r- 2)/r!]
Following the principle of induction, proof of the proposition is established immediately. Thus, given that the number of batches to be produced is n + 1, we can use (10) to solve for the sequence ( a l, a 2 , . . . , a , + a } recursively and stubstitute them into (9) to compute the set of optimal replenishment instants ( t*+ 1, t*, t*n - - a . . . . . tl* }. Again, it is interesting to note that by letting both p ---)0 and X ~ 0, one can obtain the results in Smith (1977, eqs. (6) and (7)).
Y', [ ( t i / x - 1 ) ( t i / x ) ( t i / X + l )
a k -- a k + 1 =
(lo)
X ( aJ(ti/X
- 1)} r - t i p / h .
Since fl > X from Assumption 4 and a~ > 0 for all k ~ I +, it follows that a , - a,+ 1 > 0 iff condition (11) holds. This completes the proof of the proposition. C o r o l l a r y . a 1 = a 2 --- a 3 . . . .
p = (h/ti)
E
iff
[(ti/X- 1)(ti/X)(ti/X + 1) •(ti/X + r - 2)/r!]
Optimal number of batches
)( { a n / (
We first present and prove a proposition, the results of which will facilitate the determination of the optimal number of batches for a given problem. Proposition 2. The sequence { a 1, a 2 .... } is strictly positive. In addition, it is strictly decreasing iff E
p < (h/ti)
[ ( t i / X - 1 ) ( t i / X ) ( t i / X + 2)
2 ~ < r { oo
•(ti/X + r - 2)/r!] ×{a,/(fl/X-1)}
~ forallnEI
+.
(11)
Proof. To prove the first part, we note that expression (9) requires that t.*+, = ( a / x )
ln{[ti
- (1 - a . ) X ] / ( t i
- X ) } > 0.
Thus the result that a . > 0 for all n ~ I + follows immediately. To prove the second part, we expand the second term on the right-hand side of expression (10) by the Binomial Theorem to obtain a , + 1 = (1 + t i p / h ) -{1 -a, +
E
[ ( t i / X - 1 ) ( t i / X ) ( t i / ) ~ + 1)
2~
-(ti/k + r- 2)/r!]
× {a/(ti/X-
1)} r },
ti/~
-- 1)} r
for all n ~ I +.
(12)
When condition (11) holds, we can follow Smith's approach to determine the optimal number of batches. Since under such a condition, the sequence { a 1, a 2. . . . } is strictly decreasing, it follows from Proposition 1 that the sequence of minimum total cost functions {F,+](ct ), F , ( a ) . . . . . Fl(a)} have increasing intercepts at the ordinate but decreasing slopes. In addition, the set of intersection points of adjacent minimum total cost functions, ( D , : F,( D , ) = F,+1( D , ) }, is strictly increasing. The { D, } are given by D, = cti(ti - X ) / [ h( a , - a , + l ) ] .
(13)
As observed by Smith, for all values of D,_ 1 ~< a ~< D, the least cost is given by F , ( a ) and the optimal number of batches is n. Clearly, if ~ ~< D~ only one batch need be made. We should bear in mind that this simple rule for determining the optimal number of batches is valid only when condition (11) holds. However, if the unit cost of prodution is negligible, i.e. p---, 0, then condition (11) always holds and the rule is always valid. Alternatively, condition (11) is always true when = 0, corresponding to the case of non-deteriorating inventory items. Thus, this rule is valid for the problem considered by Smith (1977) in which the inventory items do not decay. Furthermore, in the limiting case when condition (12) holds, the sequence of minimum total cost functions will have
T C.E. Cheng / Optimalproduction policy with decreasing demand
172
the same slopes. Thus the least cost in such a situation is given by F~(a). Hence only one batch should be made with a size equal to all future demand a/(fl - ~ ). Finally, we will briefly discuss the condition for the final batch. We see from (13) that the final batch should be made when the following condition holds:
(10) and noting that a 1 = (1 + tip~h) by definition, we obtain a I = 1,
(15)
a 2 = ½,
(16)
a, =
(17)
Substituting (15) and (16), and (16) and (17), into (13) respectively yields
a <~D 1 = c f l ( f l - X ) / [ h ( a I - a2) ] = [ c f l ( f l - X)/h] { f l / ( f l - X)) (#/x-l) which can be rewritten as
or
~ ) <~EOQ- { i l l ( r -
h)}/~/2x/V~-.
(18)
O z = 600.
(19)
Since a = 350 lies between D1 and D 2, two batches should be produced according to the optimal batching rule, i.e. n* = 2. It follows that the least cost is Fz(350), which can be determined using the result of Proposition 1 as follows:
{ a / ( fl - X) } 2 <~[2 ca/h ] { fl/( fl - X) } t]/x/2,
a/(fl-
D1 = 200,
(14)
F2 (350) = 2(100) + (½)(0.05)(350)/(1 - ½) = $217.5.
It is easy to see that the left-hand side of expression (14) is the total future demand while the fight-hand side is the product of a constant depending on the problem parameters and the standard EOQ formula. Thus the stopping rule can be stated as follows: Make the final batch with a size equal to the total future demand when it is less than the EOQ multiplied by a constant given by the second term on the right-hand side of
(14). When ?, ---, 0 the second term on the right-hand side of (14) becomes lim { fl/(fl
-
(20)
To find the optimal interval for the first batch, we use (9) to obtain t~' = 2 In 2 = 1.39 days.
(21)
Finally, we apply the standard EOQ formula to find EOQ = [2ca/hp/2 = 1183.
(22)
It follows from the stopping rule (14) that the final (second) batch should be produced with a size equal to total future demand when it is less than EOQ/(2v/2 -) = 419.
~ ) } / ] / 2 x / v ~ = ( e / 2 ) 1/2
A---,0
which, once again, is identical to the result presented in Smith (1977, eq. (10)).
An example For a given problem, the following data have been estimated: c = $100/batch, h = $ 0 . 5 / i t e m / day, p = $0.01/item, a = 350 items/day, fl = 1 item/day, 2~= 0.5 items/day. To solve this problem, we first note that p is small compared with the other costs. So we can safely assume that p--* 0 to simplify the procedure to find the optimal solution. Using equation
Conclusions We have considered an infinite-horizon inventory problem in which items are decaying at a constant rate and the demand rate follows an exponentially decreasing function. An analytical model based on dynamic programming has been developed from which closed form optimal results have been derived. We have noted that earlier results for a similar problem without item deterioration can be easily obtained from a special case of our model. We have also presented a special condition which establishes the general validity of the optimal batching rule. Finally we have provided a numerical example to demon-
T. C. E, Cheng / Optimal production policy with decreasing demand
s t r a t e h o w to a p p l y the theories to f i n d the optim a l s o l u t i o n to a g i v e n p r o b l e m .
Acknowledgements T h e a u t h o r wishes to a c k n o w l e d g e the N a t u r a l Sciences a n d E n g i n e e r i n g R e s e a r c h C o u n c i l of C a n a d a for p r o v i d i n g p a r t i a l f i n a n c i a l s u p p o r t for this research. H e is also g r a t e f u l to P r o f e s s o r S.R. W a t s o n of the M a n a g e m e n t S t u d i e s G r o u p , a n d P r o f e s s o r C. A n d r e w of the M a n u f a c t u r i n g E n gineering Group, Engineering Department, Univ e r s i t y of C a m b r i d g e for p r o v i d i n g the facilities to u n d e r t a k e this research.
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