2School of Management, University of Texas at Dallas, Richardson, Texas. 3Dept. of Industrial Engineering and Operations Research, Columbia University of ...
Periodic Review Inventory Model with Three Consecutive Delivery Modes and Forecast Updates1 Q. Feng2 , G. Gallego3 , S. P. Sethi4 , H. Yan5 , and H. Zhang6
Abstract
This paper is concerned with a periodic review inventory system with three consecutive delivery modes (fast, medium, and slow) and demand forecast updates. At the beginning of each period, the inventory level and demand information are updated, and decisions on how much to order using each of the three delivery modes are made. It is shown that there is a base-stock policy for fast and medium modes which is optimal. Furthermore, the optimal policy for the slow mode may not be a base-stock policy in general. Key Words: Inventory, Multiple Delivery Modes, Forecast Updates, Base-stock Policy.
1
This research was supported in part by a Faculty Research Grant from the University of Texas at Dallas, a RGC (Hong Kong) Competitive Earmarked Research Grant, a Distinguished Young Investigator Grant from the National Natural Sciences Foundation of China, and a Grant from the Hundred Talents Program of the Chinese Academy of Sciences. 2 School of Management, University of Texas at Dallas, Richardson, Texas. 3 Dept. of Industrial Engineering and Operations Research, Columbia University of New York, New York. 4 School of Management, University of Texas at Dallas, Richardson, Texas. 5 School of Management, University of Texas at Dallas, Richardson,Texas. 6 Institute of Applied Mathematics, Academy of Mathematics and Systems Sciences, Academia Sinica, Beijing, China.
Electronic copy available at: http://ssrn.com/abstract=1118353
1
Introduction
Sellers have recognized the importance of managing a portfolio of customers with different needs and have recognized the value of learning about customer demands in advance. As observed by Fisher et al. (Ref. 1) in the case of the apparel industry, re-grouping forecasting efforts from all sources such as firm orders received, pre-seasonal sale information, and the point of sales data, have been remarkably effective in obtaining demand information in advance. Effective use of early demand information has been a major initiative in many industries, such as the apparel industry (Ref. 1), the toy industry (Ref. 2), and the computers and electronics industry (Ref. 3). In addition, the advances in manufacturing technology, logistics service, and globalization make it possible for companies to choose from different sources to satisfy their customer needs. We consider a periodic review inventory system with three delivery modes and two demand forecast updates before demand is realized. We denote the three delivery modes as fast, medium, and slow. Orders made at the beginning of a period are delivered at the end of the current period, at the end of the next period, and at the end of the
1
Electronic copy available at: http://ssrn.com/abstract=1118353
second next period, respectively. In other words, fast, medium, and slow orders have leadtimes of one, two, and three periods, respectively. Fast orders are more expensive than medium orders, which, in turn, are more expensive than slow ones. The sequence of events is as follows. At the beginning of each period, the inventory/backlog level is reviewed, and forecasts are updated for the demands to be realized at the end of the next three periods, counting the current period as the first period. In addition, the size of the slow order issued two periods ago and the medium order issued in the previous period is also known. With these data in hand, decisions regarding the amounts to be ordered by slow, medium, and fast modes are made. At the end of the current period, the slow order issued two periods ago, the medium order issued in the previous period, and the fast order issued at the beginning of the current period are delivered. Then the demand for the current period materializes, which determines the inventory/backlog level at the beginning of the next period. Quantities ordered by slow, medium and fast modes in each period determine the total cost of ordering, inventory holding, and backlogging. The objective is to make the ordering decisions so as to
2
minimize the total cost over the problem horizon. There are a number of papers in the literature dealing with the problem of inventory management with demand information updates. Eppen et al. (Ref. 4) analyze a quick response program in a fashion buying problem by using a Bayesian approach to update demand distributions. Johnson et al. (Ref. 5) and Lovejoy (Ref. 6) model the demand process as an integrated autoregressive moving average process, and show the optimality of myopic policies under certain conditions. References 2-3 and 7-11 incorporate forecast updates and analyze optimal policies. Sethi et al. (Ref. 7) formulate a model, which allows for forecast updates for any number of future demands, albeit at some forecasting cost. Sethi et al. (Ref. 9) develop a model of forecast updates that is analogous to peeling layers of an onion. Each peeling of a layer reveals some updating information. Researchers have also looked into extending the results for classical single-mode inventory problems to allow for two consecutive delivery modes (Refs. 9-10, 12-15). Some of these allow for fixed as well as variable costs of ordering, and obtain optimal policies that generalize the well-known (s, S) policy. Scheller-Wolf et al. (Ref. 15) study a
3
Markovian production/inventory model. They prove the optimality of a base-stock policy, with the base-stock levels depending on the current state of the underlying Markov chain. Sethi et al. (Refs. 9-10) consider forecast updates based on signal observations and show that the optimal policy levels depend on the observed signal values. Inventory models with more than two delivery alternatives have not received much attention. To our knowledge, Fukuda (Ref. 16) and Zhang (Ref. 17) are the only ones who address three mode problems. Fukuda (Ref. 16) investigates the problem under an artificial assumption that the orders are placed only in every other period. Under this assumption, the problem reduces to a two-mode problem. Zhang (Ref. 17) extends Fukuda’s work to allow for three consecutive delivery modes. She takes unconstrained minimizers of the cost function as the base-stock levels and develops a heuristic procedure to estimate their values. This method does not yield an optimal policy in general. In this paper, we prove the existence of an optimal policy for the model that allows three consecutive delivery modes as well as forecast updates. We show that there exist optimal base-stock levels for the fast and medium delivery modes. These levels are
4
independent of the inventory position and the outstanding orders to be delivered by the end of the current period. But these levels depend in general on the outstanding slow order issued in the previous period and on the observed forecast updates. Moreover, under the optimal policy, the slow mode does not follow a base-stock policy in general. Given the inventory position (relevant for the fast mode), the fast mode level acts like the traditional base stock. Once the fast order is decided, it is added to the inventory position along with the slow order issued in the previous period to come up with the “inventory position relevant for the medium mode.” Given this position and the medium mode base-stock level, one can easily obtain the medium order decision. This decision is then added to the medium inventory position to obtain the slow order decision. The dependence of the optimal base-stock levels, as mentioned above, on the outstanding slow order issued in the previous period is a critical departure from the results obtained in single and two delivery mode systems. Because of the presence of the inventory position and an outstanding order as two of the states of the system, there is a priori every reason to expect that any policy expressed in terms of order-up-to lev-
5
els (which it, should be noted, can always be done) would have these levels depend on those two states. Such levels cannot be considered base-stocks and, therefore, there is no a priori reason to expect that there is an optimal base-stock policy. Thus, obtaining a structural optimal policy in the three mode case represents a contribution to the inventory literature. This discussion is further elaborated in Section 4. The remainder of this paper is organized as follows. In Section 2, we define the notation and formulate the model. In Section 3, we develop dynamic programming equations, and prove that an optimal Markov policy exists for the problem. In Section 4, we examine the structure of the optimal policy The paper is concluded in Section 5.
2
NOTATION AND MODEL FORMULATION
We consider a single product, periodic review inventory system. The dynamics of the system consists of two parts: the material flows and the information updates. The inbound material flows come from three supply sources, and the outbound material flows to customers. The updates include the initial forecast of demand for each period, its first update, its second update, and its realization at the end of the given period.
6
We introduce the following notation, and formulate precisely the model under consideration.
h1, N i = {1, 2, ..., N }, the time horizon;
(Ω, F, P ) = the probability space;
Fk = the nonnegative fast order quantity in period k, 1 6 k 6 N ;
Mk = the nonnegative medium order quantity in period k, 1 6 k 6 N ;
Sk = the nonnegative slow order quantity in period k, 1 6 k 6 N ;
cik =
s the unit ordering cost of the ith mode in period k, i = f, m, s, cfk > cm k > ck > 0;
Rki = the ith determinant (a random variable) of the demand in period k, i = 1, 2, 3; Dk = the nonnegative demand in period k modeled as a function dk (Rk1 , Rk2 , Rk3 );
Xk = the inventory/backlog level at the beginning of period k;
Yk = Xk + Mk−1 + Sk−2 = the inventory position at the beginning of period k;
XN +1 = the inventory/backlog level at the end of the last period N ;
Hk (x) = the inventory/backlog cost when Xk = x in period k;
7
HN +1 (x) = the inventory cost when XN +1 = x > 0 or penalty cost when XN +1 = x < 0.
We assume that that
E[Dk ] = E[dk (Rk1 , Rk2 , Rk3 )] < ∞, 1 6 k 6 N.
(1)
Furthermore, the inventory cost functions Hk (x) satisfies:
Hk (x) is nonnegative convex, Hk (0) = 0, and |Hk (x) − Hk (ˆ x)| 6 cH |x − xˆ|,
(2)
for some cH > 0, and k = 1, . . . , N + 1. The inventory balance equations are defined as
Xk+1 = Xk + Fk + Mk−1 + Sk−2 − dk (Rk1 , Rk2 , Rk3 ), 1 6 k 6 N,
(3)
where X1 = x1 is the initial inventory level, M0 = m0 , S−1 = s−1 , and S0 = s0 are inherited orders at the beginning of period 1, and R11 = r11 , R12 = r12 and R21 = r21 are given constants. [Figure 1 goes here] Let us explain the dynamics of (3) with the help of Figure 1. At the beginning of period k, we observe the value xk of the inventory level Xk , the value rk2 of the second 8
1 1 determinant Rk2 of Dk , the value rk+1 of the first determinant Rk+1 of Dk+1 . These
1 2 3 observations provide updated forecasts dk (rk1 , rk2 , Rk3 ) and dk+1 (rk+1 , Rk+1 , Rk+1 ) of Dk
and Dk+1 , respectively. We also know the outstanding orders Sk−2 , Mk−1 , and Sk−1 . The amount Yk = Xk + Sk−2 + Mk−1 is the inventory position available to meet the demand in period k. Given these, we can decide on the orders (Sk , Mk , Fk ). Since Fk is to be delivered at the end of the period, the total quantity available to meet the k th -period demand Dk is (Yk + Fk ). At the end of period k, the value rk3 of the random variable Rk3 is observed, which is tantamount to observing the demand Dk = dk (rk1 , rk2 , rk3 ). The difference of (Yk + Fk ) and Dk is the inventory level Xk+1 at the beginning of period (k + 1). This last statement represents a sample path of the dynamics (3). The objective is to choose a sequence of orders from the fast, medium and slow sources over time so as to minimize the total expected value of all the costs incurred in periods h1, N i. Thus the objective function is
J1 (x1 , s−1 , m0 , s0 , r11 , r12 , r21 , (F, M, S))
= H1 (x1 ) + E
( N Xh ℓ=1
) i s cfℓ Fℓ + cm , ℓ Mℓ + cℓ Sℓ + Hℓ+1 (Xℓ+1 ) 9
(4)
where (F, M, S) = ((F1 , ..., FN ), (M1 , ..., MN ), (S1 , ..., SN )) is a sequence of history-dependent or nonanticipative admissible decisions. Each of Fk , Mk , and Sk is a positive real-valued 1 2 function of the history of the demand information given by {Rℓ1 , Rℓ2 , Rℓ3 }k−1 ℓ=0 , (Rk , Rk )
1 and Rk+1 .
Let A1 be the class of all history-dependent admissible decisions, and define the value function for the problem over h1, N i with the initial inventory level x1 to be
V1 (x1 , s−1 , m0 , s0 , r11 , r12 , r21 ) =
inf
(F,M,S)∈A1
©
ª J1 (x1 , s−1 , m0 , s0 , r11 , r12 , r21 , (F, M, S)) .
(5)
It is important to note here that since the horizon length is N periods and the orders MN , SN −1 , and SN will not be delivered during the problem horizon, it is obvious that MN = SN −1 = SN = 0 in any optimal solution. So in what follows, we will still allow these orders in any feasible solution, but we will set them to zero in any optimal solution. This is equivalent to these decisions not being included in the model.
3
Dynamic Programming & Optimal Markov Policy
In this section, we use dynamic programming to study the problem. We verify whether the cost of a Markov policy obtained from the solution of the dynamic programming 10
equations equals the value function of the problem over h1, N i. First, as usual, we define the problem over hn, N i. Let
1 Jn (xn , sn−2 , mn−1 , sn−1 , rn1 , rn2 , rn+1 , (F, M, S))
= Hn (xn ) + E
( N Xh ℓ=n
) i s cfℓ Fℓ + cm , ℓ Mℓ + cℓ Sℓ + Hℓ+1 (Xℓ+1 )
(6)
where, with a slight abuse of notation, (F, M, S) = ((Fn , ..., FN ), (Mn , ..., MN ), (Sn , ..., SN )) is a history-dependent or nonanticipative admissible decisions for the problem defined 1 over periods hn, N i. That is, given xn , sn−2 , sn−1 , mn−1 , rn1 , rn2 and rn+1 as constants,
(Fn , Mn , Sn ) is a vector of positive constants. (Fk , Mk , Sk ), n 6 k < N , are positive real-valued functions of the history of the demand information from period n to period 1 2 1 k, given by {{Rℓ1 , Rℓ2 , Rℓ3 }k−1 ℓ=n , (Rk , Rk ), Rk+1 }. Define the value function associated with
the problem over hn, N i as follows:
1 Vn (xn , sn−2 , mn−1 , sn−1 , rn1 , rn2 , rn+1 )
=
inf
(F,M,S)∈An
©
ª 1 Jn (xn , sn−2 , mn−1 , sn−1 , rn1 , rn2 , rn+1 , (F, M, S)) ,
(7)
where An denotes the class of all history-dependent admissible decisions for the problem over hn, N i. 11
In view of (6), we can write the dynamic programming equation (DP equation, hereafter) corresponding to the problem as follows:
1 Uℓ (xℓ , sℓ−2 , mℓ−1 , sℓ−1 , rℓ1 , rℓ2 , rℓ+1 )
= Hℓ (xℓ ) + inf
F >0 M >0 S>0
n
s cfℓ F + cm ℓ M + cℓ S
¤ª £ 1 2 1 , Rℓ+1 , Rℓ+2 ) , + E Uℓ+1 (Ψℓ+1 (F ), sℓ−1 , M, S, rℓ+1
ℓ = 1, ..., N,
(8)
where the notation Ψℓ+1 (·) is defined as
Ψℓ+1 (F ) = xℓ + sℓ−2 + mℓ−1 + F − dℓ (rℓ1 , rℓ2 , Rℓ3 ), ℓ = 1, ...., N,
(9)
and F , M , and S are argument for minimization in (8). Based on the DP equation, we state the following theorem. Since its proof is similar to Theorem 4.2 in Sethi et al. (Ref. 9), we omit the proof here.
1 Theorem 3.1 The value functions Vk (xk , sk−2 , mk−1 , sk−1 , rk1 , rk2 , rk+1 ), 1 6 k 6 N , de-
fined in (7), satisfy the DP equation (8).
Next we discuss how we can obtain an optimal solution. From Assumption (2),
cik u + E[Hk+1 (u − dk (Rk1 , Rk2 , Rk3 ))] → ∞ as u → ∞, i = f, m, s. 12
(10)
It follows from (10) that there exists an upper bound order quantity L > 0 such that
inf
F >0 M >0 S>0
n £ ¤o s 1 2 1 cfℓ F + cm M + c S + E U (Ψ (F ), s , M, S, r , R , R ) ℓ+1 ℓ+1 ℓ−1 ℓ ℓ ℓ+1 ℓ+1 ℓ+2 =
inf
06F,M,S6L
n s cfℓ F + cm ℓ M + cℓ S
£ ¤ª 1 2 1 +E Uℓ+1 (Ψℓ+1 (F ), sℓ−1 , M, S, rℓ+1 , Rℓ+1 , Rℓ+2 ) , ℓ = 1, ..., N. By a measurable selection theorem (see page 503 of Ref. 12), there exist Borel measurable functions
1 fℓ∗ = fℓ∗ (xℓ , sℓ−2 , mℓ−1 , sℓ−1 , rℓ1 , rℓ2 , rℓ+1 ),
1 m∗ℓ = m∗ℓ (xℓ , sℓ−2 , mℓ−1 , sℓ−1 , rℓ1 , rℓ2 , rℓ+1 ),
1 6 ℓ 6 N,
(11)
1 6 ℓ 6 N − 1,
(12)
1 s∗ℓ = s∗ℓ (xℓ , sℓ−2 , mℓ−1 , sℓ−1 , rℓ1 , rℓ2 , rℓ+1 ), 1 6 ℓ 6 N − 2,
(13)
such that the infimum on the right-hand side of (8) is attained at (fℓ∗ , m∗ℓ , s∗ℓ ). Next we show that the minimizers (11)-(13) of the DP equation give rise to an optimal solution. Define
X1∗ = x1 , F1∗ = f1∗ (X1∗ , s−1 , m0 , s0 , r11 , r12 , r21 ), M1∗ = m∗1 (X1∗ , s−1 , m0 , s0 , r11 , r12 , r21 , ), 13
(14)
S1∗ = s∗1 (X1∗ , s−1 , m0 , s0 , r11 , r12 , r21 ), (15)
and ∗ ∗ ∗ ∗ 1 2 3 + Sℓ−3 − dℓ−1 (Rℓ−1 , Rℓ−1 , Rℓ−1 ), Xℓ∗ = Xℓ−1 + Fℓ−1 + Mℓ−2
∗ ∗ ∗ 1 Fℓ∗ = fℓ∗ (Xℓ∗ , Sℓ−2 , Mℓ−1 , Sℓ−1 , Rℓ1 , Rℓ2 , Rℓ+1 ),
2 6 ℓ 6 N,
∗ ∗ ∗ 1 Mℓ∗ = m∗ℓ (Xℓ∗ , Sℓ−2 , Mℓ−1 , Sℓ−1 , Rℓ1 , Rℓ2 , Rℓ+1 ),
∗ ∗ 1 ∗ , Sℓ−1 , Rℓ1 , Rℓ2 , Rℓ+1 ), , Mℓ−1 Sℓ∗ = s∗ℓ (Xℓ∗ , Sℓ−2
2 6 ℓ 6 N, (16)
2 6 ℓ 6 N,
2 6 ℓ 6 N.
(17)
(18)
(19)
Using the DP equation (8), we can prove the following result.
Theorem 3.2 Under the assumptions made in Section 2,
∗ (F ∗ , M ∗ , S ∗ ) = ((F1∗ , ..., FN∗ ), (M1∗ , ..., MN∗ −1 , 0), (S1∗ , ..., SN −2 , 0, 0))
given in (14)-(19) are optimal decisions to the problem. That is, # " N ´ X³ f 1 2 1 ∗ s ∗ ∗ H1 (x1 )+E cℓ Fℓ∗ +cm ℓ Mℓ +cℓ Sℓ +Hℓ+1 (Xℓ+1 ) =V1 (x1 ,s−1 ,m0 ,s0 ,r1 ,r1 ,r2 ).
(20)
ℓ=1
Remark 3.1 Theorems 3.1 and 3.2 establish the existence of an optimal Markov policy. That is, there exists a policy in the class of all history-dependent policies whose objective function value equals the value function defined in (5), and there, in turn, exists a Markov policy defined by (14)-(19), which provides the same value for the objective function. 14
4
Structure of the Optimal Policy
For a further analysis of the problem, traditionally, one recasts the DP equation (8) involving order quantities to those involving respective inventory positions that would be attained after the respective orders are delivered. Thus, we replace F by φ − yℓ , M by µ − (φ + sℓ−1 ), and S by σ − µ in (8), where yℓ = xℓ + mℓ−1 + sℓ−2 , so that φ, µ, and σ are the post-order inventory positions after the delivery of fast, medium and slow orders, respectively. When there are only two delivery modes, this transformation of variables changes the problem into a standard one-delivery-mode problem. As a result, such a transformation has been widely used to analyze problems with two delivery modes, see, e.g., Scheller-Wolf and Tayur (Ref. 15) and Sethi, Yan and Zhang (Ref. 9). However, when there are more than two delivery modes, the transformation does not reduce the problem to a single-delivery-mode problem. Nevertheless, to get the optimal policy, it is possible to directly analyze natural constraints required on the minimizers of the convex cost functions resulting from fast, medium and slow orders.
15
Now let us define a function Wℓ (·) as follows:
1 1 Wℓ (yℓ , sℓ−1 , rℓ1 , rℓ2 , rℓ+1 ) = Uℓ (xℓ , sℓ−2 , mℓ−1 , sℓ−1 , rℓ1 , rℓ2 , rℓ+1 ) − Hℓ (xℓ ).
(21)
The DP equation (8) can be written as follows:
1 Wℓ (yℓ , sℓ−1 , rℓ1 , rℓ2 , rℓ+1 )
=
inf
φ>yℓ µ>φ+sℓ−1 σ>µ
£ ¤ s 1 2 3 (µ − (φ + s )) + c (σ − µ) + E H (φ − d (r , r , R )) {cfℓ (φ − yℓ ) + cm ℓ−1 ℓ+1 ℓ ℓ ℓ ℓ ℓ ℓ 2 1 1 , Rℓ+2 )]}, ℓ = 1, ..., N. , Rℓ+1 +E[Wℓ+1 (µ − dℓ (rℓ1 , rℓ2 , Rℓ3 ), σ − µ, rℓ+1
(22)
Similar to the discussion preceding Theorem 3.2, there exist feasible minimizing functions
1 1 1 φ∗ℓ (yℓ , sℓ−1 , rℓ1 , rℓ2 , rℓ+1 ), µ∗ℓ (yℓ , sℓ−1 , rℓ1 , rℓ2 , rℓ+1 ), σℓ∗ (yℓ , sℓ−1 , rℓ1 , rℓ2 , rℓ+1 ), 1 6 ℓ 6 N,
such that the infimand on the right-hand side of (22) is attained at (φ, µ, σ) = (φ∗ℓ , µ∗ℓ , σℓ∗ ). ∗ ∗ ∗ ∗ Also µ∗N (·) = φ∗N (·), σN −1 (·) = µN −1 (·), and σN (·) = φN (·).
Let
Y1∗ = y1 = x1 + m0 + s−1 ,
Y2∗ = µ∗1 (Y1∗ , s0 , r11 , r12 , r21 ) − d1 (r11 , r12 , R13 ),
(23)
3 2 1 2 1 ∗ ∗ ), (24) , Rk−1 , Rk−1 , Rk1 ) − dk−1 (Rk−1 , Rk−1 − µ∗k−2 , Rk−1 , σk−2 Yk∗ = µ∗k−1 (Yk−1
for k = 3, ..., N . 16
Define with a slight abuse of notation,
∗ 1 Fℓ∗ = φ∗ℓ (Yℓ∗ , σℓ−1 − µ∗ℓ−1 , Rℓ1 , Rℓ2 , Rℓ+1 ) − Yℓ∗ ,
(25)
∗ 1 ∗ Mℓ∗ = µ∗ℓ (Yℓ∗ , σℓ−1 − µ∗ℓ−1 , Rℓ1 , Rℓ2 , Rℓ+1 ) − (Yℓ∗ + σℓ−1 − µ∗ℓ−1 + Fℓ∗ ),
(26)
1 ∗ ∗ ) − (Yℓ∗ + σℓ−1 − µ∗ℓ−1 + Fℓ∗ + Mℓ∗ ), (27) Sℓ∗ = σℓ∗ (Yℓ∗ , σℓ−1 − µ∗ℓ−1 , Rℓ1 , Rℓ2 , Rℓ+1
for ℓ = 1, ..., N, where σ0∗ − µ∗0 = s0 .
Theorem 4.1 Under the assumptions made in Section 2,
¡ ¡ ¢ ¡ ¢¢ ∗ (F ∗ , M ∗ , S ∗ ) = (F1∗ , ..., FN∗ ) , M1∗ , ..., MN∗ −1 , 0 , S1∗ , ..., SN , 0, 0 −2
(28)
defined in (25)-(27) are optimal decisions for the problem over h1, N i.
The proof follows from the DP equation (22) in the same way as the proof of Theorem 3.2 follows from the DP equation (8). We can now state our main result on the structure of the optimal policy.
Theorem 4.2 At the beginning of period k, k = 1, . . . , N , suppose that the observed 1 1 , respectively, the initial inventory position are rk1 , rk2 and rk+1 values of Rk1 , Rk2 and Rk+1
is yk , and the slow-order quantity ordered in period (k − 1) is sk−1 . Then there are 17
1 base-stock levels φ¯k (independent of yk , but dependent on sk−1 , rk1 , rk2 and rk+1 ) and µ ¯k
1 (independent of yk and sk−1 , but dependent on rk1 , rk2 and rk+1 ) such that the optimal
fast-order and medium-order quantities are:
Fk∗ = (φ¯k − yk )+ , k = 1, . . . , N,
(29)
Mk∗ = (¯ µk − yk − sk−1 − Fk∗ )+ , k = 1, . . . N − 1, MN∗ = 0.
(30)
Remark 4.1 In view of the fact that the base-stock levels φ¯k and µ ¯k depend on sk−1 , 1 rk1 , rk2 and rk+1 , (29)-(30) can be expressed equivalently as the optimal policy given in
(12).
The proof of Theorem 4.2 needs an important lemma, which we shall prove after some general discussion on base-stock policies. A fundamental characteristic of the optimal base-stock level in the classical single delivery mode inventory problem is that the level is independent of the inventory position. Since any ordering policy can be converted to an order-up-to policy simply by adding the order quantity to the inventory position, the proof of the optimality of a base-stock policy requires, therefore, that we can find order-up-to levels that are independent of 18
the inventory position. While the base-stock level must be independent of the inventory position, it may depend on time if the problem is nonstationary or one with a finite horizon, and on the states other than the inventory position if the system behavior is influenced by these, usually exogenous, states. In such cases, the system is sometimes referred to as worlddriven and the optimal policy as the state-dependent base-stock policy, even though it must be independent of the state called the inventory position. Since the inventory position is also a state variable, the terminology state-dependent base-stock levels is not quite correct. Our preference is, therefore, to use the term order-up-to levels if the levels can depend on time or any of the state variables including the inventory position, and use simply the term base-stock levels if the levels are independent of the current inventory position. A number of papers cited earlier prove also that the base-stock policy remains optimal with two consecutive delivery modes, provided that the ordering cost is linear; see Sethi et al. (Ref. 9) for example.
19
When we move from two modes to three modes, the issues become substantially more complicated because now there is an additional endogenous state variable, namely the slow order placed in the previous period. From (23), we see that the order-up-to levels φ∗ℓ , µ∗ℓ , and σℓ∗ depend, in general on yℓ , sℓ−1 and the observed demand signals. However, in Theorem 4.2, we have shown that there are levels φ¯ℓ and µ ¯ℓ , such that φ¯ℓ is independent of the inventory position yℓ , and µ ¯ℓ is independent of yℓ + sℓ−1 . It is easily seen from (30) that φ¯ℓ acts like the base-stock level in the single-mode case. Once the fast order Fℓ∗ is placed, the “inventory position relevant for the medium order” is yℓ + sℓ−1 + Fℓ∗ . If it is less than the level µ ¯ℓ , we order Mℓ∗ . Otherwise, we do not (i.e., Mℓ∗ = 0). This behavior is the natural generalization of the single mode base-stock policy. And since these levels are independent of their corresponding relevant inventory positions, φ¯ℓ and µ ¯ℓ can be called the base-stock levels for the first and the second mode, respectively.
Lemma 4.1 Let g(·) and h(·) be convex functions with x˜ and z˜ as their respective unconstrained minima, i.e., g(˜ x) = minx g(x) and h(˜ z ) = minz h(z). For given b > 0, let
20
a ˆ minimize g(x) + h(x + b), i.e., g(ˆ a) + h(ˆ a + b) = minx [g(x) + h(x + b)]. Then, for any a,
min [g(x) + h(z)]
x>a z>x+b
g(˜ x) + h(˜ z ), g(a) + h(˜ z ), = g(a) + h(a + b), g(ˆ a ∨ a) + h((ˆ a ∨ a) + b), =
½
if if if if
x˜ > a, x˜ < a, x˜ < a, x˜ > a,
z˜ > x˜ + b, z˜ > a + b, z˜ < a + b, z˜ 6 x˜ + b,
Case (i) Case (ii) Case (iii) Case (iv)
g(a ∨ x˜) + h(˜ z ∨ (a + b)), if z˜ > x˜ + b, Case I g(ˆ a ∨ a) + h((ˆ a ∨ a) + b), if z˜ < x˜ + b, Case II,
(31)
where in Case II, we can always choose a ˆ so that z˜ − b 6 a ˆ 6 x˜. In other words, x∗ = (a ∨ x˜) and z ∗ = ((a + b) ∨ z˜) in Case I and x∗ = (ˆ a ∨ a) and z ∗ = ((ˆ a ∨ a) + b) in Case II minimize g(x) + h(z) subject to the constraints x > a and z > x + b. Furthermore, if we define
(¯ x, z¯) =
½
(˜ x, z˜) (ˆ a, a ˆ + b)
or
(ˆ a, z˜)
in Case I, in Case II,
(32)
then x∗ and z ∗ can be expressed as
x∗ = a + (¯ x − a)+ ,
(33)
z ∗ = (x∗ + b) + [¯ z − x∗ − b]+ = a + b + (¯ x − a)+ + (¯ z − a − b − (¯ x − a)+ )+ . (34)
Finally, (¯ x, z¯) is independent of a.
21
Proof of Lemma 4.1. Let S = {(x, z)|x > a, z > x + b} denote the feasible set for minimization. We prove the result in each of the four cases shown also on Figure 2. [Figure 2 goes here] Case (i). Since (˜ x, z˜) ∈ S, the result holds trivially. Case (ii). For any (x, z) ∈ S, we have x > a > x˜. By convexity of g and the definition of z˜, it is obvious that
g(a) + h(˜ z ) 6 g(a) + h(z) 6 g(x) + h(z).
(35)
Case (iii). For any (x, z) ∈ S, we have x > a > x˜ and z > x + b > a + b > z˜. Then
g(a) + h(a + b) 6 g(x) + h(z).
(36)
x, z˜) Case (iv). It is easy to see from Figure 2 that a line joining any (x, z) ∈ S and (˜ will intersect the line z = x + b, that is the 450 line passing through the point (a, a + b). Let (ˆ x, xˆ + b) denote the point of intersection. Certainly, (ˆ x, xˆ + b) ∈ S. Moreover, depending on the location (see Figure 2) of (x, z), either x > xˆ > x˜ or x 6 xˆ 6 x˜ and either z > xˆ + b > z˜ or z 6 xˆ + b 6 z˜. That is, xˆ is in the middle of x and x˜, and xˆ + b is
22
in the middle of z and z˜. Therefore, we have an (ˆ x, xˆ + b) for each (x, z) ∈ S such that
g(ˆ x) + h(ˆ x + b) 6 g(x) + h(z).
(37)
g(ˆ x) + h(ˆ x + b) is clearly convex in xˆ. By the definition of a ˆ, case (iv) follows. [Figure 3 goes here] We now derive the second equality in (31). For this, we observe that Case I consists of Cases (i), (ii) and (iiia) and Case II consists of Cases (iiib), (iv)(see Figure 3).
We
then need to show that z˜ − b 6 a ˆ 6 x˜ in Case II. Note that for any x > x˜, we have z˜ 6 x˜ + b 6 x + b. Thus, h(˜ x + b) 6 h(x + b), Together with the fact that g(ˆ x) 6 g(x), we deduce that a ˆ 6 xˆ. Similarly, we can show that a ˆ > zˆ − b. Next, we show (32),(33) and (34). Note that z¯ > x¯ + b. Also from (31) and the definition of (¯ x, z¯), we have x∗ = x¯ ∨ a and z ∗ = z¯ ∨ (a + b). Then
x∗ = x¯ ∨ a = a + (x − a)∗ ,
z ∗ = z¯ ∨ (a + b) = z¯ ∨ (¯ x + b) ∨ (a + b) = z¯ ∨ (x∗ + b) = x∗ + b + (¯ z − x∗ − b)+ .
Finally, it is obvious that x˜, a ˆ, z˜ and which of the two cases I or II occurs do not 23
depend on a. Therefore, (¯ x, z¯) as defined in (32) is independent of a.
¤
Lemma 4.2 Let g(x) and h(z, w) be two convex functions with x˜ and (˜ z , w) ˜ as their respective unconstrained minima. If w˜ 6 z˜, then there exist reals x¯, z¯ and w¯ (independent of a) such that the solution of the problem min{g(x) + h(z, w)| x > a, z > x + b, w > z} is given by
x∗ = a ∨ x¯,
z ∗ = (x∗ + b) ∨ z¯,
w∗ = z ∗ ∨ w. ¯
(38)
Proof. Define w˜ c (z) = arg minw {h(z, w)|w > z}. Then from the well-known projection theorem (e.g., see Ref. 18), h(z, w˜ c (z)) = minw>z h(z, w) is a convex function in z. We take (¯ x, z¯) as suggested in Lemma 4.1 equation (32). Then the optimal solution (x∗ , z ∗ ) satisfies (38). Moreover, from the proof of Lemma 4.1, we have z¯ > z˜. Together with the fact that z ∗ > z¯ and z˜ > w˜ , we deduce that z ∗ > w. ˜ Note that for each fixed z ∗ > w, ˜ h(z ∗ , w) constrained on {w > z ∗ } is convex in w, with the constrained minimizer wc (z ∗ ) = z ∗ . If we take w ¯ = w, ˜ then w∗ = z ∗ ∨ w¯ holds.
¤
Proof of Theorem 4.2. First we show (30) for period N . We know from (2) that cfN φ+ 1 2 3 E[HN +1 (φ − dN (rN , rN , RN )] is convex in φ and it attains its unconstrained minimum.
24
Let this be attained at φ¯N . Then the constrained minimizer on the region [yN , ∞) is 1 2 given by φ∗N (yN , rN , rN ) = φ¯ ∨ yN . Note also that φ¯N is independent of yN . Thus, (30)
for period N follows from (22) for period N . Next we prove (30) for period N − 1.
1 2 1 WN −1 (yN −1 , sN −2 , rN −1 , rN −1 , rN )
= −cfN −1 yN −1 − cm N −1 sN −2 +
inf
φ>yN −1 , µ>φ+sN −2
{gN −1 (φ) + hN −1 (µ)},
(39)
where
£ ¤ 1 2 3 gN −1 (φ) = [cfN −1 − cm N −1 ]φ + E HN (φ − dN −1 (rN −1 , rN −1 , RN −1 ) , £ ¤ 1 2 3 1 2 hN −1 (µ) = cm µ + E W (µ − d (r , r , R ), r , R ) . N N −1 N −1 N −1 N −1 N −1 N N It follows from (22) for period N and the convexity of HN (·) and HN +1 (·) that gN −1 (φ) is convex in φ and hN −1 (µ) is convex in µ. Let φ˜N −1 , µ ˜N −1 and φˆN −1 be unconstrained minimizers of gN −1 (φ), hN −1 (µ) and gN −1 (φ) + hN −1 (φ + sN −2 ), respectively. These are clearly independent of yN −1 . Also define
(φ¯N −1 , µ ¯N −1 ) =
½
(φ˜N −1 , µ ˜N −1 ) (φˆN −1 , µ ˜N −1 ) 25
if µ ˜N −1 > φ˜N −1 + sN −2 if µ ˜N −1 < φ˜N −1 + sN −2
(40)
Then from (32) and (33) in Lemma 4.1, the constrained minimizer for (39) is
φ∗N −1 = yN −1 + (φ¯N −1 − yN −1 )+ , µ∗N −1 = φ∗N −1 + sN −2 + (¯ µN −1 − φ∗N −1 − sN −2 )+ .
It follows that the optimal ordering quantities for period N − 1 are
FN∗ −1 = φ∗N −1 − yN −1 = (φ¯N −1 − yN −1 )+ , MN∗ −1 = µ∗N −1 − φ∗N −1 − sN −2 = (¯ µN −1 − yN −1 − FN∗ −1 − sN −2 )+ .
Next we prove (30) for period N − 2. We rewrite (22) as
1 2 1 WN −2 (yN −2 , sN −3 , rN −2 , rN −2 , rN −1 )
= −cfN −2 yN −2 − cm N −2 sN −3 +
inf
φ>yN −2 µ>φ+sN −3
{gN −2 (φ) + hN −2 (µ)} ,
(41)
where
£ ¤ 1 2 3 gN −2 (φ) = [cfN −2 − cm N −2 ]φ + E HN −1 (φ − dN −2 (rN −2 , rN −2 , RN −2 )) , hN −2 (µ) =
[cm N −2
−
£
csN −2 ]µ
µ
+ inf csN −2 σ +
E WN −1 (µ −
σ>µ
1 2 3 dN −2 (rN −2 , rN −2 , RN −2 ), σ
26
−
¶ ¤
1 2 1 µ, rN −1 , RN −1 , RN )
.
Since the infimand is jointly convex in (σ, µ), it is easy to show that hN −2 (µ) is convex in µ. Also note that gN −2 (φ) is convex in φ. Hence the result for period N − 2 follows by repeating the argument for period N − 1 in (41). Repeating this procedure, we can prove the result for any period ℓ ∈ h1, N − 3i. ¤ Thus, we have found a structural form of the optimal inventory replenishment policy with three delivery modes and demand information revisions, i.e., the optimal ordering decisions for the fast and medium modes are characterized by critical numbers, independent of the inventory position, known as the base stocks. However, for period k, the base-stock level for the fast mode is a function of the slow mode decision in period (k − 1). Moreover, the optimal ordering policy for the third mode may not in general be a base-stock policy. Indeed it is not as shown in reference 19. The discussion can be extended to infinite horizon problems. Similar to the approach used in Sethi et al. (Ref. 9), one can show that the result carries through for both discounted-cost and long-run average cost minimization problems.
27
5
Concluding Remarks
In this paper we consider a discrete-time, periodic review inventory system with three consecutive supply modes and demand information updates. We demonstrate that the optimal inventory replenishment policy for the first two modes–fast and medium–is a state-dependent base-stock policy. However, the optimal policy for the third or the slow mode does not follow a base-stock policy in general. In Feng et al. (Ref. 19), we provide further characterization of the optimal policy, as well as extend these results to the general case of multiple delivery modes. Gallego et al. (Ref. 20) apply these results to average cost problems with no forecast updates, and develop computational optimization as well as heuristic procedures for solving them. Finally, the paper opens up a number of future research avenues. On the theoretical side, it is of interest to extend our results to allow for fixed ordering cots, as was done in Sethi, Yan and Zhang (Ref. 10) for the case of two delivery modes. On the practical side, research leading to concrete forecast updating schemes that are suitable for implementation would be of enormous value.
28
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2. BARNES-SCHUSTER, D., BASSOK, Y., and ANUPINDI, R., Coordination and Flexibility in Supply Contracts with Options, Manufacturing & Service Operations Management, Vol. 4, pp. 171-207, 2002.
3. YAN, H., LIU, K., and HSU, A., Optimal Ordering in a Dual-Supplier System with Demand Forecast Updates, Production and Operations Management, Vol. 12, pp. 30-45, 2003 .
4. EPPEN, G. D., and IYER, A. V., Improved Fashion Buying with Bayesian Updates, Operations Research, Vol. 45, 805-819, 1997.
5. JOHNSON, O., and THOMPSON, H., Optimality of Myopic Inventory Policies for Certain Depended Processes, Management Science, Vol. 21, pp. 1303-1307,
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1975.
6. LOVEJOY, W. S., Myopic Policies for Some Inventory Models with Uncertain Demand Distributions, Management Science, Vol. 36, pp. 724-738, 1990.
7. SETHI, S. P., and SORGER, G., A theory of Rolling Horizon Decision Making, Annals of Operations Research, Vol. 29, pp. 387-416, 1991.
8. HUANG, Y., SETHI, S. P., and YAN, H., Purchase Contract Management with Demand Information Update, Working paper, The University of Texas at Dallas, 2002.
9. SETHI, S. P., YAN, H., and ZHANG, H., Peeling Layers of an Onion: a Periodic Review Inventory Model with Multiple Delivery Modes and Forecast Updates,
Journal of Optimization Theory and Applications, Vol. 108, pp. 253-281, 2001.
10. SETHI, S. P., YAN, H., and ZHANG, H., Inventory Models with Fixed Costs, Multiple Delivery Modes and Forecast Updates, Operations Research, Vol. 51, pp. 321-328, 2003.
30
¨ ¨ Integrating Replenishment Decisions with Advance 11. GALLEGO, G., and OZER, O., Demand Information, Management Science, Vol. 47, pp. 1344-1360, 2001.
12. BENSOUSSAN, A., CROUHY, M., and PROTH, J-M, Mathematical Theory of Production Planning, North-Holland, New York, NY, 1983.
13. WHITTEMORE, A. S., and SAUNDERS, S. C., Optimal Inventory under Stochastic Demand with two Supply Options, SIAM Journal on Applied Mathematics, Vol. 32, pp. 293-305, 1977.
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15. SCHELLER-WOLF, A., and TAYUR, S., A Markovian Dual-Source ProductionInventory Model with Order Bands, GSIA Working Paper, #1998-E200, Carnegie Mellon University, 1998.
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31
17. ZHANG, V. L., Ordering Policies for an Inventory System with Three Supply Modes, Naval Research Logistics, Vol. 43, pp. 691-708, 1996.
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19. FENG, Q., GALLEGO, G., SETHI, S. P., YAN, H., and ZHANG, H., Optimality and Nonoptimality of Base-stock Policies in Inventory Problems with Multiple
Delivery Modes, Working Paper, University of Texas at Dallas, Richardson, TX, 2003.
20. G. Gallego, S.P. Sethi, Z. Wang, H. Yan, H. Zhang, and Y. Huang, Stationary Policies for Multiple Procurement Modes, Working Paper, Columbia University, New York, NY, 2003.
List of Figures Figure 1: A time-line of the system dynamics and ordering decisions Figure 2: Cases (i)-(iv) and details of Case (iv) Figure 3: Solutions in Cases I and II 32
Observed: 3 1 (rk−1 , rk2 , rk+1 ) Demand Dk−1 realized Review of Xk Demands updated: dk (rk1 , rk2 , Rk3 ) 1 2 3 dk+1 (rk+1 , Rk+1 , Rk+1 ) Orders Placed: Fk , Mk , Sk
Observed: 2 1 (rk3 , rk+1 , rk+2 ) Demand Dk realized Review of Xk+1 Demands updated: 1 2 3 dk+1 (rk+1 , rk+1 , Rk+1 ) 1 2 3 dk+2 (rk+2 , Rk+2 , Rk+2 ) Orders Placed: Fk+1 , Mk+1 , Sk+1
Period k k
Delivered: Fast order Fk−1 Medium order Mk−2 Slow order Sk−3
Observed: 3 2 1 (rk+1 , rk+2 , rk+3 ) Demand Dk+1 realized Review of Xk+2 Demands updated: 1 2 3 dk+2 (rk+2 , rk+2 , Rk+2 ) 1 2 3 dk+3 (rk+3 , Rk+3 , Rk+3 ) Orders Placed: Fk+2 , Mk+2 , Sk+2
Period k + 1 (k + 2)
k+1 Delivered: Fast order Fk Medium order Mk−1 Slow order Sk−2
33
Delivered: Fast order Fk+1 Medium order Mk Slow order Sk−1
z Case (i) (x, z)
Case (ii) S
(ˆ x, x ˆ + b)
(a, a + b)
(a + b) Case (iiia)
(˜ x, z˜)
Case (iv)
Case (iiib)
b
x 0
a
34
z
(ˆ a, a ˆ + b), a ˆ>a (˜ x, x ˜ + b) Case I (a, z˜) (˜ x, z˜) a ˆ>a S
(a + b) (a, a + b)
(˜ x, z˜)
a ˆ
