- Email: [email protected]
Relating the multiple supply problem to quantity flexibility contracts
Operations Research Letters 35 (2007) 767 – 772
Operations Research Letters www.elsevier.com/locate/orl
Relating the multiple supply problem to quantity flexibility contracts Özgür Yazlali, Feryal Erhun∗ Department of Management Science and Engineering, Stanford University, Stanford, CA 94305, USA Received 21 February 2006; accepted 2 January 2007 Available online 14 January 2007
Abstract We consider two classes of problems in the supply chain management literature: the multiple supply problem and quantity flexibility contracts. We identify the similarities between these two problems and show that for both the total operating cost of the manufacturer is convex with respect to its arguments under mild conditions. © 2007 Elsevier B.V. All rights reserved.
Keywords: Demand uncertainty; Dual supply problem with general leadtimes; Maximum capacity limit; Minimum order commitment; Dynamic programming
There has been increased interest in the literature investigating supply decisions in the presence of several sources, as many companies in today’s global economy seek alternative supply tools to meet their demand. In this research, we explore an important class of these multiple sourcing models (i.e., single-product, single-stage, deterministic leadtimes) and explain the analytical structure of these problems. We first formulate the multiple supply problem in its general form: a manufacturer has to satisfy random demand for a finite planning horizon using several capacitated supply options, each with a different ∗ Corresponding author.
E-mail address: [email protected] (F. Erhun). 0167-6377/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.orl.2007.01.002
leadtime. Our modeling setting (partially or fully) generalizes many inventory models in the literature; examples include (i) single supply problems such as in [5,6], (ii) dual supply problems with consecutive leadtimes such as in [3,4,10,12,13,15,20,21], (iii) dual supply problems with general leadtimes such as in [14,18,19], and (iv) other multiple supply problems (i.e., three or more suppliers with or without consecutive leadtimes) such as in [7–9,23]. We then show that the structural formulation of the multiple supply problem has characteristics similar to that of quantity flexibility contracts (e.g., [1,2,16]) where the manufacturer has a single supplier (with a positive leadtime), but has the flexibility to modify her previous orders. We note that there are several multi-stage models in the literature (such as [11] for the multiple supply problem and
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Ö. Yazlali, F. Erhun / Operations Research Letters 35 (2007) 767 – 772
[17] for the quantity flexibility contracts) with settings closely related to ours; nevertheless, the results in this paper do not readily apply to these models because of our single-stage focus. One of the main differences between the multiple supply problem and the quantity flexibility contract is due to the bounds on the manufacturer’s ordering decisions. In the former, it is natural to assume that order bounds for each supply option are predetermined (e.g., as a part of contracts that are signed at the beginning of a planning horizon); hence, they are independent of the pipeline inventory. On the other hand, in the latter, the flexibility to modify previously given orders is usually a function of incoming orders (e.g., the manufacturer may have the flexibility to increase/decrease incoming orders up to 10%) in a rolling-horizon fashion (i.e., at each period a new order is added to the pipeline). Our contribution is to show that when ordering/modification costs and inventory related costs are convex, the total operating cost of the manufacturer (as a function of the current state of the pipeline inventory as well as the current and future order quantities and order bounds) is convex with respect to its arguments. Furthermore, our paper links the two different research streams (namely the literature on multiple supply problems and that on quantity flexibility contracts) by showing that the two problem settings are similar, and consequently, the objective functions of both problems have common properties with respect to their arguments. Both of these problems are difficult to solve due to the high dimensionality of state and decision variables; however, our goal in this paper is to show that certain functional properties, which are commonly used in proving many well-known results in the literature, apply to these complex settings as well. Moreover, our convexity results improve the efficiency of a computational approach in solving both problems. Because of the similarities in their problem settings, the heuristic approaches developed for one problem setup (e.g., in [2,14,18]) may prove useful for the other. The organization of the paper is as follows. We first formulate the multiple supply problem in its general form in Section 1 and present our structural results. We present a special case of linear ordering costs in Section 1.1. In Section 2, we then discuss the quantity flexibility contract scenario.
1. Multiple supply problem In this section we assume that a manufacturer has a set of supply options with leadtimes in L ⊂ {l, l + 1, . . . , L}, where l and L are the leadtimes of the fastest and slowest supply options, respectively. In order to differentiate the alternative sources we use the term supply option i, i ∈ L. The manufacturer has to meet a random demand Dn in period n, n = 0, 1, . . . , N, using these resources. We assume that the demand distribution is independent (but not necessarily identical) between periods. Because of the leadtime difference, the manufacturer starts ordering at period −L, whereas her last orders are given at period N − l. The sequence of events at any period is as follows: • The manufacturer observes the pipeline inventory (xn0 , xn1 , . . . , xnL−1 ), where xni 0, i = 1, . . . , L − 1, is the inventory in the pipeline that will be available after i periods and xn0 is the on-hand inventory at the beginning of period n before the ordering decisions. • She then places her orders considering the capacity restrictions of the supply options. That is, she chooses the order quantity qni from supply option i, i ∈ L, such that 0 min qni Mni < ∞, where min and Mni are the minimum and maximum capacity restrictions of supply option i at period n, respectively. • Finally, the demand occurs and the state is updated. We assume that the manufacturer incurs a cost for excess inventory and unsatisfied demand at the end of a period. In our problem formulation, we represent the state of the pipeline inventory at period n with xn = (xn(l) , xnl+1 , xnl+2 , . . . , xnL−1 ),
(1)
(l) where xn = li=0 xni is the sum of on-hand inventory and first l periods of pipeline inventory. The vector of ordering decisions is qn = (qni )i∈L , whereas the vectors of minimum and maximum capacity restrictions are mn = (min )i∈L and Mn = (Mni )i∈L , respectively. The dynamic programming (DP) formulation of the
Ö. Yazlali, F. Erhun / Operations Research Letters 35 (2007) 767 – 772
manufacturer’s problem is as follows, where superscript ms stands for multiple supply: Jnms (xn ) =
min
mn qn Mn
Gms n (xn , qn ),
for n = −L, −L + 1, . . . , N − l, where Gms Cni (qni ) + EHn (yn(l) − Dn(l) ) n (xn , qn ) = i∈L ms + EJn+1 (yn(l) − Dn + ynl+1 ,
ynl+2 , . . . , ynL ), and yn(l) = xn(l) + qnl , i xn + qni , i ∈ L\{l, L}, i yn = qnL , i = L, i ∈ {l, l + 1, . . . , L}\L. xni , The first term in the objective function denotes the total ordering costs, where Cni (qni ) is a convex function of qni . Apart from the convexity of these ordering cost functions, there is no other assumption that orders the supply options from more to less expensive. Hence, for example, a faster supply option may also be the cheaper alternative with this assumption. Similar to the ordering costs, we use a convex function Hn (·) to rep(l) resent the inventory related costs, and Dn denotes the convolution of l + 1 periods of demand (starting with (l) period n), i.e., Dn = Dn ∗ Dn+1 ∗ · · · ∗ Dn+l . Because the manufacturer observes demand between periods 0 and N, we assume that D−L =D−L+1 =· · ·=D−1 =0. ms (xN−l+1 ) = 0 for For convenience, we define JN−l+1 all xN−l+1 ; i.e., there are no obligations regarding the unsatisfied demand or leftover pipeline inventory at the end of the planning horizon. This assumption does not affect our results, but considerably simplifies the notation. It is important to note that the dimension of the state in the DP formulation is a function of the difference between the leadtimes of the fastest and slowest options, and it is independent of other intermediate supply options. Given the state of the pipeline inventory, however, the complexity of finding the optimal ordering decisions depends on all the available supply options. Our main result in Theorem 1 shows that the costto-go function Jnms (·) is jointly convex in the pipeline inventory xn and the vectors of order bounds n and
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n , where n = (mn , mn+1 , . . . , mN−l ) and n = (Mn , Mn+1 , . . . , MN−l ) (i.e., the vectors representing the minimum and maximum order bounds of the supply options, respectively, until the end of the planning horizon including the bounds of the current period). Theorem 1. Jnms (xn , n , n ), n = −L, −L + 1, . . . , N − l, is jointly convex in its arguments. Proof. We start our proof with an observation. If g(x, y) = f (x + y) and f (·) is a convex function, g(x, y) is then jointly convex with respect to (x, y); as for (x, y), (x, ˆ y), ˆ and some ∈ [0, 1], the following holds: g(x, y) + (1 − )g(x, ˆ y) ˆ = f (x + y) + (1 − )f (xˆ + y) ˆ f ((x + y) + (1 − )(xˆ + y)) ˆ = f ((x + (1 − )x) ˆ + (y + (1 − )y)) ˆ = g((x + (1 − )x), ˆ (y + (1 − )y)). ˆ In order to prove Theorem 1 we use induction. Supms (x pose that Jn+1 n+1 , n+1 , n+1 ) is jointly convex ms (·) is with respect to its arguments. Consequently, Jn+1 also a convex function of (xn , qn , n+1 , n+1 ), which means that Gms n (·), as a sum of convex and linear functions, is convex with respect to (xn , qn , n+1 , n+1 ) as well. We next introduce the following variables to make the notation simpler: qn∗ = arg qˆ ∗n = arg
min
Gms n (xn , qn , n+1 , n+1 )
min
ˆ n+1 ), Gms xn , qn , ˆ n+1 , n (ˆ
mn qn Mn ˆn ˆ n qn M m
ˆ n . We can then for some xn , xˆ n , n , ˆ n , n , and show that the equalities and inequalities below hold for any ∈ [0, 1]: ˆ n) Jnms (xn , n , n ) + (1 − )Jnms (ˆxn , ˆ n , ms ∗ = Gn (xn , qn , n+1 , n+1 ) ˆ n+1 ) + (1 − )Gms (ˆxn , qˆ ∗ , ˆ n+1 , n
n
ˆˆ ˆˆ n , q∗ + (1 − )qˆ ∗ , ˆˆ n+1 , Gms n+1 ) n (x n n ∗ ˆ ˆ ms ˆ ˆ n+1 ) Gn (xˆ n , qˆˆ n , ˆ n+1 , ˆ ˆˆ = Jnms (xˆˆ n , ˆ n , n ),
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ˆ where xˆˆ n = xn + (1 − )ˆxn , ˆ n = n + (1 − )ˆ n , ˆˆ ˆ n = n + (1 − )n , and ∗ qˆˆ n = arg
min
ˆˆ ˆˆ n qn M m n
ˆˆ ˆˆ n , qn , ˆˆ n+1 , Gms n+1 ). n (x
The first inequality is by the convexity of the function Gms n (·). The last inequality follows from the fact that ˆˆ ˆ ˆ n qn∗ +(1−)qˆ ∗n M m n . We complete the proof by ms observing that JN −l+1 (·) is a constant function, thus satisfying the convexity property. Theorem 1 shows that the convexity results of many models in the inventory management literature (e.g., the single supply problem when L = l and the dual supply problem with consecutive leadtimes when L = l + 1), from which the well-known optimal policies (such as the base-stock policy) follow, extend to the multiple supply problem setting. This convexity result can be trivially extended to the discounted-cost infinite horizon problem. The importance of the results of Theorem 1 is two-fold. First, the convex structure of the cost function enables the use of efficient search techniques in finding the optimal solution, which is quite difficult due to the inherent complexity of the multiple supply problem (i.e., multi-dimensional state and decision vectors). Second, Theorem 1 proves that the cost function is jointly convex not only in the ordering decisions and the pipeline vector, but also in the order bound vectors. Although in this paper these bound vectors are assumed to be a part of a predetermined contract, the results of Theorem 1 are particularly important when these bounds themselves become decision variables. For example, at the beginning of the planning horizon, the manufacturer can have the option of determining these contract terms including the bounds and prices as in [22]. We next discuss the simpler linear ordering costs case and present a thorough discussion of how to transform the convexity results of Theorem 1 into the wellknown results in the inventory management literature. 1.1. A special case—linear ordering costs Because the ordering cost function Cni (qni ) is potentially a non-linear function of the order size qni , our formulation is based on the actual orders rather than the inventory positions after placing the orders.
On the other hand, when the ordering costs are linear, i.e., Cni (qni ) = ci qni , we can formulate the problem with either set of decision variables, the latter of which is commonly used in the literature. Under the linear ordering costs assumption, the reduced form of the problem is as follows: Jnms (xn ) =
min
˜n ˜ n y˜ n x˜ n +M x˜ n +m
yn )} − x˜ n · c˜ T , {Gms n (˜
n = −L, −L + 1, . . . , N − l, where Gms yn ) = y˜ n · c˜ T + EHn (yn(l) − Dn(l) ) n (˜ ms + EJn+1 (yn(l) − Dn + ynl+1 ,
ynl+2 , . . . , ynL ). We represent the state of the pipeline with x˜ n = (xn , 0) and use y˜ n as the vector of pipeline inventory after the placement of the orders: y˜ n = (l) (yn , ynl+1 , ynl+2 , . . . , ynL ). The vector of ordering costs is c˜ = (cl , cl+1 , . . . , cL ), where ci = 0 for i ∈ {l, l + 1, . . . , L}\L and the superscript T de˜n= notes the vector transpose operation. Similarly, m L l l+1 L ˜ (mln , ml+1 n , . . . , mn ) and Mn = (Mn , Mn , . . . , Mn ), i i where mn = Mn = 0 for i ∈ {l, l + 1, . . . , L}\L. Once we formulate the problem with the inventory positions under the linear ordering costs assumption, we can easily replicate the well-known inventory models in the literature. For example, the optimality of the modified base-stock policy directly follows for the single supply problem; hence, we omit the details here. For the dual supply problem with consecutive lead(l+1) (l) = yn + ynl+1 . Note that the times, we define zn (l+1) specifies a sequence of events for definition of zn the placement of orders, from the fastest to the slowest. We can then rewrite Gms yn ) as follows: n (˜ Gms yn ) = cl yn(l) + cl+1 (zn(l+1) − yn(l) ) n (˜ ms + EH (yn(l) − Dn(l) ) + EJn+1 (zn(l+1) − Dn )
= (cl − cl+1 )yn(l) + EH (yn(l) − Dn(l) ) ms + cl+1 zn(l+1) + EJn+1 (zn(l+1) − Dn ). (2)
Because the right-hand side of Eq. (2) is convex and separable, a two-level modified base-stock policy follows as in [21].
Ö. Yazlali, F. Erhun / Operations Research Letters 35 (2007) 767 – 772
Finally, for the triple supply problem with consecu(l+2) (l+1) (l) = = yn + ynl+1 and zn tive leadtimes, define zn (l+1) l+2 ms zn +yn . Then we can rewrite Gn (˜yn ) as follows: Gms yn ) = cl yn(l) + cl+1 (zn(l+1) − yn(l) ) n (˜ + cl+2 (zn(l+2) − zn(l+1) ) + EH (yn(l) − Dn(l) ) ms + EJn+1 (zn(l+1) − Dn , zn(l+2) − zn(l+1) )
= (c − c )yn(l) + EH (yn(l) − Dn(l) ) + (cl+1 − cl+2 )zn(l+1) + cl+2 zn(l+2) ms + EJn+1 (zn(l+1) − Dn , zn(l+2) − zn(l+1) ). l
771
to canceling. The coefficients in ∈ [0, 1] and in ∈ [0, ∞) define the flexibility that the manufacturer has in modifying the incoming orders. Because the order bounds are defined with respect to the pipeline inventory, we use the following state variable in the DP formulation of this section: xn = (xn(l) , xnl , xnl+1 , . . . , xnL−1 ),
l+1
ms (·) is a twoIn this case, the argument of Jn+1 dimensional vector. It is important to note that for all multiple supply problems (except single supply and dual supply with consecutive leadtimes), the argument ms (·) is a multi-dimensional vector. Hence, the of Jn+1 existence of a base-stock policy can be guaranteed only for the fastest supply option, which is because (l) only the terms with yn can be separated from the rest of the terms in the objective function as in the above example. In [8], a counter-example is provided that shows that base-stock policies are not optimal for this problem.
2. Quantity flexibility contracts Now suppose that the manufacturer has a single supplier with a leadtime of L periods. Although the leadtime is fixed, we further assume that the manufacturer has the flexibility to modify (some or all of) her previous orders, which are in the pipeline. In this scenario, the modification decisions can be interpreted as (potentially negative) orders placed to pseudo-suppliers with shorter leadtimes than the original supplier. Hence, this interpretation transforms the problem into a multiple supply problem. At any period, the manufacturer has two types of decisions:
which now includes xnl in addition to the definition of Eq. (1) in Section 1. Although any convex ordering cost is still sufficient for our convexity results, we further use the following cost structure in order to prevent the manufacturer from profiting by selling the pipeline inventory back to the supplier: i i c qn if qni 0, i i Cn (qn ) = d i qni otherwise, where mini∈L {ci } > maxi∈L {d i }. Notice that d i can possibly be negative, which means that the manufacturer has to pay for canceling orders. The following is the DP formulation of the manufacturer’s problem, where superscript qf stands for quantity flexibility: qf
Jn (xn ) =
min
−in xni qni in xni ,i∈L\L L L 0 mL n qn Mn
qf
Gn (xn , qn ),
for n = −L, −L + 1, . . . , N − l, where qf Cni (qni ) + EHn (yn(l) − Dn(l) ) Gn (xn , qn ) = i∈L qf
+ EJn+1 (yn(l) − Dn + ynl+1 , ynl+1 , ynl+2 , . . . , ynL ), (l)
yn and yni , i ∈ {l, l + 1, . . . , L}, are defined in the same way as in Section 1. Similar to Section qf 1, JN−l+1 (xN−l+1 ) = 0 for all xN−l+1 . Theorem 2 shows that a convexity result similar to Theorem 1 of Section 1 holds for the order modification problem.
L L • a regular order of qnL , where 0 mL n qn Mn , and • modification decisions, qni , i ∈ L\L, where −in xni qni in xni .
Theorem 2. Jn (xn ) is convex with respect to its arguments.
A positive modification implies an increase in the order size, whereas a negative modification corresponds
Proof (Sketch). Because the proof is similar to the proof of Theorem 1, we discuss only a brief sketch
qf
772
Ö. Yazlali, F. Erhun / Operations Research Letters 35 (2007) 767 – 772 qf
here. When it is assumed that Jn+1 (xn+1 ) has the deqf Gn (xn , qn )
sired property, it is trivial to show that is jointly convex with respect to its arguments. The critical part of the proof is to observe that the lower bound −in xni and the upper bound in xni are linear in xni , qf for all i ∈ L\L; hence, the convexity of Gn (·) with respect to xn is preserved under minimization. Acknowledgements This research was funded in part by NSF Grant #NSF/CAREER-0547021. We gratefully acknowledge partial support from GM/Stanford Collaborative Research Laboratory in Work Systems and Stanford University Office of Technology Licensing (OTL) Research Incentive Fund awards. We are indebted to an anonymous referee for valuable and constructive suggestions, which greatly improved the paper. References [1] R.Anupindi,Y. Bassok, Supply contracts with quantity commitments and stochastic demand, in: S. Tayur, R. Ganeshan, M. Magazin (Eds.), Quantitative Models for Supply Chain Management, Kluwer Academic Publishers, New York, 1998, pp. 197–233. [2] Y. Bassok, R. Anupindi, Analysis of supply contracts with commitments and flexibility, Working Paper, Northwestern University, Evanston, IL, 2005. [3] E. Bulinskaya, Some results concerning optimal inventory policies, Theory Probab. Appl. 9 (1964) 389–403. [4] K. Daniel, A delivery-lag inventory model with emergency, in: D. Gilford, M. Shelly (Eds.), Multistage Inventory Models and Techniques, Stanford University Press, New York, 1963, pp. 32–46. [5] A. Federgruen, P. Zipkin, An inventory model with limited production capacity and uncertain demands I: the average cost criterion, Math. Oper. Res. 11 (1986) 193–207. [6] A. Federgruen, P. Zipkin, An inventory model with limited production capacity and uncertain demands II: the discounted cost criterion, Math. Oper. Res. 11 (1986) 208–215. [7] Q. Feng, G. Gallego, S.P. Sethi, H. Yan, H. Zhang, Periodicreview inventory model with three consecutive delivery modes and forecast updates, J. Optim. Theory Appl. 124 (1) (2005) 137–155.
[8] Q. Feng, G. Gallego, S.P. Sethi, H. Yan, H. Zhang, Optimality and nonoptimality of base-stock policy in inventory problems with multiple delivery modes, J. Indus. Manage. Optim. 2 (1) (2006) 19–42. [9] Q. Feng, G. Gallego, S.P. Sethi, H. Yan, H. Zhang, Are basestock policies optimal in inventory problems with multiple delivery modes? Oper. Res., to appear. [10] Y. Fukuda, Optimal policies for the inventory problem with negotiable leadtime, Manage. Sci. 10 (1964) 690–708. [11] D.G. Lawson, E.L. Porteus, Multistage inventory management with expediting, Oper. Res. 48 (6) (2000) 878–893. [12] A. Scheller-Wolf, S. Tayur, Managing supply chains in emerging markets, in: S. Tayur, R. Ganeshan, M. Magazin (Eds.), Quantitative Models for Supply Chain Management, Kluwer Academic Publishers, Dordrecht, 1998, pp. 703–737. [13] A. Scheller-Wolf, S. Tayur, A Markovian dual-source production-inventory model with order bands, Working Paper, Tepper School of Business, Carnegie Mellon University, Pittsburgh, PA, 1998. [14] A. Scheller-Wolf, S. Veeraraghavan, G. van Houtum, Inventory models with expedited ordering: single index policies, Working Paper, Tepper School of Business, Carnegie Mellon University, Pittsburgh, PA, 2003. [15] S.P. Sethi, H. Yan, H. Zhang, Inventory models with fixed costs, forecast updates, and two delivery modes, Oper. Res. 51 (2) (2003) 321–328. [16] S.P. Sethi, H. Yan, H. Zhang, Quantity flexibility contracts: optimal decisions with information updates, Decision Sci. 35 (4) (2004) 691–712. [17] A.A. Tsay, W.S. Lovejoy, Quantity flexibility contracts and supply chain performance, Manuf. Serv. Oper. Manage. 1 (2) (1999) 89–111. [18] S. Veeraraghavan, A. Scheller-Wolf, Now or later: a simple policy for effective dual sourcing in capacitated systems, Oper. Res., to appear. [19] A.S. Whittemore, S.C. Saunders, Optimal inventory under stochastic demand with two supply options, SIAM J. Appl. Math. 32 (2) (1977) 293–305. [20] J. Yang, X. Qi, Y. Xia, A production-inventory system with Markovian capacity and outsourcing option, Oper. Res. 53 (2) (2005) 328–349. [21] Ö. Yazlali, F. Erhun, Dual supply inventory problem with order bounds and unrestricted ordering costs, Working Paper, Department of Management Science and Engineering, Stanford University, Stanford, CA, 2006. [22] Ö. Yazlali, F. Erhun, Managing demand uncertainty with dual supply contracts, Working Paper, Department of Management Science and Engineering, Stanford University, Stanford, CA, 2006. [23] V.L. Zhang, Ordering policies for an inventory system with three supply modes, Naval Res. Logistics 43 (1996) 691–708.
Operations Research Letters www.elsevier.com/locate/orl
Relating the multiple supply problem to quantity flexibility contracts Özgür Yazlali, Feryal Erhun∗ Department of Management Science and Engineering, Stanford University, Stanford, CA 94305, USA Received 21 February 2006; accepted 2 January 2007 Available online 14 January 2007
Abstract We consider two classes of problems in the supply chain management literature: the multiple supply problem and quantity flexibility contracts. We identify the similarities between these two problems and show that for both the total operating cost of the manufacturer is convex with respect to its arguments under mild conditions. © 2007 Elsevier B.V. All rights reserved.
Keywords: Demand uncertainty; Dual supply problem with general leadtimes; Maximum capacity limit; Minimum order commitment; Dynamic programming
There has been increased interest in the literature investigating supply decisions in the presence of several sources, as many companies in today’s global economy seek alternative supply tools to meet their demand. In this research, we explore an important class of these multiple sourcing models (i.e., single-product, single-stage, deterministic leadtimes) and explain the analytical structure of these problems. We first formulate the multiple supply problem in its general form: a manufacturer has to satisfy random demand for a finite planning horizon using several capacitated supply options, each with a different ∗ Corresponding author.
E-mail address: [email protected] (F. Erhun). 0167-6377/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.orl.2007.01.002
leadtime. Our modeling setting (partially or fully) generalizes many inventory models in the literature; examples include (i) single supply problems such as in [5,6], (ii) dual supply problems with consecutive leadtimes such as in [3,4,10,12,13,15,20,21], (iii) dual supply problems with general leadtimes such as in [14,18,19], and (iv) other multiple supply problems (i.e., three or more suppliers with or without consecutive leadtimes) such as in [7–9,23]. We then show that the structural formulation of the multiple supply problem has characteristics similar to that of quantity flexibility contracts (e.g., [1,2,16]) where the manufacturer has a single supplier (with a positive leadtime), but has the flexibility to modify her previous orders. We note that there are several multi-stage models in the literature (such as [11] for the multiple supply problem and
768
Ö. Yazlali, F. Erhun / Operations Research Letters 35 (2007) 767 – 772
[17] for the quantity flexibility contracts) with settings closely related to ours; nevertheless, the results in this paper do not readily apply to these models because of our single-stage focus. One of the main differences between the multiple supply problem and the quantity flexibility contract is due to the bounds on the manufacturer’s ordering decisions. In the former, it is natural to assume that order bounds for each supply option are predetermined (e.g., as a part of contracts that are signed at the beginning of a planning horizon); hence, they are independent of the pipeline inventory. On the other hand, in the latter, the flexibility to modify previously given orders is usually a function of incoming orders (e.g., the manufacturer may have the flexibility to increase/decrease incoming orders up to 10%) in a rolling-horizon fashion (i.e., at each period a new order is added to the pipeline). Our contribution is to show that when ordering/modification costs and inventory related costs are convex, the total operating cost of the manufacturer (as a function of the current state of the pipeline inventory as well as the current and future order quantities and order bounds) is convex with respect to its arguments. Furthermore, our paper links the two different research streams (namely the literature on multiple supply problems and that on quantity flexibility contracts) by showing that the two problem settings are similar, and consequently, the objective functions of both problems have common properties with respect to their arguments. Both of these problems are difficult to solve due to the high dimensionality of state and decision variables; however, our goal in this paper is to show that certain functional properties, which are commonly used in proving many well-known results in the literature, apply to these complex settings as well. Moreover, our convexity results improve the efficiency of a computational approach in solving both problems. Because of the similarities in their problem settings, the heuristic approaches developed for one problem setup (e.g., in [2,14,18]) may prove useful for the other. The organization of the paper is as follows. We first formulate the multiple supply problem in its general form in Section 1 and present our structural results. We present a special case of linear ordering costs in Section 1.1. In Section 2, we then discuss the quantity flexibility contract scenario.
1. Multiple supply problem In this section we assume that a manufacturer has a set of supply options with leadtimes in L ⊂ {l, l + 1, . . . , L}, where l and L are the leadtimes of the fastest and slowest supply options, respectively. In order to differentiate the alternative sources we use the term supply option i, i ∈ L. The manufacturer has to meet a random demand Dn in period n, n = 0, 1, . . . , N, using these resources. We assume that the demand distribution is independent (but not necessarily identical) between periods. Because of the leadtime difference, the manufacturer starts ordering at period −L, whereas her last orders are given at period N − l. The sequence of events at any period is as follows: • The manufacturer observes the pipeline inventory (xn0 , xn1 , . . . , xnL−1 ), where xni 0, i = 1, . . . , L − 1, is the inventory in the pipeline that will be available after i periods and xn0 is the on-hand inventory at the beginning of period n before the ordering decisions. • She then places her orders considering the capacity restrictions of the supply options. That is, she chooses the order quantity qni from supply option i, i ∈ L, such that 0 min qni Mni < ∞, where min and Mni are the minimum and maximum capacity restrictions of supply option i at period n, respectively. • Finally, the demand occurs and the state is updated. We assume that the manufacturer incurs a cost for excess inventory and unsatisfied demand at the end of a period. In our problem formulation, we represent the state of the pipeline inventory at period n with xn = (xn(l) , xnl+1 , xnl+2 , . . . , xnL−1 ),
(1)
(l) where xn = li=0 xni is the sum of on-hand inventory and first l periods of pipeline inventory. The vector of ordering decisions is qn = (qni )i∈L , whereas the vectors of minimum and maximum capacity restrictions are mn = (min )i∈L and Mn = (Mni )i∈L , respectively. The dynamic programming (DP) formulation of the
Ö. Yazlali, F. Erhun / Operations Research Letters 35 (2007) 767 – 772
manufacturer’s problem is as follows, where superscript ms stands for multiple supply: Jnms (xn ) =
min
mn qn Mn
Gms n (xn , qn ),
for n = −L, −L + 1, . . . , N − l, where Gms Cni (qni ) + EHn (yn(l) − Dn(l) ) n (xn , qn ) = i∈L ms + EJn+1 (yn(l) − Dn + ynl+1 ,
ynl+2 , . . . , ynL ), and yn(l) = xn(l) + qnl , i xn + qni , i ∈ L\{l, L}, i yn = qnL , i = L, i ∈ {l, l + 1, . . . , L}\L. xni , The first term in the objective function denotes the total ordering costs, where Cni (qni ) is a convex function of qni . Apart from the convexity of these ordering cost functions, there is no other assumption that orders the supply options from more to less expensive. Hence, for example, a faster supply option may also be the cheaper alternative with this assumption. Similar to the ordering costs, we use a convex function Hn (·) to rep(l) resent the inventory related costs, and Dn denotes the convolution of l + 1 periods of demand (starting with (l) period n), i.e., Dn = Dn ∗ Dn+1 ∗ · · · ∗ Dn+l . Because the manufacturer observes demand between periods 0 and N, we assume that D−L =D−L+1 =· · ·=D−1 =0. ms (xN−l+1 ) = 0 for For convenience, we define JN−l+1 all xN−l+1 ; i.e., there are no obligations regarding the unsatisfied demand or leftover pipeline inventory at the end of the planning horizon. This assumption does not affect our results, but considerably simplifies the notation. It is important to note that the dimension of the state in the DP formulation is a function of the difference between the leadtimes of the fastest and slowest options, and it is independent of other intermediate supply options. Given the state of the pipeline inventory, however, the complexity of finding the optimal ordering decisions depends on all the available supply options. Our main result in Theorem 1 shows that the costto-go function Jnms (·) is jointly convex in the pipeline inventory xn and the vectors of order bounds n and
769
n , where n = (mn , mn+1 , . . . , mN−l ) and n = (Mn , Mn+1 , . . . , MN−l ) (i.e., the vectors representing the minimum and maximum order bounds of the supply options, respectively, until the end of the planning horizon including the bounds of the current period). Theorem 1. Jnms (xn , n , n ), n = −L, −L + 1, . . . , N − l, is jointly convex in its arguments. Proof. We start our proof with an observation. If g(x, y) = f (x + y) and f (·) is a convex function, g(x, y) is then jointly convex with respect to (x, y); as for (x, y), (x, ˆ y), ˆ and some ∈ [0, 1], the following holds: g(x, y) + (1 − )g(x, ˆ y) ˆ = f (x + y) + (1 − )f (xˆ + y) ˆ f ((x + y) + (1 − )(xˆ + y)) ˆ = f ((x + (1 − )x) ˆ + (y + (1 − )y)) ˆ = g((x + (1 − )x), ˆ (y + (1 − )y)). ˆ In order to prove Theorem 1 we use induction. Supms (x pose that Jn+1 n+1 , n+1 , n+1 ) is jointly convex ms (·) is with respect to its arguments. Consequently, Jn+1 also a convex function of (xn , qn , n+1 , n+1 ), which means that Gms n (·), as a sum of convex and linear functions, is convex with respect to (xn , qn , n+1 , n+1 ) as well. We next introduce the following variables to make the notation simpler: qn∗ = arg qˆ ∗n = arg
min
Gms n (xn , qn , n+1 , n+1 )
min
ˆ n+1 ), Gms xn , qn , ˆ n+1 , n (ˆ
mn qn Mn ˆn ˆ n qn M m
ˆ n . We can then for some xn , xˆ n , n , ˆ n , n , and show that the equalities and inequalities below hold for any ∈ [0, 1]: ˆ n) Jnms (xn , n , n ) + (1 − )Jnms (ˆxn , ˆ n , ms ∗ = Gn (xn , qn , n+1 , n+1 ) ˆ n+1 ) + (1 − )Gms (ˆxn , qˆ ∗ , ˆ n+1 , n
n
ˆˆ ˆˆ n , q∗ + (1 − )qˆ ∗ , ˆˆ n+1 , Gms n+1 ) n (x n n ∗ ˆ ˆ ms ˆ ˆ n+1 ) Gn (xˆ n , qˆˆ n , ˆ n+1 , ˆ ˆˆ = Jnms (xˆˆ n , ˆ n , n ),
770
Ö. Yazlali, F. Erhun / Operations Research Letters 35 (2007) 767 – 772
ˆ where xˆˆ n = xn + (1 − )ˆxn , ˆ n = n + (1 − )ˆ n , ˆˆ ˆ n = n + (1 − )n , and ∗ qˆˆ n = arg
min
ˆˆ ˆˆ n qn M m n
ˆˆ ˆˆ n , qn , ˆˆ n+1 , Gms n+1 ). n (x
The first inequality is by the convexity of the function Gms n (·). The last inequality follows from the fact that ˆˆ ˆ ˆ n qn∗ +(1−)qˆ ∗n M m n . We complete the proof by ms observing that JN −l+1 (·) is a constant function, thus satisfying the convexity property. Theorem 1 shows that the convexity results of many models in the inventory management literature (e.g., the single supply problem when L = l and the dual supply problem with consecutive leadtimes when L = l + 1), from which the well-known optimal policies (such as the base-stock policy) follow, extend to the multiple supply problem setting. This convexity result can be trivially extended to the discounted-cost infinite horizon problem. The importance of the results of Theorem 1 is two-fold. First, the convex structure of the cost function enables the use of efficient search techniques in finding the optimal solution, which is quite difficult due to the inherent complexity of the multiple supply problem (i.e., multi-dimensional state and decision vectors). Second, Theorem 1 proves that the cost function is jointly convex not only in the ordering decisions and the pipeline vector, but also in the order bound vectors. Although in this paper these bound vectors are assumed to be a part of a predetermined contract, the results of Theorem 1 are particularly important when these bounds themselves become decision variables. For example, at the beginning of the planning horizon, the manufacturer can have the option of determining these contract terms including the bounds and prices as in [22]. We next discuss the simpler linear ordering costs case and present a thorough discussion of how to transform the convexity results of Theorem 1 into the wellknown results in the inventory management literature. 1.1. A special case—linear ordering costs Because the ordering cost function Cni (qni ) is potentially a non-linear function of the order size qni , our formulation is based on the actual orders rather than the inventory positions after placing the orders.
On the other hand, when the ordering costs are linear, i.e., Cni (qni ) = ci qni , we can formulate the problem with either set of decision variables, the latter of which is commonly used in the literature. Under the linear ordering costs assumption, the reduced form of the problem is as follows: Jnms (xn ) =
min
˜n ˜ n y˜ n x˜ n +M x˜ n +m
yn )} − x˜ n · c˜ T , {Gms n (˜
n = −L, −L + 1, . . . , N − l, where Gms yn ) = y˜ n · c˜ T + EHn (yn(l) − Dn(l) ) n (˜ ms + EJn+1 (yn(l) − Dn + ynl+1 ,
ynl+2 , . . . , ynL ). We represent the state of the pipeline with x˜ n = (xn , 0) and use y˜ n as the vector of pipeline inventory after the placement of the orders: y˜ n = (l) (yn , ynl+1 , ynl+2 , . . . , ynL ). The vector of ordering costs is c˜ = (cl , cl+1 , . . . , cL ), where ci = 0 for i ∈ {l, l + 1, . . . , L}\L and the superscript T de˜n= notes the vector transpose operation. Similarly, m L l l+1 L ˜ (mln , ml+1 n , . . . , mn ) and Mn = (Mn , Mn , . . . , Mn ), i i where mn = Mn = 0 for i ∈ {l, l + 1, . . . , L}\L. Once we formulate the problem with the inventory positions under the linear ordering costs assumption, we can easily replicate the well-known inventory models in the literature. For example, the optimality of the modified base-stock policy directly follows for the single supply problem; hence, we omit the details here. For the dual supply problem with consecutive lead(l+1) (l) = yn + ynl+1 . Note that the times, we define zn (l+1) specifies a sequence of events for definition of zn the placement of orders, from the fastest to the slowest. We can then rewrite Gms yn ) as follows: n (˜ Gms yn ) = cl yn(l) + cl+1 (zn(l+1) − yn(l) ) n (˜ ms + EH (yn(l) − Dn(l) ) + EJn+1 (zn(l+1) − Dn )
= (cl − cl+1 )yn(l) + EH (yn(l) − Dn(l) ) ms + cl+1 zn(l+1) + EJn+1 (zn(l+1) − Dn ). (2)
Because the right-hand side of Eq. (2) is convex and separable, a two-level modified base-stock policy follows as in [21].
Ö. Yazlali, F. Erhun / Operations Research Letters 35 (2007) 767 – 772
Finally, for the triple supply problem with consecu(l+2) (l+1) (l) = = yn + ynl+1 and zn tive leadtimes, define zn (l+1) l+2 ms zn +yn . Then we can rewrite Gn (˜yn ) as follows: Gms yn ) = cl yn(l) + cl+1 (zn(l+1) − yn(l) ) n (˜ + cl+2 (zn(l+2) − zn(l+1) ) + EH (yn(l) − Dn(l) ) ms + EJn+1 (zn(l+1) − Dn , zn(l+2) − zn(l+1) )
= (c − c )yn(l) + EH (yn(l) − Dn(l) ) + (cl+1 − cl+2 )zn(l+1) + cl+2 zn(l+2) ms + EJn+1 (zn(l+1) − Dn , zn(l+2) − zn(l+1) ). l
771
to canceling. The coefficients in ∈ [0, 1] and in ∈ [0, ∞) define the flexibility that the manufacturer has in modifying the incoming orders. Because the order bounds are defined with respect to the pipeline inventory, we use the following state variable in the DP formulation of this section: xn = (xn(l) , xnl , xnl+1 , . . . , xnL−1 ),
l+1
ms (·) is a twoIn this case, the argument of Jn+1 dimensional vector. It is important to note that for all multiple supply problems (except single supply and dual supply with consecutive leadtimes), the argument ms (·) is a multi-dimensional vector. Hence, the of Jn+1 existence of a base-stock policy can be guaranteed only for the fastest supply option, which is because (l) only the terms with yn can be separated from the rest of the terms in the objective function as in the above example. In [8], a counter-example is provided that shows that base-stock policies are not optimal for this problem.
2. Quantity flexibility contracts Now suppose that the manufacturer has a single supplier with a leadtime of L periods. Although the leadtime is fixed, we further assume that the manufacturer has the flexibility to modify (some or all of) her previous orders, which are in the pipeline. In this scenario, the modification decisions can be interpreted as (potentially negative) orders placed to pseudo-suppliers with shorter leadtimes than the original supplier. Hence, this interpretation transforms the problem into a multiple supply problem. At any period, the manufacturer has two types of decisions:
which now includes xnl in addition to the definition of Eq. (1) in Section 1. Although any convex ordering cost is still sufficient for our convexity results, we further use the following cost structure in order to prevent the manufacturer from profiting by selling the pipeline inventory back to the supplier: i i c qn if qni 0, i i Cn (qn ) = d i qni otherwise, where mini∈L {ci } > maxi∈L {d i }. Notice that d i can possibly be negative, which means that the manufacturer has to pay for canceling orders. The following is the DP formulation of the manufacturer’s problem, where superscript qf stands for quantity flexibility: qf
Jn (xn ) =
min
−in xni qni in xni ,i∈L\L L L 0 mL n qn Mn
qf
Gn (xn , qn ),
for n = −L, −L + 1, . . . , N − l, where qf Cni (qni ) + EHn (yn(l) − Dn(l) ) Gn (xn , qn ) = i∈L qf
+ EJn+1 (yn(l) − Dn + ynl+1 , ynl+1 , ynl+2 , . . . , ynL ), (l)
yn and yni , i ∈ {l, l + 1, . . . , L}, are defined in the same way as in Section 1. Similar to Section qf 1, JN−l+1 (xN−l+1 ) = 0 for all xN−l+1 . Theorem 2 shows that a convexity result similar to Theorem 1 of Section 1 holds for the order modification problem.
L L • a regular order of qnL , where 0 mL n qn Mn , and • modification decisions, qni , i ∈ L\L, where −in xni qni in xni .
Theorem 2. Jn (xn ) is convex with respect to its arguments.
A positive modification implies an increase in the order size, whereas a negative modification corresponds
Proof (Sketch). Because the proof is similar to the proof of Theorem 1, we discuss only a brief sketch
qf
772
Ö. Yazlali, F. Erhun / Operations Research Letters 35 (2007) 767 – 772 qf
here. When it is assumed that Jn+1 (xn+1 ) has the deqf Gn (xn , qn )
sired property, it is trivial to show that is jointly convex with respect to its arguments. The critical part of the proof is to observe that the lower bound −in xni and the upper bound in xni are linear in xni , qf for all i ∈ L\L; hence, the convexity of Gn (·) with respect to xn is preserved under minimization. Acknowledgements This research was funded in part by NSF Grant #NSF/CAREER-0547021. We gratefully acknowledge partial support from GM/Stanford Collaborative Research Laboratory in Work Systems and Stanford University Office of Technology Licensing (OTL) Research Incentive Fund awards. We are indebted to an anonymous referee for valuable and constructive suggestions, which greatly improved the paper. References [1] R.Anupindi,Y. Bassok, Supply contracts with quantity commitments and stochastic demand, in: S. Tayur, R. Ganeshan, M. Magazin (Eds.), Quantitative Models for Supply Chain Management, Kluwer Academic Publishers, New York, 1998, pp. 197–233. [2] Y. Bassok, R. Anupindi, Analysis of supply contracts with commitments and flexibility, Working Paper, Northwestern University, Evanston, IL, 2005. [3] E. Bulinskaya, Some results concerning optimal inventory policies, Theory Probab. Appl. 9 (1964) 389–403. [4] K. Daniel, A delivery-lag inventory model with emergency, in: D. Gilford, M. Shelly (Eds.), Multistage Inventory Models and Techniques, Stanford University Press, New York, 1963, pp. 32–46. [5] A. Federgruen, P. Zipkin, An inventory model with limited production capacity and uncertain demands I: the average cost criterion, Math. Oper. Res. 11 (1986) 193–207. [6] A. Federgruen, P. Zipkin, An inventory model with limited production capacity and uncertain demands II: the discounted cost criterion, Math. Oper. Res. 11 (1986) 208–215. [7] Q. Feng, G. Gallego, S.P. Sethi, H. Yan, H. Zhang, Periodicreview inventory model with three consecutive delivery modes and forecast updates, J. Optim. Theory Appl. 124 (1) (2005) 137–155.
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