Optimal Dual Sourcing Strategy with Capacity Constraint and Fixed Bilateral Adjustment Costs

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1 Optimal Dual Sourcing Strategy with Capacity Constraint and Fixed Bilateral Adjustment Costs Lijian Lu Columia University Awi Federgruen Columia University Feruary 1, 2016 Zhe Liu Columia University Astract This paper studies a finite horizon, single product, periodic review inventory system with two supply nodes. Supply nodes are differentiated y their fixed and variale costs, delivery lead times, as well as constraints on the order sizes. The uyer faces interesting tradeoffs: Procuring from one of the supplier, referred to onshore supplier, involves a high variale cost ut a shorter delivery delay, while procuring from the other supplier, offshore supplier, involves a low variale cost ut a longer delivery delay. Each order from oth suppliers incurs a fixed cost as well as a constraint on the maximum order size. Excess inventories could e salvaged at the end of each period with a fixed cost or e carried over to the next period. We show that the cost-to-go function satisfies a certain convexity, called (CK 1, K 2 )- convexity introduced in this paper. We also show that the (CK 1, K 2 )-convexity is preserved in certain types of optimizations and derive structural results for solutions of these optimizations involving (CK 1, K 2 )-convex functions. Based on these structural results, we partially characterize the optimal sourcing and salvaging decisions. Keywords: Dual sourcing, capacity constraints, convexity, fixed cost 1 Introduction Offshore outsourcing/production has een increasingly popular in the past one and half decades. The enefits of offshore sourcing include lower costs, etter availaility of skilled people, and lower foreign corporate tax rate. For example, aout two-thirds of Apple s $97.6 illion cash pile is offshore in year However, the cost savings of the offshore reduces the inventory flexiility in a sense that the delivery takes a longer time and the procurement has capacity limits due to the financial udget or the regulation of trading in the foreign countries. In contrast, the onshore procurement in the local market is fast ut incurring higher cost. outsourcing motivates this paper. How to trade-off etween the onshore production and the offshore

2 This paper studies the inventory replenishment policy in a capacitated inventory system with two supply nodes, a slow one (offshore supplier) and a fast node (onshore supplier). An order placed from the fast supplier at the eginning of a period is delivered at the end of the period, whereas an order from the slow node is delivered at the end of the next period. The procurement from each node incurs oth a variale cost per unit and a fixed cost per order. There is limits on the order size for each order from either slow node or fast node due to suppliers production capacity or the udget constraint. Leftover inventory at the end of each period can either e carried over to the next period incurring inventory holding cost, or e salvaged immediately with a fixed cost. We egin with the analysis for the corresponding no-capacity inventory system y introducing a concept called strong (K 1, K 2 )-convexity. The strong (K 1, K 2 )-convexity includes several commonly used convexity as special cases, such as, the convex function is strong (0, 0)-convex function, and the strong K-convex function is a strong (K, 0)-convex function. Besides, we show that the strong (K 1, K 2 )- convexity is preserved under expectation and linear comination with non-negative weight. Using these two properties, we show that the cost-to-go functions for the multiple period inventory system without capacity limits are strong (K 1, K 2 )-convex. We characterize the optimal sourcing strategy from each supplier and the salvaging decision ased on properties of the strong (K 1, K 2 )-convexity. In particular, the ordering strategy from the fast supplier at each period can e characterized y four parameters (,, s, s), where ( s s). The ordering decisions are partially determined y (, ), namely, no order when the inventory level is larger than, ordering up-to a constant level when it is elow, and either purchasing or staying put when its value is etween and. The (s, s) characterize the threshold to salvage or not, i.e., always selling down-to a constant level when the starting inventory level is larger then s, staying put when it is elow s, and either selling or staying put when it is etween s and s. The overall inventory policy for fast node is a comination of the aove two ordering and salvaging policy. As for the slow node, the ordering policy has a similar structure to the ordering policy for the fast note. We then generalize the aove convexity concept to the (CK 1, K 2 )-convexity. We show that the (CK 1, K 2 )-convexity is preserved under linear transformations and is also preserved under certain optimizations wherey the decision variale is constrained from aove. We show that the cost-to-go function for a multi-period capacitated inventory system with two supplier nodes are (CK 1, K 2 )-convex. (For conciseness and ease of presentation, the main ody of the paper focuses on the case when each replenishment order from the fast supplier has a maximum size limit. The case when orders from oth the fast supplier and the slow supplier have maximum size limit is included in Section 6.) Therefore, we could partially characterize the optimal sourcing strategy and the ordering quantities from each supply 2

3 w/o Capacity with Capacity Onshore Supplier Optimal Inventory Position Buy up to Stay Salvage down to Optimal Inventory Position Buy C f units Buy up to Stay Salvage C v units Starting Inventory Position Starting Inventory Position Offshore Supplier Optimal Inventory Position Buy up to Stay Starting Inventory Position Optimal Inventory Position Buy C s units Stay Starting Inventory Position Figure 1: Structure of the Optimal Policies. (Fixed cost equals to 1$ per order and maximum size limit equals to 10 units per order.) node on the asis of properties of the (CK 1, K 2 )-convexity. The optimal policies are similar, ut take a more complicated structure, to the aove inventory system without a capacity constraint on the order size. For the fast node, the parameter space of the starting inventory level is divided into five regions, characterized y (,, s, s). The optimal policy for the fast node is fully characterized in the three regions, ordering with the maximum size when the starting inventory level is elow, staying put when it is etween and s, and salvaging down-to a constant level (or with the maximum quantity) when it is aove s. In the other two regions, (, ) and (s, s), the optimal policy is either staying put or ordering and salvaging, respectively. Finally, we show how our model could e generalized to other important scenarios, such as including the terminal value at the end of horizon as well as constraints on the maximum order size for oth fast and slow nodes and a constraint on the size of salvaging quantity. This paper makes the following contriutions. (i) We introduce a new convexity concept, i.e., strong (K 1, K 2 )-convexity, and we show that the strong (K 1, K 2 )-convexity is preserved under linear transformations, therefore, its properties are also preserved in many applications in the context of dynamic programming. We show that the 3

4 cost-to-go function for a multiple-period inventory replenishment prolem with two supplier nodes, different delivery lead time, fixed-cost associated with ordering or salvaging, is a strong (K 1, K 2 )- convex function. We derive the properties of solutions to certain type of optimizations with a strong (K 1, K 2 )-convex function. Applying these properties of strong (K 1, K 2 )-convexity, we could therefore characterize the optimal inventory replenishment and inventory salvaging policy for such a system. (ii) We generalize the aove strong (K 1, K 2 )-convexity to (CK 1, K 2 )-convexity, which is also preserved under linear transformation. In addition, the (CK 1, K 2 )-convexity is preserved under certain optimization when the decision variale is constrained from aove. We characterize the structure of optimal solutions for these optimizations that involves a (CK 1, K 2 )-convex function. (iii) We show that the cost-to-go function for a capacitated, multi-period, dual sourcing inventory system satisfies the (CK 1, K 2 )-convexity. Applying the properties of the (CK 1, K 2 )-convexity, we could partially characterize the optimal sourcing and inventory salvaging strategy for this inventory system. (iv) Finally, we generalize the aove (CK 1, K 2 )-convexity to the (C 1 K 1, C 2 K 2 )-convexity. We show that the cost-to-go function for the aove system with an additional constraint that each salvaging order has a maximum size limit is (C 1 K 1, C 2 K 2 )-convex. The remainder of this paper is organized as follows. Section 2 provides review on the related literatures. The model is introduced in Section 3. Section 4 develops properties of a strong (K 1, K 2 )-convexity and applies it to study the inventory system without capacity limit, and Section 5 studies the capacitated inventory system using properties of (CK 1, K 2 )-convexity. Section 6 concludes the paper with several discussions. All proofs except that of theorems are provided in the Appendix. 2 Literature Review Our paper is closely related to the literature on the inventory management. In most inventory systems, finished products are outsourced from outside suppliers and sold to end customers through retailers. The prolem of interest is how to replenish inventory in the logistic network, see Zipkin (2000) for a comprehensive review. Song and Zipkin (1993, 1996) have shown that the ase stock policy is optimal when ordering cost is linear in order quantity, Scarf (1960), Veinott and Wagner (1965), and Sethi and Cheng (1997) show that the (s, S) policy is optimal when a fixed-ordering cost is incurred with each order, wherey, an order is placed to rise inventory level up-to S when the inventory level at the eginning 4

5 is less than s and stay put otherwise. Chen and Lamrecht (1996), Gallego and Scheller-Wolf (2000), and Chen (2004) extend the aove settings to incorporate the capacity constraints for each ordering quantity. However, these papers are confined themselves to unilateral inventory adjustment, namely, inventory could only e procured from suppliers and unused inventory is carried over to the next period. We consider ilateral inventory adjustments that the uyer could raise inventory from multiple suppliers and/or reduce inventory level y salvaging partial of inventory to the market. Our paper is also closely related to the growing literature dealing with inventory control with multiple suppliers. Most of this literature considers inventory systems where all suppliers have negligile fixed costs ut are differentiated instead in their variale costs and their delivery delay. The tradeoff firms face is whether to source from the slow ut cheap supplier or from the fast ut expensive supplier; see for example Moinzadeh and Schmidt (1991), Song and Zipkin (2009). Sethi et al. (2003) extends the aove results to the scenarios when ordering from oth fast supplier and slow one incurs a fixed cost. Zhang et al. (2012) study the dual sourcing with heterogeneous suppliers, whereas the uyer tradeoff etween a supplier with higher variale cost against a supplier with a lower variale cost ut with a fixed cost and capacity limit for each order. However, they assume that the suppliers deliveries have identical and negligile delays. We consider an inventory system where the inventory could e replenished from fast and/or slow suppliers that each order incurs a fixed cost and has a maximum order limit. In additional, the excess inventory could e salvaged incurring a fixed cost per salvaging order or e carried over to the next period. Tale 1: Summary of Commonly Used Convexity Concept Definition Related papers convex f(x + a) f(x) + a [f(y) f(y )], Song and Zipkin (1993, 1996, 2009) for any y x, a 0, > 0 Moinzadeh and Schmidt (1991) K-convex f(x + a) + K f(x) + a [f(x) f(x )], Scarf (1960), Sethi and Cheng (1997) for any a 0, > 0 Sethi et al. (2003) CK-convex f(x + a) + K f(x) + a [f(x) f(x )], Gallego and Scheller-Wolf (2000) for any a [0, C], > 0 strong K-convex f(x + a) + K f(x) + a [f(y) f(y )], Gallego and Scheller-Wolf (2000) for any y x, a 0, > 0 strong CK-convex f(x + a) + K f(x) + a [f(y) f(y )], Gallego and Scheller-Wolf (2000) for any y x, a [0, C], > 0 Chen and Lamrecht (1996), Chen (2004) (K 1, K 2)-convex f(x + a) + K 1 f(x) + a [f(x) f(x ) K2], Semple (2007) for any a 0, > 0 Ye and Duenyas (2007) strong (K 1, K 2)-convex f(x + a) + K 1 f(x) + a [f(y) f(y ) K2], This paper for any y x, a 0, > 0 strong (CK 1, K 2)-convex f(x + a) + K 1 f(x) + a [f(y) f(y ) K2], This paper for any y x, a [0, C], > 0 strong (C 1K 1, C 2K 2)-convex f(x + a) + K 1 f(x) + a [f(y) f(y ) K2], This paper for any y x, a [0, C 1], (0, C 2] Another field that is also closely related to our paper is the capacity expansion. Most papers in 5

6 the literature of capacity expansion assume that capacity is durale, see Van Mieghem (2003, 2007). Ye and Duenyas (2007) consider a finite-horizon period-review, single-product model with two-sided fixed-capacity adjustment cost. They introduce the (K 1, K 2 )-concavity to show the optimal strategy for capacity expansion. In particular, the capacity strategy at each period can e characterized in a similar way to the ordering policy for the fast node in a inventory system without capacity limits. A same result is otained y Semple (2007) with the concept of the weak (K 1, K 2 )-concavity. However, these papers focus exclusively on single supplier/production which does not have the capacity limit for quantity purchased or salvaged in each order. Tale 1 summarizes concepts of various convexities used in existing literatures and the relationship etween our paper with the aove literatures. 3 Model Formulation This paper studies a capacitated inventory system with two supply nodes, an offshore supplier and an onshore supplier. The procurement from oth suppliers incurs oth a variale cost per unit and a fixed cost per order. Order from the fast supplier has a capacity limit due to suppliers production capacity or the udget constraints. Leftover inventory at the end of each period can either e carried over to the next period incurring inventory holding cost or e salvaged immediately with a fixed cost. An order placed at the fast supplier at the eginning of a period is delivered at the end of the period, whereas an order from the slow supplier is delivered at the end of the next period. The framework of our model and the flow of order delivery are illustrated in Figure 2. For the convenience of analysis, the periods are numered in a ackward way. Figure 2: Model Onshore Supplier f q n f qn 1 f q 1 N n n-1 n Offshore Supplier s qn q s n 1 s q1 6

7 For each period n = 1, 2,..., N, let K s n = a fixed cost associates with each order from the slow supplier, with a unit cost c s n K f n = a fixed cost with each order from the fast supplier, with a unit cost c f n K v n = a fixed cost with each order salvaged, with a unit revenue c v n q s n = order quantity that is ordered from the slow supplier q f n = order quantity that is ordered from the fast supplier (if it is positive), or is salvaged in the local market (if it is negative) α = the discount rate, α [0, 1] L n (y) = expected one-period cost with y units of on-hand inventory at the eginning f n (x) = the cost-to-go function for a n period prolem with an initial on-hand inventory x There is capacity limits on each orders from fast supplier, namely, an expediting quantity from the fast supplier can not exceed its capacity C, i.e, q f n C. Without loss of generality, we assume c f n c v n and c s n αc v n 1, for any n. The first one is intuitive and it merely represents that the unit procurement cost is larger than the unit salvage value. The second one requires that the unit purchasing cost should e no less than the unit salvage value discounted one period later. These two conditions exclude the aritrage opportunity, namely, a negative infinite cost or positive infinity profit can e achieved y purchasing infinite orders from the slow supplier when c s n < αc v n 1 and salvaging all these amounts at the end of next period. In our analysis, we assume that the one-period inventory cost L n ( ) is a convex function, for e.g., most inventory literatures assume that the inventory cost includes the inventory holding cost and the acklog cost as L n (y) = h n E[y D n ] + + p n E[y D n ]. As a start point, we assume that f 0 (x) = 0, and we will relax this assumption later to include salvaging value or the penalty cost at the end of horizon when its inventory is positive or negative, respectively. Using the approach of the standard dynamic programming, one can easily show that the cost-to-go functions satisfy the following dynamic equations: Given the inventory position (on-hand inventory + pipeline inventory from the slow node ordered one period earlier acklog) at the eginning of period n is x, the cost-to-go functions satisfy f n(x) = f v n(x) = min {Knδ(q s n) s + c s nq s qn 0, s qn [0,C] f n + Knδ(q f n) f + c f nqn f + L n (x + qn) f + αef n 1 (x + qn f + qn s D n )}, (1) min {Knδ(q s n) s + c s nq s qn 0, s qn 0 f n + Knδ( q v n) f + c v nqn f + L n (x + qn) f + αef n 1 (x + qn f + qn s D n )}, (2) 7

8 and f n (x) = min{f n(x), f v n(x)}. (3) If we change the decision variales y y = x + q f n and z = x + q f n + q s n, we get an equivalent formulation as follows: f n(x) = min z y, y [x,x+c] {Ks nδ(z y) + c s n(z y) + K f nδ(y x) + c f n(y x) + L n (y) + αef n 1 (z D n )} = min y [x,x+c] {Kf nδ(y x) + c f n(y x) + g n (y)}, (4) f v n(x) = min z y, y x {Ks nδ(z y) + c s n(z y) + K v nδ(x y) + c v n(y x) + L n (y) + αef n 1 (z D n )} = min y x {Kv nδ(x y) + c v n(y x) + g n (y)}, (5) f n (x) = min{f n(x), f v n(x)}, where (6) g n (y) = L n (y) + f s n(y) and f s n(y) = min z y {Ks nδ(z y) + c s n(z y) + αef n 1 (z D n )}. (7) In the remaining sections of our paper, we will study the properties of the cost-to-go functions given y Equations (4)-(7) and after that, we characterize the inventory policy for such systems. To that end, we first study the system without capacity limit associated with each order and then we characterize the optimal policies for systems with capacity limit associated with each order from the fast node. In a later section, Section 6, we show how our analysis could e extended to cases when orders from the slow node has a maximum order size and when each salvage also has a maximum size. 4 Systems Without Capacity Limits This section investigates the case when there is no capacity limit for orders from the fast supplier. Alternatively, one can think of the case when C =. Before we give the analysis for optimal inventory management policy, we first introduce the concept of a strong (K 1, K 2 )-convex function. Definition 1 A real value function f(x) is called strong (K 1, K 2 )-convex, if it satisfies f(x + a) + K 1 f(x) + a [f(y) f(y ) K 2], for any y x, a 0, > 0. (8) It is easy to verify that the strong (K 1, K 2 )-convexity include the convexity and the strong K-convexity as special cases with K 1 = K 2 = 0 and K 2 = 0, respectively, see Tale 1. An equivalent definition for the strong (K 1, K 2 )-convexity is f(x + a) + K 1 f(x) + a sup [f(y) f(y ) K 2 ], for any a 0, > 0. y x 8

9 To characterize the optimal inventory policy, we first study the properties of a strong (K 1, K 2 )-convex function. These properties will then e used to show that cost-to-go functions in the dynamic programming are strong (K 1, K 2 ) convex. Using the properties of strong (K 1, K 2 )-convexity, we are then ale to characterize the optimal inventory policy for the un-capacitated inventory systems. The following definition from Porteus (1971) is used later to characterize the properties of cost-to-go functions. Definition 2 A function F (x) is non-k-decreasing if F (x) F (y) + K for all x y. A function F (x) is non-k-increasing if F (x) + K F (y) for all x y. 4.1 Properties of (K 1, K 2 )-Convexity In this section, we will study the following optimization prolem f 1 (x) = min y x {K 1 δ(y x) + c(y x) + g(y)}, f 2 (x) = min y x {K 2 δ(x y) + v(y x) + g(y)}, f(x) = min{f 1 (x), f 2 (x)}. (9) In particular, we will study the properties of the aove value function and characterize the solution to the aove minimization prolem when g is a strong (K 1, K 2 )-convex function. The properties of strong (K 1, K 2 )-convexity are summarized in the following lemma. Lemma 1 (a) If f(x) is strong (K 1, K 2 )-convex, then f(x a) is also strong (K 1, K 2 )-convex for any a. Moreover, for any random variale X, Ef(x X) is also strong (K 1, K 2 )-convex provided Ef(x X) <. () If f(x) is strong (K 1, K 2 )-convex, then f(x) is strong (K 1, K 2 )-convex for any K 1 K 1, K 2 K 2. (c) If f(x) and g(x) are strong (K 1, K 2 )-convex, strong (G 1, G 2 )-convex, respectively, then for any α, β 0, αf(x) + βg(x) is strong (αk 1 + βg 1, αk 2 + βg 2 )-convex. The eauty of normal convexity or K-convexity is its preservation of convexity or K-convexity under dynamic minimization. In the following proposition, we show that strong (K 1, K 2 )-convexity is also preserved under minimization. Proposition 1 If a function g(x) is strong (K 1, K 2 )-convex, then are also strong (K 1, K 2 )-convex. f 1 (x) = min y x {K 1δ(y x) + g(y)}, f 2 (x) = min y x {K 2δ(x y) + g(y)}, f(x) = min{f 1 (x), f 2 (x)}, 9

10 Furthermore, we can get a stronger result that shows the preservation of optimization of (K 1, K 2 )- convexity with linear operations. Namely, Corollary 1 If a function g(x) is strong (K 1, K 2 )-convex, then are also strong (K 1, K 2 )-convex, for any c, v. f 1 (x) = min y x {K 1δ(y x) + c(y x) + g(y)}, f 2 (x) = min y x {K 2δ(x y) + v(y x) + g(y)}, Different to Proposition 1, the function f(x) = min{f 1 (x), f 2 (x)} may not e a strong (K 1, K 2 )-convex function depending on the value of c and v. In the prolem of our interest, c v is satisfied, and we can show f(x) = min{f 1 (x), f 2 (x)} is indeed a strong (K 1, K 2 )-convex function. The proof needs some properties of the solutions to the aove two minimization prolems in Corollary 1, so it is presented later in Proposition 3. To solve the aove minimization prolems, we first show that the value functions satisfy a certain monotonicity in the following lemma. Lemma 2 f 1 (x) + cx is non-k 1 -decreasing, f 2 (x) + vx is non-k 2 -increasing. Now, we are ready to characterize the solutions to the minimization f(x) = min{f 1 (x), f 2 (x)} where f 1, f 2 are given in Corollary 1. Lemma 3 If c v, then all uy regions are to the left of all sell region. Saying it in other words, (i) If it is optimal to sell at x, then it is never optimal to purchase at y for any y x; (ii) If it is optimal to purchase at x, then it is never optimal to sell at y for any y x; Let { } B = sup argmin y {cy + g(y)}, = inf{x : Ã 1 (x) 0}, = sup{x : Ã1 (x) < 0}, (10) { } S = inf argmin y {vy + g(y)}, s = sup{x : Ã 2 (x) 0}, s = inf{x : Ã 2 (x) < 0}, (11) where g 1 (x) = K 1 + inf {c(y x) + g(y)}, Ã y x g 2 (x) = K 2 + inf {v(y x) + g(y)}, Ã y x 1 (x) = g 1 (x) g(x), 2 (x) = g 2 (x) g(x), Oviously, y the definitions, we have f 1 (x) = min{g(x), g 1 (x)}, Ã 1 (x) < 0 for any x <, Ã 1 (x) 0 for any x >, f 2 (x) = min{g(x), g 2 (x)}, Ã 2 (x) < 0 for any x > s, Ã 2 (x) 0 for any x < s. 10

11 The following lemma characterizes the relationship etween these points, which are critical to characterize the structure of optimal policy. Lemma 4 The critical points have the following orders (i) s s. (ii) B S s. (iii) If K 1 K 2, B; otherwise if K 1 K 2, s S. Now, we are ready to characterize the solution to (9) and its value function in the following proposition. Proposition 2 The structure of the optimal solution to (9) and its corresponding optimal value function has the following structure (i) if K 1 K 2 B, x < {B, x}, x [, ) y (x) = x, x [, s) {S(x), x}, x [s, s) S, x s K 1 + g(b) + c(b x), x < min{k 1 + g(b) + c(b x), g(x)}, x [, ), f(x) = g(x), x [, s) min{ g 2 (x), g(x)}, x [s, s) K 2 + g(s) + v(s x), x s (12) (ii) if K 2 K 1 B, x < {B(x), x}, x [, ) y (x) = x, x [, s) {S, x}, x [s, s) S, x s K 1 + g(b) + c(b x), x < min{ g 1 (x), g(x)}, x [, ), f(x) = g(x), x [, s) min{k 2 + g(s) + v(s x), g(x)}, x [s, s) K 2 + g(s) + v(s x), x s (13) where B(x) = argmin y x {g(y) + cy} and S(x) = argmin y x {g(y) + vy}. The next proposition shows that the function f(x) also preserves the (K 1, K 2 )-convexity. Proposition 3 (Preservation) If g(x) is strong (K 1, K 2 )-convex, f(x) is also strong (K 1, K 2 )-convex. We have thus shown that the strong (K 1, K 2 )-convexity is indeed preserved in a minimization that has a form of (9). With the help of these results, we are now ready to present the main results for the un-capacitated inventory systems. 11

12 4.2 Optimal Policies In this section, we employ properties of the strong (K 1, K 2 )-convexity otained in the previous section to study inventory systems that do not have capacity constraints on each order. Regarding to the fixed cost, we assume (A1) αk f n 1 Ks n K f n and K v n αk v n 1. The former condition says that the fixed cost associated with the slow order is smaller than the fixed cost with the fast order ut is larger than the one period discounted fixed cost with the fast order. Similarly, the second condition means that the fixed cost associated with the salvaging is larger than the one period discounted fixed salvaging cost. These assumptions guarantee the strong (K 1, K 2 )-convexity of cost-to-go functions are preserved in the dynamic programming, and they are not uncommon in the literature, for e.g., Sethi et al. (2003). Theorem 1 Assume that (A1) holds. f n (x) is a strong (Kn, f Kn)-convex v function for any n 0. Proof. Oviously, the result holds for n = 0. Next, we estalish the proof y induction. Assuming that f n 1 (x) is (K f n 1, Kv n 1 ) convex, we next prove that f n(x) is (K f n, K v n) convex. By Lemma 1, we know that αef n 1 (y D) is a (αk f n 1, αkv n 1 )-convex function. Since αkf n 1 Ks n, according to Lemma 1, αef n 1 (y D) is also a (Kn, s αkn 1 v ) convex function. Thus, y Corollary 1, f s n(y) = max z y {Ks nδ(z y) + c s n(z y) + αef n 1 (z D n )}, is (K s n, αk v n 1 ) convex. Since L n(y) is convex function, g n (y) = L n (y)+f s n(y) is also (K s n, αk v n 1 ) convex y Lemma 1. By Lemma 1, g n (y) is also (K f n, K v n) convex since αk v n 1 Kv n and K s n K f n. Therefore, y Corollary 1 and Proposition 3, f n(x), f v n(x), f n (x) given y equations (4), (5), (6), respectively, are also (K f n, K v n) convex. The optimal inventory replenishment policy could e determined in a similar way to Proposition 2. In particular, for n = 1, 2,..., N, let { } Bn s = sup argmin z {c s nz + αef n 1 (z D n )}, s n = inf{x : Ã s n(x) 0}, s n = sup{x : Ã s n(x) < 0}, (14) { } Bn f = sup argmin y {c f ny + g n (y)}, f n = inf{x : Ã f n(x) 0}, f n = sup{x : Ã f n(x) < 0}, (15) Sn v = inf { argmin y {c v ny + g n (y)} }, s v n = sup{x : Ã v n(x) 0}, s v n = inf{x : Ã v n(x) < 0}.(16) 12

13 Here g n (y) is given y (7) and g s n(y) = K s n + inf z y {cs n(z y) + αef n 1 (z D n )}, à s n(y) = g s n(y) αef n 1 (y D n ), g f n(x) = K f n + inf y x {cf n(y x) + g n (y)}, g v n(x) = K v n + inf y x {cv n(y x) + g n (y)}, Oviously, y aove definitions, one has à f n(x) = g f n(x) g n (x), à v n(x) = g v n(x) g n (x), f s n(y) = min{αef n 1 (y D), g s n(y)}, à s n(y) < 0, y < s n, à s n(y) 0, y > s n, f n(x) = min{g n (x), g f n(x)}, à f n(x) < 0, x < f n, à f n(x) 0, x > f n, f v n(x) = min{g n (x), ḡ n2 (x)}, Ā n2 (x) < 0, x > s v n, Ā n2 (x) 0, x < s v n. Similar to Lemma 4, it can e proved that: For any n = 1, 2,..., N, we have (i) f n f n s v n s v n and s n min{ s n, B s n}; (ii) f n B f n S v n s v n; (iii) f n B f n if K f n αk v n 1, sv n S v n if K f n αk v n 1, and s n B s n if K s n αk v n 1. Theorem 2 The structure of the optimal inventory replenishment/salvaging policy has the following structure: For any n = 1, 2,..., N, 1. (The Onshore Supplier and Salvaging Policy) Given the starting inventory level x, the optimal ordering decision with the fast node and the optimal salvaging decision are characterized as follows: (i) if Kn f Kn, v the optimal policy is determined y Bn, f yn(x) = x < f n {Bn, f x}, x [ f n, f n) x, x [ f n, s v n) {Sn(x), v x}, x [s v n, s v n) Sn, v x s v n (ii) if Kn v Kn, f the optimal policy could e characterized y Bn, f x < f n {Bn(x), f x}, x [ f n, f n) yn(x) = x, x [ f n, s v n) {S n, v x}, x [s v n, s v n) Sn, v x s v n, (17), (18) 13

14 2. (The Offshore Supplier) Given the inventory level after ordering is placed from fast node or after salvaging, the optimal ordering decision with the slow node is as follows: Bn, s y < s zn(y) n = and zn(y) = {B s n, y}, y [ s n, s n) y, y s n Bn, s y < s n {Bn(y), s y}, y [ s n, s n) y, y s n if K s n αk v n 1, (19) if K s n < αk v n 1. where B s n(y) = argmin z y {c s n(z y) + αef n 1 (z D n )}, B f n(x) = argmin y x {c f ny + g n (y)} and S v n(x) = argmin y x {c v ny + g n (y)}. The optimal policy with the slow node has a similar structure with the well-known (s, S) policy. An order is placed to raise the inventory up to a constant level B s n when the starting inventory is elow level s n, and no order is placed when its starting inventory is aove a certain level s n. While when the starting inventory is in the middle etween s n and s n, the optimal ordering policy is only partially characterized, namely, either staying put or ordering with quantity equals to an local (or gloal) point. The ordering policy for the fast node and the optimal salvaging policy could e characterized similarly y parameters ( f n, f n, B f n) and (s v n, s v n, S v n), respectively. With the optimal policies characterized in the aove proposition, the cost-to-go functions could e determined in a similar way to Proposition 2. 5 Systems with Capacity Limits To analyze the optimal inventory policy for the aove capacitated dual-sourcing inventory systems, we first give a definition, called strong (CK 1, K 2 )-convexity, that incorporates capacity limit constraint into the strong (K 1, K 2 )-convexity defined in the Definition 1. Definition 3 A real value function f(x) is called strong (CK 1, K 2 )-convex, if it satisfy f(x + a) + K 1 f(x) + a [f(y) f(y ) K 2], y x, a [0, C], > 0. (20) Our next Lemma shows that most properties of strong (K 1, K 2 ) convexity still hold for the strong (CK 1, K 2 )-convexity. Namely, strong (CK 1, K 2 ) convexity is preserved under linear operations, expectation, and optimization, see Lemma 1, Corollary 1, Propositions 1 and 3. Lemma 5 (a) If f(x) is strong (CK 1, K 2 )-convex, then f(x a) is also strong (CK 1, K 2 ) for any a. Moreover, for any random variale X, Ef(x X) is also strong (CK 1, K 2 )-convex provided Ef(x X) <. 14

15 () If f(x) is strong (CK 1, K 2 )-convex, then f(x) is strong (C K 1, K 2 )-convex for any K 1 K 1, K 2 K 2, and C C. (c) If f(x) and g(x) are strong (CK 1, K 2 )-convex, strong (CG 1, G 2 )-convex, respectively, then for any α, β 0, αf(x) + βg(x) is strong ( ) C(αK 1 + βg 1 ), αk 2 + βg 2 -convex. Oviously, a strong (K 1, K 2 )-convex is also strong (CK 1, K 2 )-convex for any C 0, and a convex function is also strong (CK 1, K 2 )-convex for any K 1 0, K 2 0, C 0. We first study a variant of minimization prolem (9): f 1 (x) = min x y x+c {K 1 δ(y x) + c(y x) + g(y)}, f 2 (x) = min y x {K 2 δ(x y) + v(y x) + g(y)}, f(x) = min{f 1 (x), f 2 (x)}. (21) Similar to Lemma 3, the next lemma shows that the sell region is always to the right of purchase region. Lemma 6 If c v, then all uy regions are to the left of all sell regions. An important issue of the convexity optimization is preservation of the convexity after one-step optimization? The next lemma shows that the strong (CK 1, K 2 ) convexity is indeed preserved after one-step optimization. Lemma 7 (Preservation) If a function g(x) is strong (CK 1, K 2 )-convex, then f 1 (x) = min {K 1δ(y x) + c(y x) + g(y)}, y [x,x+f ] f 2 (x) = min {K 2δ(x y) + v(y x) + g(y)}, y x are also strong (CK 1, K 2 )-convex for any F C. (CK 1, K 2 )-convex if c v. Furthermore, f(x) = min{f 1 (x), f 2 (x)} is strong Similar to Theorem 1, our next theorem shows that the cost-to-go functions in a capacitated inventory system with two supply nodes are strong (CK 1, K 2 )-convex. Theorem 3 Assume that (A1) holds. f n (x) is a strong (CK f n, K v n)-convex function for any n 0. Proof. The result holds trivially for n = 0. Next, we prove the result y induction. Assuming that f n 1 (x) is strong (CK f n 1, Kv n 1 ) convex, we show that f n(x) is also strong (CK f n, K v n) convex. Lemma 5, we know that αef n 1 (y D) is a strong (CαK f n 1, αkv n 1 )-convex function. Since αkf n 1 Kn, s according to Lemma 5, αef n 1 (y D) is also a strong (CKn, s αkn 1 v )-convex function. Thus, y Lemma 7 (with F = ), f s n(y) = max z y {Ks nδ(z y) + c s n(z y) + αef n 1 (z D n )}, 15 By

16 is strong (CK s n, αk v n 1 ) convex. Since L n(y) is a convex function, g n (y) = L n (y) + f s n(y) is also strong (CK s n, αk v n 1 ) convex y Lemma 5. Applying Lemma 5 once again, g n(y) is strong (CK f n, K v n) convex since αk v n 1 Kv n and K s n K f n. Thus, y Lemma 7, f n(x), f v n(x), f n (x) are also strong (CK f n, K v n) convex. In the following, we turn to characterize the optimal policies. Let { } ˆB n s = sup argmin z {c s nz + αef n 1 (z D n )}, ˆs n = inf{x : Â s n(x) 0}, ˆ s n = sup{x : Â s n(x) < 0}, (22) { } ˆB n f = sup argmin y {c f ny + g n (y)}, ˆf n = inf{x : Â f n(x) 0}, ˆ f n = sup{x : Â f n(x) < 0}, (23) Ŝn v = inf { argmin y {c v ny + g n (y)} }, ŝ v n = sup{x : Â v n(x) 0}, ŝ v n = inf{x : Â v n(x) < 0}.(24) Where ĝ s n(y) = K s n + inf z y {cs n(z y) + αef n 1 (z D n )}, Â s n(y) = ĝ s n(y) αef n 1 (y D n ), Oviously, one has ĝn(x) f = Kn f + inf y [x,x+c] {cf n(y x) + g n (y)}, ĝ v n(x) = K v n + inf y x {cv n(y x) + g n (y)}, Â f n(x) = ĝ f n(x) g n (x), Â v n(x) = ĝ v n(x) g n (x). f s n(y) = min{αef n 1 (y D n ), ĝ s n(y)}, Â s n(y) < 0, y < ˆ s n, Â s n(y) 0, y > ˆ s n, f n(x) = min{g n (x), ĝ f n(x)}, Â f n(x) < 0, x < ˆ f n, Â f n(x) 0, x > ˆ f n, f v n(x) = min{g n (x), ĝ v n(x)}, Â v n(x) < 0, x > ŝ v n, Â v n(x) 0, x < ŝ v n. Similar to Lemma 4, it can e shown that: For any n = 1, 2,..., N, we have (i) ˆ f n ˆ f n ŝ v n ŝ v n, and ˆs n min{ˆ s n, ˆB n}; s (ii) ˆ f n ˆB n f Ŝv n ŝ v n, (iii) ˆ f n ˆB n f if Kn f Kn, v Ŝv n ŝ v n if Kn v Kn, f and ˆ s n ˆB n s if Kn s αkn 1 v. The optimal policies in the capacitated inventory system have a same structure to those for uncapacitated inventory system characterized in Theorem 2. The ordering policy with the slow supplier and the salvaging decisions are exactly same to those in Theorem 2. The only difference is for the ordering quantity with the fast supplier, i.e., the order quantity from the fast node has to e less than its maximum capacity limit C, rather than raise its inventory up-to its gloal maximum in the uncapacitated inventory system. Our next theorem provides a condition under which it is optimal to order with a maximum quantity from the fast node when the starting inventory is elow ˆ f n C and raise up the inventory aove ˆ f n with an order from the fast supplier. 16

17 Theorem 4 Assume that K v n = 0. It is optimal to order with maximum size C from the fast supplier when the eginning inventory is elow ˆ f n C, and raise the inventory aove ˆ f n when the eginning inventory is etween ˆ f n C and ˆ f n. Proof. Since it is optimal to order something from the fast supplier when the eginning inventory is lower than ˆ f n, it suffices to show that the function H n (y) c f ny + g n (y) is decreasing when y < ˆ f n. For any y 1 < y 2 < ˆ f n, y (23), we have Âf n(y 2 ) < 0, i.e., g n (y 2 ) > ĝ f n(y 2 ) = K f n + c f nz 0 + g n (y 2 + z 0 ) or H n (y 2 ) > K f n + H n (y 2 + z 0 ) for some z 0 (0, C]. By Theorem 3 and Lemma 5, H n (y) is strong (CK f n, K v n)-convex, therefore, with a = z 0, = y 2 y 1 in (20), one has H n (y 2 ) > K f n + H n (y 2 + z 0 ) H n (y 2 ) + z 0 y 2 y 1 [H n (y 2 ) H n (y 1 ) K v n]. Or equivalently, H n (y 2 ) < H n (y 1 ) since K v n = 0. We have thus shown that the function H n (y) c f ny + g n (y) is strictly decreasing when y < ˆ f n. Remark 1 For any K v n > 0, it can e shown that the function c f ny + g n (y) is decreasing on y 0, therefore, it is optimal to order as much as possile whenever ordering from the fast supplier when the starting inventory is less than C. 6 Discussions We conclude our paper in this section with several discussions. In particular, we show that our analysis could e generalized to the following scenarios. First, we show that our model could extend to include a salvage value for the left-over inventory or a penalty cost for unsatisfied demands at the end of horizon. We also show that our analysis is easily extended to incorporate a capacity limit for order with the slow supplier and/or maximum size to e salvaged in each order. Salvage Value or Penalty Cost at the End of Horizon We first relax the assumption of no-value at the end of horizon. That is, we consider the case when the leftover inventory has a salvage value, or when the unsatisfied demand incurs penalty cost, or oth, at the end of period. For example, the leftover inventory can e resold to a second market or the undelivered customer demand requires to e cleared at the end of horizon. All our results still hold provided the end horizontal value function is a (K f 0, Kv 0 )-convex function. For example, the leftover inventory can e salvaged at unit price c v 0 with a fixed cost Kv 0, and the uncleared demand requires to e cleared from expediting supplier at unit price c f 0 with fixed cost Kf 0. In this case, the end-period value function is 17

18 given y } min {K v0 f 0 (x) = δ(x) cv0 x, 0, if x 0, K f 0 δ( x) + cf 0 ( x), otherwise. It is not difficult to show that f 0 (x) given aove is a strong (K f 0, Kv 0 )-convex function, therefore, is also a strong (F K f 0, Kv 0 )-convex function for any F 0. Thus, all our analysis in the previous sections, such as Theorems 1 and 3, hold without any modification. Capacity limits for Orders with Offshore Supplier and Salvaging Order In this section, we show how our analysis in the aove sections could e extended to cases when each order with the slow suppliers and/or quantities to e salvaged each time also have capacity limit, referred to as C s and/or C v, respectively. We refer the capacity limit for orders with the fast supplier to as C f. The dynamic formulations (4) (7) require slight modifications to incorporate these capacity limits as follows: fn(x) = min {Knδ(z s y) + c s n(z y) + Knδ(y f x) + c f n(y x) + L n (y) + αef n 1 (z D n )} z [y,y+c s ], y [x,x+c f ] = min y [x,x+c f ] {Knδ(y f x) + c f n(y x) + g n (y)}, (25) fn(x) v = min z [y,y+c s ], y [x C v,x] {Ks nδ(z y) + c s n(z y) + Knδ(x v y) + c v n(y x) + L n (y) + αef n 1 (z D n )} = max y [x C v,x] {Kv nδ(x y) + c v n(y x) + g n (y)}, (26) f n (x) = max{f n(x), f v n(x)}, g n (y) = L n (y) + max z [y,y+c s ] {Ks nδ(z y) + c s n(z y) + αef n 1 (z D n )}. (28) We also modify the strong (CK 1, K 2 )-convexity in Definition 3 as follows, (27) Definition 4 A real value function f(x) is called strong (C 1 K 1, C 2 K 2 )-convex, if it satisfy f(x + a) + K 1 f(x) + a [f(y) f(y ) K 2], y x, a [0, C 1 ], (0, C 2 ]. (29) Most properties for a strong (CK 1, K 2 )-convexity, such properties in Lemmas 5 7, could also e estalished for the strong (C 1 K 1, C 2 K 2 )-convexity. For example, Lemma 5() could e stated as follows: If f(x) is (C 1 K 1, C 2 K 2 )-convex, then f(x) is also (D 1 K 1, D 2K 2 )-convex for any K i K i and D i C i, i = 1, 2. With help of these properties, we show that the cost-to-go functions are strong (C 1 K 1, C 2 K 2 )-convex functions in the following Theorem. Theorem 5 Assume that (A1) holds. f n (x) is a strong (CK f n, C v K v n)-convex function for any n 0, where C = min{c f, C s }. 18

19 As a concluding remark, our model is also applicale to cases when capacity limits are non-stationary, for e.g., the capacity limit for the fast supplier, the slow supplier, and the salvaging order is time dependent, denoted as C f n, C s n, C v n, respectively. It is not difficult to verify that all our results, such as Theorem 5, still hold provided that min{cn, f Cn} s min{c f n 1, Cs n 1 } and Cv n Cn 1 v. Recall that our period is numered in a ackward way, the aove two conditions represent settings where a firm is given a larger maximum size limit per order y its suppliers and also a larger capaility in salvaging its left-over inventory in the market as the firm stays in usiness for a longer time. In practice, these conditions might e a result of the improved relationship etween the firm and its suppliers, enhanced liquidity conditions, and so on. References Chen, S., M. Lamrecht X-Y and and modified (s, S) policy. Oper. Res., 44, Chen, S The Infinite horizon periodic review prolem with setup costs and capacity constraints: A partial characterization of optimal policy. Oper. Res., 52, Gallego, G., A. Scheller-Wolf Capacitated inventory prolems with fixed order costs: Some optimal policy structure. Eur. J. of Oper. Res., 126, Moinzadeh, K., C. Schmidt An (S 1, S) inventory system with emergency orders. Oper. Res., 39, Porteus, E On the optimality of generalized (s, S) policies. Management Sci., 17, Scarf, H Optimality of (s, S) policies in dynamic inventory prolem. in Mathematical Methods in Social Sciences. J. Arrow, S. Karlin and P. Suppes (eds.). Standford University Press, Stanford, CA. Semple, J Note: Generalized notions of concavity with an application to capacity management. Oper. Res., 55, Sethi, S., F. Cheng Optimality of (s, S) policies in inventory models with Markovian demand. Oper. Res., 45, Sethi, S., H. Yan, H. Zhang Inventory models with fixed costs, forecast updates, and two delivery modes. Oper. Res., 51, Song, J., P. Zipkin Inventory control in a fluctuating demand environment. Oper. Res., 41, Song, J., P. Zipkin Inventory control with information aout supply conditions. Management Sci., 42,

20 Song, J., P. Zipkin Inventories with multiple supply sources and networks of queues with overflow ypasses. Management Sci., 55, Van Mieghem, J Risk mitigation in newsvendor networks: resource diversification, flexiility, sharing, and hedging. Management Sci., 53, Veinott, A., H. Wagner Computing optimal (s, S) inventory policies. Management Sci., 11, Ye, Q., I. Duenyas Optimal capacity investment decision with two-sided fixed capacity ajustment cost. Oper. Res., 55, Zhang, W., Z. Hua, S. Benjaafar Optimal inventory control with dual-sourcing, heterogeneous ordering costs and order size constraints. Production and Oper. Management, 21, Zipkin, P Foundations of Inventory Management, McGraw-Hill, New York. 20

21 Appendix: Proofs Proof of Lemma 1. The proof of part (a) and part () follows immediately from (20) in the Definition 1. Next, we prove part (c). Assuming that f( ) is strong (K 1, K 2 )-convex and g( ) is strong (G 1, G 2 )- convex, y (20), for any y x, a 0, > 0, one has f(x + a) + K 1 f(x) + a [f(y) f(y ) K 2], and g(x + a) + G 1 g(x) + a [g(y) g(y ) G 2]. Thus, for any α, β 0, one has ( α[f(x + a) + K 1 ] + β[g(x + a) + G 1 ] α f(x) + a ) ( [f(y) f(y ) K 2] + β g(x) + a ) [g(y) g(y ) G 2]. Let F (x) = αf(x) + βg(x), the aove inequality is equivalent to F (x + a) + αk 1 + βg 1 F (x) + a [F (y) F (y ) (αk 2 + βg 2 )]. This completes the proof of the part (c). Proof of Proposition 1. (I) First, we show f 1 (x) is strong (K 1, K 2 )-convex. For y x, a 0, > 0, let 1 = K 1 + f 1 (x + a) f 1 (x) a [f 1(y) f 1 (y ) K 2 ]. It suffices to show 1 0. To this end, we consider the following four cases. (a) f 1 (x + a) = g(x + a), f 1 (y ) = g(y ). In this case, we have 1 = K 1 + g(x + a) f 1 (x) a [f 1(y) g(y ) K 2 ] K 1 + g(x + a) g(x) a [g(y) g(y ) K 2] 0, where, the first inequality is y the definition of f 1 (x), and the second inequality follows from the definition of strong (K 1, K 2 )-convexity of g( ). () f 1 (x + a) = g(x + a), f 1 (y ) = g(y + u) + K 1. In this case, we get 1 = K 1 + g(x + a) f 1 (x) a [f 1(y) g(y + u) K 1 K 2 ]. Considering the following two sucases. 21

22 (.1) f 1 (y) g(y + u) + K 1 + K 2. Oviously, 1 K 1 + g(x + a) f 1 (x) 0. (.2) f 1 (y) > g(y + u) + K 1 + K 2. First, we show u. Assume u >, then g(y + u) + K 1 + K 2 < f 1 (y) g(y + u) + K 1, i.e., K 2 < 0. This contradicts the fact that K 2 0, therefore, u. Thus, 1 K 1 + g(x + a) f 1 (x) a u [f 1(y) g(y + u) K 1 K 2 ] K 1 + g(x + a) g(x) a u [g(y) g(y + u) K 2] 0 (c) f 1 (x + a) = K 1 + g(x + a + u), f 1 (y ) = g(y ), we have 1 = K 1 + K 1 + g(x + a + u) f 1 (x) a [f 1(y) g(y ) K 2 ] K 1 + g(x + a + u) g(x + u) a [g(y) g(y ) K 2] 0. (d) f 1 (x + a) = K 1 + g(x + a + u), f 1 (y ) = K 1 + g(y + w), we have 1 = K 1 + K 1 + g(x + a + u) f 1 (x) a [f 1(y) g(y + w) K 1 K 2 ]. We consider the following two sucases. (d.1) f 1 (y) f(y + w) + K 1 + K 2, we get 1 K 1 + K 1 + g(x + a + u) f 1 (x) K 1 0. (d.2) f 1 (y) > g(y + w) + K 1 + K 2, first, we show w. Otherwise, assume w >, we have g(y + w) + K 1 + K 2 < f 1 (y) K 1 + g(y + w), i.e., K 2 < 0, this contradicts the fact that K K 1 + K 1 + g(x + a + u) f 1 (x) a w [f 1(y) g(y + w) K 1 K 2 ] K 1 + g(x + a + u) g(x + u) a w [g(y) g(y + w) K 2] 0. Therefore, we have shown that f 1 (x) is strong (K 1, K 2 )-convex. (II) Next, we prove f 2 (x) is strong (K 1, K 2 )-convex. Similarly, for y x, a 0, > 0, let 2 = K 1 + f 2 (x + a) f 2 (x) a [f 2(y) f 2 (y ) K 2 ]. we consider the following four cases to show

23 (a) f 2 (x + a) = g(x + a), f 2 (y ) = g(y ), same to the aove Case (I)(a), we can show 2 0. () f 2 (x + a) = g(x + a), f 2 (y ) = g(y u) + K 2, we get 2 = K 1 + g(x + a) f 2 (x) a [f 2(y) g(y u) K 2 K 2 ] K 1 + g(x + a) g(x) a [g(y u) g(y u) K 2] 0, where the first inequality is ecause f 2 (x) g(x) and f 2 (y) g(y u) + K 2, and the last inequality follows from the strong (K 1, K 2 )-convexity. (c) f 2 (x + a) = g(x + a u) + K 2, f 2 (y ) = g(y ), we get 2 = K 1 + K 2 + g(x + a u) f 2 (x) a [f 2(y) g(y ) K 2 ]. we consider two sucases, (c.1) K 1 + K 2 + g(x + a u) f 2 (x), we otain 2 a [f 2(y) g(y ) K 2 ] 0. where the last inequality follows from the definition of f 2 (y). (c.2) K 1 + K 2 + g(x + a u) < f 2 (x), in this case, we can show a u, since otherwise, f 2 (x) K 2 + f(x + a u), i.e., K 1 < 0. This contradicts the fact K 1 0. Thus, a u. Thus 2 K 1 + K 2 + g(x + a u) f 2 (x) a u [f 2 (y) g(y ) K 2 ] K 1 + g(x + a u) g(x) a u [g(y) g(y ) K 2 ] 0. where, the first inequality follows from the fact that f 2 (y) g(y ) K 2 0. (d) f 2 (x + a) = g(x + a u) + K 2, f 2 (y ) = g(y w) + K 2, we have 2 = K 1 + K 2 + g(x + a u) f 2 (x) a [f 2(y) g(y w) K 2 K 2 ]. Noted that f 2 (y) g(y w) K 2 K 2 0, we consider two sucases. (d.1) K 1 + K 2 + g(x + a u) f 2 (x), oviously, 2 0. (d.2) K 1 + K 2 + g(x + a u) < f 2 (x), it is easy to show a u, thus 2 K 1 + g(x + a u) g(x) a u [g(y w) g(y w) K 2 ] 0 (III) Finally, we will show f(x) is also strong (K 1, K 2 )-convex. For y x, a 0, > 0, let = K 1 + f(x + a) f(x) a [f(y) f(y ) K 2]. we will consider the following four cases to show 0. 23

24 (a) f(x + a) = f 1 (x + a), f(y ) = f 1 (y ), we have = K 1 + f 1 (x + a) f(x) a [f(y) f 1(y ) K 2 ] K 1 + f 1 (x + a) f 1 (x) a [f 1(y) f 1 (y ) K 2 ] 0. () f(x + a) = f 1 (x + a), f(y ) = f 2 (y ), = K 1 +f 1 (x+a) f(x) a [f(y) f 2(y ) K 2 ]. We consider two cases (.1) f 2 (y ) = g(y ). In this case, since g(y ) f 1 (y ), thus, f(y ) = f 1 (y ) y case (a), 0. (.2) f 2 (y ) = K 2 + g(y u). = K 1 + f 1 (x + a) f(x) a [f(y) g(y u) K 2 K 2 ] K 1 + f 1 (x + a) f(x) a [g(y u) g(y u) K 2], where the inequality follows from f(y) f 2 (y) g(y u) + K 2. When f 1 (x + a) = g(x + a), y strong (K 1, K 2 )-convexity of g( ) and f(x) g(x), 0. Otherwise, if f 1 (x + a) = K 1 + g(x + a + w), we have K 1 + K 1 + g(x + a + w) g(x) a [g(y u) g(y u) K 2] 0. (c) f(x + a) = f 2 (x + a), f(y ) = f 1 (y ). = K 1 +f 2 (x+a) f(x) a [f(y) f 1(y ) K 2 ]. We consider two cases (c.1) f 1 (y ) = g(y ). In this case, since g(y ) f 2 (y ), thus, f(y ) = f 2 (y ) y the following case (d), 0. (c.2) f 1 (y ) = K 1 + g(y + w). = K 1 + f 2 (x + a) f(x) a [f(y) g(y + w) K 1 K 2 ]. Noted that f(y) g(y + w) + K 1 + K 2. When f 2 (x + a) = g(x + a), f(x) f 1 (x) K 1 + g(x + a), thus, 0. Otherwise, if f 2 (x + a) = K 2 + g(x + a u), we have f(x) K 1 + K 2 + g(x + a u), thus, 0. (d) f(x + a) = f 2 (x + a), f(y ) = f 2 (y ), similar to case (a), we get = K 1 + f 2 (x + a) f(x) a [f(y) f 2(y ) K 2 ] K 1 + f 2 (x + a) f 2 (x) a [f 2(y) f 2 (y ) K 2 ] 0. 24

25 Proof of Corollary 1. By Lemma1, cy + g(y) and vy + g(y) is strong (K 1, K 2 )-convex, thus, y Proposition1, f 1 (x) + cx, f 2 (x) + vx are strong (K 1, K 2 )-convex. Again, y Lemma1, f 1 (x), f 2 (x) are strong (K 1, K 2 )-convex. Proof of Lemma 2. We need to prove f 1 (x) + cx K 1 + f 1 (y) + cy for any x y, this is trivial for y = x, therefore, we only need to show x < y. Assume z(y) = argmin z y {K 1 δ(z y) + cz + g(z)}. we have f 1 (x) + cx K 1 + cz(y) + g(z(y)) K 1 + K 1 δ(z(y) y) + cz(y) + g(z(y)) = K 1 + f 1 (y) + cy. Similarly, we show f 2 (x) + vx + K 2 f 2 (y) + vy, for any x y. Proof of Lemma 3. We prove case (i) y contradiction. Suppose there exist x < y such that it is optimal to sell to s at x and optimal to purchase to at y, where s < x < y <. Then we have g(x) > K 2 + g(s) + v(s x), g(y) > K 1 + c( y) + g(), (since it is optimal to sell to s at x), (it is optimal to purchase to at y), y (K 1, K 2 ) convexity of g( ), we have K 1 + g() g(y) y x s [g(x) g(s) K 2] > v( y) c( y) This is a contradiction, since g(y) > K 1 + c( y) + g(). Thus, we have proved the case (i). Case (ii) can e proved in the same way. Proof of Lemma 4. (i). First, we show s y contradiction. Assume > s, y definition of and s in (10) and (11), there exists x, y such that s < x < y <, Ã 2 (x) < 0 and Ã1(y) < 0. Saying it in other ways, it is optimal to sell at x, and to purchase at y. This violates the Lemma 3. Next, we turn to show y contradiction. Assume >, y definition of, for any z (, ), Ã 1 (z) < 0, this contradicts the definition of. Similarly, it is easy to prove s s. In fact, we can prove a stronger results. Let ˆ = inf{x : f1 (x) f 2 (x)}, ŝ = sup{x : f 1 (x) f 2 (x)}. (A-1) 25