A stochastic model for risk management in global ... - Semantic Scholar

European Journal of Operational Research 182 (2007) 164–173 www.elsevier.com/locate/ejor

Production, Manufacturing and Logistics

A stochastic model for risk management in global supply chain networks Mark Goh

a,c,* ,

Joseph Y.S. Lim b, Fanwen Meng

c

a

Department of Decision Sciences, National University of Singapore, 1 Business Link, Singapore 117592, Singapore Department of Finance and Accounting, National University of Singapore, 1 Business Link, Singapore 117592, Singapore The Logistics Institute – Asia Pacific, National University of Singapore, 7 Engineering Drive 1, Singapore 117574, Singapore b

c

Received 11 February 2004; accepted 1 August 2006 Available online 5 January 2007

Abstract With the increasing emphasis on supply chain vulnerabilities, effective mathematical tools for analyzing and understanding appropriate supply chain risk management are now attracting much attention. This paper presents a stochastic model of the multi-stage global supply chain network problem, incorporating a set of related risks, namely, supply, demand, exchange, and disruption. We provide a new solution methodology using the Moreau–Yosida regularization, and design an algorithm for treating the multi-stage global supply chain network problem with profit maximization and risk minimization objectives. 2006 Elsevier B.V. All rights reserved. Keywords: Supply chains; Risk Management; Stochastic programming; Moreau–Yosida regularization

1. Introduction For many global supply chain networks that can comprise hundreds of companies with over several tiers of suppliers and intermediate customers, there are numerous presenting risks to consider and tackle. Generally, these risks can be classified into two types: risks arising from within the supply chain network and risks external to it. For the former, the attributes are due to the interaction between firms across the entire supply chain network. This set of internal risks can encompass supply risk, demand risk, and trade credit risk for instance. Indeed, Zsidisin (2002) defines supply risk as the potential occurrence of an incident associated with the inbound supply from individual supplier failures or the supply market in which its outcomes would result in the inability of the purchasing firm to meet demand or threaten customer well-being and safety. External risks, on the other hand, arise from the interactions between the supply chain network and its environment, such as international terrorism, and natural disasters like SARS. For the rest of this paper, we define supply chain risk management as the identification and management of risks within the supply network and *

Corresponding author. Tel.: +65 68743014; fax: +65 67792621. E-mail addresses: [email protected] (M. Goh), [email protected] (J.Y.S. Lim), [email protected] (F. Meng).

0377-2217/$ - see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2006.08.028

M. Goh et al. / European Journal of Operational Research 182 (2007) 164–173

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externally through a co-ordinated approach amongst supply chain members to reduce supply chain vulnerability as a whole. The literature on some of these risks and their effects can be found in Lee and Wolfe (2003), Sheffi (2001), Martha and Subbakrishna (2002), Cohen and Mallik (1997), Huchzermeier (2000, 2005), and Zsidisin (2003). At present, with the growing emphasis on globalization, business process outsourcing and the need to control terrorism, there is a stronger impetus to understand and handle the supply chain vulnerabilities already mentioned in the academic and trade literature. In this regard, Paulsson (2003) has presented a simple review of the research on supply chain risk management. However, no detailed strategic approach nor quantitative model to handling the risks involved in a supply chain has been discussed in his paper. As such, there is a need to model the challenging multi-stage stochastic global supply chain network which operates under an environment of uncertainty and is filled with many forms of risks. Hence, this paper attempts to fill this gap by offering a new solution methodology for minimising the different forms of risk resident in a supply chain and maximizing the usual profit objective. Hopefully, through this paper, the interested reader can extend the new solution methodology to the practical world of managing the twin objectives of profit maximization and risk minimisation in the supply chain. In what follows, Section 2 reviews some quantitative approaches to managing risk in global supply chain networks using stochastic models. Section 3 then provides a proposed multi-stage stochastic model for risk management in the context of global supply chain networks, encompassing supply, demand, exchange rate, price, and tax risks. We investigate some properties of this model. The solution methodology and an algorithm for determining the optimal solution is provided in Section 4. Section 5 concludes the paper. 2. Stochastic models in supply chain networks On the mathematical models used to depict supply chain networks operating profit maximization and risk minimization, Huchzermeier (1991) has developed a stochastic dynamic programming formulation to determine the option value of operational flexibility within a multi-country plant network, and used a trinomial approximation to capture the variance–covariance matrix of the underlying stochastic process. The profit function under consideration is derived by solving a multi-product, multi-stage supply chain model. The main results of this work include: (i) a global manufacturing and distribution logistics network can provide a multinational firm with a robust hedge against exchange rate uncertainty and demand risk; (ii) operational flexibility can effectively reduce the firm’s downside risk and enhance its shareholder value; (iii) stochastic recourse using assembly and distribution logistics postponement allows the firm to mitigate against market risk. Kogut and Kulatilaka (1994) have also developed a stochastic dynamic programming model that explicitly treats the supply chain as the equivalent of purchasing an option whose value depends on the spot exchange rate. They consider a two-country production switching model with a simple production function. The single exchange rate process is assumed to be mean-reverting which is then modelled as a discrete time Markov chain. They analyze the hysteresis effect in the presence of switching costs and explicitly determine the hysteresis band. Next, Kouvelis and Sinha (1995) provide a model that allows for switching of production modes. They formulate a profit maximizing, multi-period stochastic dynamic program which allows the firm to switch between different modes of production within a planning horizon. They conclude that a depreciating home currency favors an export policy while an appreciating home currency favors a joint venture or wholly owned subsidiary. Huchzermeier and Cohen (1996) establish a modelling framework that integrates the network flow and option evaluation approaches to global supply chain modelling. Their model maximizes the discounted, expected, global after-tax profit for a multinational firm in terms of a numeraire currency. They provide a methodology for quantifying the risks and returns of flexible global manufacturing strategies. Their work clearly demonstrates how flexibility in a facility network with excess capacity can provide real options to hedge exchange rate fluctuations in the longer term. Dasu and Li (1997) have since investigated the optimal policies of a firm operating plants in different countries subject to exchange rate variability. They formulate a two-country, single market, stochastic dynamic programming model where the combined capacity of the plants exceed the single product, deterministic demand. Under a linear or stepwise switching cost regime, they conclude that irrespective of the variable

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product cost function being concave or piecewise linear convex, the optimal policy is always of the two level barrier types, that is, each plant operates at either a minimum or a maximum output level. Cohen and Huchzermeier (1999a) propose a lattice programming model for the evaluation of the benefit of operational and managerial flexibility under price/exchange rate risk and demand uncertainty. The model consists of a multi-stage supply chain network where real options can be deployed and exercised periodically, contingent on prevailing demand and price/exchange rate scenarios. Their main results include: (i) the value of operational flexibility can be exploited effectively through global coordination, transfer pricing and knowledge transfer; (ii) the option value of managerial flexibility can be captured through distribution logistics postponement and/or stochastic recourse. Agrawal and Seshadri (2000) also demonstrate that intermediaries in supply chains help to reduce the financial risk faced by retailers. They show that a menu of mutually beneficial contracts can be designed to simultaneously: (i) induce every risk averse retailer to select a unique contract from it, (ii) maximize the distributor’s expected profit and (iii) raise the order quantity of the retailers to the expected value maximizing quantity. Kouvelis and Rosenblatt (2002) have studied the design of global facility networks and presented a mixed integer programming model that captures essential design tradeoffs of such networks and explicitly incorporates government subsidies trade tariffs and taxation issues. Their work demonstrates the pervasive effects of subsidized financing, tariffs, regional trade rules, and taxation in shaping the manufacturing and distribution network of global firms. Recently, Smith and Huchzermeier (2003) have studied the global supply chain and risk optimization, and showed how real options add value to global manufacturing firms. Nagurney et al. (2003) have also developed a framework for the modelling, analysis and computation of solutions to global supply chains. They consider three tiers of decision-makers: manufacturers, intermediaries, and consumers and establish a variational inequality formulation based on the equilibrium conditions of this model, deriving some analytical properties. 3. Proposed model In this section, we attempt to develop a unified multi-stage stochastic model of a typical global supply chain network, consisting of as many risks as possible for the problem discussed in Cohen and Huchzermeier (1999b). We will also investigate the features of this stochastic model. 3.1. Problem description We consider an international facility location and distribution logistics planning problem involving one product within the global supply chain network such as Red Bull, Heinz’s ketchup and Levi’s jeans. Typically, the firm’s objective is to maximize its global after-tax profit subject to capacity constraints in each plant and demand requirements in each market. The firm thus needs to make an open/shut decision of plants together with the corresponding shipment quantities from such plants to targeted markets taking into account the attendant uncertainties in market demand, volatility in exchange rates, differing country tax rates, and varying import tariffs at different ports of call even within a country. For this paper, we assume that the firm’s decision plan is re-cast at the beginning of each period. 3.2. Notation Let i denote the index of the m plants, j the index of n market regions. Let Mi represent the capacity of plant i and suppose that the operating cost, denoted by Ci, for opening manufacturing plant i is fixed (i = 1, 2, . . . , m). For each j, let pj denote the price quoted in the currency of the markets. And, for each i, let ci denote the variable cost, mi the markup/transfer price, and ti :¼ (ti1, ti2, . . . , tin) the distribution logistics costs from plant i to the markets, which are all quoted in the local currency of the plants. Let Dj represent the demand of market j, qEP and qEM denote the exchange rates in plants and markets respectively, qD i j ij denote export/import tariffs, and qCT denote country tax rates, where i = 1, 2, . . . , m and j = 1, 2, . . . , n. i Let the decision variables zi denote the open or shut decision variable of plant i with zi = 0 or 1, and xij denote the shipment of the product distributed from plant i to market j, i = 1, 2, . . . , m, j = 1, 2, . . . , n.


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3.3. Model development Suppose at the end of each period, the realizations of the two random variables, demand risk and exchange rate risk, can be approximated by discrete scenarios. Then, one can define the set of all scenarios N each of which is a combination of the end-of-period demand and/orPexchange rate realizations with the probability Prob(1) for any demand-exchange rate scenario 1 2 N and 12N Probð1Þ ¼ 1. For this problem, Cohen and Huchzermeier (1999b) have formulated a two-stage stochastic model as follows: " " ## m n n o X X X CT EM EP D EP EP EP max Probð1Þ 1 qi qj1 pj qi1 1 þ qij ci qi1 mi qi1 tij xij1 qi1 C i zi 12N

s:t:

n X

i¼1

xij1 6 M i zi ;

j¼1

i ¼ 1; 2; . . . ; m; 81 2 N;

j¼1 m X

ð1Þ xij1 ¼ Dj1 ;

j ¼ 1; 2; . . . ; n; 81 2 N;

i¼1

xij1 P 0; zi 2 f0; 1g;

i ¼ 1; 2; . . . ; m; j ¼ 1; 2; . . . ; n; 81 2 N; i ¼ 1; 2; . . . ; m:

Now, if we assume that the random variables in the above problem are demand, exchange rate, country tax rate, and import tariff, then for simplicity of description, one can define a multi-dimensional random variable EP E CT D E EP EM EM vector f as follows f = (D, q , q , q ), where D = (D , D , . . . , D ), q ¼ q ; . . . ; q , 1 2 n 1 m ; q1 ; . . . ; qn CT D D D D , and q . For the decision variables x ; . . . ; q ¼ q ; q ; . . . ; q and z , let y = (x qCT ¼ qCT ij i 11, 1 m 11 12 mn x12, . . . , xmn, z1, . . . , zm). Now, define " # m n n o X X f CT EM EP D EP EP f EP ^ 1q q pj q 1 þ q ci q mi q tij x q C i zi : ð2Þ f ðy ; fÞ :¼ i

i¼1

j

i

ij

i

i

ij

i

j¼1

Here yf represents xf11 ; xf12 ; . . . ; xfmn ; z1 ; . . . ; zm as the random variable f is only associated with the shipments xij. Then, the expected after-tax profit objective function is stated as follows: Ef f^ ðy f ; fÞ:

ð3Þ

Next, we assume that the uncertainty mentioned earlier is represented by a set of distinct realizations. In addition, noticing that under some simple operations, the constraints in (1) can be expressed in the form of h(u) 6 0, where h is a convex function and u is a variable vector in some appropriate finite dimensional vector space. With some manipulations and relaxation, the stochastic model for this multi-period supply chain network problem can be formulated as a multi-stage stochastic convex programming (MSSCP) in the following compact form: min

En f0 ðxn ; nÞ

s:t:

xn 2 F0 ðnÞ 8n 2 S; ! g t ; 8n; g 2 S; t ¼ 1; . . . ; T 1; xnt ¼ xgt if n t ¼ !

ð4Þ

where the random vector n :¼ (n1, . . . , nT) comprises subvectors nt with nt = ft, which are observed at stages t = 1, 2, . . . , T. We also make the convention that n1 is a deterministic vector, which implies that there is no uncertainty at the present stage, i.e., stage 1. Let S denote the set of all scenarios, and assume that s, the cardinality of S, is finite. Suppose each scenario n is now associated with a convex objective function f0(Æ; n) and a convex feasible region F0 ðnÞ. It is evident that, for the problem under consideration, f0 ð; nÞ ¼ f^ ð; nÞ as defined in (2) is a linear convex function, and F0 ðnÞ is a convex feasible set derived from that of (1) under some relaxation and reformulation.

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PT Now, assume that xnt is a kt-vector and let xn :¼ xn1 ; . . . ; xnT , k ¼ t¼1 k t . Generally, F0 ðnÞ can be defined as follows: F0 ðnÞ ¼ fxn 2 Rk jfi ðxn ; nÞ 6 0; i ¼ 1; 2; . . . ; m0 g;

ð5Þ

where fi(Æ; n) : Rk ! R are convex. ! The second set of constraints, xnt ¼ xgt , if n t ¼ ! g t ; 8n; g 2 S; t ¼ 1; . . . ; T 1, in model (4), are said to be nonanticipativity constraints (see Birge and Louveaux, 1997), which reflect the fact that decisions made at tth stage need only depend on the observations made up to stage t. These constraints ensure that the scenarios with the same past will have identical decisions up to that period. To simplify the exposition, we denote Y x ¼ ðxn : n 2 SÞ 2 Rsk ; f ðxÞ ¼ En f0 ðxn ; nÞ; F ¼ F0 ðnÞ Rsk : ð6Þ n2S

Let r denote s · k, h denote m0 · s, and Ax = a represent the nonanticipativity constraints where A 2 Rq·r and a 2 Rq. Then, the MSSCP of (4) can be represented in a compact form as follows: min

f ðxÞ

s:t:

Ax ¼ a; fi ðxÞ 6 0;

ð7Þ i ¼ 1; 2; . . . ; h;

where F can be rewritten as F ¼ fx 2 Rr jfi ðxÞ 6 0; i ¼ 1; 2; . . . ; hg. 4. Solution methodology For the problem under consideration, f in (7) is a linear convex function. It is known that, in the multi-stage stochastic programming, f and F are often separable into scenarios in the following sense: x ¼ ðxI 1 ; . . . ; xI k Þ where xI j is a subvector of x, f and fi (i = 1, 2, . . . , h) can be divided into k groups, so that for i 2 Ij, the function P fi depends only on the subvector xI j , i.e., fi ðxÞ ¼ f~ i ðxI j Þ. Further, f can be written as f ðxÞ ¼ ki¼1 f~ 0i ðxI i Þ, where f~ i , f~ 0i are the corresponding functions under consideration. It is noted that the objective function here is separable since f in (7) is a linear function. However, the nonanticipativity constraints Ax = a is not separable. Thus, we first seek to relax the constraint Ax = a by considering the Lagrangian-dual problem of (7) as follows: minfuðuÞju 2 Rq g;

ð8Þ

where uðuÞ ¼ maxff ðxÞ þ uT ðAx aÞjx 2 Fg:

ð9Þ

Note that (8) is a nonsmooth, convex and unconstrained problem due to the nondifferentiability of u. In order to overcome this nonsmoothness, many techniques have presented in the literature. One common method is to add a self-concordant barrier function b(x) = Enb(xn; n) to the objective function of (9) as follows: uðl; uÞ ¼ maxff ðxÞ lbðxÞ þ uT ðAx aÞjx 2 intFg;

l > 0;

ð10Þ

where l is a parameter and b(Æ; n) is defined on the interior of F0 ðnÞ. Let us now recall the notion of self-concordant functions: Definition 4.1. Let Q Rn be convex and F : Q ! R, f : Q ! R be convex C3 functions. (i) F is said to be an a-self-concordant function if for any x 2 int Q, 2 jr3 F ðxÞ½h; h; hj 6 pffiffiffi ðr2 F ðxÞ½h; hÞ3=2 a

8h 2 Rn :

F is said to be strongly self-concordant if F(x) tends to infinity for any sequence of x converging to the boundary of Q.


169

(ii) F is said to be a h-self-concordant barrier if F is a strongly 1-self-concordant function and for any x 2 int Q jrF ðxÞT hj 6 h1=2 ðr2 F ðxÞ½h; hÞ1=2

8h 2 Rn :

(iii) f is said to be b-compatible with a h-self-concordant barrier F if for any x 2 int Q jr3 f ðxÞ½h; h; hj 6 bð3r2 f ðxÞ½h; hÞð3r2 F ðxÞ½h; hÞ1=2

8h 2 Rn :

Throughout this paper, we make the following assumptions: Assumption 4.1. Matrix A has full row rank, F is a compact convex set and int F 6¼ ;. In the following analysis, we adopt the logarithmic barrier function b as follows: bðxÞ ¼

h X

lnðfi ðxÞÞ:

i¼1

It is known that b is self-concordant with parameter h. Proposition 4.1. Let f be defined in (4). Then, for any b > 0, f : F ! R is b-compatible with the h-selfconcordant log-barrier b. Proof. The proposed result follows directly from Definition 4.1 and noticing that f is an affine function. h Under Assumption 4.1 and by Proposition 4.1, it is well known that the maximization problem in (10) has a unique solution x(l, u). By Meng et al. (2004), the optimal value function u(l, u) is strictly convex and twice continuously differentiable. Hence, the problem minfuðl; uÞju 2 Rq g

ð11Þ

has a unique solution u(l) for any fixed l > 0. Moreover, by Zhao (2005), u(l) tends to an optimal solution u* of (8) as l ! 0 and x(l, u) tends to an optimal solution of (4) as l ! 0 and u ! u*. It is noted that the above log-barrier based scheme is very successful in global convergence. To speed up the local convergence, we consider another smoothing approach using the so-called Moreau–Yosida regularization of u as follows: minfwðuÞju 2 Rq g; where

1 2 wðuÞ ¼ minq uðvÞ þ kv uk ; v2R 2k

ð12Þ

k > 0:

ð13Þ

It is known that the set of minimizers of the problem (12) is exactly the set of minimizers of problem (8), and w is continuously differentiable with the gradient gðuÞ ¼ rwðuÞ ¼ kðu pðuÞÞ 2 ouðpðuÞÞ;

ð14Þ

where p(u) denotes the unique solution of (13) and ou denotes the Clarke generalized Jacobian of u (see Hiriart-Urruty and Lemarchal (1993) for more properties on the Moreau–Yosida regularization). We now incorporate the above log-barrier technique into the Moreau–Yosida regularization to solve problem (4). Our algorithm is composed of two main parts: (i) solving the log-barrier problem (11) using Newton’s method as long as the chosen parameter l does not exceed the threshold and satisfies certain accuracy of the solution; (ii) otherwise, solving the regularized problem using some generalized Newton’s method in which part (i) would be treated as a subprogram. As problem (13) is a nonsmooth problem, it is difficult to derive a unique solution. We now resort to derive an approximate solution of p(u) by solving the following subproblem: 1 2 wðl; uÞ ¼ minq uðl; vÞ þ kv uk ; k > 0: ð15Þ v2R 2k

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This problem is a combination of Moreau–Yosida regularization to the log-barrier function. We should point out that, in our algorithm, (15) only serves to derive some approximate solution, denoted by p(u) with > 0, which is used in the established generalized Newton’s method for solving (12). In the first part of the algorithm, the Newton direction for solving (11) is

1 DLog u ¼ r2uu uðl; uÞ ru uðl; uÞ; ð16Þ where

1 ru uðl; uÞ ¼ Axðl; uÞ a; r2uu uðl; uÞ ¼ A r2uu lbðxðl; uÞÞ AT qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T and we set .ðl; uÞ :¼ l1 ðDLog uÞ r2uu uðl; uÞDLog u. In the second part of the algorithm, the Newton direction used for solving (15) is as follows:

1 DReg v ¼ r2vv /ðl; v; uÞ rv /ðl; v; uÞ; 1 kv 2k

ð17Þ

2

where /ðl; v; uÞ ¼ uðl; vÞ þ uk , and

1 1 1 r2vv /ðl; v; uÞ ¼ A r2vv lbðxðl; vÞÞ AT þ I; rv /ðl; v; uÞ ¼ Axðl; vÞ a þ ðv uÞ: k k Set qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dðl; v; uÞ :¼ l1 ðDReg vÞT r2vv /ðl; v; uÞDReg v: We now state the algorithm as follows: Algorithm ^ > 0; v0 ; u0 ; k > 0. Let k = 0. Step 1. Choose e0 > 0, ^0 :¼ lðe0 Þ > 0, c 2 ð0; 1Þ; 0 > 0; l0 > 0; l Step 2. (Solve the log-barrier Lagrange-dual problem). 2.1. Let u = uk. Solve smooth problem (10) and obtain x(lk, u). 2.2. Construct the Newton direction DLogu by using (16). 2.3. Choose a step size j P 0. Set u+ = u + j DLogu. 2.4. If .(lk, u+) 6 k. Set u(k+1) = u+ and go to Step 2.5. Otherwise, set u = u+ and go to Step 2.1. ^, stop. Otherwise, if lk 6 ^k , set lk = l0, go to Step 3; otherwise, set lk+1 = clk. Set k = k + 1 2.5. If lk 6 l and go to Step 2.1. Step 3. Let v = vk, u = uk. 3.1. Maximize m(lk, v, Æ) and obtain x(lk, v). 3.2. Construct the Newton direction DRegv by using (17). 3.3. Choose a step size a P 0. Set v+ = v + aDRegv. 3.4. If d(lk, v+, u) 6 k. Set v(k+1) = v+ and go to Step 4. Otherwise, set v = v+ and go to Step 3.1. Step 4. If lk 6 ^k , set lk = l0, go to Step 5. Otherwise, set lk+1 = clk. Set k = k + 1 and go to Step 3. Step 5. Let pek ðuk Þ ¼ vðkþ1Þ . Compute gek ðuk Þ ¼ 1k ðuk pek ðuk ÞÞ, choose Vk = (1/k + j)I where j is a small positive number and compute the search direction k d k ¼ V 1 k gek ðu Þ:

Step 6. Choose a step size sk > 0 (0 < sk 6 1), set uk+1 = uk + skdk. Choose a scalar 0 < ek+1 < ek. Let k = k + 1, go to Step 3. Next, we investigate the convergence analysis of the above algorithm. We note that the algorithm can either terminate at Step 2 depending on the choice of initial parameters or complete the whole set of steps. Hence, one can adjust the convergence of the algorithm during implementation, according to the specific practical problem provided. Mifflin (1977) has introduced an important type of Lipschitz functions–semismooth functions. In order to study the superlinear convergence of Newton method for solving nonsmooth equations, Qi and Sun (1993) have extended the notion of semismoothness to vector valued functions. Here we are interested in the semi-


171

smoothness of g(Æ) = $w(Æ), which is a key condition for the superlinear convergence of an approximate generalized Newton method designed in Fukushima and Qi (1996) for solving nonsmooth convex optimization problems. There are several equivalent ways for defining semismoothness. Here, we adopt the following definition. Definition 4.2. Let U : X X ! Y be a locally Lipschitz continuous function on the open set X, where X and Y are finite dimensional vector spaces. U is semismooth at a point z 2 X if (i) U is directionally differentiable at z; and (ii) for any Dz ! 0 and V 2 oU(z + Dz) Uðz þ DzÞ UðzÞ V ðDzÞ ¼ oðkDzkÞ:

ð18Þ

Let f(x) = wTx + b, where w 2 Rr and b 2 R. Hence, u(u) can be written as T

uðuÞ ¼ maxfðAT u wÞ xjx 2 Fg uT a b:

ð19Þ

In the following analysis, for simplicity, we will consider the case in which the domain of u, denoted by dom u, is the whole space, i.e., dom u = Rq. Furthermore, it is not hard to show that dom u = Rq if and only if fAT u wju 2 Rq g Fb , where Fb denotes the barrier cone of the convex set F defined by Fb ¼ fz 2 Rr j there exists b such that zT x 6 b for all x 2 Fg:

ð20Þ

We now derive the following result with respect to the semismoothness of the gradient of the Moreau–Yosida regularization. Proposition 4.2. The gradient g in (14) of the Moreau–Yosida regularization of u is semismooth on Rq. Proof. As F is a convex set, it is easy to see that u(u) is affine on Rq since the objective function in (19) is affine. From (14), to show the semismoothness of g, we only need to show that of p(Æ). We now consider the epigraph of u: epiðuÞ ¼ fðu; tÞ 2 Rq Rjt P uðuÞg:

ð21Þ q

Then, we have epi(u) as a polyhedral convex set of R · R due to the linearity of u. Thus, the projection mapping, Pepi(u)(Æ), on the epigraph of u is semismooth on Rq · R. On the other hand, by Theorem 4 in Meng et al. (2005), it is known that for u0 2 Rq, p(Æ) is semismooth at u0 if Pepi(u)(Æ) is semismooth at (u0, t0 1/k) where t0 = u(u0). Hence, according to the above arguments, p(Æ) is semismooth at u0 for any u0 2 Rq. Thereby, g is semismooth on Rq. h Proposition 4.3. Suppose that sk ! 1 as k ! 1. Then, any accumulation point of {uk} is either an optimal solution of (11) or an optimal solution of problem (12) as ek ! 0. Proof. Without loss of generality, we assume that x* is an accumulation point of {uk}. For simplicity, we use {uk} to represent the corresponding convergence subsequence. Then, it follows from the algorithm that {uk} is generated either by implementing Step 2 or Steps 3–6. It is clear Step 2 is actually the standard Newton’s method and therefore u* is an optimal solution of (11) for the first case. For the latter it follows that case, is bounded which limk!1 gek !0 ðuk Þ ¼ gðu Þ. According to choice of Vk in the algorithm, it is evident that V 1 k implies the boundedness of {dk} for all k. Hence, there exists a convergent subsequence, still denoted by {dk} for convenience, such that dk ! d* as k ! 1. Then, by assumption, we have lim ðukþ1 uk Þ ¼ lim sk d k ¼ d ¼ 0:

k!1

k!1

As {Vk} is bounded, thus lim gk ðuk Þ ¼ lim V k d k ¼ 0: k!1

k!1

Hence, we have gðu Þ ¼ limk!1 gk ðuk Þ ¼ 0. Thus, u* is an optimal solution of (12). h

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Let x(l, u) denote the unique solution of problem (10). Then, it is well known that x(l, u) tends to an optimal solution of original problem (7) when u converges to an optimal solution of (11) and l ! 0. Hence, it follows from Proposition 4.3 that xk :¼ x(lk, lk) tends to an optimal solution of (7) when lk ! 0 and uk tends to an optimal solution of (11). Note that when ek ! 0, then lk ! 0 since lk is determined by e in the algorithm. In practice, one can choose lk = ik where i 2 (0, 1). Similarly, given l, u, let v(l, u) denote the solution of (15), x(l, v) be the maximizer corresponding to the optimal value function u(l, v), and set x(l, u) :¼ x(l, v(l, u)). Note that in this case x(l, u) converges to an optimal solution of (7) when l ! 0 and u tends to an optimal solution of (11). Thus, for {uk} and lk generated by the whole algorithm, xk converges to a solution of the original MSSCP as uk ! u*, k ! 0 where u* denotes the optimal solution of (11). In Theorem 5.2 of Fukushima and Qi (1996), there are two additional assumptions required: (i) g is semismooth; (ii) the nonsingularity of B-subdifferential of g. For the problem considered in this paper, it is clear that the latter condition is redundant as u is piecewise affine. By Proposition 4.2, we have g(u) is semismooth on Rq. Then, with the similar arguments followed by Theorem 5.2 of Fukushima and Qi (1996), we derive the following result. Theorem 4.1. Suppose u* is an optimal solution of (12). Assume that ek = o(kg(uk1)k2) and sk = 1 for all large k. Then, kuk+1 u*k = o(max{kuk1 u*k, kuk u*k}). That is, {uk} tends to u* at least 2-step superlinearly. 5. Concluding remarks In this paper, we have extended Cohen and Huchzermeier’s (1999b) two-stage stochastic model to a multistage stochastic model for supply chain networks by providing a general formulation of the multi-stage supply chain network problem operating under a scenario of a variety of risks. Next, we incorporate the properties and well-developed techniques found in stochastic programming to provide a solution methodology and to design an algorithm for obtaining the optimal solution to the original supply chain network problem with profit maximization and risk minimization objectives. This algorithm can be used to smooth nonsmooth functions for rapid convergence, and can therefore be used to solve some practical supply chain network problems. Future research will focus on the application of the solution to a large scale supply chain network. We defer this discussion to a later paper. References Agrawal, V., Seshadri, S., 2000. Risk intermediation in supply chains. IIE Transactions 32, 819–831. Birge, J.R., Louveaux, F., 1997. Introduction to Stochastic Programming. Springer-Verlag. Cohen, M.A., Huchzermeier, A., 1999a. Global supply chain network management under price/exchange rate risk and demand uncertainty. In: Muffato, M., Paswar, K.S. (Eds.), Logistics in the Information Age. SGE Ditorali, pp. 219–234. Cohen, M.A., Huchzermeier, A., 1999b. Global supply chain management: A survey of research and applications. In: Tayur, S., Ganeshan, R., Magazine, M. (Eds.), Quantitative Models for Supply Chain Management. Kluwer Academic Publishers, pp. 669–702. Cohen, M.A., Mallik, S., 1997. Global supply chains: Research and applications. Production and Operations Management 6, 193– 210. Dasu, S., Li, L., 1997. Optimal operating policies in the presence of exchange rate variability. Management Science 43, 705–727. Fukushima, M., Qi, L., 1996. A global and superlinear convergent algorithm for nonsmooth convex minimization. SIAM Journal on Optimization 6, 1106–1120. Hiriart-Urruty, J.B., Lemarchal, C., 1993. Convex Analysis and Minimisation Algorithms II. Springer-Verlag, Berlin. Huchzermeier, A., 1991. Global manufacturing strategy planning under exchange rate uncertainty. PhD Thesis, University of Pennsylvania, Pennsylvania. Huchzermeier, A., 2000. The real option value of operational and managerial flexibility in global supply chain networks. In: Frenkel, M., Hommel, U., Rudolf, M. (Eds.), Risk Management: Challenge and Opportunity. Springer Verlag, pp. 181–201. Huchzermeier, A., 2005. The real option value of operational and managerial flexibility in global supply chain networks. In: Frenkel, M., Hommel, U., Rudolf, M. (Eds.), Risk Management: Challenge and Opportunity, second ed. Springer Verlag, Berlin, pp. 609–629. Huchzermeier, A., Cohen, M.A., 1996. Valuing operational flexibility under exchange rate uncertainty. Operations Research 44, 100– 113. Kogut, B., Kulatilaka, N., 1994. Operating flexibility, global manufacturing, and the option value of a multinational network. Management Science 10, 123–139.


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