Feb 25, 1998 - Abstract. We investigate a tool which small manufacturers engaging in international supply contracts can use to dampen the variations in their ...
Reducing International Risk through Quantity Contracts Alan Scheller-Wolf Sridhar Tayur GSIA, Carnegie Mellon University Pittsburgh, PA 15213-3890, USA. February 25, 1998 Abstract
We investigate a tool which small manufacturers engaging in international supply contracts can use to dampen the variations in their income caused by exchange rate
uctuations. The method we study involves contractual speci cations for a minimum purchase quantity with the buyer. The particular case we consider is that of a smaller manufacturer providing a product to a large overseas corporation, where transactions are conducted in the manufacturer's domestic currency. The large corporation employs a periodic review inventory policy with back orders, incorporating unit, holding and penalty costs which may be statedependent. The state is the current exchange rate, which varies in a Markovian manner according to one of three considered models (random walk, mean reverting, and momentum). Demands for products arrive randomly and may also be state dependent, to a degree determined by the rate of pass through of exchange rate e ects. We rst establish that the optimal policy for the corporation is an order up to policy with state dependent thresholds. After establishing this, we use computer experimentation to investigate the e ect increasing the minimum purchase quantity has. This minimum order quantity transfers some of the cash ow variance to the customer, which in turn drives the corporation's holding/penalty costs up. To counteract this, the corporation will expect a discount from the smaller supplier. This gives rise to a mean/standard deviation frontier for the supplier's cash ow. We also study the e ects of increasing exibility in order quantity, increased supplier capacity, and the interactions between these two factors.
GSIA Working Paper # 1997-42
1 Introduction As the number of companies engaging in international trade grows, the importance of developing coping strategies to deal with exchange rate uctuations and their e ects becomes increasingly apparent. The giants of international trade, large multinational corporations, have developed an assortment of techniques which they use to dampen the variations in cash ow caused by such uctuations. These include forward buying and hedging in the nancial markets, management of currency choice and timing of transactions, and using parallel sources of cash ow. Additionally, using operational strategies (`real options') to counteract exchange rate variability have become common; see [5], [11], [16]. Regrettably, smaller players in the international arena often do not have the resources to utilize these techniques. This is particularly unfortunate, as these smaller companies are as a rule less able to weather disruptions in their cash ow than their larger cousins. We look then for a technique which smaller manufacturers might use to dampen the variations in their income caused by exchange rate uctuations. The method we study involves contractual speci cations with the buyer, a practice being followed in some industries domestically. The particular case we consider is that of a smaller manufacturer providing a product to a large overseas multinational corporation. Transactions are conducted in the manufacturer's domestic currency; therefore the consideration of exchange rate e ects in determining an optimal policy must be undertaken by the larger corporation. The smaller manufacturer seeks a method by which some of the cash ow variance inherent in international trade can be transferred to the larger customer, who can in turn deal with its e ects by engaging in some of the practices mentioned above. In exchange for accepting a larger portion of this variance the multinational corporation will expect a discount from the supplier. This gives rise to a mean/standard deviation frontier for the supplier's cash ow: A series of pairs of values he may choose from, in which he can sacri ce a portion of his expected pro ts in exchange for a reduction in the expected variance of his cash ow. Our model for the multinational corporation (or the customer) is that of a company employing a periodic review inventory process with back orders, having unit, holding and penalty costs which may be state-dependent. The state is the current exchange rate, which varies in a Markovian manner. Demands for products arrive randomly and may also be state dependent { the mechanism for this will be discussed further below. The corporation may place orders for the product from the supplier each period, up to a capacity C , which we assume arrive with no lead time. Then demand is revealed, costs are incurred, and we 1
move to the next period. The supplier charges a xed unit price c for the product, but due to the exchange rate the customer sees a state dependent price ci . Demands are satis ed by stock on hand if present; if not they are backlogged and supplied when stock arrives. For incurring such a backlog, the customer must pay a penalty cost pi , where the subscript i denotes the period we are in. The current exchange rate may have an e ect on competition, and thus the penalty costs may be related to the state. If the corporation holds positive stock on hand from period to period it pays a holding cost hi per unit; as the unit price may vary from state to state, so too may the holding cost. We consider the average cost objective for the in nite horizon model. There is some disagreement as to the proper way to model exchange rate behavior. (This is well known.) Within the framework of a nite state Markov chain, we will consider three di erent models of its evolution: The random walk, mean reverting behavior, and a momentum model. The reader is referred to [17] for further explanation of these models. To incorporate these into our computational work, we will assume that the state space is governed by a ve-state Markov chain, with a di erent transition matrix for each of the three models we are experimenting with. The speci c transition matrices are illustrated in Appendix A. It should be noted that in theory a larger state space could have been used with no substantial change in the work. We chose a ve-state chain as it proved sucient to illustrate the behavior of the system without being overly cumbersome. As described above, the model is essentially that of [15], generalized to a Markovian, rather than a cyclical setting1 . To this model we add one further element, a mandatory minimum order quantity, Q; see [1], [2] for related models in a nite horizon, stationary setting. This is a quantity which must be purchased by the customer every period, regardless of state, inventory position, or demand. The constant purchase of this amount reduces the variance in the cash ow of the supplier, transferring this randomness to the customer. This has the e ect of increasing the customer's expected holding/shortage costs. Therefore we assume that in exchange for the agreement to purchase this xed order quantity, the supplier lowers the unit price in a manner which keeps the customer's overall expected inventory/holding plus order costs constant. This in turn reduces the supplier's expected cash ow to some degree, giving rise to a mean/standard deviation frontier. It remains to be discussed how we model the e ect of exchange rates on demand, and the rationale behind it. Depending upon the nature of the market and the multinational
1 Thus, we generalize the models in [3],[8], [9] and complement the models in [12] (lost sales) and [18] (with a xed order cost but no capacity constraint or Q).
2
corporation's position in it, we propose three di erent degrees of e ect, or pass through. If the corporation is in e ect a monopoly providing an essential good or service, a change in exchange rates causing the price of the product to rise will have little to no e ect on demand, as the consumers have no choice but to purchase the product. We term this situation where demand is independent of state as the case of zero pass through. A more common situation is that where a change in exchange rates a ecting the price of the product does e ect demand. Depending upon such factors as product di erentiation, the concentration of foreign rms to domestic rms, non-tari barriers, and of course social dynamics (fashion and \mass psychology") the impact of exchange rates on demand can vary. We consider two such models here, which we call partial and full pass through, respectively. These in turn re ect the situations where demand does vary, but has a greater or lesser degree of stickiness (possibly due to the fact that price does) in the multinational corporation's country. For a fuller discussion of the causes and e ects of pass through, see [6], [10], and [19]. In summary then, we consider the problem of a smaller manufacturer supplying a product to a large foreign multinational corporation, where transactions are undertaken in the manufacturer's domestic currency. The multinational follows a periodic review inventory policy with back orders and state-dependent unit, holding, and penalty costs. The state varies as the exchange rate does, evolving as a nite state Markov chain with a transition matrix taking the form of a random walk, mean reverting, or momentum process. The demand is subject to variation as the state changes, according to a level of zero, partial, or full pass through. The corporation may order at most C units form the supplier in any period, which arrive with zero lead time. In addition, there is a speci ed non-negative minimum order quantity, Q, which the corporation agrees to order every period. In exchange for this the corporation receives a discount on the unit price. The exporter thus has dual objectives, keeping his mean cash ow high while reducing its variance. We begin by formally de ning our problem, in Section 2. Then we establish the form of the optimal policy for the corporation, given a xed price structure, C and Q. It turns out that the corporation will in this situation follow a modi ed state-dependent order up to policy: If the inventory is above a certain level, zi , Q they order the minimum amount, Q. If it is between zi , C and zi , Q, they order up to zi. If it is below zi , C , they order the maximum amount, C . This is done for the nite horizon in Section 3 and the in nite horizon in Section 4. The stipulated minimum order quantity in uences the order-up-to levels, zi , which in 3
turn a ect the average cost of the policy. Therefore if the manufacturer wishes to receive a contractual agreement to increase Q, which will increase his customer's costs, he must drop his unit price. The exact amount of decrease is not immediately apparent though: Once he decreases his price the customer's order up to levels change and so too does his average cost. Therefore an iterative procedure must be implemented to nd the reduction in price which will ensure that a customer following an optimal policy for an increased Q will encounter the same average costs as if Q were zero. We use In nitesimal Perturbation Analysis (IPA; see [13], and [14]) to establish both the order up to levels and then the price structure which will maintain the customer's average cost. The validation of this is done in Section 5. We then move to a presentation and discussion of our various experimental results in Section 6. Here we consider the e ect of adopting di erent values of Q for each of the demand models and pass through assumptions, illustrating the e ects with mean/standard deviation frontier plots. We also consider cases where we assume the wrong model for the exchange rates (for example we optimize under the assumption of a random walk, when in fact the exchange rate evolve in a mean reverting manner). In addition, we look at the e ect varying the capacity has on the customer, and how this allows the supplier to increase his unit price without increasing the costs of his customer. We also consider models where the capacity is a function of the minimum order size, Q. Finally, we study the e ects of conducting transactions using supplier's domestic currency. We conclude with topics for further investigation in Section 7.
2 De nition of the Model We begin with some de nitions and assumptions which we will be using in the paper. All of the de nitions below are made from the point of reference of the multinational corporation; i.e., xn is the inventory the corporation has on hand at the start of period n.
2.1 De nitions xn : The inventory at the beginning of period n. yn : The inventory after ordering but before demand arrives in period n. C: A non-negative maximum order capacity. 4
Q: A minimum order size; 0 Q < C .
in : The state of the system in period n. We assume the possible states of in form a nite
state, irreducible (hence positive recurrent) Markov chain with transition probabilities pij which depend only on i. We assume that the chain has K states, and de ne S def = f1; : : : ; K g as the state space of the chain.
ci : A nite, non-negative, state-dependent ordering cost per item. pi: A nite, non-negative, state-dependent penalty cost per item; pmax def = maxfi2S g fpi g: hi : A nite, non-negative state-dependent holding cost per item; hmax def = maxfi2S g fhi g: Di : A non-negative state-dependent demand distribution, which is independent of time. We denote a generic value sampled from the distribution Di by di . We assume E[d4i ] < 1 for all i 2 S .
2.2 Assumptions 1. If we denote the stationary probabilities of the embedded Markov chain and the means of the demand distributions by i and i respectively, for 1 i K , we assume (1)
K X
Q
zi will lead to a suboptimal solution for all N , and hence can lead to a solution no better than ordering the minimum, in the limit as N !1.
Lemma 4.8 If for a given period i, ci < pi, then under an optimal order-up-to policy, zi 0. 15
Proof : Follows from the fact that in this case it is cheaper to order up to a level where you have satis ed all outstanding demand than to keep the orders backlogged.
We can also establish weaker sucient conditions for all zi to be nite.
Lemma 4.9 If pi > 0 for some i, then there is an optimal policy having zj > ,1; 8j 2 S . Proof : Assume we are given a solution where zj = ,1. Let H def = fijzi > ,1g, the set of indices where the order up to level is not negative in nity. We look at three di erent cases.
1. In this case we assume:
X i2 H
Ci +
X j 2H c
Qj
0 and i > 0. Therefore as solution Z has a nite optimal value, solution can not be optimal. 2. In this case we assume:
X i2H
Ci +
X j 2H c
Qj =
X i2K
ii :
Here we decrease Q slightly while keeping the order up to levels the same. This places us in category 1, and hence this solution with the decreased Q has an average cost of +1. As the value of the solution is convex with respect to Q (as the nite horizon solution is, Proposition 3.4), this implies that the solution value at the original Q is likewise +1, again a contradiction. 3. Here
X i2H
Ci +
X j 2H c
Qj >
X i2K
ii :
We examine a particular sample path, !. Assume that the inventory is currently at some large negative level, ,M , and that we are currently in state j with zj = ,1. For ease, we will denote the time period we are currently in as time zero. We thus 16
will order Q according to this optimal policy . We will examine the cost of ordering Q + 1 units instead of Q. Under the assumption that we order Q units at time zero, let be the rst time along ! that the inventory returns to a level greater than mini2H fzi g. Note from Assumption 1 that H is nonempty. Let l def = minfz ; 0g. If l = 0, then rede ne as the rst time the inventory reaches a nonnegative level, if we order Q + 1 units in period zero. As the inventory process has positive drift and nite second moment, is nite with P probability one. Up until time , we save on average Ki=1 pii per period by not having to pay a shortage cost for the extra unit we order. Once is reached, the di erence in our cost due to ordering this extra unit can be bounded by:
fv(x ; i ) , v(x , 1; i )g = ;
max
def
f,1 0. As all states communicate, and we assume that P r1 (d , C ) > r) > 0 for all r and some N 1, there is a path there is a path such that P( Nn=1 in r having positive probability from (x0 ; i) to a point, w, where under Zx0 full capacity will be used. As (x0 ; i) is positive recurrent, the process eventually return to (x0 ; i) from w under Zx0 . This being the case, de ne policy Zx0 ;i;j; as equal to policy Zx0 at all points save for the rst one where Zx0 uses full capacity. At this one single point, Zx0 ;i;j will use one unit less than full capacity. It will then follow Zx0 until the process would return to (x0 ; i) had Zx0 been used exclusively. Then, under Zx0;i;j the process will at this time reach (x0 , 1; i). This procedure can be repeated times, until with positive probability the inventory reaches (x0 , ; i). From here, by assumption, we can reach (x0 ; j ) using policy Zx0 . The complement of the above assumption is that all paths from (x0 ; i) go to (x0 , ; j ), for positive . In this case we use our assumption that all states communicate and that P(PNi=1r2 (Q , di) > r) > 0 for all r and some N2r . This implies that there is a path from (x0 ; i) to a point, w, where under Zx0 we will order the minimal amount possible, Q. In this case we modify Zx0 to order Q + 1 the rst time this happens, and to then follow Zx0 from then on, calling this slightly modi ed policy Zx0 ;i;j . This will lead to (x0 + 1; i) with positive probability, and repeating this times will eventually get the inventory to the level (x0 + ; i), from which point we can reach (x0 ; j ) with positive probability under Zx0 . By construction, we will assume that the policyZx0;i;j reverts to policy Zx0 after the state (x0 ; j ) is reached.
We now use the conditions of [7] to verify the form of the optimal policy.
Theorem 4.2 For the model described previously, assuming discrete demand , the up-to 2
policy is optimal for average cost criterion.
2 The vanishing discount approach in [4] can potentially be used to prove this result for continuous demands.
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Proof : The previously de ned model comprises a Markov process with denumerable state space. We de ne a space of policies to be considered. First, de ne G , which consists of policies (possibly randomized) such that for all x > Ai , we do not produce more than Q. From Lemma 4.7, we know that any policy is dominated by a policy in G , and therefore nding an optimal policy is equivalent to nding an optimal policy in G . We expand the space of policies to be considered by rst de ning the countable set K def = f,1; : : : ; A g x f1; : : : K g, where A def = maxfi=1:::K gfAi g. We then de ne the new space of policies F to be the union of G and the sets of policies fZx jx 2 Kg and fZx0 ;i;j jx0 ; i; j 2 Kg We will verify that conditions 1 through 5 of Federgruen, Schweitzer, and Tijms [1983] hold for policies in F and the set K. Assumptions 1 and 4 are satis ed for all Markov chains. Assumption 3 is satis ed as our cost functions are bounded from below. Assumption 2a requires that the expected time and cost from any state (x; i) until the rst visit to K be nite under all policies in F . First, for policies in F but not in G , we know that, like policy Z , they are positive recurrent and thus have nite expected cost. Next, for policies in G , consider any initial x0 (the state i is irrelevant). As for any n, yn A implies that xn 2 K, we will examine the time and until yn reaches this set. There are two cases: (i) x0 > A : Under any policy in F , the time until the inventory process reaches K def = K can be modeled as a busy period of a GI=GI=1 queue. This queue is constructed by treating as service and interarrival times the cumulative order quantity and demand between visits to a positive recurrent state i. More speci cally, if the sequence of arrival epochs to recurrent state i are ft0 ; t1 ; : : :g, then the service times to the constructed queue are de ned as Gi def = Q(t1 , ti,1 ), and the interarrival times are def Pti+1 ,1 Ti = j =ti dj . This queue has exceptional rst service having distribution Sa x0 ; a:s:, and negative P drift, due to our assumption that Ki=1 i E[di ] > Q. Therefore as the Gi have moments of all orders, so too does K . (ii) x0 2 K : In this case we can model the time to return as a busy period for a queue with exceptional rst service having Sa bounded by Q. This will again be nite by the reasoning above.
In both cases (i) and (ii) the expected cost can be bounded by (hmax + pmax )fC (jx0 j +
jA j) K + Q K2 g def = W . This is nite as the corresponding moments of K are nite. 19
Therefore both criteria of 2a are satis ed. For 2b, we must show that all states in K communicate under some policy in F . Starting at any point (x0 ; i), to reach state (x1 ; j ), assuming both states are in K, we can follow policy Zx1 to reach (x1 ; l), for some l in S . If j is di erent from l, we can then follow Zx1 ;l;j which will reach (x1 ; j ), following Lemma 4.1. Having shown that assumptions 1-4 are satis ed, we know that the system of equations has a solution. We now verify condition 5, which ensures that the optimal solution is attained at the minimizing point of the right hand side of the optimality equation. Condition 5 states that the sum of the expected times and costs (under any policy in F ) from each point along a path from any state (x; i) until the process reaches a state in K is nite. We will work rst with the expected time. Note that if we start at any state (x; i), and we denote the time to reach K from here under a given policy and sample path 2 ! as x;i(!), then the expected sum of times along the path is bounded by E[ x;i ]. Similar to the proof of condition 2a, for policies in G we can model this amount of time as a busy period in a GI=GI=1 queue with negative drift and exceptional rst service. Therefore the 2 necessary condition for E[ x;i ] < 1 for any given (x; i) is E[G3i ] < 1, which is true. For policies in F but not in G , we can use reasoning similar to Lemma 4.1 to show that again 2 E[d3i ] < 1 ensures E[ x;i ] < 1. The expected sum of the costs from points along a path can be bounded by E[W K], which requires E[ K3 ] to be nite, which holds as E[d4i ] is assumed nite for all i 2 S . It follows from Theorem 4.2 that the optimal policy can be determined by solving the following set of equations:
vs = amin fG(s; a) , g + 2A(s)
X t2I
pst (a)vt :g
Unfortunately, to solve such an equation is no trivial matter. Instead, to compute the optimal up to values, we will use simulation based optimization; speci cally In nitesimal Perturbation Analysis (IPA) to obtain gradient estimates. The method we follow is similar to that in [13] and [14].
5 Optimization As the basis for IPA we need recursions for both the process and some key derivatives. From Lemma 3.1 we know that Jn ; In ; and vn are convex; therefore they are also continuous. This implies that one-sided derivatives exist at all points. We will use right derivatives. 20
5.1 Derivative Recursions We rst establish simulation recursions. Di erentiating the simulation recursions yields the derivative recursions, the validity of which are shown in Theorem 5.1. Let z1 ; z2 ; : : : zK be the base stock levels, with the (assumed unique) maximum of these being zmax which occurs at period m. Each period has the following sequence: Production following the base stock policy with minimum order Q; demand occurs; costs are incurred. Let n denote the period index, Xn the inventory level, and Sn the shortfall (not to be confused with S , the state space of the embedded Markov chain). In this case Sn is de ned as the di erence between zmax and Xn , (after production but before demand in period n). We de ne Un as the amount ordered in period n. We start the system with S0 = zmax , z1 , assuming we have arbitrarily numbered the periods so that we begin in period type 1 with initial inventory X0 = z1 . Let n def = zmax ,zin , where in is the period type we are in at time n. We then have the following recursion for Sn :
(
Q if Sn + dn , Q < n+1 Sn+1 = max(SSn++ddn,,C; ) otherwise: n n n+1 Di erentiating with respect to some zi other than zmax :
8 > Sn0 if Sn + dn , Q < n+1 < 0 Sn0 if Sn + dn , C > n+1 Sn+1 = > : ,I fi = in+1 g otherwise:
Di erentiating with respect to zmax :
8 > Sn0 if Sn + dn , Q < n+1 < 0 Sn+1 = > Sn0 if Sn + dn , C > n+1 : I fin+1 6= zmax g otherwise:
The value of Un+1 can be determined by:
8 > Q if Sn + dn , Q < n+1 < Un+1 = > C if Sn + dn , C > n+1 : Sn + dn , n+1 otherwise:
Therefore di erentiating with respect to some zi other than zmax :
8 > 0 if Sn + dn , Q < n+1 < 0 Un+1 = > 0 if Sn + dn , C > n+1 : Un0 + I fi = in+1 g otherwise: 21
Di erentiating with respect to zmax :
8 > 0 if Sn + dn , Q < n+1 < Un0 +1 = > 0 if Sn + dn , C > n+1 : Sn0 , I fzmax = in+1 otherwise: The cost in period n is Cn = hn(zmax , Sn , dn)+ + pn(zmax , Sn , dn ), + cn Un:
The average cost over N periods is
C N = N1
N X n=1
Cn ;
where the in nite horizon counterpart C 1 is arrived at by letting N go to in nity in C N . Di erentiating Cn with respect to some zi other than zmax :
Cn0 =
(
,hnSn0 + cn Un0 if Sn + dn < zmax
pnSn0 + cnUn0 if Sn + dn > zmax :
Di erentiating with respect to zmax :
Cn0 =
(
hn , hn Sn0 + cn Un0 if Sn + dn < zmax pn Sn0 , pn + cnUn0 if Sn + dn > zmax :
The justi cation of the derivatives is given by
Theorem 5.1 For the above model, for any up-to policy with vector fz ; : : : ; zK g of up-to 1
levels:
1. Finite Horizon (a) The right derivatives Sn0 , Un0 and Cn0 exist with probability one. (b) E[Sn0 ] = E[Sn ]0 ; E[Un0 ] = E[Un ]0 ; E[Cn0 ] = E[Cn ]0 . P (c) N1 Nn=1 Cn0 is a valid estimate for E[C N ]0 . 2. Discounted In nite Horizon Results analogous to the above hold as well in this case. 3. In nite Horizon Average Cost (a) The Markov chain (Un ; Sn ; in ) is positive regenerative with regeneration point (; 0; m)
22
(b) The process (Un ; Un0 ; Sn ; Sn0 ; in ) is a positive regenerative Markov chain with (Sn0 ; Un0 2 f,1; 0; 1g). P P (c) limN !1 N1 Nn=1 Cn0 is a valid estimate for the derivative of limN !1 N1 Ni=1 Cn . Proof :
1. (a) Follows from Lemma 3.1. (b) We can rewrite the recursion for Sn as:
Sn+1 = Sn + dn , Q + minf(Sn + dn , Q , n+1 )+ ; C , Qg: This makes it clear that both Cn and Sn are compositions of Lipschitz functions, and therefore are themselves Lipschitz. Arguments similar to those in [14] can be used to show that these functions have integrable moduli, and thus Lemma 3.2 of that same paper can be applied to yield the desired result. (c) Follows from the fact that Cn is Lipschitz, the Dominated Convergence Theorem, and Lemma 3.2 of [14]. 2. These results follow directly from the nite horizon results using Lemma 4.3 and Theorem 4.4 of [14]. 3. (a) Note that under all policies in F , the inventory process returns in nitely often to the interval (zmin ; zmax ). Therefore to prove positive recurrence of (Sn ; in ), we must show that from any state within this interval we return to (0; m) with probability one, within a nite amount of time. This being established, as the Un are i.i.d., the overall process, and hence Cn, is positive regenerative at (0; m). As the demand is continuous and ergodic, and due to assumptions (1) and (3) we can nd a path with positive probability Pm such that state m communicates with itself in l 1 steps where: P X def = ln=1 (din , Q) is positive and falls into an open interval A def = (0; A1 ) with positive probability. P Y def = ln,=11 (din , Q) is positive and falls into an open interval B def = (0; B1 ) with positive probability. Note that Pm is independent of the state (x0 ; i) we begin in. We will assume that l 2; if l = 1 the proof simpli es. Let def = (C , Q)=4. Also de ne A2 def = minfA1 ; 2g: 23
Assume we start in state i. From an initial state (zmin ; i), due to (2) and the assumption that our demand is continuous, we can nd a path which will take us to an interval of the form (si ; si + ) and into state m with probability i > 0, where si > zmax . Note that this implies that from any initial state (x0 ; i) where x0 2 (zmin ; zmax ) there is a path with probability no smaller than i which will take to a state (w; m) where w 2 (si + (x0 , zmin ); si + (x0 , zmin ) + ). Note that as si is xed, w si + zmax , zmin + is also xed. Therefore for any w as above, we can nd a nite integer Lw;i such that Lw;i is the minimal integer which satis es w , Lw;iA2 , B1 zmax , Q , (C , Q)=2. Note also that as w is bounded and A2 is strictly positive, Lw;i is also bounded, i.e. Lw;i < Li for all w in the prescribed interval. Let pX def = P(A2 , (=Lw;i ) X A2 ), and pY def = P(B1 , Y B2 ): As demand is continuous and Lw;i Li < 1; pX > 0 and pY > 0. Starting from (w; m), with probability no less than (Pm )Li;w (pX )Li;w pY (Pm pX )Li pY > 0; w will be mapped into the interval (zmax , Q , (C , Q)=2 , a; zmax , Q , (C , Q)=2 , a + 2), where a = w modulo 2 < 2 = (C , Q)=2: This implies that zmax , Q , (C , Q)=2 , a zmax , Q , (C , Q) = zmax , C . Also, zmax , Q , (C , Q)=2 , a + 2 zmax , Q , (C , Q)=2 + (C , Q)=2 = zmax , Q. Therefore this action will take the inventory into an interval contained in (zmax , C; zmax , Q) and the state into state m, from which we can order and reach (zmax ; m). Therefore starting from (x0 ; i) for x0 2 (zmin ; zmax ), we reach (0; m) with probability no less than Pi def = i (Pm pX )Li pY > 0: We can repeat this procedure for each i 2 S , and let P def = minfi2S g fPi g > 0: Therefore from any (x0 ; j ) in (zmin ; zmax ) we return to (0; m) with probability no less than P > 0. As we return to the interval (zmin ; zmax ) in nitely often, the shortfall process is positive recurrent in the state (0; m). (b) Follows form (a) and the fact that Sn0 and Cn0 are nite state Markov chains. (c) Follows proof of Theorem 4.7 in [14], in light of (b) and arguments similar to those in section 1 of this proof.
24
6 Experimental Results 6.1 Procedure
The computer experiments were run in C++ on a personal computer with a Pentium 486 processor. For each given set of parameters, an initial optimal solution was found via IPA for Q = 0. We found the nal solution to be essentially insensitive to the initial solution, so for all cases we began at the point f20:1; 20; 20; 20; 20g. We simulated a ve state Markov chain for the exchange rate level, according to one of the three models: Random walk, mean reverting, or momentum. The matrices for these cases can be seen in Appendix A. We took p = 20 and h = 5 for all periods. The demand was chosen to be taken from a Gamma(2,15) distribution, for the base case (so expected base demand was 30). For partial pass through the pass through coecient, , was chosen to be 0:4; for full 0:8. In cases where pass through was present, in state three we had base demand, and in other states we increased or decreased the mean demand by 0:2 for states two and four respectively. We doubled this e ect for states one and ve. The unit cost vector was f16; 18; 20; 22; 24g for the Q = 0 case; it of course changed as we found solutions for higher Q values. Given xed values for item costs and C and Q, the algorithm used IPA analysis to converge to a set of optimal order up to levels, zi . It began with a step size equal to fty, and ran until either the step size was less than 0:0001 or two million data points had been used. After a ve thousand point initialization, it used iterations of ve thousand points to estimate costs and derivatives of cost with respect to the z values. It then updated the z values, recording the best found up to that point. If it went ve iterations without an improvement in cost, the best values were returned to and the step size halved. After converging to a value, a run of one million points with a ve thousand point initialization was used to estimate the mean inventory/backorder cost and ordering cost of this solution. This procedure actually appeared in a larger loop. This is the loop where Q is varied. Given a larger value of Q than zero, using the optimal ordering cost for Q = 0, the program searched for a vector of order up to values which would return a total cost close to that when Q = 0. It did so until the rst time the costs were within fty cents of each other, or it had iterated fteen times from the rst time the costs were within two dollars of each other. After each iteration, the di erence in costs was calculated, and the item costs scaled accordingly. We began with a step size of 0:8, to reduce the chance of overshoot. Each time there was no improvement, the step size was scaled by 0:8 and we returned to the best value. If there was no improvement for three iterations the algorithm was forced to step 25
Mean 600
500
400
Pass Through
300
None Partial Full
200
100
200
300
STD
Figure 1: Random Walk Model with Di erent Q Values. and the step size scaled back up. If termination was caused by iterating fteen times within two dollars, the minimum di erence point was used for the solution. After nding the solution for the larger Q value (and also for the initial solution), the variance of the item cost for the solution was estimated by evaluating one million points with a ve thousand point initialization. The cost used for the mean cost was that found from the one million point evaluation of the z solution described above. This procedure was performed for Q values of zero, ve, ten, fteen, twenty, twenty- ve, twenty-seven, twenty-eight, and twenty-nine.
6.2 Results for Variable Minimum Order Quantities In Figures 1 - 3 we see the mean/standard deviation frontier formed by increasing the minimum order quantity, Q, while keeping the customer's total cost constant, for each of the three di erent exchange rate and pass through models. In each plot the value of Q increases as we move left; from a value of zero through the values of 5, 10, 15, 20, 25, 27, and 28. This causes the simulated estimates of the expectation and standard deviation of income per period to decrease. Looking to Figure 1, we see that in the random walk model with zero pass through and Q = 0 we have a mean income of $599:36 with a standard deviation of $324:76, the point in the upper right-hand corner of the graph. As Q changes a concave mean/standard 26
Mean 600
500
400
Pass Through
300
None Partial Full
200
100
200
300
STD
Figure 2: Mean Reverting Model with Di erent Q Values. deviation frontier is formed, which indicates that we can decrease our risk with virtually no impact on our expected income, for relatively small values of Q. For example, for this case with Q = 15, the mean income is reduced by only two dollars, to $597:41, while the deviation decreases by slightly more than ten percent, to $290:48. This holds true for all types of pass through and exchange rate models we considered. Once we increase Q to a level above twenty, the expected income begins to decrease rapidly. Note in particular that the nal (leftmost) three points on each of the nine plots are for the Q values of 25, 27 and 28. (For the momentum model with full pass through, Q = 28 does not appear. The increase in holding/penalty costs for this case is such that in order to keep the manufacturer's costs level, the supplier would have to have negative price levels.) For every exchange rate model and pass through level, the decrease in income is greater in this span of three units than in the previous span of 25. For the momentum model, this drives the income to a level below the bottom of the graph for the partial and full pass through cases. This is caused by the proximity of the minimum order quantity and the expected demand, which equals thirty. Having these two parameters so close increases instability in the system, which drives up the customer's inventory/penalty costs. To compensate for this the producer must reduce his unit price considerably, which cuts into his pro ts. Turning to individual exchange rate models, one is struck rst by the insensitivity to pass through which the mean-reverting model demonstrates, as seen by the virtually identical 27
Mean 600
500
400
Pass Through
300
None Partial Full
200
100
200
300
STD
Figure 3: Momentum Model with Di erent Q Values. paths in Figure 2, traced by increasing Q for each of the three pass through values. This is as one would expect, as this model spends the majority of its time in the mean or middle state, where pass through has no e ect. This is in contrast to the momentum model, seen in Figure 3, where the proportion of time the rates spend in the extremely high or low state is larger. In this case a de nite pattern of domination by the zero pass through case can be seen: Both income and variance tend to be higher for equal values of Q when compared to the other two pass through models. This becomes acute as Q grows large, due to the inherent instability in the momentum model. This same e ect is also apparent, to a lesser degree, in the random walk model (Figure 1). We believe that this increased income for the zero pass through case follows from the insensitivity of end demand to exchange rate, which forces the customer to maintain high stocks even when the price he must pay the producer is high. If he could expect a decrease in demand he could buy less at these times when the exchange rate didn't favor him, which would decrease his costs and the producer's income. This is seen as pass through increases. We hypothesize that the di erence in standard deviation simply re ects the larger mean income (and hence order size as well).
Remarks.
(1) We also experimented to see how Q a ects the MNC's cost variance. We found that the change in variance is not monotone: there is an interval where the variance goes down because the increase in holding cost variance is more than o set by decrease in 28
purchasing cost variance. A future work explores in detail various risk attitudes of the MNC. (2) To isolate the e ects of exchange rate uctuations on supplier's revenue variance, we also experimented with the special case when the end demand has no variance in any period, but depends on the exchange rate. Not surprisingly, the supplier's variance comes down. The bene t of Q at this lower variance remains, but is not as signi cant as when there is end demand variance.
6.3 Comparison of Exchange Rate Models Using Figures 1-3 one may also compare the di erent exchange rate models for the same pass through parameter. In general, they indicate that less volatile the exchange rate model, the better it is for the producer: the mean/variance trade-o curve is higher for the mean reverting model as compared to the momentum model. The bene t of Q, however, appears uniform: it helps reduce the variance both in the momentum model as well as the mean reverting model. One exception of note is for the zero pass through case, where the simulated income is often greatest for the momentum model. This again is due, we believe, to the fact that in this case the customer is often forced to order when the unit price has been in ated by exchange rates. This drives his costs, and the producer's pro ts, up on average.
6.4 Variable Minimum Order Quantities and Capacities Figures 4 and 5 show the e ect varying capacity and the minimum order quantity have on the momentum model with zero and full pass through, respectively. We have altered the unit costs to keep the customer's total average cost constant across all points in each gure. The di erent lines correspond to di erent capacities, evaluated for values of Q which increase as before, as we move right to left. From these graphs it is apparent that while increasing capacity does increase the expected income, it likewise has the e ect of increasing the producer's risk. Looking rst to Figure 4, where the darker circle in the plot of C = 40 represents two data points, we see that all three plots have the same general shape. Furthermore, while increasing the capacity from forty to fty units can increase the expected income 29
Mean 600
500
400
Capacity
300
40 50 60
200
100
200
300
STD
Figure 4: Momentum Model, No Pass Through, with Di erent Q and C Values.
Mean 600
500
400
Capacity
300
40 50 60
200
100
200
300
STD
Figure 5: Momentum Model, Full Pass Through, with Di erent Q and C Values.
30
appreciably, from $534:40 to $605:62 for Q = 0, it likewise increases the standard deviation of income from $235:95 to $337:54. So as one would expect, the increase in expected pro ts does not come without a cost, that of augmented risk. This e ect is also present when we increase from fty to sixty units of capacity; the incremental increase in risk is smaller, but so too are the supplemental pro ts. This exposes an often overlooked e ect of increasing capacity: By giving the customer a greater choice in the amount ordered, the producer is exposing himself to greater uctuation in cash ow, and thus greater risk. It must be remembered that this is in addition to any risk undertaken due to capital expenditure on the producer's part to increase capacity. So in e ect, the uncertainty incurred through the increase of capacity is twofold; that brought on by the probable investment necessary to upgrade or expand production, coupled with the increase in exibility a orded to the customer. In such a situation it becomes vital for the producer to hedge against such cash ow
uctuations, especially when considering the larger debt load he may have incurred. Contractual minimum order quantities can play an extremely important role in this e ort. By simultaneously increasing the minimum order quantity with the capacity, the producer is able to expand his production while keeping his cash ow variance under control. This allows him to give his customers a greater sense of security through a greater ceiling on capacity, while also exacting a concession from them which will dampen the producer's own risk. This point is reinforced by the data illustrated in Figure 5, where the bene ts which can be reaped by securing an agreement for a minimum order quantity are clearly shown. If the producer increases his capacity from forty with no minimum order quantity to fty while also securing an agreement for a minimum order quantity of 21 units (shown as a square on the plot of C = 50 in Figure 5), he can increase his expected income from $478:93 to $556:37 while having his standard deviation decrease from $207:63 to $201:10. He is thus able to improve both his expected cash ow and his risk position, without passing on any greater expected costs to his customer. This type of a technique appears to be more of an option for the full pass through model, where there is a greater vertical separation between the three plots than in the zero pass through case. Indeed, the manufacturer could attain the same dual bene t by increasing capacity from fty to sixty, and the minimum order quantity from zero to fteen. 31
Mean 600
500
400
Capacity
300
40 50 60
200
100
200
300
STD
Figure 6: Mean Reverting Model, Full Pass Through, with Di erent Q and C Values. To test the theory that the pass through was a signi cant factor in determining the bene ts of this strategy we experimented with the most stable of our models, that of mean reverting exchange rates. In Figure 6 we see that once again, the bene t of increasing the minimum order quantity along with capacity is pronounced for the high pass through case. In this gure the dominating solution, denoted again with a square, is for Q = 22. It would appear then that the variability of the full pass through model makes the increase in capacity quite valuable to the customer, who in exchange for this increased capacity might be willing to accept a larger increase in unit price. This in turn grants the producer a substantial increase in pro ts.
32
7 Conclusions In this paper we have sought to study international supply contracts, in the hopes of providing a way for smaller, less nancially exible companies to reduce variances in their cash
ow caused by exchange rate uctuations. We propose the idea of contractually setting a minimum order quantity per period, which in e ect transfers some of the uncertainty to the customer in exchange for a reduction in unit price. Our experiments show that this can be a very e ective procedure, appreciably reducing risk with only a slight decrease in average income. We also analyzed the behavior of the di erent exchange rate models, from which it appears that for nearly all cases, models with greater inherent stability favor the manufacturer. These would be, in descending order, the mean reverting, random walk, and then the momentum model. An exception to this comes in the case of zero pass through, where for relatively small values of Q the inherent instability of the momentum model can drive the producer's pro ts up. Finally, we illustrated the e ect varying capacity and minimum order quantities together can have on the mean and standard deviation of the manufacturer's income. It was shown that by securing an agreement for a larger minimum order quantity while at the same time increasing capacity, the producer is able to o set some of the increased risk inherent in adding capacity. Furthermore, in certain situations it might be possible for the producer to increase his expected income while at the same time reducing his risk. There are many further problems which can be considered: Multiple suppliers, using a vehicle currency (which is neither the supplier's or customer's domestic currency), multiple products, lead times, specifying that a total amount must be bought over larger intervals (rather than in each period) and the like.
A Transition Matrices A.1 Random Walk
For the random walk the transition matrix was: 2 :5 :5 0 0 66 :5 0 :5 0 66 0 :5 0 :5 64 0 0 :5 0 0 0 0 :5 33
0 0 0 :5 :5
3 77 77 : 75
A.2 Mean Reverting For the mean reverting model the transition matrix was:
2 66 66 64
3 :2 :8 0 0 0 :1 :3 :6 0 0 777 0 :1 :8 :1 0 77 : 0 0 :6 :3 :1 5 0 0 0 :8 :2
A.3 Momentum For the momentum model the transition matrix was: 2 :8 :2 0 0 0 66 :6 :3 :1 0 0 66 0 :5 0 :5 0 64 0 0 :1 :3 :6 0 0 0 :2 :8
3 77 77 : 75
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