A two-level hedging point policy for controlling a manufacturing system with time-delay, demand uncertainty and extra capacity

European Journal of Operational Research 176 (2007) 1528–1558

www.elsevier.com/locate/ejor

Production, Manufacturing and Logistics

A two-level hedging point policy for controllinga manufacturing system with time-delay, demand

uncertainty and extra capacity

Felix T.S. Chan a,*, Zheng Wang b, Jie Zhang c

a Department of Industrial and Manufacturing Systems Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong, Chinab Research Institute of Automation, Southeast University, Nanjing, Jiangsu 210096, PR China

c Computer Integrated Manufacturing Institute, Shanghai Jiao Tong University, Shanghai 200030, PR China

Received 26 June 2004; accepted 20 October 2005Available online 18 January 2006

Abstract

This paper focuses on the production control of a manufacturing system with time-delay, demand uncertainty andextra capacity. Time-delay is a typical feature of networked manufacturing systems (NMS), because an NMS is com-posed of many manufacturing systems with transportation channels among them and the transportation of materialsneeds time. Besides this, for a manufacturing system in an NMS, the uncertainty of the demand from its downstreammanufacturing system is considered; and it is assumed that there exist two-levels of demand rates, i.e., the normal oneand the higher one, and that the time between the switching of demand rates are exponentially distributed. To avoid thebacklog of demands, it is also assumed that extra production capacity can be used when the work-in-process (WIP)cannot buffer the high-level demands rate. For such a manufacturing system with time-delay, demand uncertaintyand extra capacity, the mathematical model for its production control problem is established, with the objective of min-imizing the mean costs for WIP inventory and occupation of extra production capacity. To solve the problem, a two-level hedging point policy is proposed. By analyzing the probability distribution of system states, optimal values of thetwo hedging levels are obtained. Finally, numerical experiments are done to verify the effectiveness of the control policyand the optimality of the hedging levels.� 2005 Published by Elsevier B.V.

Keywords: Control; Production; Networked manufacturing systems; Time-delay; Hedging point policy

0377-2217/$ - see front matter � 2005 Published by Elsevier B.V.doi:10.1016/j.ejor.2005.10.032

* Corresponding author.E-mail address: [email protected] (F.T.S. Chan).

mailto:[email protected]

F.T.S. Chan et al. / European Journal of Operational Research 176 (2007) 1528–1558 1529

1. Introduction

Time-delay is a distinctive feature of the dynamics of a networked manufacturing system (NMS). AnNMS is composed of many manufacturing systems that are geographically distributed and linked throughtransportation or distribution channels, which form a supply chain network. Time-delay exists in an NMS,because the transportation of materials from one manufacturing system to another will take a period oftime. This is different from a shop floor manufacturing system, in which the time-delay between neighboredmachines is usually neglected, though it exists, because the time spent on material flowing is usually shorterthan the time spent on machining.

Most of the existing researches on the NMS usually focus on its system architecture and the coordina-tion mechanism and less on its dynamics. For example, Montreuil et al. (2000) proposed the NetMan stra-tegic framework for enabling networked manufacturing. The two fundamental factors of this frameworkare the NetMan centers and the NetMan networks. The former are building blocks of the whole frameworkand missions are assigned to them to complete business processes. The latter is constructed and organizedby the former in a flexible and dynamic way; and the coordination rules and mechanisms are designed toguarantee all the NetMan center work cooperatively (Montreuil et al., 2000; Frayret et al., 2001; Cloutieret al., 2001). Wang et al. (2005) propose an autonomous system architecture integrating both the global andlocal coordination based on the mobile agent technology. Actually, the mobile agent technology has beensuccessfully used in the negotiation between the distributed intellectual manufacturing systems (Shin andJung, 2004). Some coordination mechanisms applied to supply chain networks, such as the multi-agentapproach, the genetic optimization method and the analysis hierarchical process, can also be used to coor-dinate the production, ordering and delivery process in the NMS (Chan and Chan, 2004; Chan et al., 2004).

Compared to the coordination mechanisms, the dynamics and control of the NMS has not been paidmuch attention to. In a previous paper, we establish a hybrid Petri-nets model of the NMS, which cansimultaneously describe its continuous part, i.e., the production process, and its discrete part, i.e., the order-ing and delivery process (Wang et al., 2005). However, this is just an original work since the control of theNMS is not included. In this paper, the production control for a single manufacturing system in the NMS isresearched. The reason why we focus on a single manufacturing system is that the global and centralizedcontrol of such a complex system is very difficult to realize. However, the control on a single manufacturingsystem is practical and tractable.

Many results have been obtained on the issue of shop floor control, which can be extended to the NMSwith the consideration of time-delay. For the control of shop floor manufacturing systems, the dynamicalprogramming method is usually used (Gershwin, 1994; Maimon et al., 1998). In this method, the Hamilton-Jacob-Bellman (HJB) equation can be obtained and then the hedging point control policy is derived formthe HJB equation. The original work by Kimemia and Gershwin (1983) presented the general form of hedg-ing point policy. Based on that, a lot of researches have been done on the control of different types of man-ufacturing systems. For example, for a single machine and single part-type manufacturing system, thehedging point control has a very concise formulation (Gershwin, 1994). Akella and Kumar (1986) and Bie-lecki and Kumar (1988) obtained the closed form solution to the hedging point optimization problem forthis simplest system based on the assumption of Markovian property of the transition of machine states,i.e., operational or failure. Furthermore, Hu and Xiang (1993) proposed an improved method of optimizingthe hedging point by proving the equivalency of the single part-type and single machine manufacturing sys-tem and the M/M/1 queuing system. For multiple machines and single part-type manufacturing systems,van Ryzin et al. (1993) researched the optimal control problem for the tandem production line composedof two machines and proposed an approximate two-boundary policy. For this type of system, Gershwin(1997, 2000) proposed a kind of hedging point policy that assigns an independent hedging point for eachmachine and takes the constraints from the production rates of its upstream and downstream machines intoconsideration. For the single machine and multiple part-types manufacturing system, the switching curve

1530 F.T.S. Chan et al. / European Journal of Operational Research 176 (2007) 1528–1558

control policy and the prioritized hedging point policy is often used (Srivastan, 1993; Perkins and Srikant,1997). For the former, Srivastan and Dallery (1998) analyzed the stability condition of the manufacturingsystem under this control; for the latter, Perkins and Srikant (1998) proposed the sufficient and necessaryconditions for the optimality of the just-in-time production policy. For the situation where the demand rateis uncertain, Feng and Xiao (2002) and Perkins and Srikant (2001) also proved that the hedging point pol-icy is optimal.

Besides the time-delay, two more features will be considered in this paper, i.e., the demand uncertaintyand the using of extra production capacity. The former always exists and the latter is for avoiding the back-log of demands. Usually, a manufacturing system in the NMS can only know the demand rates from itsimmediately downstream manufacturing systems. Here, for simplicity, we assume that a manufacturing sys-tem has only one immediately downstream manufacturing system and only the single product-type case isconsidered. The demand rate is often uncertain. In this paper, we consider a simple situation wherethe demand rate of the immediately downstream manufacturing system has only two-levels: usually thedemand rate is on the low level, but sometimes it is on the high level. Both levels are constants, and thetime between the switching of demand rates is exponentially distributed. Usually, the production capacitythat a manufacturing system assigns to a certain product is only sufficient to meet the low demand rate. Soit can be called the normal capacity. When the demand rate is on the high level and cannot be buffered bythe work-in-process (WIP), the system can utilize extra production capacity in order to avoid the backlog ofdemands. Of course, the occupation of extra production capacity will bring extra cost. On the other hand, ahigher WIP buffering the high demand rate will bring more inventory cost. So there is a tradeoff between thetwo types of costs. The objective of the production control is to minimize the total costs for the WIP andthe occupation of extra capacity.

For such a manufacturing system with time-delay, demand uncertainty and extra capacity, the existingcontrol policy of the shop floor manufacturing system cannot be directly applied to the NMS because oftheir different dynamics. Therefore, a practical two-level hedging point policy will be proposed, where thetwo hedging levels are to deal with the two demands rates by using extra capacity, and a simple methodfor forecasting system states are applied to deal with the time-delay.

The remainder of this paper is organized as follows. In Section 2, we propose a mathematical formula-tion for the production control problem of a manufacturing system with time-delay, demand uncertaintyand extra capacity, and a new two-level hedging point policy is proposed to solve this problem. In Section3, the probability distributions of the system states are analyzed and then the values of the hedging levelsare optimized. In Section 4, the results of some numerical experiments are presented to verify the effective-ness of the control policy. Section 5 is the conclusion part.

2. Production control of a single manufacturing system in the NMS

In this paper, only the single product-type case is researched. For a manufacturing system in the NMS,let x(t) denote the level of its WIP or backlog of demands at time t; u(t), its production rate at time t; L, thetime-delay between delivery of the materials from the manufacturing system and the receiving of the mate-rials by its immediately downstream manufacturing system; and D(t), the demand rate from its immediatelydownstream system at time t. So the dynamic equations of this manufacturing system is

xðt þ LÞ ¼xðtÞ þ

R t0

uðsÞds�R tþL

t DðsÞds; if t < L;

xðtÞ þR t

t�L uðsÞds�R tþL

t DðsÞds if t P L;

(

where x(t) P 0 means the WIP level of the manufacturing system; and x(t) < 0 means the backlog of thedemands from its immediately downstream manufacturing system.


In this paper, we will analyze the system’s performance on its steady state. Therefore, only the situationthat t is much larger than L is considered. So the above equation can be simplified as follows:

xðt þ LÞ ¼ xðtÞ þZ t

t�LuðsÞds�

Z tþL

tDðsÞds. ð1Þ

Note that the demand rate is uncertain. Suppose that the demand rate has two-levels. The first level is thenormal demand rate denoted by D1 and the second one is the high demand rate denoted by D2, where D1

and D2 are constants satisfying D2 > D1, and suppose that the time between demand rate switching is expo-nentially distributed.

From Eq. (1), it can be seen that the control after time t, i.e., u(s) for s > t, does not influence the state attime t + L, i.e., x(t + L). This is the influence of the time-delay. For time-delay systems, the receding hori-zon control is often used (Kwon et al., in press). However, in this research, a different control policy will beintroduced including a simple method for forecasting future states.

Suppose that the materials that are supplied to the manufacturing system are always sufficient, i.e., noinput starvation will happen. Then the constraint on the production rate u(t) is the production capacity ofthe system. The production capacity that can be assigned to the product is also on two-levels. The first oneis the normal production capacity, which can be used if the demand rate is on the low level. The second oneis the extra production capacity, which can be borrowed from outside and assigned to the product if thedemand rate is on the high level and cannot be buffered by the WIP Inventory. The purpose of using extraproduction capacity is to avoid the backlog of demands. Let l1 and l2 denote the maximum productionrates corresponding to the normal and extra production capacities, respectively. So the constraint can bedescribed as follows:

uðtÞ 6 l1; if only the normal production capacity is used; ð2ÞuðtÞ 6 l2; if the normal and extra production capacities are used; ð3Þ

where D1 < l1 < D2 < l2 holds.The costs of the production process are composed of two parts: the first part is the costs for the WIP; and

the second part is the cost for the occupation of extra production capacity. Note that if the extra capacity isused, its cost must be higher than that for normal capacity.

Because of the use of extra capacity, the backlog of demands can be avoided, i.e., x(t) P 0 holds. Let c

denote the costs for one unit of WIP and k the cost for one time unit of occupying extra capacity. Letg[x(t), u(t)] be the cost function as follows:

g½xðtÞ; uðtÞ� ¼ g1½xðtÞ� þ g2½uðtÞ� ¼ cxðtÞ þ k sgn½uðtÞ � l1�; ð4Þ

where sgn[u(t) � l1] = 1 if u(t) > l1 and sgn[u(t) � l1] = 0 if u(t) 6 l1. Then the objective function for theoptimal production control of the manufacturing system is

limT!1

1

TEZ T

0

g½xðsÞ; uðsÞ�dsjxð0Þ;Dð0Þ� �

; ð5Þ

where [x(0), D(0)] is the system’s initial state and T is a large positive number denoting the end of the timeinterval to be considered. Therefore, the optimal production control problem of this manufacturing systemcan be formulated as follows:

Min limT!1

1

TEZ T

0

g½xðsÞ; uðsÞ�dsjxð0Þ;Dð0Þ� �

s.t. Eqs. (1)–(3).


Because of the time-delay, the control at time t, i.e., u(t), will influence the system state after a period oftime, namely, at time t + L. Therefore, according to the principle of the state feedback control, the controlu(t) depends on the forecasted future state at time t + L, which is denoted by x̂ðt þ LÞ. In this paper, weassume that the time-delay is always smaller than the time between the switching of demand rates. Thisassumption is reasonable because the speed of material transportation is usually faster than that of demandfluctuation. So a simple forecasting method is applied by assuming D(s) = D(t) for all s 2 (t, t + L]. There-fore, according to Eq. (1) we have

x̂ðt þ LÞ ¼ xðtÞ þZ t

t�LuðsÞds� LDðtÞ. ð6Þ

Hedging point policy is simple and practical so that it can be conveniently applied in controlling the pro-duction rate. There exist many different formulations of the hedging point policy. Here, since there are twodemand rate levels and two corresponding production rate levels, a two-level hedging point policy is definedas follows:

uðtÞ ¼

l1; if DðtÞ ¼ D1 and x̂ðt þ LÞ < Z1;

D1; if DðtÞ ¼ D1 and x̂ðt þ LÞ ¼ Z1;

0; if DðtÞ ¼ D1 and x̂ðt þ LÞ > Z1;

l2; if DðtÞ ¼ D2 and x̂ðt þ LÞ < Z2;

D2; if DðtÞ ¼ D2 and x̂ðt þ LÞ ¼ Z2;

l1; if DðtÞ ¼ D2 and x̂ðt þ LÞ > Z2;

8>>>>>>>>>>>><>>>>>>>>>>>>:

ð7Þ

where Z1 and Z2 are the hedging levels corresponding to D1 and D2, respectively, and Z2 < Z1.Zhang et al. (2001) and Tan (2002) also proposed double hedging point policy for controlling the man-

ufacturing systems with demand uncertainty. But their control policies are different from ours. For exam-ple, in Eq. (21), the high hedging level is for the low demand rate, but in Tan (2002) the high hedging level isfor the high demand rate. This is because the problem we deal with is different from that in Tan (2002)—there exists extra capacity that can be used in the problem defined in this paper.

3. Optimization of the hedging point

The state of a single manufacturing system in an NMS is composed of two parts. The first part is theWIP level and the second part is the demand rate. For the first part, i.e., x(t), its dynamics is describedby Eq. (1). For the second part, i.e., D(t), we have assumed that the time between demands rate switchingis exponentially distributed. We also suppose that the mean time between D2 changing into D1 and D1

changing into D2 is p, and that the mean time between D1 changing into D2 and D2 changing into D1 isq. Therefore, in a time interval dt, the probability that the demand state changes from D1 to D2 is aboutpdt and the probability that the demand state changes from D1 to D2 is about q dt. Now we analyze thebehavior of the WIP when it is in the steady state.

3.1. The WIP in the steady state

Suppose that at time ti, i = 1, 3, 5, . . . , 2n � 1, . . ., the demand rate changes from D1 to D2; and that attime ti, i = 2, 4, 6, . . . , 2n, . . ., the demand rate changes from D2 to D1. And suppose that L < t2n � t2n�1 andL < t2n+1 � t2n. According to Eqs. (1) and (6), it can be obtained that


x̂ðt þ LÞ � xðt þ LÞ ¼Z tþL

tDðsÞds� LDðtÞ; ð8Þ

which can be rewritten as

x̂ðtÞ � xðtÞ ¼Z t

t�LDðsÞds� LDðt � LÞ. ð9Þ

If t 2 [t2n+1, t2n+1 + L), then we have (1) t � L 2 [t2n+1 � L, t2n+1); (2) D(t � L) = D1; (3) D(s) = D1 ifs 2 [t � L, t2n+1) and (4) D(s) = D2 if s 2 [t2n+1, t]. Therefore, according to Eq. (9), we have

x̂ðtÞ � xðtÞ ¼Z t2nþ1

t�LD1 dsþ

Z t

t2nþ1

D2 ds� LD1 ¼ ðD2 � D1Þðt � t2nþ1Þ. ð10Þ

If t 2 [t2n+1 + L, t2n+2), then we have (1) t � L 2 [t2n+1, t2n+2 � L); (2) D(t � L) = D2 and (3) D(s) = D2

for all s 2 [t � L, t] � [t2n+1, t2n+2). Therefore, according to Eq. (9), we have

x̂ðtÞ � xðtÞ ¼Z t

t�LD2 ds� LD2 ¼ 0. ð11Þ

Similarly, it can be obtained that

x̂ðtÞ � xðtÞ ¼ ðD1 � D2Þðt � t2nþ2Þ; 8t 2 ½t2nþ2; t2nþ2 þ LÞ; ð12Þx̂ðtÞ � xðtÞ ¼ 0; 8t 2 ½t2nþ2 þ L; t2nþ3Þ. ð13Þ

For the WIP state x(t) and the forecasted WIP state x̂ðtÞ, the following Theorem holds:

Theorem 1. In the steady state, the average values of forecasted and actual WIP level are equal, i.e., for T0

that is large enough,

limT!1

1

T � T 0

Z T

T 0

½x̂ðtÞ � xðtÞ�dt ¼ 0. ð14Þ

Proof. See Appendix A. h

Therefore, we can substitute x(t) by x̂ðtÞ when computing the mean of production cost (i.e., the objectivefunction) based on analyzing the probability distribution of x̂ðtÞ. From Eqs. (10) and (12), it can be seenthat x̂ðtÞ is discontinuous at time t2n+1 + L and t2n+2 + L.

Since the purpose of using extra capacity is to avoid the backlog of demands, it is reasonable to supposethat Z2 P 0 and Z1 � Z2 > L(D2 � D1). Based on that, the following theorem will guarantee the non-neg-ativity of the forecasted WIP level.

Theorem 2. Suppose that Z1 � Z2 > L(D2 � D1), L < t2n+1 � t2n and L < t2n+2 � t2n+1. Then in the steady

state, the forecasted WIP level exists between Z2 and Z1, i.e., Z2 6 x̂ 6 Z1. Furthermore, in the steady state, if

the demand rate is D1, then the forecasted WIP level meets Z2 þ LðD2 � D1Þ < x̂ < Z1; if the demand rate isD2, then the forecasted WIP level meets Z2 < x̂ < Z1 � LðD2 � D1Þ.

Proof. See Appendix B. h

3.2. Probability distribution of system states

Let f ½x̂ðtÞ;Dðt � LÞ� denote the probability density function of the state ½x̂ðtÞ;Dðt � LÞ�. ForZ2 þ LðD2 � D1Þ < x̂ðtÞ < Z1 and D(t � L) = D1, it satisfy the following partial differential equation:


of ½x̂ðtÞ;D1�ox̂ðtÞ ðl1 � D1Þ ¼ �f ½x̂ðtÞ;D1�p þ f ½x̂ðtÞ � LðD2 � D1Þ;D2�q. ð15Þ

For Z2 < x̂ðtÞ < Z1 � LðD2 � D1Þ and D(t � L) = D2, it satisfy the following partial differential equation:

of ½x̂ðtÞ;D2�ox̂ðtÞ ðl1 � D2Þ ¼ �f ½x̂ðtÞ;D2�qþ f ½x̂ðtÞ þ LðD2 � D1Þ;D1�p. ð16Þ

Please see Appendix C(1) and (2) for the derivation of Eqs. (15) and (16), respectively. We can suppose thatf ½x̂;Di� has the form of

f ½x̂;Di� ¼ Ai expfax̂g; i ¼ 1; 2. ð17Þ
Substitute Eq. (17) to Eqs. (15) and (16) and it can be obtained that
A1aðl1 � D1Þ ¼ �A1p þ A2 expf�aLðD2 � D1Þgq; ð18ÞA2aðl1 � D2Þ ¼ �A2qþ A1 expfaLðD2 � D1Þgp. ð19Þ

From Eqs. (18) and (19), it can be obtained that

a ¼ qD2 � l1

� pl1 � D1

; ð20Þ

A2 ¼ K1A1 ¼ A1

l1 � D1

D2 � l1

expfaLðD2 � D1Þg; ð21Þ

where a and K1 ¼ l1�D1

D2�l1expfaLðD2 � D1Þg are constants. Since l1 > D1 and D2 > l1, K1 > 0 holds.

For x̂ðtÞ ¼ Z1 and D(t � L) = D1, the probability that the system state is at (Z1, D1) is

P ½x̂ ¼ Z1;D1� ¼A1ðl1 � D1Þ

pexpfaZ1g. ð22Þ

For x̂ðtÞ ¼ Z2 and D(t � L) = D2, the probability that the system state is at (Z2, D2) is

P ½x̂ ¼ Z2;D2� ¼K1A1ðD2 � l1Þ

qexpfaZ2g. ð23Þ

Please see Appendix C(3) and (4) for the derivation of Eqs. (22) and (23), respectively. Obviously,P ½x̂ ¼ Z1;D2� ¼ P ½x̂ ¼ Z2;D1� ¼ 0 holds.

Eqs. (17), (22) and (23) must satisfy the following condition:

P ½x̂ ¼ Z1;D1� þ P ½x̂ ¼ Z2;D2� þZ Z1

Z2þLðD2�D1ÞA1 expfax̂gdx̂þ

Z Z1�LðD2�D1Þ

Z2

K1A1 expfax̂gdx̂ ¼ 1. ð24Þ

Substitute Eqs. (17), (22) and (23) into Eq. (24), it can be obtained that

A1½H 1 expfaZ1g þ H 2 expfaZ2g� ¼ 1; ð25Þ
where
H 1 ¼l1 � D1

pþ 1

aþ K1

aexpf�aLðD2 � D1Þg

and

H 2 ¼K1ðD2 � l1Þ

q� 1

aexpfaLðD2 � D1Þg �

K1

a

are constants. Then it can be obtained that

A1 ¼ ½H 1 expfaZ1g þ H 2 expfaZ2g��1. ð26Þ


3.3. Mean value of production costs

The production costs consist of two parts: the first part is the cost for WIP; and the second part is thecost for the occupation of extra production capacity. Note that (1) according to Theorem 2, in the steadystate, if Z2 P 0, no backlog of demands will exist; (2) in the steady state the extra production capacity isused only when the system state is (Z2, D2). Therefore the mean value of production costs is

E½gðx̂; uÞ� ¼ E½g1ðx̂Þ þ g2ðuÞ� ¼ E½cx̂þ ksgnðu� l1Þ�

¼ cZ Z1

Z2þLðD2�D1Þx̂f ðx̂;D1Þdx̂þ c

Z Z1�LðD2�D1Þ

Z2

x̂f ðx̂;D2Þdx̂þ cZ1Pðx̂ ¼ Z1;D1Þ

þ cZ2P ðx̂ ¼ Z2;D2Þ þ kP ðx̂ ¼ Z2;D2Þ¼ ½H 1 expfaZ1g þ H 2 expfaZ2g��1½ða1Z1 þ b1Þ expfaZ1g þ ða2Z2 þ b2Þ expfaZ2g�; ð27Þ

where

a1 ¼ c1

aþ K1 expf�aLðD2 � D1Þg

aþ ðl1 � D1Þ

p

� �¼ cH 1;

a2 ¼ cK1ðD2 � l1Þ

q� 1

aexpfaLðD2 � D1Þg �

K1

a

� �¼ cH 2;

b1 ¼ �cK1 expf�aLðD2 � D1Þg

aLðD2 � D1Þ þ

1

a

� �� c

a2

and

b2 ¼ �c expfaLðD2 � D1Þg

aLðD2 � D1Þ �

1

a

� �þ cK1

a2þ kK1ðD2 � l1Þ

q

are constants.

3.4. Optimization of hedging point

In the steady state, minimizing the objective function (6) is equivalent to minimizing E½gðx̂; uÞ�. ForE½gðx̂; uÞ�, the following theorem holds.

Theorem 3. The hedging point ðZ�1; Z�2Þ that can minimize E½gðx̂; uÞ� must satisfy Z�2 ¼ 0.

Proof. See Appendix D. h

According to Theorem 3, Eq. (27) becomes

E½gðx̂; uÞ� ¼ ða1Z1 þ b1Þ expfaZ1g þ b2

H 1 expfaZ1g þ H 2

. ð28Þ

Then

oE½gðx̂; uÞ�oZ1

¼ expfaZ1gH 1a1 expfaZ1g þ H 2aa1Z1 þ H 2a1 þ aðH 2b1 � H 1b2Þ

½H 1 expfaZ1g þ H 2�2. ð29Þ

Since a1 = cH1 and a2 = cH2, the above equation becomes


¼ expfaZ1gcH 21

expfaZ1g þ aH 2Z1=H 1 þ H 2=H 1 þ aðH 2b1 � H 1b2Þ=cH 21

½H 1 expfaZ1g þ H 2�2.


Let

G3ðZ1Þ ¼ expfaZ1g þaH 2

H 1

Z1 þH 2

H 1

þ ac

H 2b1

H 21

� b2

H 1

� �. ð30Þ

Then we have oG3(Z1)/oZ1 = a exp{aZ1} + aH2/H1 and o2G3ðZ1Þ=oZ2

1 ¼ a2 expfaZ1g > 0. So oG3(Z1)/oZ1

increases monotonically.If oG3ðZ1Þ=oZ1jZ1¼LðD2�D1Þ P 0, then oG3(Z1)/oZ1 P 0 holds for all the Z1 > L(D2 � D1) (see Fig. 1a),

i.e., G3(Z1) increases monotonically. So if G3(L(D2 � D1)) > 0, then G3(Z1) P 0 holds (i.e.,oE½gðx̂; uÞ�=oZ1 > 0 holds) for all Z1 > L(D2 � D1) (see Fig. 2a). Therefore E½gðx̂; uÞ� increases monotoni-cally with respect to Z1 and ðZ�1; Z�2Þ ¼ ðLðD2 � D1Þ; 0Þ is the optimal hedging point (see Fig. 2b). If

Fig. 1a. oG3ðZ1Þ=oZ1jZ1¼LðD2�D1Þ P 0.

Fig. 1b. oG3ðZ1Þ=oZ1jZ1¼LðD2�D1Þ < 0.

Fig. 2b. E½gðx̂; uÞ� increases monotonically with respect to Z1.

Fig. 2a. oG3ðZ1Þ=oZ1jZ1¼LðD2�D1Þ P 0 and G3(L(D2 � D1)) > 0.


G3(L(D2 � D1)) < 0, then there exists a unique Z�1 such that G3ðZ�1Þ ¼ 0, and thereforeoE½gðx̂; uÞ�=oZ1jZ1¼Z�

1¼ 0 (see Fig. 3a). The Z�1 can be obtained by solving the equation G3(Z1) = 0 by the

Newton iteration algorithm. Then ðZ�1; Z�2Þ ¼ ðZ�1; 0Þ is the optimal hedging point (see Fig. 3b).

If oG3ðZ1Þ=oZ1jZ1¼LðD2�D1Þ < 0, there exists Z��1 ¼lnð�H2=H1Þ

a > LðD2 � D1Þ such that oG3ðZ1Þ=oZ1jZ1¼Z��i¼ 0

in light of the monotony of oG3(Z1)/oZ1 (see Fig. 1b). If Z1 < Z��1 , then oG3(Z1)/oZ1 < 0 and G3(Z1)decreases monotonically; if Z1 > Z��1 , then oG3(Z1)/oZ1 > 0 and G3(Z1) increases monotonically. ThenZ��1 is the point that minimizes G3(Z1).

If G3ðZ��1 ÞP 0, then G3(Z1) P 0 holds (i.e., oE½gðx̂; uÞ�=oZ1 > 0 holds) for all Z1 > L(D2 � D1) (seeFig. 4a). So ðZ�1; Z�2Þ ¼ ðLðD2 � D1Þ; 0Þ is the optimal hedging point (see Fig. 4b).

If G3ðZ��1 Þ < 0 and G3(L(D2 � D1)) < 0, then there exists a unique Z�1ð2 ½Z��1 ;þ1ÞÞ such that G3ðZ�1Þ ¼ 0,and therefore oE½gðx̂; uÞ�=oZ1jZ1¼Z�

1¼ 0 (see Fig. 5a). Z�1 can be obtained by solving the equation G3(Z1) = 0

by the Newton iteration algorithm. Then ðZ�1; Z�2Þ ¼ ðZ�1; 0Þ is the optimal hedging point (see Fig. 5b).

Fig. 3a. oG3ðZ1Þ=oZ1jZ1¼LðD2�D1Þ P 0 and G3(L(D2 � D1)) < 0.

Fig. 3b. oE½gðx̂; uÞ�=oZ1jZ1¼Z�1¼ 0 and ðZ�1; 0Þ minimizes E½gðx̂; uÞ�.


If G3ðZ��1 Þ < 0 and G3(L(D2 � D1)) P 0, then there exists two solutions meeting G3(Z) = 0 (see Fig. 6a).The first solution Z�11 2 ½LðD2 � D1Þ; Z��1 Þ and the second solution Z�12 2 ðZ��1 ;þ1Þ. If Z1 2 ½LðD2 � D1Þ; Z�11Þ,then G3(Z1) > 0, i.e., oE½gðx̂; uÞ�=oZ1 > 0; and if Z1 2 ðZ�11; Z

��1 Þ, then G3(Z1) < 0, i.e., oE½gðx̂; uÞ�=oZ1 < 0.

Therefore Z�11 is a local maximum point of E½gðx̂; uÞ�. If Z1 2 ðZ��1 ; Z�12Þ, then G3(Z1) < 0, i.e.,oE½gðx̂; uÞ�=oZ1 < 0; and if Z1 2 ðZ��12;þ1Þ, then G3(Z1) > 0, i.e., oE½gðx̂; uÞ�=oZ1 > 0. Therefore Z�12 is alocal minimum point of E½gðx̂; uÞ�. Z�12 can be obtained by solving the equation G3(Z1) = 0 in the domainðZ��1 ;þ1Þ by the Newton iteration algorithm. Then Z�1 ¼ LðD2 � D1Þ if E½gðx̂; uÞ�jZ1¼LðD2�D1Þ 6

E½gðx̂; uÞ�jZ1¼Z�12

; Z�1 ¼ Z�12 if E½gðx̂; uÞ�jZ1¼LðD2�D1Þ > E½gðx̂; uÞ�jZ1¼Z�12

; and ðZ�1; Z�2Þ ¼ ðZ�1; 0Þ is the optimal

hedging point (see Fig. 6b).

( )−

Fig. 4a. oG3ðZ1Þ=oZ1jZ1¼LðD2�D1Þ < 0 and G3(Z1) P 0 holds for all Z1 > L(D2 � D1).

( )−

( )[ ]

Fig. 4b. E½gðx̂; uÞ� increases monotonically with respect to Z1.


4. Numerical experiments

In this section, numerical experiments are done to verify the conclusions drawn by theoretical analysis,i.e., the theorems presented in the previous sections.

Example 1. Suppose that the parameters of a manufacturing system with time-delay, demand uncertaintyand extra capacity in a NMS are as follows. The time-delay is L = 5 time units; the low demand rate,D1 = 40.0 products per time unit; the high demand rate, D2 = 90.0 products per time unit; the lowproduction rate, l1 = 50.0 products per time unit; the high production rate, l2 = 100.0 products per timeunit; the probability that the low demand rate changes to the high demand rate in a time unit, p = 0.02; theprobability that the high demand rate changes to the low demand rate in a time unit, q = 0.1.

The actual and forecasted WIP states, i.e., x and x̂, in the time interval [10 001, 10 200] and for(Z1, Z2) = (250.0, 0.0) and (Z1, Z2) = (400.0, 100.0), are shown in Figs. 7 and 8, respectively. Fig. 7 shows

Fig. 5a. G3ðZ��1 Þ < 0 and G3(L(D2 � D1)) < 0.

Fig. 5b. oE½gðx̂; uÞ�=oZ1jZ1¼Z�1¼ 0 and ðZ�1; 0Þ minimizes E½gðx̂; uÞ�.


that in the steady state, the forecasted WIP states are in the domain [Z2, Z1]. That is the conclusion ofTheorem 2. However, Fig. 8 shows that the WIP at some time are below Z1. This is because the error causedby the discrete time index in the simulation, which make the system state change in a discrete way instead ofa continuous way.

According to the theoretical analysis, Z�1 ¼ 250:0 and Z�2 ¼ 0:0 are the optimal hedging point. To verifythis result, 30 simulation experiments are done, each of which corresponds to different hedging point. Theseexperiments are divided into three groups: in the first group, Z2 = 0 is fixed and Z1 changes, i.e., thehedging points (Z1, Z2) are on the line Z2 = 0; in the second group, Z1 and Z2 both change while keepingZ1 � Z2 = L(D2 � D1), i.e., the hedging points (Z1, Z2) are on the line Z1 = Z2 + L(D2 � D1); in the thirdgroup, the hedging points are chosen randomly in the open domain {(Z1, Z2)jZ2 > 0 andZ1 > Z2 + L(D2 � D1)}. For each of the 30 hedging points, 10 simulations are done and the averagevalues of the objective function are calculated. The results are listed in Table 1.

Fig. 6a. G3ðZ��1 Þ < 0 and G3(L(D2 � D1)) P 0.

Fig. 6b. Z�1 ¼ LðD2 � D1Þ if E½gðx̂; uÞ�jZ1¼LðD2�D1Þ 6 E½gðx̂; uÞ�jZ1¼Z�12

; Z�1 ¼ Z�12 if E½gðx̂; uÞ�jZ1¼LðD2�D1Þ > E½gðx̂; uÞ�jZ1¼Z�12

.


Besides Example 1, seven other examples with different parameters are also calculated and simulated.For each example, thirty hedging points are experimented and the results of simulation and the theoreticalanalysis are compared. The results are shown in Tables 2–8.

The results of the eight examples show that the conclusion of Theorem 3 is correct, i.e., the optimalhedging point must be on the line Z2 = 0. Usually, the optimal hedging point is Z�1 ¼ LðD2 � D1Þ andZ�2 ¼ 0. Actually, from Tables 1–8 it can be seen that no matter how the hedging points change, along theline Z2 = 0 (corresponding to the first four columns in each table), along the line Z1 = Z2 + L(D2 � D1)(corresponding to the middle four columns in each table), or in the open domain {(Z1, Z2)jZ2 > 0 andZ1 > Z2 + L(D2 � D1)} (corresponding to the last four columns in each table), ðZ�1; Z�2Þ ¼ ðLðD2 � D1Þ; 0Þis always the optimal one, whose corresponding objective function value is always the minimum. FromTables 1–8 it can also been seen that for a fixed hedging point, relative differences between the theoreticalvalue of the objective function, i.e., Egðx̂; uÞ, and the simulation result are quite small, and so are the

0

50

100

150

200

250

300

1 11 21 31 41 51 61 71 81 91 101 111 121 131 141 151 161 171 181 191

Time

Wor

k in

Pro

cess

x x̂

Fig. 7. The forecasted work-in-process ðx̂Þ and the actual work-in-process (x) of a single manufacturing system if Z1 = 250.0 andZ2 = 0.0.

0

50

100

150

200

250

300

350

400

450

1 11 21 31 41 51 61 71 81 91 101 111 121 131 141 151 161 171 181 191

Time

Wor

k in

Pro

cess

x x̂

Fig. 8. The forecasted work-in-process ðx̂Þ and the actual work-in-process (x) of a single manufacturing system if Z1 = 400.0 andZ2 = 100.0.


Table 1The theoretical and simulation results of the system with parameters L = 5, D1 = 40.0, D2 = 90.0, l1 = 50.0, l2 = 100.0, p = 0.02 and q = 0.1

(Z1, Z2) Egðx̂; uÞ Egðx̂; uÞ Eg(x, u) (Z1, Z2) Egðx̂; uÞ Egðx̂; uÞ Eg(x, u) (Z1, Z2) Egðx̂; uÞ Egðx̂; uÞ Eg(x, u)Theoretical Simulation Simulation Theoretical Simulation Simulation Theoretical Simulation Simulation

(250.0, 0.0) 224.9999 225.7131 225.7159 (250.0, 0.0) 224.9999 225.7131 225.7159 (250.0, 0.0) 224.9999 225.7131 225.7159(260.0, 0.0) 232.8647 233.6382 233.6029 (260.0, 10.0) 234.9999 235.7131 235.7188 (270.0, 10.0) 242.8647 243.5390 243.5442(270.0, 0.0) 240.6306 241.4637 241.4649 (270.0, 20.0) 244.9999 245.7131 245.7155 (290.0, 20.0) 260.6306 261.4459 261.4463(280.0, 0.0) 248.3043 249.2497 249.2177 (280.0, 30.0) 254.9999 255.7131 255.7192 (310.0, 30.0) 278.3043 279.1475 279.1545(290.0, 0.0) 255.8920 256.9081 256.9074 (290.0, 40.0) 264.9999 265.7130 265.7151 (330.0, 40.0) 295.8920 296.8884 296.8862(300.0, 0.0) 263.3994 264.5058 264.4707 (300.0, 50.0) 274.9999 275.7130 275.7196 (350.0, 50.0) 313.3994 314.3954 314.4043(310.0, 0.0) 270.8317 271.9864 271.9841 (310.0, 60.0) 284.9999 285.7130 285.7147 (370.0, 60.0) 330.8317 331.9508 331.9459(320.0, 0.0) 278.1939 279.4044 279.3560 (320.0, 70.0) 294.9999 295.7131 295.7198 (390.0, 70.0) 348.1939 349.2892 349.2989(330.0, 0.0) 285.4903 286.7151 286.7021 (330.0, 80.0) 304.9999 305.7131 305.7145 (410.0, 80.0) 365.4903 366.5897 366.5908(340.0, 0.0) 292.7251 294.0128 293.9683 (340.0, 90.0) 314.9999 315.7131 315.7200 (430.0, 90.0) 382.7251 384.0124 384.0194

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(350.0, 0.0) 308.3333 315.3493 315.3563 (350.0, 0.0) 308.3333 315.3493 315.3563 (350.0, 0.0) 308.3333 315.3493 315.3563(360.0, 0.0) 315.8461 323.1548 323.0328 (360.0, 10.0) 318.3333 325.3493 325.3562 (370.0, 10.0) 325.8461 332.9687 332.9815(370.0, 0.0) 323.3855 330.9670 330.9666 (370.0, 20.0) 328.3333 335.3493 335.3565 (390.0, 20.0) 343.3855 350.9628 350.9687(380.0, 0.0) 330.9529 338.9313 338.8161 (380.0, 30.0) 338.3333 345.3493 345.3562 (410.0, 30.0) 360.9529 368.6687 368.6885(390.0, 0.0) 338.5495 346.7986 346.7961 (390.0, 40.0) 348.3333 355.3793 355.3814 (430.0, 40.0) 378.5495 386.9464 386.9293(400.0, 0.0) 346.1764 354.7229 354.6271 (400.0, 50.0) 358.3333 365.3372 365.3529 (450.0, 50.0) 396.1764 404.3044 404.3798(410.0, 0.0) 353.8344 362.6610 362.6651 (410.0, 60.0) 368.3333 375.2260 375.2524 (470.0, 60.0) 413.8344 423.0096 422.9698(420.0, 0.0) 361.5244 370.7618 370.6572 (420.0, 70.0) 378.3333 385.3750 385.3621 (490.0, 70.0) 431.5244 440.2980 440.3411(430.0, 0.0) 369.2468 378.9135 378.8784 (430.0, 80.0) 388.3333 395.5569 395.5314 (510.0, 80.0) 449.2468 459.1327 459.0801(440.0, 0.0) 377.0023 386.8745 386.7987 (440.0, 90.0) 398.3333 405.3103 405.3462 (530.0, 90.0) 467.0023 476.2651 476.3866

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(600.0, 0.0) 516.6666 519.0729 519.0665 (600.0, 0.0) 516.6666 519.0729 519.0665 (600.0, 0.0) 516.6666 519.0729 519.0665(610.0, 0.0) 524.5701 526.2656 526.2233 (610.0, 10.0) 526.6666 528.3804 528.5275 (630.0, 10.0) 542.4489 545.4370 545.3956(620.0, 0.0) 532.4489 534.9495 534.9401 (620.0, 20.0) 536.6666 539.0729 539.0443 (660.0, 20.0) 568.1407 570.7557 570.7232(630.0, 0.0) 540.3050 543.3919 543.1744 (630.0, 30.0) 546.6666 549.8134 549.6192 (640.0, 30.0) 554.5701 556.7317 556.6419(640.0, 0.0) 548.1407 550.7687 550.7562 (640.0, 40.0) 556.6666 559.0729 559.0681 (690.0, 40.0) 595.9576 597.7410 597.8339(650.0, 0.0) 555.9576 557.9616 557.9373 (650.0, 50.0) 566.6666 568.2868 568.4555 (670.0, 50.0) 582.4489 585.4958 585.4517(660.0, 0.0) 563.7578 566.4810 566.4677 (660.0, 60.0) 576.6666 579.0729 579.0431 (680.0, 60.0) 592.4489 594.9493 594.9161(670.0, 0.0) 571.5427 574.8553 574.6325 (670.0, 70.0) 586.6666 589.9014 589.6871 (700.0, 70.0) 610.3050 612.7670 612.6654(680.0, 0.0) 579.3141 582.0589 582.0444 (680.0, 80.0) 596.6666 599.0729 599.0689 (740.0, 80.0) 643.7578 646.4665 646.4543(690.0, 0.0) 587.0734 589.1084 589.0968 (690.0, 90.0) 606.6666 608.2048 608.3929 (720.0, 90.0) 630.3050 632.5100 632.5704

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(500.0, 0.0) 442.8571 449.3256 449.3452 (500.0, 0.0) 442.8571 449.3256 449.3452 (500.0, 0.0) 442.8571 449.3256 449.3452(510.0, 0.0) 449.4898 453.0142 452.5082 (510.0, 10.0) 452.8571 459.6979 459.5874 (550.0, 10.0) 481.4334 485.6177 485.6071(520.0, 0.0) 456.4939 460.2776 460.0037 (520.0, 20.0) 462.8571 469.3256 469.3448 (530.0, 20.0) 469.4898 472.3419 472.3745(530.0, 0.0) 463.8217 469.0942 469.1159 (530.0, 30.0) 472.8571 478.9065 479.0317 (560.0, 30.0) 493.8217 499.3404 499.3333(540.0, 0.0) 471.4334 475.7420 475.5249 (540.0, 40.0) 482.8571 489.3256 489.3453 (580.0, 40.0) 511.4334 515.5277 515.5216(550.0, 0.0) 479.2956 484.3404 484.1773 (550.0, 50.0) 492.8571 499.7899 499.6524 (560.0, 50.0) 499.4898 502.4121 502.4497(560.0, 0.0) 487.3798 493.1866 493.1538 (560.0, 60.0) 502.8571 509.3256 509.3458 (590.0, 60.0) 523.8217 529.7080 529.5733(570.0, 0.0) 495.6613 500.5909 500.5260 (570.0, 70.0) 512.8571 518.8220 518.9728 (600.0, 70.0) 533.8217 539.3461 539.3323(580.0, 0.0) 504.1189 509.6008 509.5409 (580.0, 80.0) 522.8571 529.3256 529.3450 (620.0, 80.0) 551.4334 555.5262 555.5194(590.0, 0.0) 512.7340 519.4482 519.3323 (590.0, 90.0) 532.8571 539.8675 539.7071 (650.0, 90.0) 577.3798 582.7350 582.9106

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(1200.0, 0.0) 1099.9999 1145.8605 1145.8076 (1200.0, 0.0) 1099.9999 1145.8605 1145.8076 (1200.0, 0.0) 1099.9999 1145.8605 1145.8076(1210.0, 0.0) 1109.0070 1154.0112 1153.9257 (1210.0, 10.0) 1109.9999 1154.4428 1154.3898 (1240.0, 10.0) 1136.9094 1184.0266 1183.9741(1220.0, 0.0) 1117.9765 1163.1385 1163.1435 (1220.0, 20.0) 1119.9999 1164.1719 1164.2420 (1260.0, 20.0) 1155.8063 1205.0485 1204.9927(1230.0, 0.0) 1126.9094 1175.5361 1175.4343 (1230.0, 30.0) 1129.9999 1177.2880 1177.2353 (1250.0, 30.0) 1147.9765 1193.4373 1193.3843(1240.0, 0.0) 1135.8063 1183.4958 1183.4338 (1240.0, 40.0) 1139.9999 1185.8605 1185.8076 (1290.0, 40.0) 1184.6678 1231.4329 1231.3802(1250.0, 0.0) 1144.6678 1191.5921 1191.4689 (1250.0, 50.0) 1149.9999 1194.4235 1194.3704 (1270.0, 50.0) 1167.9765 1216.0189 1215.9663(1260.0, 0.0) 1153.4948 1203.8000 1203.6646 (1260.0, 60.0) 1159.9999 1207.6238 1207.4450 (1300.0, 60.0) 1195.8063 1241.8416 1241.7913(1270.0, 0.0) 1162.2877 1212.7613 1212.7092 (1270.0, 70.0) 1169.9999 1217.3068 1217.2536 (1320.0, 70.0) 1214.6678 1262.8561 1262.8038(1280.0, 0.0) 1171.0474 1220.7223 1220.6380 (1280.0, 80.0) 1179.9999 1225.8605 1225.8076 (1380.0, 80.0) 1268.4691 1317.2825 1317.3017(1290.0, 0.0) 1179.7743 1228.5707 1228.5148 (1290.0, 90.0) 1189.9999 1234.4058 1234.3527 (1350.0, 90.0) 1243.4948 1293.4319 1293.3789

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(900.0, 0.0) 827.2727 867.2656 867.2853 (900.0, 0.0) 827.2727 867.2656 867.2853 (900.0, 0.0) 827.2727 867.2656 867.2853(910.0, 0.0) 836.2769 878.8669 878.8292 (910.0, 10.0) 837.2727 879.2403 879.2403 (940.0, 10.0) 864.2029 907.7073 907.7031(920.0, 0.0) 845.2534 887.2371 887.2416 (920.0, 20.0) 847.2727 888.0859 888.1059 (930.0, 20.0) 856.2769 896.4516 896.5029(930.0, 0.0) 854.2029 895.5823 895.6173 (930.0, 30.0) 857.2727 896.9731 896.9879 (970.0, 30.0) 893.1259 935.1277 935.1428(940.0, 0.0) 863.1259 907.2776 907.2532 (940.0, 40.0) 867.2727 909.1134 909.0891 (990.0, 40.0) 912.0229 957.0452 957.0034(950.0, 0.0) 872.0229 917.0155 916.9755 (950.0, 50.0) 877.2727 919.2891 919.2711 (970.0, 50.0) 895.2534 935.9616 936.0198(960.0, 0.0) 880.8946 925.3125 925.3032 (960.0, 60.0) 887.2727 928.1650 928.1383 (1000.0, 60.0) 923.1259 966.3433 966.3299(970.0, 0.0) 889.7413 933.5856 933.6210 (970.0, 70.0) 897.2727 936.9438 936.9593 (1020.0, 70.0) 942.0229 987.0671 987.0149(980.0, 0.0) 898.5637 943.1875 943.2104 (980.0, 80.0) 907.2727 947.0126 947.0382 (1010.0, 80.0) 934.2029 975.5233 975.5610(990.0, 0.0) 907.3623 954.9591 954.9161 (990.0, 90.0) 917.2727 959.3163 959.2976 (1030.0, 90.0) 953.1259 997.5117 997.4913

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(1200.0, 0.0) 1100.0000 1156.2050 1156.1594 (1200.0, 0.0) 1100.0000 1156.2050 1156.1594 (1200.0, 0.0) 1100.0000 1156.2050 1156.1594(1210.0, 0.0) 1109.0041 1164.4049 1164.4328 (1210.0, 10.0) 1110.0000 1164.7823 1164.8377 (1230.0, 10.0) 1127.9807 1186.6340 1186.5640(1220.0, 0.0) 1117.9807 1173.6838 1173.7019 (1220.0, 20.0) 1120.0000 1174.4955 1174.5478 (1260.0, 20.0) 1155.8531 1216.0166 1215.9583(1230.0, 0.0) 1126.9302 1186.2252 1186.1273 (1230.0, 30.0) 1130.0000 1187.6371 1187.5119 (1250.0, 30.0) 1147.9807 1203.9963 1203.9967(1240.0, 0.0) 1135.8531 1194.3824 1194.3575 (1240.0, 40.0) 1140.0000 1196.2050 1196.1811 (1270.0, 40.0) 1166.9302 1226.2427 1226.1447(1250.0, 0.0) 1144.7502 1202.5232 1202.5510 (1250.0, 50.0) 1150.0000 1204.7635 1204.8205 (1290.0, 50.0) 1185.8531 1242.9362 1242.9910(1260.0, 0.0) 1153.6218 1215.0888 1214.9968 (1260.0, 60.0) 1160.0000 1217.9885 1217.8645 (1210.0, 60.0) 1114.5435 1166.7245 1166.7265(1270.0, 0.0) 1162.4686 1224.2589 1224.1595 (1270.0, 70.0) 1170.0000 1227.6553 1227.5290 (1300.0, 70.0) 1196.9302 1253.1427 1253.1378(1280.0, 0.0) 1171.2910 1232.3585 1232.3096 (1280.0, 80.0) 1180.0000 1236.2050 1236.1584 (1330.0, 80.0) 1224.7502 1282.4900 1282.5201(1290.0, 0.0) 1180.0895 1240.4416 1240.4681 (1290.0, 90.0) 1190.0000 1244.7458 1244.8037 (1370.0, 90.0) 1261.2910 1320.8904 1320.9465

F.T

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Ch

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eta

l./

Eu

rop

ean

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al

of

Op

eratio

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lR

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17

6(

20

07

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52

8–

15

58

1549

Table 8The theoretical and simulation results of the system with parameters L = 20, D1 = 90.0, D2 = 160.0, l1 = 110.0, l2 = 170.0, p = 0.002, q = 0.01


(1400.0, 0.0) 1183.3333 1249.8980 1249.8980 (1400.0, 0.0) 1183.3333 1249.8980 1249.8980 (1400.0, 0.0) 1183.3333 1249.8980 1249.8980(1410.0, 0.0) 1191.6309 1257.4394 1257.4545 (1410.0, 10.0) 1193.3333 1258.4363 1258.4667 (1430.0, 10.0) 1209.9235 1278.6914 1278.6911(1420.0, 0.0) 1199.9235 1269.2109 1269.1910 (1420.0, 20.0) 1203.3333 1271.3364 1271.3359 (1450.0, 20.0) 1228.2114 1294.8768 1294.9117(1430.0, 0.0) 1208.2114 1277.6935 1277.6772 (1430.0, 30.0) 1213.3333 1281.3716 1281.3402 (1480.0, 30.0) 1254.7726 1324.2273 1324.2111(1440.0, 0.0) 1216.4944 1285.2240 1285.2237 (1440.0, 40.0) 1223.3333 1289.8980 1289.8978 (1460.0, 40.0) 1239.9235 1305.8589 1305.8779(1450.0, 0.0) 1224.77 1292.7470 1292.7625 (1450.0, 50.0) 1233.3333 1298.4173 1298.4488 (1500.0, 50.0) 1274.7726 1345.6286 1345.6232(1460.0, 0.0) 1233.0461 1301.1808 1301.1998 (1460.0, 60.0) 1243.3333 1308.4029 1308.4016 (1530.0, 60.0) 1301.3148 1370.7059 1370.6999(1470.0, 0.0) 1241.3148 1313.0111 1312.9948 (1470.0, 70.0) 1253.3333 1321.3902 1321.3577 (1560.0, 70.0) 1327.8382 1399.7529 1399.7363(1480.0, 0.0) 1249.5789 1320.5243 1320.5241 (1480.0, 80.0) 1263.3333 1329.8980 1329.8981 (1550.0, 80.0) 1321.3148 1393.0375 1393.0211(1490.0, 0.0) 1257.8382 1328.0304 1328.0465 (1490.0, 90.0) 1273.3333 1338.3993 1338.4313 (1600.0, 0.0) 1364.3429 1436.8451 1436.8577

1550F

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differences between the simulation values of Egðx̂; uÞ and Eg(x, u). That proves the conclusion of Theorem1, namely, the forecasted WIP level x̂ can be used to evaluate the criteria performance instead of the actualWIP level x.

5. Conclusions

Time-delay and demand uncertainty make the control of a NMS different from the control of shop floormanufacturing systems. This research focuses on the production control of a single manufacturing systemin an NMS, which is the beginning of the research on the control of the whole NMS. In the dynamic equa-tion of a single manufacturing system in the NMS, the time-delay is explicitly described. The demand uncer-tainty is simplified by assuming that there are two demand rate levels and that the time between demandrate switching is exponentially distributed.

To control such a manufacturing system with time-delay and demand uncertainty, a two-level hedgingpoint policy is proposed. The high hedging level is corresponding to the normal demand rate; and the lowhedging level is corresponding to the high demand rate. To avoid the backlog of demands when the demandrate is on the higher level and cannot be buffered by the WIP, extra production capacity is allowed to use,which brings the extra production cost. The objective of the production control problem is to minimize thetotal costs for WIP inventory and extra capacity. For this problem, a two-level hedging point control policyis proposed, as well as the methods for optimizing the hedging point so as to minimize the total productioncosts. Some insights can be obtained by the analysis of this control policy and the results of numericalexperiments:

1. The two hedging levels can determine the bounds of steady-state WIP from both above and below, whichthen control the influence of demand uncertainty and avoid the backlog.

2. The differences between the two hedging levels are determined by both the time-delay and demand levels,and L(D2 � D1) is the lower bound of this difference. Since D2 � D1 cannot be controlled, decreasing thetime-delay L is the only way to decrease the difference. From Theorem 3, it is also the only way todecrease the high hedging level so as to reduce the total production cost. Therefore, in the control ofNMS, fastening the speed of material transportation is an important issue.

3. The theoretical analysis and the numerical experiments show that the optimal low hedging level is zeroand that in many situations the optimal high hedging level is equal to the difference between the high andlow demand rates multiplied by the time-delay, i.e., Z�1 ¼ LðD2 � D1Þ. However, it is not always true (seethe analysis in Section 3 and Figs. 1–6). The main reason for Z�1 6¼ LðD2 � D1Þ is that the cost for extracapacity is so high, such that Z�1 must be higher so as to increase the WIP and then decrease the time ofoccupying extra capacity.

Many other problems of the production control of NMS are sill open for further research. For example,

1. In this paper, only a single manufacturing system in the NMS is considered. However, an NMS is com-posed of multiple manufacturing systems, and the time-delays between every two manufacturing systemsare different. In this connection, a new problem is raised: how to control and coordinate the productionof multiple manufacturing systems in an NMS with different time-delays so that the production processof the whole NMS can be synchronized.

2. In this paper, only the single product-type case has been investigated. In general, a manufacturing systemproduces many types of products. If the case of multiple product-types is considered, there arises a sec-ond new problem of assigning the limited production capacity to various product types such that theuncertain demands for all types of products can be satisfied.


These two problems are direct extensions of the results in this paper and will be the second and thirdsteps for future researches on the control of NMS. In these two proposed problems, the first one is moredifficult, because it will deal with the nature of the dynamics of NMS and its control, i.e., the synchroni-zation of a complex network. The synchronization conditions and pinning control policy proposed byWang (2002) and Wang and Chen (2002a,b,c) provide us the direction for solving this problem.

Acknowledgments

This research is partially supported by the National Natural Science Foundation of China under grant50375098 and the PROCORE- France/Hong Kong Joint Research Scheme.

Appendix A. Proof of Theorem 1

Suppose that T0 6 2n + 1, 2n + 2, 2n + 3, . . . , 2n + m 6 T. If m = 2k + 1, i.e., an odd integer, thenaccording to Eqs. (10)–(13) we have

Z T

T 0

½x̂ðtÞ � xðtÞ�dt ¼Z t2nþ1

T 0

½x̂ðtÞ � xðtÞ�dtþZ t2nþ1þL

t2nþ1

½x̂ðtÞ � xðtÞ�dtþZ t2nþ2

t2nþ1þL½x̂ðtÞ � xðtÞ�dt

þZ t2nþ2þL

t2nþ2

½x̂ðtÞ � xðtÞ�dtþZ t2nþ3

t2nþ2þL½x̂ðtÞ � xðtÞ�dtþ � � �

þZ t2nþm�2þL

t2nþm�2

½x̂ðtÞ � xðtÞ�dtþZ t2nþm�1

t2nþm�2þL½x̂ðtÞ � xðtÞ�dtþ

Z t2nþm�1þL

t2nþm�1

½x̂ðtÞ � xðtÞ�dt

þZ t2nþm

t2nþm�1þL½x̂ðtÞ � xðtÞ�dtþ

Z T

t2nþm


¼Z t2nþ1

T 0

½x̂ðtÞ � xðtÞ�dtþZ t2nþ1þL

t2nþ1

ðD2 �D1Þðt� t2nþ1Þdtþ 0

þZ t2nþ2þL

t2nþ2

ðD1 �D2Þðt� t2nþ2Þdtþ 0þ � � � þZ t2nþm�2þL

t2nþm�2

ðD2 �D1Þðt� t2nþm�2Þdtþ 0

þZ t2nþm�1þL

t2nþm�1

ðD1 �D2Þðt� t2nþm�1Þdtþ 0þZ T

t2nþm


¼Z t2nþ1

T 0

½x̂ðtÞ � xðtÞ�dtþ L2

2ðD2 �D1Þ þ 0þ L2

2ðD1 �D2Þ þ 0þ � � � þ L2

2ðD2 �D1Þ

þ 0þ L2

2ðD1 �D2Þ þ 0þ

Z T

t2nþm


¼Z t2nþ1

T 0

½x̂ðtÞ � xðtÞ�dtþZ T

t2nþm

½x̂ðtÞ � xðtÞ�dt.

SinceR t2nþ1

T 0½x̂ðtÞ � xðtÞ�dt and

R Tt2nþm½x̂ðtÞ � xðtÞ�dt are bounded, it can be obtained that

limT!1

1

T � T 0

Z T

T 0

½x̂ðtÞ � xðtÞ�dt ¼ limT!1

1

T � T 0

Z t2nþ1

T 0

½x̂ðtÞ � xðtÞ�dt þZ T

t2nþm


( )¼ 0.


If m = 2k, i.e., an even integer, then similarly, according to Eqs. (10)–(13) we have
Z T
T 0

½x̂ðtÞ � xðtÞ�dt ¼Z t2nþ1

T 0

½x̂ðtÞ � xðtÞ�dt þZ t2nþ1þL

t2nþ1

½x̂ðtÞ � xðtÞ�dt þZ t2nþ2

t2nþ1þL½x̂ðtÞ � xðtÞ�dt

þZ t2nþ2þL

t2nþ2

½x̂ðtÞ � xðtÞ�dt þZ t2nþ3

t2nþ2þL½x̂ðtÞ � xðtÞ�dt þ � � �

þZ t2nþm�3þL

t2nþm�3

½x̂ðtÞ � xðtÞ�dt þZ t2nþm�2

t2nþm�3þL½x̂ðtÞ � xðtÞ�dt þ

Z t2nþm�3þL

t2nþm�3


þZ t2nþm�1

t2nþm�2þL½x̂ðtÞ � xðtÞ�dt þ

Z T

t2nþm�1


¼Z t2nþ1

T 0


t2nþm�1

½x̂ðtÞ � xðtÞ�dt.

SinceR t2nþ1

T 0½x̂ðtÞ � xðtÞ�dt and

R Tt2nþm�1

½x̂ðtÞ � xðtÞ�dt are bounded, it can be obtained that

limT!1

1

T � T 0

Z T

T 0

½x̂ðtÞ � xðtÞ�dt ¼ limT!1

1

T � T 0

Z t2nþ1

T 0


t2nþm�1


( )¼ 0.

Similarly, for T0 6 2n, 2n + 1, 2n + 2, . . . , 2n + m 6 T, it can also be obtained that

limT!1

1

T � T 0

Z T

T 0

½x̂ðtÞ � xðtÞ�dt ¼ 0.

Appendix B. Proof of Theorem 2

From Eq. (6), it can be obtained that

x̂ðtÞ ¼ xðt � LÞ þZ t�L

t�2LuðsÞds� LDðt � LÞ

¼ xð0Þ þZ t�2L

0

uðsÞds�Z t�L

0

DðsÞdsþZ t�L


¼ xð0Þ þZ t�dt�2L

0

uðsÞds�Z t�dt�L

0

DðsÞdsþZ t�2L

t�dt�2LuðsÞds

�Z t�L

t�dt�LDðsÞdsþ

Z t�L


¼ xðt � dt � LÞ þZ t�dt�L

t�dt�2LuðsÞds� LDðt � dt � LÞ þ LDðt � dt � LÞ

�Z t�L

t�dt�LDðsÞdsþ

Z t�L

t�dt�LuðsÞds� LDðt � LÞ

¼ x̂ðt � dtÞ þ L½Dðt � dt � LÞ � Dðt � LÞ� þZ t�L

t�dt�L½uðsÞ � DðsÞ�ds

¼ x̂ðt � dtÞ þ L½Dðt � dt � LÞ � Dðt � LÞ� þ ½uðt � dt � LÞ � Dðt � dt � LÞ�dt þ oðdtÞ. ðB:1Þ

Eq. (B.1) is equivalent to

x̂ðt þ dtÞ ¼ x̂ðtÞ þ L½Dðt � LÞ � Dðt þ dt � LÞ� þ ½uðt � LÞ � Dðt � LÞ�dt þ oðdtÞ. ðB:2Þ


According to the two-level hedging point policy (see Eq. (7)) and the dynamics of the system, the systemstates x and x̂ will reach the hedging levels Z1 or Z2.

Suppose that x̂ðtÞ ¼ Z1 and t 5 t2n+1 + L. Then D(t � L) = D(t + dt � L) = D1 if dt is very small.Since x̂ðtÞ ¼ Z1, u(t � L) = D1 must hold in the light of Eq. (7). Then Eq. (B.2) becomesx̂ðt þ dtÞ ¼ x̂ðtÞ þ oðdtÞ. Let dt! 0, it can be obtained that dx̂ðtÞ=dt ¼ 0. That means the control policycan keep x̂ on the hedging level Z1 if x̂ðtÞ ¼ Z1 and t 5 t2n+1 + L.

If x̂ðtÞ ¼ Z1 and t = t2n+1 + L, then D(t � L) = D1 and D(t + dt � L) = D2. Since x̂ðtÞ ¼ Z1, accordingto Eq. (7) we have u(t � L) = D1. Then Eq. (B.2) becomes x̂ðt2nþ1 þ Lþ dtÞ ¼ Z1 � L½D2 � D1��½D2 � D1�dt þ oðdtÞ. Let dt ! 0, it can be obtained that x̂ððt2nþ1 þ LÞ þ 0Þ ¼ Z1 � L½D2 � D1�. That meansthe control policy can make x̂ drop down by L[D2 � D1] at t2n+1 + L when the demand rate changes fromD1 to D2. Because Z1 � Z2 > L[D2 � D1], x̂ððt2nþ1 þ LÞ þ 0Þ > Z2 holds.

If Z2 < x̂ðtÞ < Z1 � L½D2 � D1� and t 5 t2n+2 + L. Then D(t � L) = D(t + dt � L) = D2 if dt is verysmall. Since Z2 < x̂ðtÞ < Z1 � L½D2 � D1�, according to Eq. (7) it can be got that u(t � L) = D1. Then Eq.(B.2) becomes x̂ðt þ dtÞ ¼ x̂ðtÞ þ ½l1 � D2�dt þ oðdtÞ. Let dt! 0, it can be obtained that dx̂ðtÞ=dt ¼l1 � D2 < 0. That means the control policy can drive x̂ down towards the hedging level Z2 ifZ2 < x̂ðtÞ < Z1 � L½D2 � D1� and t 5 t2n+2 + L.

If Z2 < x̂ðtÞ < Z1 � L½D2 � D1� and t = t2n+2 + L, then D(t � L) = D2 and D(t + dt � L) = D1.Since Z2 < x̂ðtÞ < Z1 � L½D2 � D1�, according to Eq. (7) it can be got that u(t � L) = l1. Then Eq. (B.2)becomes x̂ðt2nþ2 þ Lþ dtÞ ¼ x̂ðt2nþ2 þ LÞ þ L½D2 � D1� þ ½l1 � D1�dt þ oðdtÞ. Let dt! 0, it can beobtained that x̂ððt2nþ2 þ LÞ þ 0Þ ¼ x̂ðt2nþ2 þ LÞ þ LðD2 � D1Þ. That means the control policy can make x̂increase by L(D2 � D1) if Z2 < x̂ðtÞ < Z1 � L½D2 � D1� and t = t2n+2 + L when the demand ratechanges from D2 to D1. Since Z2 < x̂ðt2nþ2Þ < Z1 � L½D2 � D1�, we have Z2 þ LðD2 � D1Þ <x̂ððt2nþ2 þ LÞ þ 0Þ < Z1.

If x̂ðtÞ ¼ Z2 and t 5 t2n+2 + L, then D(t � L) = D(t + d t � L) = D2 if dt is very small. Since x̂ðtÞ ¼ Z2,according to Eq. (7) it can be got that u(t � L) = D2. Then Eq. (B.2) becomes x̂ðt þ dtÞ ¼ x̂ðtÞ þ oðdtÞ. Letdt! 0, it can be obtained that dx̂ðtÞ=dt ¼ 0. That means the control policy can keep x̂ on the hedging levelZ2 if x̂ðtÞ ¼ Z2 and t 5 t2n+2 + L.

If x̂ðtÞ ¼ Z2 and t = t2n+2 + L. Then D(t � L) = D2 and D(t + dt � L) = D1. Since x̂ðtÞ ¼ Z2, accordingto Eq. (7) we have u(t � L) = D2. Then Eq. (B.2) becomes x̂ðt2nþ2 þ Lþ dtÞ ¼ Z2 þ L½D2 � D1�þ½D2 � D1�dt þ oðdtÞ. Let dt ! 0, it can be obtained that x̂ððt2nþ2 þ LÞ þ 0Þ ¼ Z2 þ LðD2 � D1Þ. That meansthe control policy can make x̂ increase by L(D2 � D1) if x̂ðtÞ ¼ Z2 and t = t2n+2 + L when the demand ratechanges from D2 to D1. Since Z2 < Z1 � L[D2 � D1], we have x̂ððt2nþ2 þ LÞ þ 0Þ < Z1.

If Z2 þ L½D2 � D1� < x̂ðtÞ < Z1 and t 5 t2n+3 + L. Then D(t � L) = D(t + dt � L) = D1 if dt is verysmall. Since Z2 þ L½D2 � D1� < x̂ðtÞ < Z1, according to Eq. (7) it can be got that u(t � L) = l1. Then Eq.(B.2) becomes x̂ðt þ dtÞ ¼ x̂ðtÞ þ ½l1 � D1�dt þ oðdtÞ. Let dt! 0, it can be obtained that dx̂ðtÞ=dt ¼l1 � D1 > 0. That means the control policy can drive x̂ up towards the hedging level Z1 ifZ2 þ L½D2 � D1� < x̂ðtÞ < Z1 and t 5 t2n+3 + L.

If Z2 þ L½D2 � D1� < x̂ðtÞ < Z1 and t = t2n+3 + L, then D(t � L) = D1 and D(t + dt � L) = D2.Since Z2 þ L½D2 � D1� < x̂ðtÞ < Z1, according to Eq. (7) it can be got that u(t � L) = l1. Then Eq.(B.2) becomes x̂ðt2nþ3 þ Lþ dtÞ ¼ x̂ðt2nþ3 þ LÞ þ L½D1 � D2� þ ½l1 � D1�dt þ oðdtÞ. Let dt! 0, it canbe obtained that x̂ððt2nþ3 þ LÞ þ 0Þ ¼ x̂ðt2nþ3 þ LÞ � LðD2 � D1Þ. That means the control policy canmake x̂ decrease by L(D2 � D1) if Z2 þ L½D2 � D1� < x̂ðtÞ < Z1 and t = t2n+3 + L when the demand ratechanges from D1 to D2. Since Z2 þ L½D2 � D1� < x̂ðt2nþ3Þ < Z1, we have Z2 < x̂ððt2nþ3 þ LÞ þ 0Þ <Z1 � LðD2 � D1Þ.

From the above analysis, it can be concluded that in the steady state (1) the forecasted WIP level mustexist between Z2 and Z1, i.e., Z2 6 x̂ 6 Z1; (2) if the demand rate is D1, then the forecasted WIP value meetsZ2 þ LðD2 � D1Þ < x̂ < Z1; (3) if the demand rate is D2, then the forecasted WIP value meetsZ2 < x̂ < Z1 � LðD2 � D1Þ.


Appendix C. Computing the steady-state probability distribution

(1) Derivation of Eq. (15): For Z2 þ LðD2 � D1Þ < x̂ðtÞ < Z1 and D(t � L) = D1, if D(t � dt � L) = D1,then according to Eq. (7), we have u(t � dt � L) = l1. So according to Eq. (B.1) in Appendix B, we havex̂ðtÞ ¼ x̂ðt � dtÞ þ ðl1 � D1Þdt þ oðdtÞ. If D(t � dt � L) = D2, then according to Eq. (7), we also haveu(t � dt � L) = l1. So according to Eq. (B.1) in Appendix B, we have x̂ðtÞ ¼ x̂ðt � dtÞ þ LðD2 � D1Þþðl1 � D2Þdt þ oðdtÞ. Therefore,

f ½x̂ðtÞ;D1� ¼ f ½x̂ðt� dtÞ;D1�ð1� pdtþ oðdtÞÞ þ f ½x̂ðt� dtÞ;D2�qdt

¼ f ½x̂ðtÞ � ðl1�D1Þdtþ oðdtÞ;D1�ð1� pdtþ oðdtÞÞ þ f ½x̂ðtÞ � LðD2�D1Þ

� ðl1�D2Þdtþ oðdtÞ;D2�qdtþ oðdtÞ

¼ f ½x̂ðtÞ;D1� � f ½x̂ðtÞ;D1�pdt� of ½x̂ðtÞ;D1�ox̂ðtÞ ðl1�D1Þdtþ f ½x̂ðtÞ � LðD2 �D1Þ;D2�qdtþ oðdtÞ.

Let dt! 0, then Eq. (15) can be obtained.(2) Derivation of Eq. (16): For Z2 < x̂ðtÞ < Z1 � LðD2 � D1Þ and D(t � L) = D2, if D(t � dt � L) = D2,

then according to Eq. (7), we have u(t � dt � L) = l1. So according to Eq. (B.1) in Appendix B, we havex̂ðtÞ ¼ x̂ðt � dtÞ þ ðl1 � D2Þdt þ oðdtÞ. If D(t � dt � L) = D1, then according to Eq. (7), we also haveu(t � dt � L) = l1. According to Eq. (B.1) in Appendix B, x̂ðtÞ ¼ x̂ðt � dtÞ þ LðD1 � D2Þþðl1 � D1Þdt þ oðdtÞ holds. Therefore,

f ½x̂ðtÞ;D2� ¼ f ½x̂ðt� dtÞ;D2�ð1� qdtþ oðdtÞÞ þ f ½x̂ðt� dtÞ;D1�pdt

¼ f ½x̂ðtÞ � ðl1�D2Þdtþ oðdtÞ;D2�ð1� qdtþ oðdtÞÞ þ f ½x̂ðtÞ � LðD1�D2Þ

� ðl1�D1Þdtþ oðdtÞ;D1�pdtþ oðdtÞ

¼ f ½x̂ðtÞ;D2� � f ½x̂ðtÞ;D2�qdt� of ½x̂ðtÞ;D2�ox̂ðtÞ ðl1�D2Þdtþ f ½x̂ðtÞ þ LðD2�D1Þ;D1�pdtþ oðdtÞ.

Let dt! 0, then Eq. (16) can be obtained.(3) Derivation of Eq. (22)

P ½x̂ ¼ Z1;D1�

¼ P ½x̂ ¼ Z1;D1�ð1� pdtÞ þ ð1� pdtÞZ Z1

Z1�ðl1�D1Þ dtf ½x̂;D1�dx̂þ qdt

Z Z1�LðD2�D1Þ

Z1�LðD2�D1Þ�ðl1�D2Þ dtf ½x̂;D2�dx̂

¼ P ½x̂ ¼ Z1;D1�ð1� pdtÞ þ ð1� pdtÞZ Z1

Z1�ðl1�D1Þ dtA1 expfax̂gdx̂þ oðdtÞ

¼ P ½x̂ ¼ Z1;D1�ð1� pdtÞ þ A1

að1� pdtÞfexpfaZ1g � expfaðZ1 � ðl1 � D1Þdtgg þ oðdtÞ

¼ P ½x̂ ¼ Z1;D1�ð1� pdtÞ þ A1 expfaZ1gðl1 � D1Þdt þ oðdtÞ.

Let dt! 0, then Eq. (22) can be obtained.(4) Derivation of Eq. (23)

P ½x̂ ¼ Z2;D2�

¼ P ½x̂ ¼ Z2;D2�ð1� qdtÞ þ ð1� qdtÞZ Z2�ðl1�D2Þ dt

Z2

f ½x̂;D2�dx̂þ pdtZ Z2þLðD2�D2Þ�ðl1�D1Þ dt

Z2þLðD2�D2Þf ½x̂;D1�dx̂


¼ P ½x̂ ¼ Z2;D2�ð1� qdtÞ þ ð1� qdtÞZ Z2�ðl1�D2Þ dt

Z2

A2 expfax̂gdx̂þ oðdtÞ

¼ P ½x̂ ¼ Z2;D2�ð1� qdtÞ þ ð1� qdtÞK1A1

aexpfa½Z2 � ðl1 � D2Þdt�g � expfaZ2gf g þ oðdtÞ

¼ P ½x̂ ¼ Z2;D2�ð1� qdtÞ � K1A1 expfaZ2ga

aðl1 � D2Þdt þ oðdtÞ.

Let dt ! 0, then Eq. (23) can be obtained.

Appendix D. Proof of Theorem 3

Since Z1 � Z2 P L(D2 � D1), the definition domain of the function E½gðx̂; uÞ� is X = {(Z1, Z2)jZ2 P 0and Z1 P Z2 + L(D2 � D1)}.

If the optimal hedging point ðZ�1;Z�2Þ is in the open domain {(Z1, Z2)jZ2 > 0 and Z1 > Z2 + L(D2 � D1)},then it must satisfy


¼ oE½gðx̂; uÞ�oZ2

¼ 0. ðD:1Þ

Let G1(Z1, Z2) = H1 exp{aZ1} + H2 exp{aZ2} and G2(Z1, Z2) = (a1Z1 + b1)exp{aZ1} + (a2Z2 + b2)exp{aZ2}. Then E½gðx̂; uÞ� ¼ G2ðZ1; Z2Þ=G1ðZ1; Z2Þ. Therefore,


¼ oG2ðZ1; Z2ÞoZ1

G1ðZ1; Z2Þ �oG1ðZ1; Z2Þ

oZ1

G2ðZ1; Z2Þ� ��

½G1ðZ1; Z2Þ�2.

Let oE½gðx̂; uÞ�=oZ1 ¼ 0, then it can be obtained that

oG2ðZ1; Z2ÞoZ1

G1ðZ1; Z2Þ ¼oG1ðZ1; Z2Þ

oZ1

G2ðZ1; Z2Þ. ðD:2Þ

Similarly, let oE½gðx̂; uÞ�=oZ2 ¼ 0 and it can be obtained that

oG2ðZ1; Z2ÞoZ2

G1ðZ1; Z2Þ ¼oG1ðZ1; Z2Þ

oZ2

G2ðZ1; Z2Þ. ðD:3Þ

From Eqs. (D.2) and (D.3), it can be obtained that

oG1ðZ1; Z2Þ=oZ1

oG1ðZ1; Z2Þ=oZ2

¼ oG2ðZ1;Z2Þ=oZ1

oG2ðZ1;Z2Þ=oZ2

. ðD:4Þ

Namely,

H 1

H 2

¼ a1 þ aða1Z1 þ b1Þa2 þ aða2Z2 þ b2Þ

. ðD:5Þ

Because a1 = cH1 and a2 = cH2, it can be obtained from Eq. (D.5) that

Z2 ¼ Z1 þb1

a1

� b2

a2

¼ Z1 þ h; ðD:6Þ

where h = b1/a1 � b2/a2. From Eq. (D.2), it can be obtained that

½a1 expfaZ1g þ ða1Z1 þ b1Þa expfaZ1g�½H 1 expfaZ1g þ H 2 expfaZ2g�¼ H 1a expfaZ1g½ða1Z1 þ b1Þ expfaZ1g þ ða2Z2 þ b2Þ expfaZ2g�. ðD:7Þ


Substitute Eq. (D.6) into Eq. (D.7), we have

½a1 expfaZ1g þ ða1Z1 þ b1Þa expfaZ1g�½H 1 expfaZ1g þ H 2 expfaZ1 þ ahg�¼ H 1a expfaZ1g½ða1Z1 þ b1Þ expfaZ1g þ ða2Z1 þ a2hþ b2Þ expfaZ1 þ ahg�.

Since a1 = cH1, a2 = cH2 and h = b1/a1 � b2/a2, the above equation becomes

cH 1ðH 1 þ H 2 expfahgÞ ¼ 0. ðD:8Þ
However, usually Eq. (D.8) does not hold. Therefore, Eq. (D.1) does not hold. Namely, the optimal hedg-ing point ðZ�1; Z�2Þ is not in the open domain {(Z1, Z2)jZ2 > 0 and Z1 > Z2 + L(D2 � D1)}.
If the optimal hedging point ðZ�1; Z�2Þ is on the line Z1 = Z2 + L(D2 � D1), then Eq. (27) becomes

E½gðx̂; uÞ� ¼ ½a1Z2 þ a1LðD2 � D1Þ þ b1� expfaZ2g expfaLðD2 � D1Þg þ ða2Z2 þ b2Þ expfaZ2gH 1 expfaZ2g expfaLðD2 � D1Þg þ H 2 expfaZ2g

¼ ½a1 expfaLðD2 � D1Þg þ a2�Z2 þ ½a1LðD2 � D1Þ þ b1� expfaLðD2 � D1Þg þ b2

H 1 expfaLðD2 � D1Þg þ H 2

.

Because a1 = cH1 and a2 = cH2, the above equation becomes

E½gðx̂; uÞ� ¼ cþZ2 þ½a1LðD2 � D1Þ þ b1� expfaLðD2 � D1Þg þ b2

H 1 expfaLðD2 � D1Þg þ H 2

.

Obviously, it can be known from the above equation that if the optimal hedging point is on the lineZ1 = Z2 + L(D2 � D1), the value of Z2 must be zero.

The last situation is the optimal hedging point is neither in the open domain {(Z1, Z2)jZ2 > 0 andZ1 > Z2 + L(D2 � D1)} nor on the line Z1 = Z2 + L(D2 � D1). So it must on the line Z2 = 0.

Summarizing the above analysis, it can be concluded that the optimal hedging point ðZ�1; Z�2Þ must satisfyZ�2 ¼ 0.

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A two-level hedging point policy for controlling a manufacturing system with time-delay, demand uncertainty and extra capacity