《系统工程》课程教学资源（英文文献）Modeling for logistics system

MODELINGFORLOGISTICS SYSTEMXIU-QING LIU, LIANG SUNSchool of transportation and vehicle engineering, Shandong Univers ity of technology,Zibo, 255049, ChinaE-MAIL: liuxiuqinggsdut.edu.cnAbstract:The modelingmethods available cannot describe impacts of adjustment ability oflogistics system on pay off and cost of logistics system. In view of such a fact, theorem of adjustmentability of logistics system in supply chain circumstance on condition that payoff of supply chain is aunique increasing process was proposed. A sequence of MGF (moment generating function) ofout-of-goods risk process for logistics system is a martingale. The model we proposed can overcomeshortcomings of previous ones which can't describe characteristics of adjustment ability of logisticssystem At last, Simulations results coinc ide with theoretical analysis.Keywords: Logistics engineering, logistics system, martingale1.IntroductionOne of the major challenges of logistics system modeling is to provide a valid explanation asimpacts of adjustment ability of logistics system on pay offand cost ofSupply chain. Computer simulation models using. for example, Petri nets approach have beendeveloped these last years. However, there generally consist in a case-by- case study for which acorrect description goes through the program itself, making them not well suited for exchange andthus difficult to replicate or to modify. On other hand, previous studies fail to take into accountimpacts of adjustment ability of logistics system on pay off and cost of supply chain in a quantitativeway.Liet al proposed a model for supply chain using AHCP [1]. Cui et al used GSPN to establish amodel for supply chain [2].Ma et al further presented a model for supply chain management [3]. Zhouet al applied MAPN technique to analysis system-of-systems of supply chain [4]. Zhu et al proposed ageneral structure named CLsim to describe the system-ofsystems of logistics [5]Cohen and Leedeveloped an analysis model for the performance of supply chain [6]. Lorant A. Tavasszy et al furtherdeveloped a decision support system to describe system-ofsystems of logistics of Holland. They

MODELING FOR LOGISTICS SYSTEM XIU-QING LIU, LIANG SUN School of transportation and vehicle engineering, Shandong University of technology, Zibo, 255049, China E-MAIL: liuxiuqinggsdut.edu.cn Abstract: The modeling methods available cannot describe impacts of adjustment ability of logistics system on pay off and cost of logistics system. In view of such a fact, theorem of adjustment ability of logistics system in supply chain circumstance on condition that payoff of supply chain is a unique increasing process was proposed. A sequence of MGF (moment generating function) of out-of-goods risk process for logistics system is a martingale. The model we proposed can overcome shortcomings of previous ones which can't describe characteristics of adjustment ability of logistics system. At last, Simulations results coincide with theoretical analysis. Keywords: Logistics engineering; logistics system; martingale 1.Introduction One of the major challenges of logistics system modeling is to provide a valid explanation as impacts of adjustment ability of logistics system on pay off and cost of Supply chain. Computer simulation models using, for example, Petri nets approach have been developed these last years. However, there generally consist in a case-by- case study for which a correct description goes through the program itself, making them not well suited for exchange and thus difficult to replicate or to modify. On other hand, previous studies fail to take into account impacts of adjustment ability of logistics system on pay off and cost of supply chain in a quantitative way. Li et al proposed a model for supply chain using AHCP [1]. Cui et al used GSPN to establish a model for supply chain [2].Ma et al further presented a model for supply chain management [3]. Zhou et al applied MAPN technique to analysis system-of-systems of supply chain [4]. Zhu et al proposed a general structure named CLsim to describe the system-of-systems of logistics [5].Cohen and Lee developed an analysis model for the performance of supply chain [6]. Lorant A. Tavasszy et al further developed a decision support system to describe system-of-systems of logistics of Holland. They

describe system-ofsystems of logistics through 3 levels: Production, storage and multi-modetransportation They solve problem of inventory cost control using a comprehensive cost function [7]Cohn and Barnhart used Composite-variable technique to model for system-ofsystems of serviceparts logistics [8]. Rupesh Kumar Pati focused on modeling for system-ofsystems of reverse logisticsusing goal-programming [9]. A micro-simulation model of system-of-systems of national freight ofNorwayand Sweden was proposedby Gerardde Jong[10]1.Definitionoflogistics system2.1 Logistics systemLogistics system is an aggregated and complicated system that integrated forward and reversesflow and storage of goods, services, and related information between the point of origin and point ofconsumption Logistics system is a part ofsupply chain2.2 Adjustme nt ability of logistics systemAdjustment ability of logistics system refers to having products available when they are neededby customers. It's important to recognize that different products have different sensitivities to time andquantity.For example,three-day late delivery of perishable items likely has more seriousconseguences thanthree-daylate delivery of non-perishable items.The numberof items can'tmeetrequirements ofcustomer, which will has serious consequence even if items arrived in time.2.Modeling logistics system3.1Preliminary concepts>Definition of moment generatingfunctionIfX is a random variable, moment generating function of X is as follows:E(e")=/e*dF(x)(1)provided that E(e)k Likewise,momentgeneratingfunctionofX isasfollows:m(r)=Eerr(2)>Adjustme nt coefficientA discrete-time martingale is a discrete-time stochastic process (ie.,a sequence of random variables)X1, X2, X3... that satisfies for all nE(X,D<8(4)E(X IX...X,)=X.(5)ie., the conditional expected value of the next observation, given all the past observations, is equal to

describe system-of-systems of logistics through 3 levels: Production, storage and multi-mode transportation. They solve problem of inventory cost control using a comprehensive cost function [7]. Cohn and Barnhart used Composite-variable technique to model for system-of-systems of service parts logistics [8]. Rupesh Kumar Pati focused on modeling for system-of-systems of reverse logistics using goal-programming [9]. A micro-simulation model of system-of-systems of national freight of Norway and Sweden was proposed by Gerard de Jong [10]. 1. Definition of logistics system 2.1 Logistics system Logistics system is an aggregated and complicated system that integrated forward and reverses flow and storage of goods, services, and related information between the point of origin and point of consumption. Logistics system is a part of supply chain. 2.2 Adjustment ability of logistics system Adjustment ability of logistics system refers to having products available when they are needed by customers. It's important to recognize that different products have different sensitivities to time and quantity. For example, three-day late delivery of perishable items likely has more serious consequences than three-day late delivery of non-perishable items. The number of items can't meet requirements of customer, which will has serious consequence even if items arrived in time. 2. Modeling logistics system 3.1 Preliminary concepts  Definition of moment generating function If X is a random variable, moment generating function of X is as follows:  Adjustment coefficient A discrete-time martingale is a discrete-time stochastic process (i.e.,a sequence of random variables) X1, X2, X3. that satisfies for all n i.e., the conditional expected value of the next observation, given all the past observations, is equal to

thelastobservationSomewhat more generally, a sequence Y1, Y2, Y3 ... is said to be a martingale with respect toanothersequenceX1,X2,ifforallnE(Y, DDemanding process D (t) is a unique increasing process, that is to say, Where c is apositive constant.3c>0,Vt≥0,D(t)=ct(10)> Supplying process S(t) is defined as follows:NS(0)=X三WhereN(t) is a Poisson process which parameter is X. N(t) indicate that cargo were transported N(t)times in time interval [0,t]Xk represent Xk cargo unit was transported in k-th time.(Xk) is a sequence of independent identical istribution(N(t) is independent on (Xk)S(t) describe that total cost of logistics system increase when cargoes were transported in order tomeet requirements of customer.>riskprocessLet R(t)-S(t)-P(t) be a adjustment process of logistics systemLet U(t)-u-R(t)be a risk process refers to being out of items at the same timethere is a willingbuyerfor it. U indicate actual demand of supply chain U(t) refers to risk of stock out. R (t)< u indicate thatitems available can't meet requirements of customer.3.Analysis of the model

the last observation. Somewhat more generally, a sequence Y1, Y2, Y3 . is said to be a martingale with respect to another sequence X1, X2, if for all n Similarly, a continuous-time martingale with respect to the stochastic process X, is a stochastic process Y, such that for all t 3.2 Basic assumption  Demanding process D (t) is a unique increasing process, that is to say, Where c is a positive constant.  Supplying process S(t) is defined as follows: Where N(t) is a Poisson process which parameter is X. N(t) indicate that cargo were transported N(t) times in time interval [0,t]. Xk represent Xk cargo unit was transported in k-th time. {Xk} is a sequence of independent identical istribution. {N(t)} is independent on {Xk} S(t) describe that total cost of logistics system increase when cargoes were transported in order to meet requirements of customer.  risk process Let R(t)=S(t)-P(t) be a adjustment process of logistics system. Let U(t)=u-R(t) be a risk process refers to being out of items at the same time there is a willing buyer for it. U indicate actual demand of supply chain. U(t) refers to risk of stock out. R (t)< u indicate that items available can't meet requirements of customer. 3. Analysis of the model

Lemma 4.1 if X, Yare random variable, thenE[E(X Y)] = E(X)1) Let X, Y be continue random variables.E[E(X|Y)]=E(X|Y=y)-f)dy=x.f(xy)dx1f,)dy-)ddyfr)=f(xy）)=x-fx(x)dx=E(x)2) As for X, Y are discrete random variables, the results follows by (1)Lemma4.2 Let Sn=X,+ X2 + ...+Xn .N is a uncertain positive integer. (Xk) is a sequence ofindependent identical distribution We getMs, (t)=E(es)=E[E(e|N))=E(E[erX+X++J) = E(M (0)*)=E[eWnMr(]=M[nMx()]Theorem 1 demanding and supplying process are subject to1ct=-log Eer()Aunder the condition that payoff of supply chain is unique increasingf()= m(r)-1f(r)=m(r)-1rrObviously, r is a unique solution ofPROOF.LetBy lemma 4.1 and lemma 4.2, we getben=be2=[M, (0)]"=M[InM(0)]=ex(m(r-)Thus,1C=-logEe'srMoregenerality.1-logEen(t)ct=rTheorem2e-r(isamaringale underthe conditionthat-log EersC=7

Lemma 4.1 if X, Y are random variable, then E[E(X Y)] = E(X) 1) Let X, Y be continue random variables. 2) As for X, Y are discrete random variables, the results follows by (1) Lemma4.2 Let SN=X1+ X2 + . +XN .N is a uncertain positive integer. {Xk} is a sequence of independent identical distribution. We get Theorem 1 demanding and supplying process are subject to under the condition that payoff of supply chain is unique increasing. PROOF. Let Obviously, r is a unique solution of By lemma 4.1 and lemma 4.2, we get

PROOFGiven S=S(1)E(e-(/3,)=e0E(e(-Uo)/3,)=e-woE(e-ro-a)=e-ruE(er(s(-s(-)=e-(o)Where,referstoall informationon R(s):se[0,]Specially,Ee-rue-"WegetEe-rtr()=e-mThat is to say,Ee'r(s-0)=1Thus,1-log EersC=74. Simulation resultsWe used the simulation software package MissRdPCby IXI and all simulations were carried on apersonnelcomputer.Cost of logistics system is subject to P(4)(Poission distribution which X =4). Cost of logistics systemis unique increasing with a positive constant 6.Initial cost of logistics system is 80. The simulationwent on 120 time units. Analysis of the result is as follows:

PROOF 4. Simulation results We used the simulation software package MissRdP©by IXI and all simulations were carried on a personnel computer. Cost of logistics system is subject to P(4)(Poission distribution which X =4). Cost of logistics system is unique increasing with a positive constant 6.Initial cost of logistics system is 80. The simulation went on 120 time units. Analysis of the result is as follows:

Fig.3payoffof logisticssystemfordifferentronconditionthat s(t) issubject toP(4)Fig.+cost relatedto logistics for differentronconditionthatC-6Fig.3 gives the developing trends ofpayoff for different values ofr.As expected, payoff increasewhenrincrease.Thoseresults are inagreementwiththeorem1The cost related to logistics from 0.03 to 7 is plotted in Fig.4.The impacts of logistics system oncost isn't so high when r-1.36. The impacts of logistics system on cost is significant as r>1.36.Thoseresults reflected that logistics system play an important role in supply chain with the developing ofscience and technology. Those results coincide with previous studies [9-10]5.ConclusionWe have investigated the relationship between payoff and cost for a kind of logistics system Wehave shown that characteristics of adjustment ability of logistics system. MGF of risk process oflogisticssystemisamartingalethroughadjustmentabilityoflogisticssystem.Our future work is as follows:We'll present relationship between payoff and adjustment ability of logistics system whendistributionofpayoff is a random stochastic process

Fig.3 gives the developing trends of payoff for different values of r. As expected, payoff increase when r increase .Those results are in agreement with theorem 1. The cost related to logistics from 0.03 to 7 is plotted in Fig.4.The impacts of logistics system on cost isn't so high when r=1.36. The impacts of logistics system on cost is significant as r>1.36.Those results reflected that logistics system play an important role in supply chain with the developing of science and technology. Those results coincide with previous studies [9-10]. 5. Conclusion We have investigated the relationship between payoff and cost for a kind of logistics system. We have shown that characteristics of adjustment ability of logistics system. MGF of risk process of logistics system is a martingale through adjustment ability of logistics system. Our future work is as follows: We'll present relationship between payoff and adjustment ability of logistics system when distribution of payoff is a random stochastic process