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sample average approximation and dual decomposition

Production,Manufacturing and Logistics

Supply chain design under uncertainty using sample average approximation and dual decomposition q

Peter Schütz a,*,Asgeir Tomasgard a,b ,Shabbir Ahmed c

Department of Industrial Economics and Technology Management,Norwegian University of Science and Technology,7491Trondheim,Norway b

SINTEF Technology &Society,7465Trondheim,Norway c

School of Industrial and Systems Engineering,Georgia Institute of Technology,Atlanta,GA 30332,USA

a r t i c l e i n f o Article history:

Received 4October 2007Accepted 21November 2008

Available online 7December 2008Keywords:

Supply chain design

Stochastic programming

Sample average approximation Dual decomposition

a b s t r a c t

We present a supply chain design problem modeled as a sequence of splitting and combining processes.We formulate the problem as a two-stage stochastic program.The ?rst-stage decisions are strategic loca-tion decisions,whereas the second stage consists of operational decisions.The objective is to minimize the sum of investment costs and expected costs of operating the supply chain.In particular the model emphasizes the importance of operational ?exibility when making strategic decisions.For that reason short-term uncertainty is considered as well as long-term uncertainty.The real-world case used to illus-trate the model is from the Norwegian meat industry.We solve the problem by sample average approx-imation in combination with dual https://www.doczj.com/doc/ef17133947.html,putational results are presented for different sample sizes and different levels of data aggregation in the second stage.

1.Introduction

Supply chain management includes design of,planning for and operation of a network of suppliers,production facilities,warehouses,and distribution centers in order to satisfy customer demand.Strategic decisions regarding the design of supply chains affect the ability to ef?ciently serve customer demand.The design decisions should therefore not be taken without considering the effect on the operational decisions (Lee and Billington,1992).In this paper we examine the impact different modeling choices of the supply chain operations have on the strategic decisions regarding its design and structure.

The importance of supply chain design was recognized already in the early 1970s (see,e.g.Geoffrion and Graves,1974).The early mod-els however assume the parameters that in?uence the design decisions to be deterministic.For long planning horizons this assumption is unlikely to hold.Demands,prices for raw materials,components and ?nished products,locations of markets,etc.are usually highly uncer-tain over the lifetime of the supply chain and thus require a supply chain that is robust and ?exible enough to cope with the challenges of a changing environment.These challenges lead to an increased interest in stochastic programming models over the past 10–15years.

Traditionally,the research literature focused on the facility location component of supply chain management.Schütz et al.(2008)is a recent example of the facility location approach with uncertain demand,non-convex,non-concave ?rst-stage costs and convex second-stage costs.For a broader overview over stochastic facility location,we refer to the reviews by Louveaux (1993)and Snyder (2006).

We consider a model which covers several levels of the supply chain in contrast to the classical one level facility location models or location–allocation models.In this class of models,there is less literature on stochastic models.MirHassani et al.(2000)present a solution method for a supply chain design problem under uncertain demand that is based on scenario analysis.They also discuss the use of Benders decomposition in a parallel implementation.Lucas et al.(2001)consider a similar capacity planning problem,but develop a solution meth-od based on Lagrangean relaxation and scenario analysis.In Alonso-Ayuso et al.(2003),a supply chain planning model is presented with binary ?rst-stage decisions and continuous second-stage decisions.They use an algorithmic approach based on branch-and-?x to solve their problem.Alonso-Ayuso et al.(2005)present several models for supply chain design and production planning and scheduling.Differ-ent formulations for the problems are discussed.They also provide computational experience for the supply chain design problem.Santoso et al.(2005)consider a supply chain design problem and a solution method based on sample average approximation (SAA)and Benders

This research is sponsored by the Research Council of Norway through the Smartlog project.The authors also wish to thank Dash Optimization for sponsoring the licenses for the Xpress runtime libraries used in the computational cluster.*Corresponding author.Tel.:+4773551371;fax:+4773590260.

E-mail addresses:peter.schutz@iot.ntnu.no ,peter.schutz@sintef.no (P.Schütz).European Journal of Operational Research 199(2009)

409–419

Contents lists available at ScienceDirect

European Journal of Operational Research

j o u r n a l h o m e p a g e :w w w.e l s e v i e r.c o m /l o c a t e /e j o

https://www.doczj.com/doc/ef17133947.html,putational results from a real-world case are also presented.Our paper extends this literature both in terms of solution method and model scope,as explained below.

Our case is from the Norwegian meat industry,based on cooperation with Nortura (www.nortura.no )and studies the effect on the stra-tegic decisions of including the operation of the supply chain under uncertain and highly variable demand.The case is based on work car-ried out in 2005and 2006in restructuring their production network.In Tomasgard et al.(2005),the operational aspects of the same supply chain are modeled,but no computational results are presented.That paper does not look into strategic decisions regarding locations and capacities;the supply chain structure is ?xed.Schütz et al.(2008)present a stochastic one level facility location model for the same supply chain.There,the operational part is aggregated into a single time period and the model handles only a single commodity and only the ?rst part of the supply chain.

The main contribution of this paper is to model and solve a stochastic multi-commodity supply chain design problem with a detailed description of operational consequences from the strategic decisions.It has the same level of detail on the operational side as Tomasgard et al.(2005)and handles in addition strategic decisions regarding location and capacities at every level in the supply chain.Because de-mand in the meat industry can exhibit huge variations over short periods,the supply chain is not only subject to long-term demand uncer-tainty,but also considerable short-term demand uncertainty.The long-term uncertainty covers trends and the development of national demand for meat products over a longer time horizon,whereas the short-term uncertainty deals with weekly demand variations.We study in this paper alternative models where the level of detail used to describe the operational decisions varies and discuss the impact on the strategic decisions.To our knowledge,our paper is the ?rst one to examine whether or not the level of aggregation in the second stage affects the ?rst-stage solution in supply chain design models.Our analysis also examines how the use of a stochastic model in?uences the decisions as compared to a deterministic model.

Modeling operational decisions in the second stage increases the model size considerably.For our model we can only solve problem instances with three or four scenarios when using a single processor computer with 6gigabyte of memory.To solve the model we combine sample average approximation (SAA)(Kleywegt et al.,2001)with dual decomposition (Car?e and Schultz,1999).A second contribution of the paper is therefore to investigate how the approach of combining SAA with dual decomposition scales when distributed processing on a higher number of processors is used to increase the number of scenarios in the formulation.We also examine if increasing the number of scenarios have effect on the quality of the ?rst-stage solution.

In Section 2,we present a supply chain model for the Norwegian meat industry.The scenario generation procedure and the different cases for uncertain demand are discussed in Section 3.In Section 4,we then give a generalized two-stage stochastic programming formu-lation for a supply chain design problem where the supply chain is modeled as a sequence of splitting and combing processes.The solution scheme is presented in Section https://www.doczj.com/doc/ef17133947.html,putational results and an analysis of these results follow in Section 6.We conclude in Section 7.2.The supply chain

The levels and the material ?ows of the supply chain used in our case are depicted in Fig.1.Slaughtering takes livestock as input and produces carcasses and intestines.It is followed by a two-step cutting process where the carcasses are split into smaller parts.The fourth level is processing where the intermediate products from the previous levels are blended into ?nished products like sausages,ground meat,etc.The ?nished products are then shipped to the distribution centers where customer demand is satis?ed.Intermediate products are sold to external customers or shipped to all downstream facilities that process or sell the product.Finished products are only sent to distribution centers.

For both the cutting and the processing levels several recipes may be chosen in order to process or produce a given product.Which rec-ipes to use at the processing level depends on the availability of the different intermediate products which in turn depends on the recipes chosen at the cutting level and eventually on the supplied livestock.A more detailed description of the supply chain and the different pro-cesses can be found in Tomasgard et al.(2002,2005).

The part of Nortura’s supply chain that we consider consists of 17slaughterhouses,18plants at the cutting level,nine processing facil-ities,and nine distribution centers.The task is to redesign the processing level of the supply chain.Internal studies in Nortura revealed overcapacities in processing that have to be reduced.Thus,the ?rst-stage decisions are to determine which of the existing nine facilities should keep its processing unit in order to satisfy uncertain future demand.We look at 13possibilities for locating different production equipment at the processing facilities.In the second-stage,we operate the production network in order to satisfy customer demand for 103?nished products.As input we use seven different animal types.For each of the animal types we have one recipe for the production of intestines at the slaughtering level.The carcasses are split into intermediate products at the ?rst cutting level.We can choose among 35recipes.At the second cutting level,we have a set of 87recipes to choose from.With these recipes,both from the slaughtering and the cutting level,we produce 72different intermediate products.All of the intermediate products can be used for further processing or can be sold to external customers.At the processing level,we can choose from 169recipes to produce 103?nished products.

The objective is to minimize the sum of annualized ?xed facility costs and the expected annual operating costs of the supply chain.The ?rst-stage costs are the ?xed facility costs.These costs include capital cost,personnel and insurance,among others.We use

annualized

410P.Schütz et al./European Journal of Operational Research 199(2009)409–419

?xed costs in order to be able to compare them with the operating costs in the second stage.For our problem instances,the second-stage costs cover the variable costs of one year operating the production network.The operating costs include production,transportation,and shortfall costs.Production costs cover both direct and indirect costs like ingredients,packaging,personnel,and energy.The transportation costs are given by contracts between Nortura and the different transportation companies.They depend both on origin and destination as well as the pallet type used for transporting the products.

Our supply chain has as a set of facilities or plants which are possible locations for different production processes.The production pro-cesses can be different production lines,technologies or production phases.A production process is either a splitting processes or a combin-ing https://www.doczj.com/doc/ef17133947.html,bining processes are common in manufacturing and assembling and often described by means of a bill of material (BoM).The BoM lists which and how many input products p i are needed to produce one unit of output product p o (see Fig.2a).The splitting pro-cesses are described in a similar manner.The process industry (e.g.the chemical and petroleum industry)uses a so-called reversed bill of material (rBoM)that de?nes in which output products p o a unit of input product p i can be split (see Fig.2b).The meat processing industry can be seen as an example of a supply chain consisting of subsequent splitting and combining processes (see,e.g.Tomasgard et al.,2002,2005).Our formulation allows for choosing from different BoMs for the same process.In our model,each BoM is assigned to the production of one speci?c output product in one speci?c process.Correspondingly each rBoM is assigned to the processing of one speci?c input product.For each location in our supply chain model we will have a set of possible production processes and for each process,a set of possible products.3.Modeling uncertainty

Here we describe the modeling of short-term and long-term uncertainty,the temporal structure of the second-stage and the different demand cases.

3.1.Short-term vs.long-term uncertainty

One of the main purposes of our analysis is to examine the importance of modeling the short-term operations and corresponding de-mand uncertainty when making strategic decisions.There can be high variations in weekly demand for meat products over the year.The products we consider in our problem instances belong to three different product groups.Weekly demand per product group is shown in Fig.3.We see that demand per product group changes by as much as 50%within 1–2weeks,demand for a single product as used in our problem may exhibit even greater variations.

We refer to weekly changes in demand as short-term variations.Some of the variations are due to seasonal changes in the demand pat-tern,but also daily variations,for example due to weather,can have a signi?cant impact on demand.These short-term variations are ne-glected when demand is aggregated in longer time periods (or even a single period)in the second stage.The term long-term uncertainty is used to describe changes in the total level of demand over a longer time horizon like 5–10years.This uncertainty is for example the market share of the company,the total market size or trends in demand for meat products.

In order to study the effect of short-term uncertainty and operational decisions on the strategic decisions,we have to generate scenarios for the short-term demand.We then combine these scenarios with scenarios for changes in the total demand level,i.e.the long-term uncer-tainty.The methods used to generate the demand scenarios are described

below.

P.Schütz et al./European Journal of Operational Research 199(2009)409–419411

Short-term supply chain planning is based on demand forecasts.To capture the short-term uncertainty we use a methodology that com-bines forecasting and scenario-generation (Nowak et al.,2007).We will brie?y describe the steps for the short-term scenario generation here:

(1)For each product,we parametrize an autoregressive process of N th order,AR(N )-process,on historical data (see,e.g.Hamilton,1994).

For our problem instances,we use an AR(1)-process.

(2)We then ?nd the distribution of the historical prediction error for each product.We assume that the error term for a given product is

i.i.d.between different time periods.The historically observed prediction errors then give us a multivariate discrete distribution for all the products’error terms.

(3)On this empirical distribution,we perform a principal component analysis.For the P products we get P principal components and

sort them by the share of variance they explain for the error distribution.

(4)We calculate the ?rst four moments of the empirical distribution of these principal components and choose the K principal compo-nents explaining the highest part of variance as stochastic in the further scenario generation process.For the P àK remaining com-ponents we use their expected value.

(5)We then generate S scenarios for the K stochastic principal components using a moment matching procedure ensuring that the ?rst

four moments in the scenarios are the same as in the historical distribution.The scenarios are equally likely.The method used here is a modi?ed version of the method developed by H?yland et al.(2003).

(6)The scenarios for the principal components are then transformed back into scenarios for the prediction error.

(7)Finally,we combine the forecasting method with the scenario tree for the prediction error to get demand scenarios rather than pure

prediction error scenarios.This procedure is illustrated in detail in Fig.4:Using a deterministic AR(N )-process as forecasting method,

predicted demand in period t +1is given by the formula ^x t t1?a tP N

i ?1b i x t t1ài .In our case,we add a realization of the error term

s t t1as represented in the scenario tree to the ?rst prediction.Thus,the ?rst prediction ^x s t t1?a tP N

i ?1b i x t t1ài t s t t1is based on his-torical data and the error scenario s t t1.When demand for several periods is predicted,the prediction may be based on both historical

demand,predicted demand,and the error scenarios as shown in Fig.4.The advantage of this method is that correlations between stochastic variables are taken care of through principal component analysis.Also,the principal component analysis makes it possible to reduce the number of stochastic variables used.In our case by using 150prin-cipal components for 821stochastic variables we still explain 93%of the variance in the multivariate error distribution.

For the long-term uncertainty we choose a different approach.For reasons of simpli?cation,we model a possible change in long-term demand level without differentiating between the different demand regions.We also assume that an increase in total demand and a de-crease in total demand are equally likely.We use a uniform distribution to describe the demand level,with todays demand as the expec-tation.The scenarios for long-term uncertainty are generated by sampling a factor from a uniform distribution on [0.5;1.5].By multiplying this factor with the demand in any demand scenario we model a change in the long-term demand level.Alternatively,a long-term trend,for example a general increase or decrease in demand level,would be easy to model by changing the interval.3.2.Demand uncertainty

To properly catch all weekly demand variations,one would have to model the second-stage with 52time periods.However,due to restrictions in both solution time and model size,this is not practical.We have to reduce the number of time periods in the second-stage by aggregating them while ensuring that we capture the effect of short-term demand variations.

Demand in the Norwegian meat market follows typical seasonal patterns:during the summer months,demand for barbecue products increases and both Easter and Christmas have distinct demand patterns.We therefore split the year in four seasons,each three

months

412P.Schütz et al./European Journal of Operational Research 199(2009)409–419

https://www.doczj.com/doc/ef17133947.html,ually,there is no correlation in demand between different seasons.We aggregate the ?rst two months of each season into one per-iod and model the last month of each season with a weekly time resolution (four weeks).This way,we capture the weekly demand vari-ations around Easter,early and late summer,and Christmas in detail.

We describe here the problem instances based on three criteria:long-term uncertainty,short-term uncertainty,and aggregation.STU The ?rst problem instance we use is only taking short-term uncertainty into consideration.We choose to generate 200scenarios for

each of the four-week periods using the scenario generation method described above.As we assume no temporal correlation between the different quarters,we can use any combination of these scenarios,resulting in a total of 2004possible demand scenar-ios.Demand in the cumulated two-month periods is represented by its expected value.No long-term uncertainty is modeled.

LTU The second case only takes long-term uncertainty into account.We use expected weekly demand to represent the demand variations

over the year.This demand data is then multiplied with a random factor drawn from a uniform distribution on [0.5;1.5]to model the long-term changes in demand level.The scenarios may be viewed as possible demand realization at a point in the future where all scenarios are equally likely.Alternatively the scenarios may be viewed as realizations of yearly demand over a set of years.No short-term uncertainty is modeled.

LSTU The third case is a combination of long-term and short-term uncertainty.We use the scenario tree from STU and multiply each sce-nario with a random factor that is uniformly distributed on [0.5;1.5].This way,we capture both effects.For each of these non-aggregated cases,we de?ne a corresponding aggregated problem instances.In these aggregated problem in-stances (ASTU,ALTU,and ALSTU),we replace the 20-period second stage problem by a single period problem.All demand,supply,and capacities are cumulated.4.Optimization model

Let us introduce the following notation for our two-stage stochastic programming formulation: Sets F c set of possible facility locations for combining processes F s set of possible facility locations for splitting processes W set of possible warehouse locations

L set of all possible locations,L ?F c [F s [W C set of customer locations

U ej Tset of upstream locations able to send products to location j ,j 2L [C

D ej Tset of downstream locations able to receive products from location j ,j 2L O c ej Tset of combining processes that can be performed at location j ,j 2F c O s ej Tset of splitting processes that can be performed at location j ,j 2F s O ej Tset of all processes that can be performed at location j ,j 2L ,O ej T?O c ej T[O s ej TO set of all processes,O ?S j 2L O ej TP set of products

P i eo Tset of input products for process o ,o 2O P o eo Tset of output products of process o ,o 2O

B eo ;p Tset of (reversed)bills of materials that can be used for processing product p in process o ,o 2O ;p 2P S set of scenarios T

set of time periods

Indices and superscripts o process index,j 2L ;o 2O ej Tb bill of materials index,o 2O ;p 2P ;b 2B eo ;p Tj ,k location indices,j ;k 2L [C p ,q product indices,p ;q 2P s scenario index,s 2S t time period index,t 2T

Parameters,constants,and coef?cients A b p o ;p i

yield of product p o in case one unit p i is processed with reversed bill of materials b ,o 2O s ;p i 2P i eo T;p o 2P o eo T;b 2B eo ;p i TB b p o ;p i amount of product p i needed to produce one unit p o using bill of materials b ,o 2O c ;p i 2P i eo T;p o 2P o eo T;b 2B eo ;p i TC os jt

capacity of process o at location j in scenario s at time t ,j 2F c [F s ;o 2O ;s 2S ;t 2T D ps jt demand for product p at customer location j in scenario s at time t ,j 2C ;p 2P ;s 2S ;t 2T

F o j ?xed cost of locating process o at location j ,j 2F c [F s ;o 2O

H ps jt penalty for not satisfying one unit of demand of product p at customer location j in scenario s at time t ,j 2C ;p 2P ;s 2S ;t 2T P ps jbt cost of processing one unit of product p at location j using (reversed)bill of material b in scenario s time t ,j 2F c [F s ;o 2O ej T;p 2P ;b 2B eo ;p T;s 2S ;t 2T

S ps jt supply of product p at location j in scenario s at time t ,j 2F c [F s ;p 2P ;s 2S ;t 2T

T ps jkt cost of transporting one unit of product p from location j to location k in scenario s at time t ,j 2L ;k 2D ej T;p 2P ,s 2S ;t 2T i ps j 0initial inventory of product p in scenario s at location j ,j 2W ;p 2P ;s 2S p s

probability of scenario s ,s 2S

P.Schütz et al./European Journal of Operational Research 199(2009)409–419413

Decision variables v ps jt inventory of product p at location j in scenario s at period t ,j 2W ;p 2P ;s 2S ;t 2T w ps jkt amount of product p transported from location j to location k in scenario s at time t ,j 2L ;k 2D ej T;p 2P ;s 2S ;t 2T

x ps jbt amount of product p processed/produced at location j using bill of material b in scenario s at time t ,j 2F c [F s ;o 2O ej T,p 2P ;b 2B eo ;p T;s 2S ;t 2T

y o j 1if process o is located at facility location j ,0otherwise,j 2F c [F c ;o 2O ej T

z ps jt

unsatis?ed demand for product p at customer location j in scenario s 2S in period t ,j 2C ;p 2P ;s 2S ;t 2T

We now model our supply chain as two-stage stochastic program with recourse.For reasons of simplicity,we denote the set of feasible combinations of ?rst-stage decisions y o j by Y .The uncertain parameters in this formulation are the costs,supply,capacity,and demand.By n ,we denote the vector of these parameters.For our problem we have n ?eP ;T ;H ;S ;C ;D T,where n s is a given realization of the uncertain parameters

min

X j 2L X

o 2O ej T

F o j y o j t

s 2S

p s Q ey ;n s T

e1T

subject to

y 2Y #f 0;1g j L jáj O j

e2T

with Q ey ;n s Tbeing the solution of the following second-stage problem:

Q ey ;n s T?min

X j 2L X p 2P X

o 2O ej TX b 2B eo ;p TX

t 2T

P ps jbt x ps

jbt tX p 2P X j 2L X k 2D ej TX

t 2T

T ps jkt w ps

jkt t

X p 2P X j 2C X

t 2T

H ps jt z ps

e3T

subject to

S ps jt tX

k 2U ej T

w ps kjt ?X o 2O s ej TX

b 2B eo ;p Tx ps jbt j 2F s ;p 2[

o 2O s ej T

P i eo T;t 2T ;e4TX o 2O s ej TX q 2P i eo TX

b 2B eo ;q T

A b p ;q áx qs

jbt ?

k 2D ej T

w ps jkt

j 2F s ;p 2[

o 2O s ej T

P o eo T;t 2T ;

e5TX

p 2P i eo TX

b 2B eo ;p T

x ps jbt 6C os jt áy o

j 2F s ;o 2O s ej T;t 2T ;e6T

S ps jt tX

k 2U ej T

w ps kjt ?

X o 2O c ej TX q 2P o eo TX

b 2B eo ;q T

B b q ;p áx qs

jbt

j 2F c ;p 2

[

o 2O c ej T

P i eo T;t 2T ;e7TX o 2O c ej TX b 2B eo ;p Tx ps jbt ?X k 2D ej T

w ps jkt j 2F c ;p 2

[

o 2O c ej T

P o eo T;t 2T ;

e8TX

p 2P o eo TX

b 2B eo ;p T

x ps jbt

C os jt

áy o j j 2F c ;o 2O c ej T;t 2T ;e9Tv ps jt à1t

X k 2U ej T

w ps kjt ?v ps

jt t

k 2D ej T

w ps jkt

j 2W ;p 2P ;t 2T ;

e10TX

k 2U ej T

w ps kjt t

z ps jt

D ps jt

j 2C ;p 2P ;t 2T ;

e11Tv ;w ;x ;z P 0

e12T

The objective function (1)is the sum of the ?rst-stage costs and the expected second-stage costs.The ?rst-stage costs represent the costs

of installing a given process at location j .The objective function of the second stage (3)consists of three parts:?rstly,the production costs,secondly,the transportation costs,and thirdly,the shortfall penalty for unsatis?ed demand.Restriction (2)de?nes the feasible set for the binary ?rst-stage variables.Constraints (4)–(6)describe the splitting processes.Constraints (4)ensure that the external supply and all prod-ucts transported into the splitting node are processed.Restrictions (5)force all produced products to be transported to a downstream node.Restrictions (6)limit production to the available capacity in the splitting node.Constraints (7)–(9)describe combining processes in a similar way.Constraints (7)ensure that all products needed in the combining process are supplied at the combining node.Constraints (8)make sure all produced products are transported to downstream nodes and restrictions (9)take care of the capacity restrictions in the combining nodes.Restrictions (10)are the mass balance constraints for the inventory.With Eq.(11),we make sure that the sum of all products transported into a demand node and shortfall is equal to customer demand.Constraints (12)are the non-negativity constraints,the indices are omitted.5.Solution scheme

For the model (1)–(12)off-the-shelf solvers can typically solve instances with 3–4scenarios (the amount of memory is in our experience the limit).A typical problem instance in a practical case would have thousands of scenarios.We handle this using sample average approx-imation (Kleywegt et al.,2001)and dual decomposition (Car?e and Schultz,1999).These procedures are described in the following subsections.

5.1.Sample average approximation

We use sample average approximation (SAA)to reduce the size of problem (1)–(12)by repeatedly solving it with a smaller set of sce-narios.We generate random samples with N

by the sample average function 1

N P N n ?1Q ey ;n n T.The problem (1)–(12)is then approximated by the following SAA problem:

414P.Schütz et al./European Journal of Operational Research 199(2009)409–419

min y 2Y

ey T:?X j 2L X

o 2O ej T

F o j y o

t1N X N n ?1

Q ey ;n n T()

:e13T

The optimal solution of (13),^y

N ,and the optimal value,v N ,converge with probability one to an optimal solution of the original problem (1)–(12)as the sample size increases (Kleywegt et al.,2001).Assuming that the SAA is solved to an optimality gap d P 0,we can estimate the sample size N needed to guarantee an e -optimal solution to the true problem with a probability of at least 1àa as

N P

3r 2max ee àd Tej L jj O jelog 2Tàlog a T

e14T

with e P d and a 2(0,1).

In (14),r 2max is related to the variability of Q ey ?;n Tat the optimal solution y ?

(see Kleywegt et al.(2001)for details).One would in practice choose N taking into account the trade-off between the quality of the solution obtained for the SAA problem and the computational effort needed to solve it.Solving the SAA problem (13)with independent samples repeatedly can be more ef?cient than increasing the sample size N .This procedure can be found in Santoso et al.(2005),but we include it here for the sake of completeness:(1)Generate M independent samples of size N and solve the corresponding SAA

min y 2Y

ey T:?X j 2L

o 2O ej T

F o j y o

j t1N X n

Q ey ;n n T(

)

:We denote the optimal objective function value by v m N and the optimal solution by ^y m N ;m ?1...M .

(2)Compute the average of all optimal objective function values from the SAA problems, v

N ;M and its variance,r 2 v

N ;M : v

N ;M ?1M X

m ?1v m

and

r 2 v

N ;M ?1eM à1TM X M m ?1

ev m N à v N ;M T2

:The average objective function value v

N ;M provides a statistical lower bound on the optimal objective function value for the original problem (1)–(12)(Norkin et al.,1998;Mak et al.,1999).

(3)Pick a feasible ?rst-stage solution y 2Y for problem (1)–(12),e.g.one of the solutions ^y m N .With that solution,estimate the objective

function value of the original problem using a reference sample N 0

N 0e y T:?X j 2L X

o 2O ej T

F o j y o j

t1N 0X N 0

n ?1

Q ey ;n n T:The estimator ~g

N 0e y Tserves as an upper bound on the optimal objective function value.The reference sample N 0is generated indepen-dently of the samples used in the SAA problems.Since the ?rst-stage solution is ?xed,one can choose a greater number of scenarios for N 0than for N as this step only involves the solution of N 0independent second-stage problems (3)–(12).We can estimate the variance of

N 0e y Tas follows:r 2N 0e y T?

eN 0

à1TN 0

X N 0n ?1

X j 2L X

o 2O ej T

F o j y o j tQ ey ;n n

Tà~g N 0e y T

(4)Compute the estimators for the optimality gap and its https://www.doczj.com/doc/ef17133947.html,ing the estimators calculated in steps 2and 3,we get

gap N ;M ;N 0e y

T?~g N 0e y Tà v N ;M and

r 2gap ?r 2N 0e y Ttr 2 v

N ;M :The con?dence interval for the optimality gap is then calculated as

~g N 0e y Tà v N ;M tz a er 2N 0e y Ttr 2 v

N ;M T1=2with z a :?U à1e1àa T,where U (z )is the cumulative distribution function of the standard normal distribution.

5.2.Dual decomposition and Lagrangean relaxation

Step (1)of the SAA algorithm outlined above involves solving a two-stage stochastic mixed-integer problem (13)with N scenarios.Even

though the number of scenarios in this problem is considerably lower than in the original problem (1)–(12),it is still a large problem.To solve each of the SAA problems,we decompose the problem in scenarios (see,e.g.Car?e and Schultz,1999).In order to do this,we intro-duce ?rst-stage variables y 1;...;y n for each scenario n =1,...,N and add non-anticipativity constraints y 1?ááá?y n to the problem (Rock-afellar and Wets,1991).For each j 2L ;o 2O ej T,we implement the non-anticipativity constraints by the equation P N n ?1K n y on

j ?0where K 1?1àN and K n

?1for n =2,...,N .

We de?ne k as the vector of Lagrangean multipliers associated with the non-anticipativity constraints and relax these.The resulting Lagrangean relaxation is

LR ek T?min y 2Y 1X N n ?1X j 2L X o 2O ej T

eF o j y on j tk o j K n y on j TtQ ey n

;n n T !(

)e15T

with Q ey n ;n n Tbeing the solution to the second-stage problem (3)–(12)given realization n of the random parameters.Note that problem (15)

is separable in scenarios.

P.Schütz et al./European Journal of Operational Research 199(2009)409–419415

We ?nd the best lower bound for our problem by solving the Lagrangean dual

LD ?max

LR ek T:

To solve LD ,we use cutting planes (Kelley,1961)in a bundle method with box constraints (see,e.g.Lemaréchal,1986).Let k denote the

superscript for the current iteration and let r k ?P N n ?1K n y

be the subgradient of (15)with respect to k k in iteration k .Further,de?ne L k ?LR ek k Tàk k ár k as the value of (15)without the Lagrangean penalty term.By D we denote the allowed change in the Lagrangean mul-tiplier.The Lagrangean multipliers are then updated solving the following linear problem:

max

k k t1;/

/e16T

subject to

8i ?1;...;k :/6L i tr i ák k t1;e17Tk k t16k k tD ;e18Tk k t1P k k àD ;e19T/2R ;k k t12R j L jáj O j :

e20T

The Langrangean multipliers are optimal once /is no longer changing.To speed up the process of ?nding the optimal Lagrangean multipliers,we solve the single-scenario subproblems only with an optimality gap c P 0in the ?rst iterations.We then reduce c while we proceed with the iterative procedure and eventually solve all the subproblems to optimality.The idea behind this is that the solutions from the ?rst iter-ations provide an initial search direction,whereas we need a better accuracy in the later iterations to determine the optimal value of the multipliers.

The solution y k produced in iteration k is in general not feasible for the SAA problem (13)as the non-anticipativity constraints may be violated.Step (3)of the SAA procedure requires a feasible solution for the original problem (1)–(12),so we use a simple heuristic to turn the infeasible Lagrangean solution into a feasible,but possibly not optimal solution.We generate a feasible solution by ?xing the binary ?rst-stage variables at 1,if they are 1in more than 50%of the optimal single-scenario solutions and 0otherwise.This heuristic may produce feasible solutions far away from the optimal solution,thus increasing the necessary number of iterations.However,it ?nds good enough solutions for our problem instances.Once the ?rst-stage variables are ?xed,we solve the second-stage problem (3)–(12)for each scenario and get an upper bound on the optimal objective function value.5.3.Quality of the solutions

We choose not to solve the Lagrangean subproblems to optimality during the ?rst iterations.During these iterations,we may not get a lower bound on the optimal objective function value.In fact,the calculated lower bound may actually be above the optimal objective func-tion value of the SAA problem.It may also be bigger than the upper bound from a feasible solution.However,we can still guarantee the quality of the feasible solution found.

If we solve all Lagrangean subproblems with an relative optimality gap c P 0and stop the SAA procedure once the relative gap between upper and lower bound estimator is less or equal to P 0,then the feasible solution providing the upper bound is a e tc t c T-optimal solution to the original SAA problem (13).

Let LR c be a the c -optimal upper bound on the solution of the Lagrangean relaxation LR (k )(15).With v N being the optimal objective

function value of (13)and ~g

N 0e y Tbeing an upper bound provided by the feasible solution y ,we get the following three cases:(1)LR c 6v N 6~g

N 0e y T:LR c is a lower bound for v N ,i.e.the solution y providing ~g N 0e y Tis -optimal for the SAA problem.(2)v N 6LR c 6~g

N 0e y T:LR c overestimates v N ,but not by more than c .The difference between LR c and ~g N 0e y Tdoes not exceed ,thus the feasible solution y

is e tc t c T-optimal for the SAA problem.(3)v N 6~g

N 0e y T6LR c :The difference between v N and LR c is at most c .This means,that y is a c -optimal solution for the SAA https://www.doczj.com/doc/ef17133947.html,putational results

The calculations were carried out on a Linux cluster,running kernel 2.6.9with each node consisting of two 1.6GHz Dual-Core Intel Xeon

5110processors and 8GB RAM.The solution scheme is implemented in C++using the library functions of the message passing interface (MPI)for distributed processing and Xpress 2006runtime libraries as solver for the SAA problems.Xpress 2006is also used to solve the linear program updating the Lagrangean multipliers.6.1.Problem instances

We choose to solve all cases using M =20SAA problems.For the SAA problems,we use sample sizes of N =20,40,and 60scenarios.Dual decomposition is combined with a simple heuristic as described in Section 5.2for the non-aggregated cases.The heuristic is used every ?ve iterations to generate a feasible solution to the SAA problem.We stop once the objective function value is within 5%of the lower bound estimator.The best feasible solution of each SAA is then stored as a candidate solution for valuation in the reference sample.For the aggre-gated cases,each SAA problem is solved to optimality,so we store the optimal solutions for evaluation in the reference sample.The size of the reference sample is set to N 0?1000scenarios.

Due to the size of the non-aggregated cases (STU,LTU,and LSTU),we use one processor core per single-scenario subproblem.The single-scenario subproblems have approx.740,000variables and 1650,00constraints.The aggregated problems (ASTU,ALTU,and ALSTU)are

416P.Schütz et al./European Journal of Operational Research 199(2009)409–419

P.Schütz et al./European Journal of Operational Research199(2009)409–419417 much smaller in size,so we can solve stochastic two-stage problems with60scenarios without having to decompose them.The aggregated problems with60scenarios have approx.2,825,000variables and926,000constraints.

Solving the reference sample with a given?rst-stage solution provides a statistical upper bound on the optimal objective function value of the original problem.For the aggregated problem instances,the statistical lower bound is provided by the average of the M optimal objective function values of the SAA problems.

For the disaggregated problem instances,we normally get an optimality gap of less than5%after5–10iterations when using a c-optimal solution of LR(k)to calculate the estimator for the lower bound on the SAA problem.As c>0during these iterations,we cannot guarantee that this estimator is a true lower bound on the optimal objective function value of the SAA problem(see Sections5.2and 5.3).To provide a valid lower bound on the SAA problem and a valid estimate for the optimality gap,we recalculate the lower bound estimator after the SAA procedure is completed using the lower bound on the objective function value of the Lagrangean relaxation.Test runs indicated that this procedure provides good enough solutions considerably faster than using the lower bound on LR(k)directly or solving LR(k)to optimality.

The statistical lower and upper bounds are shown in Table1.We compare the results of the different problem instances with the solu-tion to the expected value problem(EVP),i.e.the solution to the problem where the uncertain parameters are replaced by their expected value.We compute the upper bound provided by the EEV(see,e.g.Birge and Loveaux,1997),the expected value of the EVP solution,by ?nding the expected value of implementing the EVP?rst-stage solution for the different cases.For each disaggregated case,we also give the upper bound when using the solution from the corresponding aggregated case(ASTU,ALTU,and ALSTU).

Firstly,we note that the feasible solutions from the SAA problems give an upper bound that is approx.16%lower than the EEV.The value of the stochastic solution(VSS,see Birge and Loveaux,1997)is at least180mill NOK.Secondly,using the?rst-stage solution from the cor-responding aggregated case as a solution for the disaggregated case works better than the solution from the EVP.However,the solutions for the aggregated case still give expected results that are approx.50mill.NOK worse than the best solutions from the disaggregated cases. Thirdly,the estimator for the lower bound is increasing in the number of scenarios while its variance is decreasing.

In Table2,we present the estimator for the optimality gap as well as the upper and lower limit of the90%-con?dence interval for the best solution from solving the SAA problems with the different sample sizes.The estimator for the optimality gap is calculated by subtract-ing the lower bound(the SAA procedure)from the upper bound(the reference sample).The optimality gap for the EEV and the aggregated cases is calculated using the best lower bound from the SAA problems.

The con?dence interval for the optimality gap is getting narrower as we increase the number of scenarios in the SAA problem.This is mainly due to a smaller variance of the lower bound.Thus,increasing the number of scenarios,we can give a better guarantee with respect to how close we are to the optimal solution.The results for the optimality gap indicate that the solutions produced by our scheme are good enough to be used in a practical application.

The average CPU-time per processor core for solving a single scenario in the SAA problem varies between34min(STU,N=60)and 62min(LTU,N=60).For cases STU and LSTU the CPU-time is slightly decreasing in the number of scenarios,whereas it is slightly increas-ing for case LTU.The CPU-time for the aggregated cases is increasing in the number of scenarios per SAA problem.This is not surprising,as these problems are not decomposed.The CPU-time required for these problems increases for case ASTU from46min(N=20)to206min

Table1

Statistical lower and upper bounds of the SAA problems for M=20and N0?1000.

Case N Lower bound Upper bound

Average r LB Average r UB

STU201,182,82927841,218,2301871 401,185,93128351,217,1201947

601,186,20619201,225,7301946

EEV1,484,7901939

ASTU1,269,5701950

LTU201,288,11137,1071,384,23028,545 401,331,21828,0181,349,84028,158

601,306,37825,9711,381,64029,665

EEV1,630,18030,180

ALTU1,439,33030,175

LSTU201,334,35944,2361,327,40028,671 401,310,12425,2111,340,13028,927

601,314,99522,9011,382,57028,540

EEV1,569,50030,660

ALSTU1,383,14030,595

ASTU201,118,16023961,119,3901653 401,119,70021491,116,7601657

601,119,53014961,116,2501653

EEV1,393,7401670

ALTU201,268,77047,0031,298,21027,811 401,244,49042,3821,301,49028,011

601,277,34024,3301,305,56027,699

EEV1,513,60028,456

ALSTU201,226,48031,0671,235,24028,100 401,249,04035,0421,251,81027,826

601,251,20027,6471,246,13028,574

EEV1,443,54028,875

418P.Schütz et al./European Journal of Operational Research199(2009)409–419

Table2

Estimated optimality gap and con?dence interval with M=20and N0?1000.

Case N Estimated optimality gap90%Con?dence interval

[1000NOK]%r gap Min.%Max.%

STU2035,401 2.99335430,573 2.5840,230 3.40 4031,189 2.63343926,239 2.2136,139 3.05

6039,524 3.33273435,589 3.0043,459 3.66

EEV298,58425.172728294,65724.84302,51225.50

ASTU93,3647.03273779,425 6.7087,3047.36

LTU2096,1197.4646,81728,725 2.23163,51312.69 4018,622 1.4039,722à38,559à2.9075,804 5.69

6075,363 5.7639,42818,504 1.42109,1628.36

EEV298,96222.4641,181239,68218.00358,24326.91

ALTU108,1128.1241,17748,837 3.67167,38812.57

LSTU20à6959à0.5252,715à82,843à6.2168,926 5.17 4030,006 2.2938,371à25,231à1.9385,243 6.51

6067,575 5.1436,59214,899 1.13120,2519.14

EEV235,14117.6253,822157,66211.82312,62023.43

ALSTU48,731 3.6653,785à28,645à2.15126,2079.46

ASTU2012300.112911à2960à0.2654200.48 40à2940à0.262714à6847à0.619670.09

60à3280à0.292229à6489à0.58à71à0.01

EEV274,04024.472721270,12324.12277,95724.82

ALTU2029,440 2.3254,614à49,179à3.88108,0598.52 4057,000 4.5850,802à16,131à1.30130,13110.46

6028,220 2.2136,867à24,851à1.9581,291 6.36

EEV236,26018.5037,439182,36614.28290,15422.72

ALSTU2087600.7141,890à51,543à4.2069,063 5.63 4027700.2244,746à61,643à4.9467,183 5.38

60à5070à0.4139,635à62,125à4.9751,985 4.15

EEV192,34015.3739,852134,97210.79249,70819.96 (N=60),for case ALTU from32min(N=20)to148min(N=60),and from36min(N=20)to166min(N=60)for case ALSTU.Evaluating a given candidate solution in the reference sample requires approximately150min for the disaggregated cases when using20processor cores.The candidate solutions of the aggregated cases need10–15min of CPU-time.

6.2.Solution properties

When we compare the stochastic solutions to the expected value solution,we see that the solutions from the SAA problems open facil-ities with more capacity than the EVP solutions.The solutions of the disaggregated problem instances also install more capacity than the solutions of the aggregated cases.Case LTU is an exception,as the best solution from the SAA problems installs less capacity for ground meat than both the EVP solution and the ALTU solution.In Table3,we show the amount of capacity installed per product group for the solution providing the lowest upper bound for each case.

Opening facilities with more capacity incurs higher?xed costs.The total costs of operating the value chain however are lower for the stochastic solutions that install more capacity.The additional capacity provides a?exibility in the second stage that reduces the expected second-stage costs by more than the?rst-stage costs increase.

One of the purposes of solving several samples in the SAA approach is to?nd good candidates for?rst-stage solutions to be tested in the reference sample.The solution scheme produces19candidate solutions for case STU with N=20and N=40.When we solve this case with N=60,we get18candidate solutions.For case LTU,the number of candidate solutions produced by our heuristic decreases from20for N=20to7when N=60.Case LSTU with combined long-term and short-term uncertainty is solved with20candidate solutions for N=20,15candidate solutions for N=40,and20candidate solutions for N=60.The upper bounds from these candidate solutions do not vary a lot,indicating rather?at objective functions.The solution from the expected value problem however performs poorly in all cases. The aggregated cases are all solved to optimality and produce a single solution over all samples.The optimal solution of the aggregated cases is the same for all problem instances,independent of the number of scenarios or the type of uncertainty modeled.

7.Conclusions

In this paper,we have presented a supply chain design problem from the Norwegian meat industry.The mathematical formulation of the problem can be applied to any supply chain that consists of subsequent levels of splitting and combining processes.It is also possible to adapt the model for supply chains that consist only of splitting processes(e.g.the processing industry)or combining processes(e.g.man-ufacturing and assembly-based industries).We model both detailed operations of the supply chain and aggregated operations in the sec-ond-stage of the problem and examine the effect of this modeling choice on the?rst stage decisions.Due to the size of the non-aggregated problem instances,we use sample average approximation in combination with dual decomposition to solve our https://www.doczj.com/doc/ef17133947.html,paring the results from both types of second-stage models to their corresponding expected value problem,we see that the?rst-stage decisions from the stochastic problems result in considerably lower costs than the solution of the EVP.We also note that the?rst-stage solutions from the aggregated problem instances have higher expected costs than those from the disaggregated problem instances in the disaggregated data-sets,though not as high as the EVP solution.

P.Schütz et al./European Journal of Operational Research199(2009)409–419419

Table3

Installed capacities for the different product groups.

Case Installed capacity(tons/year)

Ground meat Sausages Cold cuts

EVP18,21614,06511,381 STU23,10023,04111,381 LTU17,60023,04111,381 LSTU18,21633,64511,381 AEVP18,21610,6047920 ASTU18,21614,1687920 ALTU18,21614,1687920 ALSTU18,21614,1687920

The results also show that our solution scheme produces good?rst-stage solutions already for small sample sizes in the SAA problem. Increasing the sample size improves the lower bound on the optimal objective function value,thus improving the quality guarantee on the optimality gap.Due to the distributed processing of the solution scheme,we can increase the size of the SAA problems by using more pro-cessors.For our problem instances,we see that increasing the number of scenarios in the SAA problems even reduces the total CPU-time,as the bigger problems produce fewer candidate solutions that have to be evaluated in the reference sample.

Even though we get good solutions for our problem instances,it might be worthwhile to investigate if a more sophisticated heuristic for ?nding feasible solutions produces even better results.We also need to reduce the runtime further in order to include more product fam-ilies and add more facilities to the supply chain design decisions.

For the end-user in the meat industry,variations of this model have been used in the strategy process to examine structural decisions.A traditional supply chain model would not give the same insight as it lacks the operational detail.As the results show:high variations at the operational level will in?uence the strategic decisions.This cannot be captured properly in aggregated models.In addition,even when it comes to pure operational models,we do not know any alternative model that handles a combination of splitting processes and combining processes.This is essential in the meat industry,in order to capture the value creation of the cutting stage and the processing stage.Finally, one of the main advantages for Nortura in practical use has been the ability to examine a set of supply chain con?gurations that are almost equally good.This way,Nortura is more?exible with respect to the future design of the supply chain,knowing that the chosen design might not be optimal,but also that it will not be far away.

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模拟联合国大会策划案

第四届模拟联合国大会策划书主办：承办：

二O 一三年十月

目录一、活动背景、意义和目标二、所需会议资源三、活动时间及地点四、会议筹备五、模拟联合国大会具体流程六、奖项设置七、经费预算

一、活动背景、意义和目标 1、什么是模拟联合国大会模拟联合国大会是一项国际性的大学生组织和活动，以拓宽学生视野，锻炼学生综合性能力为宗旨，以培养国际性人才为目标，有利于提高大学生组织管理、研究写作、公开发言、与人沟通、求同存异等多面的能力，是一项健康积极、极富教育意义的学生活动。 2、模拟联合国在常理工为丰富我校校园文化，增进我校学生对于当前重大国际议题的理解认识，提高我校学生综合能力，特举办常熟理工学院第三届模拟联合国大会。 3、活动简介学生白由组队、扮演成所代表国的外交官，参与到“联合国会议” 中，按联合国的规则讨论国际热点问题，模拟联合国会议的全过程。模拟联合国中所探讨的涉及人权、环保和社会发展等诸多面的国际议题有助于拓展学生的视野，培养学生以国际眼光看待问题的良好思维模式。同时此次活动以全中文的形式进行，让更多有兴趣的同学能够有机会参与其中。 4、活动目的和意义

哈佛大学全美模拟联合国大会的会议介绍中，第一句话就提至V: “Our primary goal is to provide students interested in exploring the difficulties and complexities of international relations with best possible simulation of diplomacy and negotiation ”。参与模拟联合国活动，最初的目的就是让参与者拥有一种对错综复杂的国际关系的基本认识，伴随着这种认识，参与者将展现出更高水平的外交谈判能力。我们认为，这也就是参与模拟联合国活动的意义和最终目的，即是在对国际社会、间关系有一种基本了解的基础上，能够通过语言表达、文件写作和白身的魅力维护本国利益，达到会议目标。并通过整个的活动，重新认识白己，认识白己的，认识到白己的在将来世界的位置。

《模拟联合国》社团课

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6、经历从实际生活中发现问题、提出问题、解决问题的过程，体会数学在日常生活中的作用，初步形成综合运用数学知识解决问题的能力。 7、了解解决植树问题的思想方法，培养从生活中发现数学问题的意识，初步培养探索解决问题有效方法的能力，初步形成观察、分析及推理的能力。 8、体会学习数学的乐趣，提高学习数学的兴趣，建立学好数学的信心。 9、养成认真作业、书写整洁的良好习惯。三、教材的编写特点: 本册实验教材具有内容丰富、关注学生的经验与体验、体现知识的形成过程、鼓励算法多样化、改变学生的学习方式，体现开放性的教学方法等特点。同时，本实验教材还具有下面几个明显的特点。 1. 改进四则运算的编排，降低学习的难度，促进学生的思维水平的提高。四则运算的知识和技能是小学生学习数学需要掌握的基础知识和基本技能。以往的小学数学教材在四年级时要对以前学习过的四则运算知识进行较为系统的概括和总结，如概括出四则运算的意义和运算定律等。对于这些相关的内容，本套实验教材在本册安排了“四则运算”和“运算定律与简便计算”两个单元。“四则运算”单元的教学内容主要包括四则混合运算和四则运算的顺序。而关于四则运算的意义没有进行概括，简化了教学内容，降低了学习的难度。

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心理学研究方法复习题一、重要概念 1、研究的效度：即有效性，它是指测量工具或手段能够准确测出所需测量的心理特质的程度。 2、内部一致性信度：主要反映的是测验内部题目之间的信度关系，考察测验的各个题目是否测量了相同的内容或特质。内部一致性信度又分为分半信度和同质性信度。 3、外推效度：实验研究的结果能被概括到实验情景条件以外的程度。 4、半结构访谈：半结构化访谈指按照一个粗线条式的访谈提纲而进行的非正式的访谈。该方法对访谈对象的条件、所要询问的问题等只有一个粗略的基本要求，访谈者可以根据访谈时的实际情况灵活地做出必要的调整，至于提问的方式和顺序、访谈对象回答的方式、访谈记录的方式和访谈的时间、地点等没有具体的要求，由访谈者根据根据情况灵活处理。 5、混淆变量：如果应该控制的变量没有控制好，那么，它就会造成因变量的变化，这时，研究者选定的自变量与一些没有控制好的因素共同造成了因变量的变化，这种情况就称为自变量混淆。 6、被试内设计：每个被试接受接受自变量的所有情况的处理。 7、客观性原则：是指研究者对待客观事实要采取实事求是的态度，既不能歪曲事实，也不能主观臆断。 8、统计回归效应：在第一次测试较差的学生可能在第二次测试时表现好些,而第一次表现好的学生则可能相反,这种情形称为统计回归效应.。统计回归效应的真正原因就是偶然因素变化导致的随机误差，以及仅仅根据一次测试结果划分高分组和低分组。 9、主体引发变量：研究对象本身的特征在研究过程中所引起的变量。 11、研究的信度：测量结果的稳定性程度。换句话说，若能用同一测量工具反复测量某人的同一种心理特质，则其多次测量的结果间的一致性程度叫信度，有时也叫测量的可靠性。 12、分层随机抽样：它是先将总体各单位按一定标准分成各种类型（或层）；然后根据各类型单位数与总体单位数的比例，确定从各类型中抽取样本单位的数量；最后，按照随机原则从各类型中抽取样本。13、研究的生态效度：生态效度就是实验的外部效度，指实验结果能够推论到样本的总体和其他同类现象中去的程度，即试验结果的普遍代表性和适用性。 14、结构访谈：又称为标准化访谈，指按照统一的设计要求，按照有一定结构的问卷而进行的比较正式的访谈，结构访谈对选择访谈对象的标准和方法、访谈中提出的问题、提问的方式和顺序、访谈者回答的方式等都有统一的要求。 15、被试间设计：要求每个被试者（组）只接受一个自变量处理，对另一被试者（组）进行另一种处理。

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模拟联合国大会优秀立场文件范例 (2011-09-07 17:31:53) 标签：这是两篇很好的立场文件，大家可以参考~ ? ? 范文一代表：陈卫杰学校：向明中学国家：马达加斯加共和国委员会：联合国经济和社会发展理事会议题：国际移民和城市发展 ? 现今，世界上很多国家都面对着国际移民与城市发展的问题。由于世界经济全球化，地缘政治发展，通讯技术革命，低价商业运输和人口结构变动等原因，国际移民的数量正在不断地增长中。国际移民很大程度上影响了各国城市发展，对目的国的城市经济、社会、环境、文化诸方面发展产生越来越显着的影响，有积极的一面，也有消极的一面。移徙与发展有着千丝万缕的联系，发展主导移徙，移徙反过来又影响发展。所以，如何使国际移民规范化，对城市发展起到积极作用，成了一个需要我们共同探讨的问题。

? 联合国《国际移徙与发展》报告指出：“在最理想的情况下，移民可以使接受国、原籍国和移徙者本人都得到好处。向外移徙与许多人最终回归一样，对于振兴国家经济发挥了决定性的作用。”为达到这种最理想的情况，国际社会都做出了很大的努力。1946年通过的《世界人权宣言》和1990年通过的《保护所有移徙工人及其家庭成员权利国际公约》为移徙者的人权保护奠定了基础。国际移民组织也通过与各国合作处理移民问题，确保移民有秩序地移居接收国。国际移徙问题全球委员会在2005年10月发布的报告中提出33条建议，旨在加强对国际移徙问题的国家、区域和全球治理。欧盟等区域集团提出了区域性倡议，有些国家制定了双边协定，来确保移民流动在有关国家互惠的情况下进行。 ? 作为一个非洲的发展中国家，马达加斯加在国际移民与城市发展问题上做出了巨大的努力。我们主张充分发挥移徙者与本地工人的互补性，促进接受国的经济。通过立法来消除种族歧视、歧视妇女、歧视移徙工人。必要时，我们创建特别移民区来保护文化多样性发展。同时，我们也设法克服与其他国家的分歧并改变外交政策的战略。 ? 马达加斯加政府提倡发达国家采取更加宽松的移民政策，这样一方面使得发展中国家可以通过侨汇增加本国的外汇收入，同时学习和引进发达国家先进的科学技术，另一方面也使得发达国家的劳工市场得到有效的补充。我们建议发达国家应该消除移民人数增加会导致目的国失业率增加的这一顾虑。在目的地国，多数不同背景的研究都表明，国际移徙的增加对总体工资水平和失业影响甚微。移徙者的职业分布与非移徙者迥异，也充分表明两者的互补性，而不是竞争性。我们也提倡发展中国家放宽投资移民的政策，这样可以实现较好的招商引资，加快目的国的城市经济发展。同时，也为发达国家拓宽全球市场提供了绝好的商机。我们同样不能忽视一个问题，无论是发达国家还是发展中国

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