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UNIVERSIDAD TÉCNICA PARTICULAR DE LOJA La Universidad Católica de Loja 73

SEGUNDO BIMESTRE Guía didáctica: Teoría de Colas

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UNIVERSIDAD TÉCNICA PARTICULAR DE LOJA La Universidad Católica de Loja74

SEGUNDO BIMESTREGuía didáctica: Teoría de Colas

Cuestiones de repaso

Como medidor de asimilación de los contenidos, desarrollaremos las siguientes cuestiones de repaso; le recomendamos que responda las preguntas de auto evaluación y para su información registre el nivel de desempeño que observo, esto le permitirá saber los temas que debe volver a revisar si su desempeño lo considera medio, y en caso de observar un desempeño malo recuerde que puede solicitar tutoría mediante el campus virtual o telefónicamente.

NO. CUESTIÓN

DESPUéS DE RESPONDER, EL DESEMPEñO hA SIDO:

MALO MEDIO MUY BIEN

9.1. ¿Quétipodecálculosimplicaunasimulación?

9.2. ¿Los lenguajes orientados a procesos se activan conlas mismas acciones que los lenguajes orientados a eventos?

9.3. ¿QuéeselGPSS?

9.4. Diga algunos paquetes del mercado que se utilizan para hacer simulación.

9.5. ¿En qué tipos de lenguajes otros usuarios escribenmodelos de simulación



Interactividad a través de los Foros de Campus Virtual

Ingrese periódicamente al campus virtual que se encuentra en la siguiente dirección: http://www.utpl.edu.ec, allí existe un link en donde dice Entorno Virtual de Aprendizaje digite su usuario y contraseña personal luego haga clic en el botón Entrar y de respuesta a la siguiente pregunta que se ha previsto como parte del foro, su aporte es importante.

• A su criterio porque los lenguajes orientados a procesos se basan en el concepto de entrada y salida con una “caja negra”.

Ejercicios

Para reforzar el nivel de conocimientos del presente capítulo se deben realizar las siguientes actividades.

• Analice las características de los dos tipos de lenguajes de simulación y establezca 2 características comunes y 2 diferencia.

Documentación adicional

Para ampliar la información del texto base se dispone de bibliografía adicional, que estará disponible como anexo en la guía de estudio o en digital, a estos últimos recursos podrá acceder a través del campus virtual.

Descripción del documento Archivo disponible en UTPLONLINE

Este documento nos hace referencia a información relacionada al capítulo, que es de vital importancia para aclarar algunos conceptos y ejemplos vertidos en el mismo. Por lo que se sugiere sea revisado conjuntamente con el texto base.

MaterialdeApoyoCap9



Datos Generales:

Texto baseANDERSON David, SWEENEY Dennis, WILLIAMS Thomas., “Métodos Cuantitativos para los negocios “, Thomson. Edición, 2004, 822 pág. ISBN; 970-686-372-9.

Capítulo 15. Simulación

Páginas 673 - 677Horas de estudio empleadas para el desarrollo del contenido

4 horas

Propósito

Elpropósitodeestecapítuloesverificaryvalidarelprocesodesimulacióndesistemasreales.

Conceptos Clave

Verificación

Proceso de determinar que el procedimiento de computadora que realiza los cálculos de la simulación es correcto desde el punto de vista lógico.

Validación

Proceso de asegurar que el modelo de simulación proporciona una representación precisa de un sistema real.

Otros Problemas de Simulación

Capítulo 10



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Cuestiones de repaso

Como medidor de asimilación de los contenidos, desarrollaremos las siguientes cuestiones de repaso; le recomendamos que responda las preguntas de auto evaluación y para su información registre el nivel de desempeño que observo, esto le permitirá saber los temas que debe volver a revisar si su desempeño lo considera medio, y en caso de observar un desempeño malo recuerde que puede solicitar tutoría mediante el campus virtual o telefónicamente.

NO. CUESTIÓN

DESPUéS DE RESPONDER, EL DESEMPEñO hA SIDO:

MALO MEDIO MUY BIEN

10.1. ¿Qué significado tiene la simulación porcomputadora?

10.2. ¿Los modelos de simulación también puedenelaborarse usando lenguajes de programación de propósito general?

10.3. ¿Qué ejecuta con frecuencia un comando en unpaquete de simulación de simulación de propósito especial?

10.4. ¿Cuáles son esos dos pasos esenciales en cualquierestudio de simulación?

10.5. Determine 2 ventajas y 2 desventajas de usar simulación

10.6. ¿Lasimulacióngarantizaunasoluciónóptima?



Interactividad a través de los Foros de Campus Virtual

Ingrese periódicamente al campus virtual que se encuentra en la siguiente dirección: http://www.utpl.edu.ec, allí existe un link en donde dice Entorno Virtual de Aprendizaje digite su usuario y contraseña personal luego haga clic en el botón Entrar y de respuesta a la siguiente pregunta que se ha previsto como parte del foro, su aporte es importante.

• Describalaimportanciadehacerverificaciónyvalidaciónenlasimulación.

Ejercicios

Para reforzar el nivel de conocimientos del presente capítulo se deben realizar las siguientes actividades.

• Lea MC en Acción de la Pág. 676 en donde se analiza sobre un Diseño de Sistemas de Manufactura en Vilpac de México.

Documentación adicional

Para ampliar la información del texto base se dispone de bibliografía adicional, que estará disponible como anexo en la guía de estudio o en digital, a estos últimos recursos podrá acceder a través del campus virtual.

Descripción del documento Archivo disponible en UTPLONLINE

Este documento nos hace referencia a información relacionada al capítulo, que es de vital importancia para aclarar algunos conceptos y ejemplos vertidos en el mismo. Por lo que se sugiere sea revisado conjuntamente con el texto base.

Se adiciona una ayuda del libro base donde hay soluciones a problemas planteados en el mismo.

También se adjunta presentaciones en Power Point que le servirán de mucha ayuda durante el estudio de esta asignatura

Material de Apoyo Cap10

Ayudasparaelsegundobimestre–LibroBase (inglés)

Segundo Bimestre


SOLUCIONARIOGuía didáctica: Teoría de Colas

Nota importante: Estas respuestas serán subidas al sistema EVA(Entorno Virtual de Aprendizaje), después de que Usted haya enviado el trabajo a distancia correspondiente a cada bimestre para ser revisada antes de cada examen bimestral, el mismo que le servirá para comparar el avance de su aprendizaje. No es necesario que las respuestas suyas sean con las mismas frases que desarrolla el profesor guía, pero no deberá salirse del contexto de lo que realmente se le pregunta.

PRIMER BIMESTRE

Solución a las cuestiones de repaso del Capítulo 1 al 5

No. TEMA RESPUESTA

DESPUéS DE RESPONDER, EL DESEMPEñO HA SIDO:

MALO MEDIO MUY BUENO

1 Estructura de un Sistema de Línea de Espera

CAPITULO 1

2 Modelos de Línea de espera de un solo canal

CAPITULO 2

3 Modelo de Línea de Espera con Canales Múltiples

CAPITULO 3

4 Algunas relaciones generales para modelos de líneas de espera

CAPITULO 4

5 Otros Modelos de Línea de Espera

CAPITULO 5

Solucionario


SOLUCIONARIO Guía didáctica: Teoría de Colas

SEGUNDO BIMESTRE

Solución a las cuestiones de repaso del Capítulo 6 al 10

No. TEMA RESPUESTA

DESPUéS DE RESPONDER, EL DESEMPEñO HA SIDO:

MALO MEDIO MUY BUENO

6 Introducción a la Simulación

CAPITULO 6

7 Simulación de Inventario

CAPITULO 7

8 Simulación de Línea de Espera

CAPITULO 8

9 Lenguajes de Simulación

CAPITULO 9

10 Otros Problemas de Simulación

CAPITULO 10

AYUDAS PRIMER BIMESTRE ( Libro base en Inglés)Chapter 14

Waiting Line Models

Case Problem 1: Regional Airlines

1. Single-Channel Waiting Line Analysis

The analysis that follows is based upon the assumptions of Poisson arrivals and exponential service times. With one call every 3.75 minutes, we have an average arrival rate of

λ = 60/3.75 = 16 calls per hour

Similarly, with an average service time of 3 minutes, we have a service rate of



µ = 60/3 = 20 calls per hour

The operating characteristics of a single channel system with λ = 16 and µ = 20 are as follows:

hours (12 minutes)

hours (15 minutes)

Operating the telephone reservation service with only one ticket agent appears unacceptable. With 80% of incoming calls waiting (Pw = 0.80) and an average waiting time of 12 minutes (Wq = 12), the company clearly needs to consider using two or more agents.

2. Multiple-Channel Waiting Line Analysis

Since Regional’s management team agreed that an acceptable service goal was to immediately answer and process at least 85% of the incoming calls, the probability of waiting must be 15% or less. Computing Pw for k = 2 agents and k = 3 agents provides the following.

For k = 2

For k = 3



Based on the value of Pw, 3 ticket agents will be required to meet the service goal. Other operating characteristics of the 3-ticket-agent system are as follows:

P0 = 0.4472

Lq = 0.0189

L = 0.8189

Wq = 0.0012 hours = 0.07 minutes

W = 0.0512 hours = 3.07 minutes

3. We would need to know the average arrival rate for each hourly period throughout the day. An analysis similar to the one above would determine the recommended number of reservation agents each hour. This information could then be used to develop full-time and part-time shift schedules which would meet the service goals.

Case Problem 2: Office Equipment, Inc.

1. λ = 1 call/50 hours = 0.02 calls per hour

2. Mean service time = travel time + repair time = 1 + 1.5 = 2.5 hours

µ = 1 / 2.5 hours = 0.4 customers per hour

3. The travel time is 1 hour. While this is considered part of the service time it actually means that the customer will be waiting during the first hour of the service time. Thus, travel time must be added to the time spent in line as predicted model in order to determine the total customer waiting time.

4. Using output from The Management Scientist, we have the following:

Probability that no customers are in the system 0.5380

Average number of customers waiting 0.2972

Average number of customers in the system 0.7593

Average time a customer spends in the waiting line 1.6082 hours*

Average time until the machine is back in operation 4.1082 hours

Probability of a wait more than one hour 0.4620

Hours a week the technician is not on service calls

(0.5380) x 40 hours = 21.5 hours

Total cost per hour for the service operation $155.93

*The average time a customer spends in the waiting line is 1.6082 hours. This is the average time for the service technician to complete all previous service call commitments and be ready to travel to the new customer. Since the average travel time is 1 hour for the service technician to reach the new customer’s office, the total customer waiting time is 1.6082 + 1 = 2.6082 hours.



Thus, the one technician is able to meet the company’s 3-hour service guideline. The total cost is $155.93 per hour.

Note that the waiting line model indicates the probability that a customer has to wait is 0.4620. Since all customers wait an average of 1-hour of travel time whenever the service technician is free, this probability is actually the probability that a customer will have to wait more than 1-hour for a service technician to arrive.

5. If the company continues to use one technician when the customer base expands to 20 customers, the average time in the waiting line will increase to 6.9454 hours. With an average travel time of 1 hour, the average total waiting time will be 6.9454 + 1 = 7.9454 hours. The total cost will be $397.78 per hour. This average total waiting time is too long and a second technician is definitely necessary. Using output from The Management Scientist, two service technicians provide the following:


Average number of customers in the waiting line 0.2104





Hours a week the technicians are not on service calls

P0 = 0.3525 (0.3525) x 2 technicians x 40 hours = 28.2 hours

P1 = 0.3525 (0.3525) x 1 technician x 40 hours = 14.1 hours

Total = 42.3 hours

Total cost per hour of service operation $275.27

*The average time a customer spends in the waiting line is 0.5581 hours. This is the average time for the service technician to complete all previous service call commitments and be ready to travel to the new customer. Since the average travel time is 1-hour for the service technician to reach the new customer’s office, the total customer waiting time is 0.5581 + 1 = 1.5581 hours. Thus, two technicians are needed to meet the company’s 3-hour service guideline when the company reaches 20 customers. The total cost is $275.27 per hour.

6. A comparison of two and three technicians with 30 customers shows that the average total waiting time with two technicians will be 2.6895 hours and the average total waiting time with three technicians will be 1.2626 hours. The hourly cost with two technicians is $391.94 and the hourly cost with three technicians is $397.08. While three technicians provide a smaller waiting time, two technicians are able to meet the 3-hour service guideline for a total lower cost. Thus, the company should continue to use two technicians when the customer base expands to 30 customers. Using output from The Management Scientist, two service technicians provide the following:




Average number of customers in the waiting line 0.9353





Hours a week the technicians are not on service calls

P0 = 0.1760 (0.1760) x 2 technicians x 40 hours = 14.08 hours

P1 = 0.2640 (0.2640) x 1 technician x 40 hours = 10.56 hours

Total = 24.64 hours

Total cost per hour of service operation $391.94

*The average time a customer spends in the waiting line is 1.6895 hours. While the average travel time is 1-hour for the service technician to reach the new customer’s office, the average total customer waiting time is 1.6895 + 1 = 2.6895 hours.

7. The OEI planning committee’s proposal anticipated that three technicians would be needed at a total cost of $397.08 per hour. Thus, the recommendation to stay with two technicians has as annual savings of (397.08 – 391.94) x 8 hours/day x 250 days/year = $10,280.

AYUDAS SEGUNDO BIMESTRE ( Libro base en Inglés)

Chapter 15

Simulation

Case Problem 1: Tri-State Corporation

With the specific financial analysis data input into cellsD3:D8, the formulas used to develop theportfolio projection spreadsheet are shown below. The rows are copied to extend the spreadsheet to the desired 30-year projection.

Beginning New Portfolio Ending

Year Age Portfolio Salary Investment Earnings Portfolio

1 =D3 =D5 =D4 =$D$7*D12 =$D$8*(C12+0,5*E12) =C12+E12+F12

2 =B12+1 =G12 =(1+$D$6)*D12 =$D$7*D13 =$D$8*(C13+0,5*E13) =C13+E13+F13

3 =B13+1 =G13 =(1+$D$6)*D13 =$D$7*D14 =$D$8*(C14+0,5*E14) =C14+E14+F14

4 =B14+1 =G14 =(1+$D$6)*D14 =$D$7*D15 =$D$8*(C15+0,5*E15) =C15+E15+F15

5 =B15+1 =G15 =(1+$D$6)*D15 =$D$7*D16 =$D$8*(C16+0,5*E16) =C16+E16+F16

1. Increasing the annual investment rate in cell D7 will generate the following 30-year portfolio projections:



Rate Projected Portfolio

5% $ 721,667

6% 815,397

7% 909,127

8% 1,002,857

A 1% increase in the annual investment rate increases the projected 30-year portfolio by $93,730. The annual investment rate would have to be increased to 8% in order to achieve the $1,000,000 goal.

2. The simulation spreadsheet that we developed placed the simulated annual salary growth rate in column D and the simulated annual portfolio growth rate in column G. The revised data input section and the cell formulas used to develop the simulation spreadsheet are as follows.

Data Input

Financial Analysis - Portfolio Projection

Age 25

Current Salary $34.000

Current Portfolio $14.500

Annual Salary Growth Rate

Minimum Rate 0%

Maximum Rate 10%

Annual Investment Rate 8%

Annual Portfolio Growth Rate

Mean Rate 10%

Standard Deviation 5%

Formula Spreadsheet – Columns D to I

Salary New Portfolio Portfolio Ending

Growth % Salary Investment Growth % Earnings Portfolio

=D4 =$D$9*E16 =DISTR.NORM.INV(ALEATORIO();$D$11;$D$12) =G16*(C16+0,5*F16) =C16+F16+H16

=ALEATORIO()*($D$8-$D$7) =(1+D17)*E16 =$D$9*E17 =DISTR.NORM.INV(ALEATORIO();$D$11;$D$12) =G17*(C17+0,5*F17) =C17+F17+H17




The simulated 30-year portfolio amounts will vary considerably from trial to trial. The uncertainty associated with the annual salary growth rate and the annual portfolio growth rate will show that there is uncertainty in reaching the $1,000,000 even if the annual investment rate is increased to 8%.



A few simulation trials will point out that the 30-year portfolio variability and the uncertainty of reaching the $1,000,000 goal. However, repeating the simulation numerous times will be necessary to provide an objective basis for estimating the probability of reaching $1,000,000.

We preformed the simulation of 1000 trials. The probability of reaching $1,000,000 was estimated to be 0.48. Thus, the simulation conclusion is that there is less than a 50% chance of reaching $1,000,000 even if an 8% investment rate is used. During the 1000 trials, the 30-year portfolio varied from $550,000 to $1,900,000. There was a 30% chance that the portfolio would not reach $900,000.

3. The simulation model suggests additional strategies should be considered to obtain a reasonably high probability of reaching the $1,000,000 portfolio goal. Increasing the annual investment rate to 9% or 10% may be worth considering. If this rate is getting too high, then extending the 30-year period by adding years may be the best strategy for reaching the $1,000,000 goal.

4. The longer 35-year period is definitely a good idea. Expanding the simulation spreadsheet by five years will show that almost every simulated 35-year portfolio exceeded $1,000,000. 1000 simulated trials with the 35-year period showed better than a 99% chance that the portfolio would exceed $1,000,000.

In fact, the extra five years increased the expected portfolio from $1,002,857 to $1,697,622, or almost $700,000. This result demonstrates the long-term advantages of investing.

5. The simulation worksheet can be used for any employee. The employee may enter his/her age, current salary, current portfolio, and any assumptions he/she cares to make about salary growth rate, investment rate, and portfolio growth rate. These assumptions in cells D3:D8 would be different for each employee and rows would be added to the worksheet to reflect the number of years appropriate for the employee.

By varying the inputs and projecting the ultimate portfolio value, the employees should be able to make investment observations such as the following:

• Begin early with an investment program. The more years the better the ending portfolio value.

• Make new contributions to a investment program at the highest possible rate. Increasing the rate 1% or 2% will have a significant impact on the long-term portfolio.

• Resist the temptation to use assets in investment programs for short-term personal expenditures. Early withdraws can be shown to have a major impact the long-term portfolio value.

• Be patient. Benefits of investment programs may not be significant for the first 5 to 10 years. It is really in the later years where a consistent investment program pays its biggest dividends.



Case Problem 2: Harbor Dunes Golf Course

The Crystal Ball simulation worksheet that we used is as follows:

Harbor Dunes Golf Course Simulation - Option 1

Available Tee Times 20

Regular Green Fee $160

Replay Green Fee $25

Cart Fee $20

Number of Option 1: $25 Green Fee + Cart Fee

AdvanceReservations

Number ofReplay RequestsProbability Probability

8 0,01 0 0,01

9 0,04 1 0,03

10 0,06 2 0,05

11 0,08 3 0,05

12 0,10 4 0,11

13 0,11 5 0,15

14 0,12 6 0,17

15 0,15 7 0,15

16 0,10 8 0,13

17 0,09 9 0,09

18 0,07 10 0,06

19 0,05

20 0,02

Cell Formulas:

Crystal Ball Model

Number of Advance Reservations Crystal Ball Custom Distribution with Data B11:C23

Number of Times Available =B3-C27

Number of Requests for a Replay Crystal Ball Custom Distribution with Data D11:E21

Number of Replays =MIN(C28,C29)

Revenue from Advance Reservations =4*(B4+B6)*C27

Revenue from Replays =4*(B5+B6)*C30

Forecast

Cell

Total Revenue =C32+C33



1. Selected statistical results:

Simulation results for various. Below is one set of possible results:

Statistics Option 1 Option 2

Mean $11,062 $11,173

Median $11,160 $11,360

Standard Deviation $ 1,802 $ 1,838

Minimum $ 5,940 $ 5,760

Maximum $14,400 $14,400

Using the mean, the options are similar, but Option 2 is preferred with a daily revenue advantage of $11,173 - $11,062 = $111.

2. Go with Option 2: The $50 per replay option.

3. Without the replay option, Harbor Dunes reported $10,240 daily revenue. Thus the Option 2 replay policy is estimated to generate an additional revenue of $11,173 - $10,240 = $933 per day. Over a 90-day spring season, the estimated revenue increase is 90 * $933 = $83,970. The replay policy is definitely worthwhile.

4. One suggestion is that Harbor Dunes wait until mid-morning to determine the afternoon replay option. If by mid-morning, the afternoon tee time reservations are relatively low, Harbor Dunes may want to offer the Option 1 replay policy which generates more demand for the afternoon tee times. However, if by mid-morning, the afternoon tee time reservations were relatively high, Harbor Dunes would want to continue the Option 2 replay policy. Simulation runs could help determine a level of advance afternoon tee time reservations that would indicate whether to implement the Option 1 or Option 2 replay policy.

Case Problem 3: County Beverage Drive-Thru

Spreadsheets patterned after those used for the ATM one- and two- channel simulations in Figure 6.15 and 6.7 may be used to solve this case problem. We recommend using one workbook with three worksheets, one for each of the following Drive-Thru designs: 1 channel with 1 clerk, 1 channel with 2 clerks and 2 channels with 2 clerks.

The1Channelwith1ClerkDesign

Use the format of the Hammondsport 1 ATM simulation model and set up the data for the 1 channel with 1 clerk design as follows:



The interarrival time for customer 1 would be provided by the cell formula

=-$B$4*LN(RAND())

The service time for customer 1 would be provided by the cell formula

=VLOOKUP(RAND(),$A$9:$C$16,3)

The summary statistics formulas should be provided at the end of the simulation data. The =COUNTIF function can be used to count the number of customers who wait more than 6 minutes and more than 10 minutes.

The1Channelwith2ClerksDesign

Use the format of the Hammondsport 1 ATM simulation model and set up the data for the 1 channel with 2 clerks design as follows:

1

2

34

5

6

7

8

9

10

11

12

13

14

1516

A B C DCounty Beverage Drive-Thru with One Channel One Clerk

Interarrival times (Exponential Distribution)Mean 6

Service Time Distribution

Lower Upper Service

Random No. Random No. Time

0.00 0.24 2

0.24 0.44 3

0.44 0.59 4

0.59 0.73 5

0.73 0.85 6

0.85 0.93 7

0.93 0.98 80.98 1.00 9

1

2

34

5

6

7

8

9

10

11

1213

A B C DCounty Beverage Drive-Thru with One Channel Two Clerks

Interarrival times (Exponential Distribution)Mean 6

Service Time Distribution

Lower Upper Service

Random No. Random No. Time

0.00 0.20 1

0.20 0.55 2

0.55 0.85 3

0.85 0.95 40.95 1.00 5

The service time for customer 1 would be provided by the cell formula

=VLOOKUP(RAND(),$A$9:$C$13,3)




The2Channelswith2ClerksDesign

Use the format of the Hammondsport 2 ATMs simulation model and set up the data as previously shown for the 1 channel with 1 clerk design. Customer interarrival times and service times are the same as shown for the 1 channel with 1 clerk design.

This part of the case is optional in that significant spreadsheet modeling skills are required to duplicate the 2-channel simulation model. Selected cell formulas area as follows:

Cell I16 = G16

Cell J16 = 0

Cell I17 = IF(I16=MIN(I16,J16),G17,I16)

Cell J17 = IF(J16=MIN(I16,J16),G17,J16)

Cells I17 and J17 can be copied to fill columns I and J.


Each design was tested with a 1000 customer simulation run. Data on the first 100 customers was discarded and summary statistics collected for a total of 900 customers.

Simulation results will vary but the approximate results are as follows:

Characteristic 1 Channel 1Clerk

1 Channel 2 Clerks

2 Channels 2Clerks

Number Waiting 640 377 167

Probability of Waiting 0.71 0.42 0.19

Average Waiting Time 6.1 1.0 0.4

Maximum Waiting Time 37.8 11.7 9.3

Utilization of Drive Thru 0.72 0.42 0.36

Number Waiting > 6 Minutes

322 26 7

Probability Waiting > 6 Minutes

0.36 0.03 0.01

Number Waiting > 10 Minutes

191 4 1

Probability Waiting > 10 Minutes

0.21 0.01 0.00

The 1 channel with 1 clerk system appears unacceptable. The mean waited time is over 6 minutes which exceeds the company guideline of 1.5 minutes. In addition, over 300 customers waiting over 6 minutes and almost 200 customers waited over 10 minutes. The company must do something to improve the service characteristics of its drive-thru operation or face the loss of substantial business.



The 1 channel with 2 clerks system appears to be the best design., The mean waiting time of approximately 1 minute is with the company’s guideline of 1.5 minutes. Relatively few customers experienced the 6 to 10 minute waiting times. The performance of the 2 channel with 2 clerks system is the best overall, but the added cost may not justify the expansion to the two channel operation.

LE/lg/27-08-09/92

cll/2011-06-15

Documents

G18904.3[1]