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WAITING LINES AND SIMULATION I. WAITING LINES (QUEUEING) : II. SIMULATION
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WAITING LINES I. Length of line: number of people in queue II. Time waiting in line III. Efficiency: waiting vs idle server IV. Cost of waiting
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I.WAITING LINES ASSUMTIONS 1) FIRST COME FIRST SERVE 2) ARRIVALS COME FROM VERY LARGE POPULATION 3) NUMBER OF ARRIVALS IS POISSON 4) SERVICE TIME IS EXPONENTIAL 5) ARRIVALS INDEPENDENT
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APPLICATIONS BANK TELLER LINE, CAR WASH INTERNET: CABLE VS PHONE LINE WAITING FOR CABLE GUY METERED FREEWAY ON RAMPS WAREHOUSE: ORDERS WAIT TO BE SHIPPED AIRPLANES WAITING TO LAND
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EXAMPLE: AUTO REPAIR ONE MECHANIC MAY NOT BE POISSON IF CUSTOMERS ARE CLUSTERED EARLY MORNING OR AFTER WORK MAY NEED TO USE SIMULATION LATER
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L=Average Length ALL customers in system Waiting AND being served
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Lq=Average Length of queue Customers waiting in line Number waiting to be served
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W=Av Time customer in system From arrival time to departure time Time waiting and being served
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Wq=Av time customer waits in queue Waiting to be served Marketing, Service operations management Customers may go to competitor if Wq big Exception: lowest price(trade off) Car dealer: Wq=0
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Interpret Wq Wq=40 minutes waiting in line W=60 minutes in system 20 minutes being served
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U=Utilization U=efficiency Probability server is busy Probability customer has to wait
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U=2/3 67% efficiency
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Po=P(zero customers in system) Po=1-U P(server is idle) P(customer does not have to wait) Here: Po =.33
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COST OF WAITING SUPPOSE EACH HOUR A CUSTOMER WAITS COSTS $10
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INTANGIBLE COST NOT ACCOUNTING COST MARKETING ESTIMATE USED FOR DECISION MAKING
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SUPPOSE MECHANIC RESIGNS TWO ALTERNATIVE ACTIONS ACT 1: MECHANIC #1, $17/HR LABOR COST, 3 CARS/HR ACT 2: MECHANIC #2, $19/HR, 4 CARS/HR 8 HRS/DAY
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MINIMIZE TOTAL COST TOTAL COST = WAITING COST + LABOR COST LABOR COST = (8)(COST/HR) WAIT COST = (#HRS WAITING)($10) AVERAGE #CARS ARRIVE/HR= 2 TOTAL #CARS/DAY = 8(2)=16
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MECHANIC #1 3 CARS/HOUR
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MECHANIC #2 4 CARS/HOUR
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WAIT COST MECHANIC#1MECHANIC#2 #SERVED/HR34 WAIT TIME.67 HR.25 HR DAILY WAIT TIME.67(16)= 10.67HR.25(16)= 4HR WAIT COST10.67(10)=$1074(10)=$40
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LABOR COST MECHANIC#1MECHANIC#2 HOURLY WAGE $17/HR$19/HR DAILY LABOR COST 8(17)=$1368(19)=$152
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Total cost MECHANIC#1MECHANIC#2 WAIT COST$107$40 LABOR COST$136$152 TOTAL COST$243$192=MIN
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HIRE SECOND MECHANIC? SIMILAR TABLE: 2 SERVERS VS 1 SERVER
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II. SIMULATION DEFINE PROBLEM DEFINE VARIABLES BUILD MODEL: IMITATE BEHAVIOR OF REAL WORLD LIST ALTERNATIVE ACTIONS RANDOM NUMBERS CHOOSE BEST ALTERNATIVE
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MONTE CARLO SIMULATION ADVANTAGES Flexibility Probabilities: Client understands model Familiar simulations: dice, board games, video games, flight simulator DISADVANTAGES No mathematical optimization (LP guarantees optimum) Trial and error Might not try best action
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EXAMPLES APOLLO 13 EMERGENCY RETURN WEATHER FORECAST SUGAR PLANTATION DECISION WHICH FIELD TO BURN
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EXAMPLE: WAIT LINE PREVIOUS SECTION RESTRICTIVE ASSUMPTIONS EXACT FORMULAS SIMULATION NO RESTRICTIVE ASSUMPTIONS ONLY APPROXIMATIONS
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EXAMPLE: WAIT LINE REFERENCE: RENDER, BARRY QUANTITATIVE ANALYSIS, P 708 BARGES ARRIVE AT PORT BARGES UNLOADED IN PORT OBJECTIVE: MINIMIZE DELAY FCFS:FIRST COME FIRST SERVED
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GIVEN: PROBABILITY DISTRIBUTIONS X1= NUMBER OF BARGES ARRIVING AT PORT X2= MAXIMUM NUMBER OF BARGES UNLOADED IN PORT
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ARRIVALS X1P(X1) O.13 1.17 2.15 3.25 4.20 5.10
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STEP1:CUMULATIVE PROB X1P(X1)P(X1<x) O.13 P(X1<0) 1.17.30P(X1<1) 2.15.45P(X1<2) 3.25.70 4.20.90 5.101
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STEP 2: RANDOM NUMBER INTERVALS X1P(X1)P(X<x)X1 RN O.13 P(X1<0)01 to 13 1.17.30P(X1<1)14 to 30 2.15.45P(X2<2)31 to 45 3.25.7046 to 70 4.20.9071 to 90 5.10191 to 00
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STEP 3: SIMULATE ARRIVALS DAYX1 RN (GIVEN) SIMULATED ARRIVALS 1060 2503 3884 4533
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MAX UNLOADED X2P(X2); GIVEN 1.05 2.15 3.50 4.20 5.10
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STEP 4: CUMULATIVE PROB X2P(X2)P(X2<x) 1.05 2.15.20 3.50.70 4.20.90 5.101
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STEP 5: RANDOM NUMBER INTERVALS X2P(X2)P(X2<x)X2 RN 1.05 01 to 05 2.15.2006 to 20 3.50.7021 to 70 4.20.9071 to 90 5.10191 to 00
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STEP 6: SIMULATE UNLOADING DAYX2 RN (GIVEN) SIMULATED MAXIMUM UNLOADED 1633 2283 3021 4744
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UNLOADED=MIN(3),(4) (1)#DE- LAYED (2) ARRIV (3) TOTAL (4)MAX UNL UNLOA DED 0003MIN(0,3 =0 0333MIN(3,3 =3 0441MIN(4,1 =1 4-1=333+3=64MIN(6,4 =4
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AVERAGE NUMBER DELAYED AV = TOTAL DELAYED = TOTAL NUMBER DAYS = ¾ = 0.75 REAL-WORLD: WOULD RE-DO SIMULATION WITH MORE WORKERS TO UNLOAD BARGES TO RE- CALCULATE AV
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