Sim i (yield mgmt)

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SIMULATION – PART IIntroduction to Simulation

and Its Application toYield Management

For this portion of the session, the learning objectives are: Receive an introduction to the technique of Simulation. Learn the meaning of Yield Management. Illustrate how simulation can be applied to Yield Management in general and the

Airplane Overbooking Problem in specific. Learn how to simulate using a Table of Random Numbers. Receive an introduction to simulation using Crystal Ball, an add-in to Excel.

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GENERAL PRINCIPLES OF SIMULATION A simulation is an experiment in which we attempt to understand how

something will behave in reality by imitating its behavior in an artificial environment that approximates reality as closely as possible. (In this course, the artificial environment will be an Excel spreadsheet within a computer.)

Within this artificial environment, a simulation conducts an experiment that would be too costly and too time-consuming to conduct in reality. A simulation uses “funny money” and just a few minutes (or seconds) of time.

Because a simulation is based on random numbers, any value obtained from a simulation is only an estimate, that is, only an approximation of the true value.

Because a simulation is based on random numbers, obtaining accurate estimates requires a simulation with a very large number of “trials” (or “runs” or “iterations”).

Because a simulation requires a very large number of trials, a simulation is best conducted on a computer.

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Yield Management is used by many businesses, such as:

Airlines

Hotels

Rental Cars

Restaurants

Yield Management encompasses a wide variety of techniques, such as maximizing profit by determining how to adjust the prices of “seats” as it gets closer and closer to the date/time when customers will use the “seats”. In this course, we will not consider this technique.

A common practice in Yield Management is overbooking, that is, confirming more reservations than the number of “seats” available.To illustrate how simulation can be applied to Yield Management, we will use an example of airplane overbooking.

A technique of Yield Management that we will consider is optimizing the number of reservations to confirm for a limited number of “seats”, where there are two types of penalties:1. A penalty for having customers who have confirmed reservations but who are unable to “occupy a seat”, and2. A penalty for having empty “seats” because of customers who are “no shows”.

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A B C D E F G H I J K L M N O P Q

Airplane OverbookingGIVEN DATA

115 Plane Capacity$400 Ticket Price (Assume this is fully refundable to those who "no-show".)$700 Cost per "Bumped" Passenger

Assume that the variable cost per passenger is 0.Assume no "stand-bys".

0.1 Probability That a Passenger with a Confirmed Reservation is a "No-Show"Probability Distribution for the Demand for Confirmed Reservations ----------> DEMAND PROB.

100-109 0.01 eachDECISION 110-119 0.02 each

127 Maximum Allowable Number of Confirmed Reservations 120-129 0.04 each130-139 0.02 each140-149 0.01 each

Decision Cell with lower limit of 115,upper limit of 149, and step size of 1

To illustrate both simulation and the airplane overbooking problem, we will consider the example below.

EXAMPLE

NOTE: As indicated in Cell A13, we will temporarily assume that the maximum allowable number of confirmed reservations is 127.

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Below is a complete summary of our example’s given data:

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A B C D E F G H I J K L M N O P Q R S




0.1 Probability That a Passenger with a Confirmed Reservation is a "No-Show"Probability Distribution for the Demand for Confirmed Reservations ----------> DEMAND PROB. ----------> DEMAND PROB.

100-109 0.01 each 100 0.01

110-119 0.02 each 101 0.01

120-129 0.04 each 102 0.01

130-139 0.02 each 103 0.01

140-149 0.01 each 104 0.01

105 0.01

106 0.01

107 0.01

108 0.01

109 0.01

110 0.02

111 0.02

112 0.02

113 0.02

114 0.02

115 0.02

116 0.02

117 0.02

118 0.02

119 0.02

120 0.04

121 0.04

122 0.04

123 0.04

124 0.04

125 0.04

126 0.04

127 0.04

128 0.04

129 0.04

130 0.02

131 0.02

132 0.02

133 0.02

134 0.02

135 0.02

136 0.02

137 0.02

138 0.02

139 0.02

140 0.01

141 0.01

142 0.01

143 0.01

144 0.01

145 0.01

146 0.01

147 0.01

148 0.01

149 0.01

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33 37 41 45 49 53 57 61 65 6932 36 40 44 48 52 56 60 64 68

11 13 15 17 19 21 23 25 27 29 31 35 39 43 47 51 55 59 63 67 71 73 75 77 79 81 83 85 87 8900 01 02 03 04 05 06 07 08 09 10 12 14 16 18 20 22 24 26 28 30 34 38 42 46 50 54 58 62 66 70 72 74 76 78 80 82 84 86 88 90 91 92 93 94 95 96 97 98 99

100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149

Random Numbers Corresponding to Demand

Demand for Confirmed Reservations

SIMULATING DEMAND

USING A TABLE OF RANDOM NUMBERS

As examples,

If RN = 07, then Demand =



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The Binomial Probability DistributionTo model the scenario where customers with confirmed reservations are “no shows”, we will use the Binomial Probability Distribution.

Suppose there will be n independent trials of an event that has two possible outcomes: Outcome 1, with probability p Outcome 2, with probability 1-p

Then, the number of the n trials that end in Outcome 1 has a Binomial Probability Distribution with parameters n and p. (Alternatively, the number of the n trials that end in Outcome 2 has a Binomial Probability Distribution with parameters n and 1-p.)

Example 1: The number of “heads” that results when you flip a coin 10 times has a Binomial Probability Distribution with parameters n=10 and p=0.50.

Example 2: If there is a 10% chance that a potential airplane passenger with a confirmed reservation is a “no show”, then the number of “no shows” that results when there are 120 confirmed reservations has a Binomial Probability Distribution with parameters n=120 and p=0.10.

For a Binomial Probability Distribution with parameters n and p, the mean is np and the variance is np(1-p).

The next slide displays the probability distributions for the Binomial Probability Distributions in Example 1 and Example 2 above.

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Example 1: Flipping a Coin Example 2: “No Shows”

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SIMULATION THE NUMBER OF “NO SHOWS”

USING A TABLE OF RANDOM NUMBERS

As examples,

For simplicity, assume

There 15 confirmed reservations.

0.1 is the probability that a person with a confirmed reservation is a “No Show”.

# of “No Shows” =



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SPREADSHEET FOR SIMULATION

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The following pages provide a summary of how to use Crystal Ball to analyze the Airplane Overbooking Problem.

13DefineAssumption

DefineDecision

DefineForecast

Copy Data Paste Data Run Preferences

Start Simulation

Stop Simulation

Reset Simulation

Single Step

Forecast Charts Create Report

New Menu Selections

OVERVIEW OF CRYSTAL BALLAfter launching Crystal Ball, you will see the following menu and toolbars, where the three menu selections and the lower toolbar have been added-in to Excel. Crystal Ball permits three types of cells:

Assumption Cells: Each Assumption Cell contains a value about which you are uncertain. (Think of the Assumption Cells as the decision problem’s independent variables or inputs.)

Forecast Cells: Each Forecast Cell is one of the spreadsheet’s “bottom lines” and contains a formula that refers directly or indirectly to at least one of the Assumption Cells. (Think of the Forecast Cells as the decision problem’s dependent variables or outputs.)

Decision Cells: Each Decision Cell is under control of the decision maker and contains a value from of a set of alternative values.

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Defining Assumption Cell A18: the Demand for Confirmed Reservations

The demand for confirmed reservations is a so-called Custom Probability Distribution.

It would be too time-consuming to manually enter the Custom Probability Distribution displayed in the Cell Range R11:S60.

Fortunately, Crystal Ball provides a way to “read in” the 50 values and the associated probabilities.

To do so, we proceed as summarized on the next slide.

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After doing so, the dialog box to the right appears, in which the Custom Probability Distribution has been “read in”. Click OK to return to the spreadsheet.

First click on Cell A18, next click the Define Assumption icon, then click Custom, and finally click OK. After doing so, the dialog box to the right appears. In this dialog box, first enter the Assumption Cell’s name as “Demand”, and then click Load Data.

After doing so, the dialog box to the right appears. In this dialog box, enter the Cell Range R11:S60, and then click OK.

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Defining Assumption Cell A20: the Number of “No Show” Reservations

To define Assumption Cell A20,

1. Click on Cell A20.

2. Click Binomial.

3. Click OK.

4. In the resulting dialog box,

A. Enter the name as Number Who “No-Show”.

B. Enter “Probability” as the cell reference =A9, and enter “Trials” as cell reference =A19.

C. Click Enter.

D. Click OK.

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After temporarily assuming that the maximum allowable of confirmed reservations is set to 127, after defining the two Assumption Cells in Cells A18 and A20, and after defining the Forecast Cell in Cell A27, we obtain the following spreadsheet:

Our goal is to determine what value in Cell A13 will maximize the mean of Cell A27.

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A B C D E F G H I J K L M N O P Q R S




0.1 Probability That a Passenger with a Confirmed Reservation is a "No-Show"Probability Distribution for the Demand for Confirmed Reservations ----------> DEMAND PROB. ----------> DEMAND PROB.

100-109 0.01 each 100 0.01

DECISION 110-119 0.02 each 101 0.01

127 Maximum Allowable Number of Confirmed Reservations 120-129 0.04 each 102 0.01

130-139 0.02 each 103 0.01

140-149 0.01 each 104 0.01

105 0.01

RESULTS OF DECISION ANNOTATIONS FOR COLUMN A 106 0.01

127 Demand for Confirmed Reservations <--- Assumption Cell: Custom with above data 107 0.01

127 Actual Number of Confirmed Reservations <--- =MIN(A13,A18) 108 0.01

13 Number With Reservations Who "No-Show" <--- 109 0.01

114 Number With Reservations Who "Show" <--- =A19-A20 110 0.02

114 Number With Reservations Who Board Plane <--- =MIN(A4,A21) 111 0.02

0 Number With Reservations Who Are "Bumped" <--- =A21-A22 112 0.02

113 0.02

$45,600 Total Revenue from Tickets <--- =A5*A22 114 0.02

$0 Total Cost of "Bumping" <--- =A6*A23 115 0.02

$45,600 TOTAL CONTRIBUTION <--- 116 0.02117 0.02

Assumption Cell: Binomial with n=A19 & p=A9

Forecast Cell: =A25-A26


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This slide and the following three slides display spreadsheets resulting from “debugging” the model by repeatedly clicking on the Single Step icon until four distinct types of scenarios are obtained.

Scenario 1: Demand > Supply & Bumping Occurs

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A B C D E F G H I J K L M N O P







RESULTS OF DECISION ANNOTATIONS FOR COLUMN A136 Demand for Confirmed Reservations <--- Assumption Cell: Custom with above data127 Actual Number of Confirmed Reservations <--- =MIN(A13,A18)

7 Number With Reservations Who "No-Show" <---120 Number With Reservations Who "Show" <--- =A19-A20115 Number With Reservations Who Board Plane <--- =MIN(A4,A21)

5 Number With Reservations Who Are "Bumped" <--- =A21-A22

$46,000 Total Revenue from Tickets <--- =A5*A22$3,500 Total Cost of "Bumping" <--- =A6*A23

$42,500 TOTAL CONTRIBUTION <---




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Scenario 2: Demand > Supply & No Bumping Occurs

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RESULTS OF DECISION ANNOTATIONS FOR COLUMN A130 Demand for Confirmed Reservations <--- Assumption Cell: Custom with above data127 Actual Number of Confirmed Reservations <--- =MIN(A13,A18)13 Number With Reservations Who "No-Show" <---

114 Number With Reservations Who "Show" <--- =A19-A20114 Number With Reservations Who Board Plane <--- =MIN(A4,A21)


$45,600 Total Revenue from Tickets <--- =A5*A22$0 Total Cost of "Bumping" <--- =A6*A23





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Scenario 3: Demand < Supply & Bumping Occurs

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Scenario 4: Demand < Supply & No Bumping Occurs

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Now that we are confident that the spreadsheet has been properly constructed, we are ready to run the simulation.

Recall that our goal is to determine the optimal value for the Maximum Number of Reservation to Confirm, that is the value for Cell A13 that maximizes the mean of the total contribution (to overhead and profit)

Although time-consuming, one way to do this would be to run the simulation 35 times, first with Cell A13 =115, then with Cell A13 =116, …, and finally with Cell A13 =149. After doing so, we could then choose the value that maximized the mean of the total contribution.

Wouldn’t it be nice if Crystal Ball could automate this process for us?

In fact, Crystal Ball can do so through its Decision Table Tool.

The next slide illustrates how to use the Decision Table Tool.

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Using Crystal Ball’s Decision Table ToolStep 1. To define the Decision Cell, first click cell and then click the Define Decision icon. The dialog box below will pop up. Within this box, enter a descriptive name for the decision and enter its lower & upper limits; then click on the radio button for Discrete and enter the Step. Finally click on OK.

Step 2. Choose the Run, Tools, Decision Table menu selection. The dialog box below (#1 of 3) will pop up. Within this box, highlight one of the Forecast Cells to be the Target Cell (i.e., the Forecast Cell whose mean value you want to optimize). Then click Next.

Step 3. In the resulting dialog box (#2 of 3), move the Decision Variable from “Available” to “Chosen” (i.e., from left to right) by first highlighting the decision variable and then clicking “>>”. Finally, click Next.

Step 4. In the resulting dialog box (#3 of 3), first enter the number of trials for each simulation and then click Start.

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Crystal Ball’s Decision Table Tool yields Rows 1-3 in the spreadsheet below. By clicking in Cell A1 on Forecast Charts, you can view any of the 35 Forecast Charts, including the one corresponding to the maximum Total Contribution, which can then be pasted into the spreadsheet.

Economy & Finance

Sim i (yield mgmt)