Chapter 14

Simulation

What Is Simulation?

• Simulation is to mimic a process by using computers.

Simulation and Decision Making

• Simulation helps pre-view a complicated business process to identify possible problems;

• It allows to try different decision alternatives, and select the best alternative as the decision.

Simulation and Uncertainty

• Simulation is particularly useful when there exist uncertainties (demand, processing time, for example) in a process with many correlated factors.

Probable Events andRandom Numbers

• Probable events:– Tossing a coin, it may land on Head (0.5) or Tail

(0.5).– Daily demand of an item can be 4 units (0.3), 5

units (0.45), or 6 units (0.25).

• Computers can randomly generate a number between 0 and 1.

• Monte Carlo Technique is using random numbers to do simulations.

Monte Carlo Process• The Monte Carlo process is an approach of

simulating a physical process with randomly generated numbers. The key idea:

Probability of a possible actual outcome =

Probability of the corresponding possible outcome in the simulation with random numbers

How “Monte Carlo” Works

Step 1. Put all possible outcomes and their probabilities in a table;

Step 2. Calculate cumulative probabilities;Step 3. Use the cumulative probabilities as

the cutting points of random number ranges.

Simulate Tossing Coins

• So, we can simulate the result of tossing a coin by generating a random number T so that if T is between 0 and 0.5, then the result is “Head”; if T is between 0.5 and 1, then the result is “Tail”.

Possible Outcome Probability Cumulative

ProbabilityHead 0.5 0.5Tail 0.5 1.0

Simulate Tossing a Dice

So,(0 to 0.1667) for “landing with 1”;(0.1667 to 0.333) for “landing with 2”;…(0.8333 to 1) for “landing with 6”.

Possible outcome Probability Cumulative Probability

1 1/6 1/6 = 0.1667

2 1/6 2/6 = 0.3333

3 1/6 3/6 = 0.5

4 1/6 4/6 = 0.6667

5 1/6 5/6 = 0.8333

6 1/6 6/6 = 1.0

Simulate Daily Demands

Daily Demand

Probability Cumulative Probability

Random Number Range

40 units 0.250 units 0.560 units 0.3

=RAND()

• =RAND() is an Excel function generating a random number between 0 and 1.

=IF()

• Syntax of =IF() function in Excel:=IF(C,a,b)It means: If condition C is true then the function value is a, otherwise the value is b.

• e.g. For tossing coins: =IF(C4>0.5, ”Head”, ”Tail”)

• =IF() function can be nested, e.g.=IF(B5<0.2, 40, IF(B5<0.7,50, 60))

=VLOOKUP()

• If we use =IF() function to represent six possible outcomes in tossing a dice case, then we have to have get =IF() function nested for five times, which is too awkward.

• =VLOOKUP() facilitates our work in that case.• Syntax of =VLOOKUP():

=VLOOKUP(value to lookup, range table,2)

Range Table for =VLOOKUP()

• 1st column contains separating points of ranges– Start from the smallest allowable number of the

ranges– In ascending order

• 2nd column lists values of =VLOOKUP corresponding to the ranges.

=VLOOKUP() for Tossing Dices (1)

• Set up the table of ranges.– Cumulative probabilities are in the 1st column,

starting from 0– Possible outcomes are in the 2nd column.

B C

5 Cumu. Prob Values6 0 17 0.1667 28 0.3333 39 0.5 4

10 0.6667 511 0.8333 612 1

=VLOOKUP() for Tossing Dices (2)

• Generate the simulation table:– In F5: =RAND()– In G5: =VLOOKUP(F5, $B$6:$C$12, 2)– Copy E5, F5, and G5 down to other rows.

E F G2 Simulation of tossing dices 3

4Tossing

#Random number

Outcome of tossing

5 1 0.618707 46 27

Simulate Demands (1)

• Possible daily demands (from past data):

Demand Probability8 0.01

9 0.0610 0.1111 0.3412 0.3113 0.1014 0.0515 0.02

Simulate Demands (2)

• Calculate Cumulative Probabilities:

Demand Probability Cumulative Probability8 0.01 0.01

9 0.06 0.0710 0.11 0.1811 0.34 0.5212 0.31 0.8313 0.10 0.9314 0.05 0.9815 0.02 1.00

Simulate Demands (3)

• Generate Vlookup range table in Excel

E F

Cumulative Probability Demand

4 0 85 0.02 96 0.07 107 0.18 118 0.52 129 0.83 13

10 0.93 1411 0.98 1512 1

Simulate Demands (4)

• Generate Simulations in Excel– In B4: =RAND() – In C4: =VLOOKUP(B4,$E$4:$F$12,2)– Copy A4, B4, C4 down to other rows

A B C

3Day #

Random number

Demand

4 1 0.575409 125 26 3

Tips of using Excel

• Use cell addresses, relative or absolute, rather than the values in the cells;

• Use Copy / Paste functions as far as possible;• Use multiple columns to decompose

complicated formulas;• Put the parameters to be changed on the top

of the spreadsheet;• Put the summary results on top.

Simulation for Decision Making• A simulation is actually a description of

day-by-day or week-by-week business operations.

• For a decision alternative, the simulation shows its effects on business quality or/and profit.

• After trying alternatives, the manager can pick one that would be best for business.

How Many Cases to Stock? (1)• Product BC-6 costs $56.95/case from the supplier, and is sold

at the price of $91.80/case. For the cases unsold at the end of a week, the store will sell them to a convenient store at price of $12.50/case. Shortage penalty cost is about $4 per case short. Possible demands of a week and their probabilities are as follows from the past records:

• Manager Wendy is considering how many cases of product BC-6 to stock at beginning of each week.

Weekly demand of BC-6 Probability

11 cases 0.45

12 cases 0.35

13 cases 0.2

Simulations of Weekly Business Operations on BC-6

Selling price ($): 91.80 per caseSalvage value ($): 12.50 per case unsolde at the end of a weekOrder cost ($): 56.95 per caseGoodwill penalty ($): 4.00 per case short

Number of cases of BC-6 to stock every week: 12 <- enter your trial order quantity in casesTotal number of cases shortage in 52 weeks: 14Total number of cases surplus in 52 weeks: 18Total profit in 52 weeks: 20,263.00 Average weekly profit of 52 weeks: 389.67

Week #

Number of cases ordered

per week

Random #

Weekly Demand

(case)

Numbr of cases sold in

the week

Revenue of the week from

regular sales ($)

Number of cases

short in the week

Goodwill penalty in the

week due to

shortage ($)

Number of cases

surplus in the week

Revenue of the week

from selling surplus at salvage price ($)

Total revenue of the

week ($)

Total cost of

the week ($)

Total profit of the week

($)

1 12 0.527107 12 12 1101.6 0 0 0 0 1101.6 683.4 418.22 12 0.951569 13 12 1101.6 1 4 0 0 1101.6 687.4 414.23 12 0.059013 11 11 1009.8 0 0 1 12.5 1022.3 683.4 338.94 12 0.967144 13 12 1101.6 1 4 0 0 1101.6 687.4 414.25 12 0.466342 12 12 1101.6 0 0 0 0 1101.6 683.4 418.26 12 0.070006 11 11 1009.8 0 0 1 12.5 1022.3 683.4 338.9

How Simulation Helps Decide How Many Cases to Stock (3)

• The worksheet each time simulates 52 weeks of business operations.

• Given the number of cases to order (green cell), Excel gives the operation results of 52 weeks (yellow cells).

• Decision maker may change the green cell, observe the outcome of 52 weeks in yellow cells, and choose the best order quantity.

Inventory Simulation (1)

• An inventory simulation simulates day-by-day transactions occurred on inventory, such as daily demand of inventoried item, number of units in stock, placing an order, length of lead time, and costs involved.

• An inventory simulation helps determine when to place an order to the supplier and how many units in an order.

Inventory Simulation of 365 days

Ordre Quantity: 150 Cost H per day: 0.05 Total holding: 718.3Reorder Point: 10 Cost S per order: 25 Total setup: 125

Cost L.S. / unit: 20 Total L.S. cost: 500Overall total: 1343.3

Total 750 15099 97.195 758 25 14366 750 3.0048 16 34

DayUnits

receivedBegin on

hand Rand # DemandLost sales

Ending on hand Order?

Order Quantity Rand #

Lead Time

Lead time Remaining

0 0 25 0 0 25 0 01 0 25 0.6682 4 0 21 N 0 0 02 0 21 0.9004 5 0 16 N 0 0 03 0 16 0.5151 4 0 12 N 0 0 04 0 12 0.3733 4 0 8 Y 150 0.5057 3 35 0 8 0.8038 5 0 3 N 0 0 26 0 3 0.9435 6 3 0 N 0 0 17 0 0 0.7776 5 5 0 N 0 0 08 150 150 0.5561 4 0 146 N 0 0 0

How Simulation Helps Make Inventory Decision (3)

• For each “re-order point” and “order quantity” tried, the simulation shows the total inventory cost (including holding cost, ordering cost, and lost sales cost) of a year.

• A good re-order point (showing when to place an order) and a good order quantity (showing many units in an order) can thus be selected from many alternatives.

Other Examples of Business Simulations with Excel

• Minutes-by-minutes waiting lines;• Gambling game;• Production in a workshop;• Transactions in a bank;• Truck transportation;• Department store operations to see

the requirements of resources.