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Please Browse to page 1, there are two project here, The Multi-channel Queue / The Able-caller problem and the
Inventoty Problem note that the Inventory problem can be solve using matlab
so please do consider using matlab for the second project unless you can handle it in java or c++ or c sharp
thanks
Please check the file
1. Project Title : Inventory problem
Prog. Language :
MATLAB / JAVA / C++
Project type : The News Dealer Inventory System
Newspapers seller buys the newspaper with 33 cents each and sells them for 50 cents
each. Newspapers not sold at the end of the day are sold as scrap for 5 cents each.
Newspapers can be purchased in bundles of 10. Thus, the seller can buy 50, 60, and so
on. There are three types of Newsday, “good”, “fair”, and “poor” with probabilities of
0.35, 0.45, and 0.20 respectively. The distribution of papers demanded on each of these
days is given in table 2.1. The problem is to determine the optimal number of papers the
seller should purchase to increase his profit.
Assumption: the profits are given by the following relationship:
Profit = [(revenue from sales) – (cost of newspapers) – (lost profit from excess
demand) + (salvage from sale of scrap papers)]
Project 2 – The News Dealer – marking criteria
Considera computer technical support center where personnel take calls and provide
service. The time between calls ranges from 1 to 4 minutes, with distribution as shown
in table 1.1. There are two technical support people – Able and Baker. Able is more
experienced and can provide service faster than Baker. The distribution of their service
times are shown in table 1.2 and 1.3.
Rule: “Able gets the call if both technical support people are idle”. The problem is to
estimate the system measures of performance in terms of:
1) The efficiency of Able.
2) The efficiency of Baker
3) The average caller delay.
Please answer the following question:
4) Do we need extra server? Why?
In case that Able is alone (without Baker), Compute the following terms:
1- The average service time.
2- The average waiting time (in the queue).
3- The maximum queue length.
4- The probability that a customer wait in the queue.
5- The portion of idle time of the server.
Moreover, you need to get some answers (with justification) for the following questions:
6- Does the theoretical average service time match with the experimental one?
7- Does the theoretical average interarrival time match with the experimental one?
Modeling and Simulation
Assignment
5 report—The Refrigerator
Complex Inventory problem
Faculty of Informatics and Computer Science
Computer Science Department
Year 3
Shaimaa Hegazy 105258
Tashreen Shaikh 105816
Dina Tarek 105462
23/12/2010
Modelling and Simulation:: Real Application Document
Refrigerator Inventory Problem Simulation Report 1
List of figures
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Refrigerator Inventory Problem Simulation Report 2
1. Problem Formulation and Objective Setting ……………………………………………………….. 3
1.1. Problem Reformulation ………………………………………………………………………………….. 3
1.1.1. Simple Queuing System Flow Operation ……………………………………………………. 5
1.2. Objectives …………………………………………………………………………………………………….. 6
2. Model Conceptualization ……………………………………………………………………………………. 7
2.1. Definition of the System ………………………………………………………………………………… 7
2.2. Type of the Model …………………………………………………………………………………………. 8
2.2.1. Static Model …………………………………………………………………………………………… 8
2.2.2. Stochastic Model …………………………………………………………………………………… 10
2.3. Equation of the Main Variables …………………………………………………………………….. 11
2.4. Flow Chart of Model ……………………………………………………………………………………. 12
3. Experimental Design ………………………………………………………………………………………… 14
3.1. Length of Simulation Run ………………………………….. Error! Bookmark not defined.
3.2. Number of replications ………………………………………. Error! Bookmark not defined.
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Refrigerator Inventory Problem Simulation Report 3
1. Problem Formulation and Objective Setting
1.1. Problem Reformulation
The problem here is to simulate a complex inventory problem, where the company deals
with refrigerators inventory problems. A company selling refrigerators maintains inventory
by conducting a review after a fixed number of day’s and making a decision about how many
to order to replenish the store’s inventory. The current policy is to order up to a level using
the following relationship:
Order Quantity = (Order-up-to-level) – (Ending inventory) + (Shortage quantity)
In order to generalize the situation the company, let us assume problem through the
following terms in Table 1.1.;
Table 1.1: Inventory terms representation
Terms/ Description Values
Order-up-to-level 11 items
Ending inventory 3 items
Review period 5 day
On 5
th
day Shortages of 2 items 13 items
On 5
th
day Shortages of 3 items 14 items
To demonstrate the situation, the order up to the level is 11 items (refrigerators), where the
ending inventory is 3 items (refrigerators) in the store. Moreover, we take the review period
constant to 5days, where inventory is checked in each period of time. Therefore, on the 5
th
of
the cycle, according to the equation (8 = 11 – 3), 8 refrigerators would be ordered from the
supplier. If there are shortages of two refrigerators on the check of fifth day the inventory that
would be ordered is 13 refrigerators. Moreover, in the same manner if the shortages of three
refrigerators occur, the ordered inventory from the supplier would be first served to the
customer who requested three refrigerators rather than putting in the inventory.
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Refrigerator Inventory Problem Simulation Report 4
Moreover, the sales cost lost is importance to notice when the customer demanded certain
items and the inventory is unavailable to fulfil the demanded refrigerators. The number of
refrigerators ordered each day is randomly distributed as shown in Table 1.2.
Table 1.2—Distribution of daily demand
Daily Demand Probability Cumulative Probability
0 0.10 0.10
1 0.25 0.35
2 0.35 0.70
3 0.21 0.91
4 0.09 1.00
Also, the number of days after the order is placed with the supplier before arrival, or lead time is
given by the distribution in Table 1.3.
Table 1.3 Distribution of Lead Time
Lead Time(days) Probability Cumulative Probability
1 0.60 0.60
2 0.30 0.90
3 0.10 1.00
Moreover, we assumed that the orders are placed at the end of the day, in this sense if the
lead time is zero, the order from the supplier is arrived on the same day would be store in the
inventory and available for distribution in the next day and so on.
Hence, to begin with the simulation the inventory level begins with 3 refrigerates, and an
order for 8 refrigerates to be arrived in 2 days.
Hence, after analysing the problem and understanding the refrigerator inventory problem,
the company owner wanted to get some of the basic statistics to improve the operations of the
of the inventory operation. These statistics are provided by simulating the average ending
inventory problem, analysing it and finally drawing conclusions from it through the
distribution chart provided.
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Refrigerator Inventory Problem Simulation Report 5
1.1.1. System Flow Operation
Figure 1.1 The Refrigerator Inventory system Flow Chart
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1.2. Objectives
The objective of the simulation is to satisfy, analyse, and fulfil the requirements asked by
the management of the inventory company. The simulation of the inventory problem will
generate the distribution of the ordered inventory demands and relatively shortage inventory
for certain set of days to approximate the average distribution of shortage inventory. The
analysis of shortage inventory units would help the inventory manager to understand
following information;
1. Understand the type of shortage condition
2. Could be helpful to understand inventory cost
3. Better understanding of plans to avoid shortage problems
4. Serve customer demands in mean time
5. Minimize the problem of lead time
6. Understand the flow of operations in inventory system
The following statistics will help to analyze the importance of necessary changes and
improvement with the average proposed simulation problem.
1. Ordered Quantity,
2. Order Up to Level
3. Ending Inventory
4. Shortage Quantity
5. Pending Order
6. Lead Time
7. Days Until ordered arrived
8. Days within each cycle
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Refrigerator Inventory Problem Simulation Report 7
2. Model Conceptualization
2.1. Definition of the System
The Refrigerator inventory distribution system is the type of Complex Inventory system
were the states of the inventory depends upon the items or inventory present in the system
respect to time.
Each state of the inventory system is affected by the events occurring in the system at
certain present state. There are others components such as attributes, activities, entities, and
system itself to be defined to generalize the Inventory system.
Following are components of the Complex Inventory system for the refrigerator inventory
problem is defined by each necessary operation in the system. Error! Reference source not
found. summarises the components below.
Table 2.1—Complex Refrigerators Inventory System components
Components
Real Representation
System Refrigerator Inventory System
Entities Refrigerator, Day
Attributes Demand, Supply, Amount of Ordered Quantity,
Shortage Quantity, Ending Inventory, Order up to
Level
Activities Selling demanded refrigerators, making profit
Events Quantity demanded at each day
State Variables 1. Amount of refrigerator quantity demanded
2. Amount of refrigerator quantity supplied to
customer
3. Amount of current inventory
4. Profit from daily sales
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2.2. Type of the Model
The refrigerator complex inventory system is a much familiarized type of simulation. It is
real life phenomenon were the companies deal with demand, sales, and profit making models
in daily life of business models. Hence, the complex inventory system is categorized into two
models as Continuous and Stochastic model.
2.2.1. Continuous Model
The refrigerator inventory system is based on the changes of the time (which is the day in
our case) changes the performance of the system. The demanded or ordered quantity varies
depending upon the demand distribution, with changing time period per day. Hence, all
inventory systems serve as buffers between time-dependent activities. While an inventory’s
outputs are solely determined by external demands, e.g. by customers ordering products, its
inputs can be controlled by inventory policies, whose optimal choice is inventory
management’s ultimate goal. A simple inventory system defines the total cost (or profit) of an
inventory system is the performance measure.
1. Carrying stock in inventory has associated cost.
2. Purchase/replenishment has order cost.
3. Not fulfilling order has shortage cost (which is the prime objective here).
Therefore, the two main probabilistic considerations for demand and lead type distribution
that affects the total performance measure of the system, which is the shortage quantity.
1. Demand Distribution
Random values are generated in order to get the demand distribution for that particular
day (shown in table 2.2 and 2.3). According to the cumulative probability the range is
selected for each type of demand, and the generated random number derives the demand
distribution for that particular day.
Table 2.2– Demand Random Range
Demand Cumulative Probability Random Digit
Assignment
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Refrigerator Inventory Problem Simulation Report 9
0 0.10 00 – 10
1 0.35 11 – 35
2 0.70 36 – 70
3 0.91 71 – 91
4 1.00 92 – 1
Table 2.3—Simulation of Demand
Day Random
Digits
Demand Type
1 84 3
2 10 0
3 74 3
4 53 3
5 17 1
6 79 3
7 91 3
8 67 2
9 89 3
10 38 2
2. Lead Time Distribution
Here, the random values are generated in order to get the possibility to decide the amount
of time would be required for supplier to supply items from the demanded period (shown in
Table 2.4). According to the cumulative probability the range is selected for each lead time
days and the generated random number derive the lead day type for each cycle of demanded
quantities. At the end of each cycle that is the period of 5 days the lead time is calculated
(shown in Table 2.5).
Table 2.4:Lead Time Distribution Random Range
Lead Time(days) Cumulative Probability Random Digit Assignment
1 0.60 00 – 60
2 0.90 61 – 90
3 1.00 91 – 1
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Table 2.5—Simulation of Lead time
Day Random Digits Type of Lead Time
1 84 2
2 10 1
3 74 2
4 53 1
5 17 1
6 79 2
7 91 3
8 67 2
9 89 2
10 38 1
2.2.2. Stochastic Model
The refrigerator inventory system could also be categorized as stochastic model as the
input of the system is chosen through the random artificial selection lead time type and
demand distribution. The output produced that is the average shortage ending inventory is
based on selecting some random and artificial values to each 25 days. The system consists of
probabilistic values that are the lead time type and demand distribution as mentioned above.
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2.3. Equations of the Main Variables
To simulate the complex inventory system many statistical factors have to be considered
to generate the required objective of the system. To construct the Refrigerator inventory
model, we must derive the following equations to get the reasonable output;
1. Order Quantity to place orders
This is to determine the required ordered quantity for placing demand to supplier,
concerning the Order up to level i.e. 11, in terms of ending inventory or shortage quantity.
Order Quantity = (Order up to Level) – (Ending Inventory) + (Shortage Quantity)
Equation 2:1—order quantity
2. Ending Inventory
This is to get the current ending inventory per day if the beginning inventory is greater
than demand.
Ending Inventory = Beginning Inventory – Demand
Equation 2:2—ending inventory
3. Shortage Quantity
This is to get the shortage quantity for per day concerning the beginning inventory is less
than demand and ending inventory is zero.
Shortage quantity =Demand – Beginning Inventory
Equation 2:3—shortage quantity
4. Beginning cycle of inventory (N)
This is to calculate the beginning of current inventory cycle over the past ending
inventory of N-1.
Beginning cycle of inventory (N) = Ending Inventory of (N – 1)
Equation 2:4—beginning cycle of inventory N
5. Average shortage inventory
It is the average shortage inventory unit for each day over each cycle.
Average shortage inventory =
Equation 2:5—average shortage inventory
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2.4. Flow Chart of Model
The flow chart represents the operation of the refrigerator complex inventory system
implemented through using programming languages i.e. C#. Many processes are interrelated
with certain input and producing some informative output, where the programs start with
selection of random numbers for each lead and demand distribution. The system at the end
output and fulfils all the required objectives to get the average shortage time through
analysing the simulation of the system. Figure 2.1 shows the model’s flow chart.
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Figure 2.1—Model’s flow chart
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3. Experimental Design
This section describes the design decisions that need to be made regarding the length of
each run and the number of replications to be made. The decisions presented here are a result
of the simulation runs already made.
The term ―run‖ is used to indicate a single trial of the simulation model. In this case, it
indicates the number of cycles (C). However the term ―replication‖ is used to refer to the
number of trials/runs (T) made for each cycle. Therefore, this section starts by deciding the
length of each run (i.e. number of cycles) which produces the most accurate result, it then
moves to deciding the number of replications (T) to make for each run.
In a single simulation run of the model, random variables are produced for each of the
lead type (I) and the demand distribution (D) for each day, and then for each trial T. The
problem with a single run is that the variables produced are not statistically independent,
since the value of Ii has some effect on the value of Ii+1. Therefore the sequence of random
variables I1, I2, I3, …, In may be ―auto-correlated‖ (Banks et al, 2001). In order to overcome
this correlation, each run is replicated T times, this ensures that variables across different
replications are statistically independent.
3.1. Length of Simulation Run
As described above, the length of simulation run refers to the number of cycles in each
trial. This section experiments the different number of cycles for a fixed number of trials
(100) and then decides on the best number of cycles to use.
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Figure 3.1—Result when run length is 5
Figure 3.2—Result when run length is 10
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Figure 3.3—Result when run length is 10
Figure 3.4—Result when run length is 20
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Figure 3.5—Result when run length is 25
Figure 3.6—Result when run length is 30
Figure 3.1 through Figure 3.6 show the resulting histograms when the number of cycles
was varied from 5 to 30. Analyses of these histograms show that with fewer number of
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Refrigerator Inventory Problem Simulation Report 18
cycles, the resulting ending inventory range (from 2 to 6) was wide; however as the number
of cycles increased the range (3 to 4) decreased hugely. Moreover, the results for number of
cycles 25 and 30 are almost the same. This means that increasing the number of cycles yields
an average ending inventory with much less variations. Therefore the number of cycles is
chosen to be 25 (since no big difference between 25 and 30).
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3.2. Number of Replication
In section 3.1, the length of simulation run was chosen to be 25 (with 5 days per cycle)
this section is concerned with identifying the necessary number of replications to be made for
each run. Replicating the runs produces more accurate result; since it produces variables that
are statistically independent across different runs. In order to experiment the effect of varying
the number of trials, the number of cycles was set to a constant (25) and the number of trials
varied. Finally the resulting histograms are analysed.
Figure 3.7—Histogram for number of trials = 100
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Figure 3.8—Histogram for number of trials = 200
Figure 3.9—Histogram for number of trials = 300
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Figure 3.10—Histogram for number of trials = 300
Figure 3.7 through Figure 3.10 show the resulting histograms when the number of trials
was changed from 100 to 300. Moreover, the average ending inventory is approximately 2.9
in all cases. This indicates that changing the number of trials does not have a significant
effect on the result.
This section discussed the effect of changing the number of cycles and the number of trials
on the results. It has been concluded that as the number of cycles increases, the result become
more stable. However changing the number of trials does not have significant effect on the
result. The number of cycles was thus chosen to be 25 and the number of trials 200. These
values are chosen as intermediary values.
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4. Results analysis and conclusion
This section shows the results of simulation, analyses them and finally draws conclusions.
Results will be shown for 25 cycles and 200 trials as mentioned in section 3.
4.1. Result analysis
Figure 4.1 shows the histogram for 25 cycles and 200 trials. The graph shown is slightly
positively skewed, indicating that the average ending inventory is usually small with an
average of about 2.9 ( 3) items. This means that according to the current inventory
conditions, the policy followed by the company—which is to make frequent reviews (at the
end of each cycle) and order accordingly, is a good policy; since the average ending
inventory is less than the expected demand (1.94).
Figure 4.1
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4.2. Conclusion
This paper discussed the refrigerator inventory system, and how its model is designed and
built to simulate it. The objective of this simulation is to analyse and fulfill requirements
asked by the inventory company management. It should help in understanding how the
inventory works (flow of operations), also understand problems to solve or prevent further
occurrences. Firstly this simulation will generate the demands ordered distribution, which
will help us deduce the remaining amount of refrigerators in the inventory (end inventory
level) or the shortage in the refrigerators’ numbers in inventory, which will then be added to
our next supply order.
As the figures show the complex refrigerators inventory system by simulation was easily
understood at any state. The company selling refrigerators when reviewing the system will be
able to locate any shortages in inventory and provide plans to avoid shortage problems,
besides minimizing lead time problems in addition to better understanding of the system,
amount of ordered refrigerators, pending orders and ending inventory level are of huge
importance to enhance the company’s quality of service.
Important inputs in this simulation model is the number of cycles the user enters to occur in
one trial (5,10,15..) and the number of trials this operation is conducted. After running the
simulation and testing it, the optimal values chosen for the number of cycles were 25 and the
number of trials was 200 where increasing them does not affect the result greatly. The
resulting average inventory was 3 items, which is more than the average daily demand of
2 items, and thus the current company’s policy of making frequent reviews (at the end of
each cycle) and ordering accordingly is not a bad one.
