2-504-09 Lecture Notes - Lecture 5: Menu Cost, Sunk Costs, Computer Network
Course 1: Introduction
Analysis: 1) Experience and intuition of the decision maker
2) To complement the intuition of the decision-maker, one can resort to
a model -> Abstraction -> Modeling (Formulation and data) -> Model
analysis (Resolution and validation) -> Interpretation
- Models: simplified representations of reality
- Benefits: easier to understand, often less expensive and deliver needed
information faster, enable one to perform experiences that would be
impossible to do in reality
- Types of models > Iconic (physical) models is a physical replica of a
system, usually based on a smaller scale from the original. Iconic models may appear to scale in
three dimensions, such as models of an airplane, car, bridge, or production line.
> Analog models: does not look like the real system but behaves like it. An analog model could
be a physical model, but the shape of the model differs from that of the actual system. Some
examples include organizational charts that depict structure, authority, and responsibility
relationships
> Symbolic (quantitative) models): Use mathematical objects
(variables, equations describing relationships among variables,
functions) to highlight the important aspects
- Example of the 8 technicians/ 8 jobs : a model is necessary to find
the optimal solution
+ A model can be used to determine which technician to send in
which city in order to minimize total travel costs
- The model will contain: > variables which correspond to the
decisions of sending a technician from a city to another city
> Mathematical functions to evaluate the cost of a set of decisions > Relationships which can
impose that the number of technicians sent is 1
- A variable is a mathematical element which can take
different values in a given set. The decision maker will give a
value to each of the variables. A solution of the model is
composed of a value for each of the variables
-> XIj = number of technicians sent from city I to city j, with I = A for Atlanta, B for Baltimore and
C for Chicago, and j=a for Austin, b for Boston and c for Colombus. These variables can take
values in the set {0; 1}
- Variables: XIj = 1 If technician I (Atlanta, B,C) is sent to job j (Austin, Boston)
0 otherwise -> Binary
-Obj. Function:
Min Z=900 XAa+ 1100 XAb+ 525 XAc+…+ 300 XCC
-Constraints
-The functions of the model will evaluate the consequences associated with different values of the
variables
- In this example, we need one cost function to compute the total solution cost + 6 functions to
compute the number of technicians departing the 6 cities
- The objective of the model is to find amongst all feasible solutions, the one which has the best
performance (lowest cost, highest profit…): objective function
- The constraints of the model: We will impose that the values given to the variables are such
that the value of the function is:
equal to ( = ) a target value, smaller than or equal to ( ≤ ) a target value, or greater than or equal
to ( ≥ ) a target value
The target value (first side of the constraint) is called the right-hand side (RHS) of the constraint)
-> If a set of values for the variables respect all the constraints, the solution is said feasible
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- Many softwares are available on the
market to solve optimization problem
(solver)
- Multiple objectives : we may want
to minimize costs and maximize a
service -> find compromise
solutions
Functions were linear (more efficient resolution process,
solution guaranteed to be
optimal, evaluating the impact
of varying a model parameter
is easier)
- Having an objective function that penalizes large gaps between
supply and demand. The simplest function that allows this kind of
penalty is to minimize the sum of squared deviations between the
available supply and demand
- We could impose to have an integer (entier) number of employees
but solving it can take time
-It may happen that the value of some parameters is not known
with certainty. Some techniques can be used:
> Sensitivity analysis can give information about the impact on the value of the optimal
solution following a change in a parameter value (analyze what is the impact of changing the
value of a single parameter on the solution)
> Stochastic programming is used to optimize the expected value of the objective function
according to the probability of the parameter values
> Simulation can evaluate the consequences of a considering different probability sets for the
parameter values
- A scenario is a set of values for a set of parameters
When we do not know the scenario that takes place when selecting the solution, it is difficult
to make the optimal decision
On the other hand, we can take a decision according to the most probable scenarios
-> Strategies: optimistic (good weather), less optimistic (bad weather+ additional employees
when the weather turns out to be good)
- Simulation is a technique used to reproduce probable situations by respecting their
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probability and by analyzing the probabilistic behavior. This technique does not yield the
optimal solution but can help evaluate the impact of a decision
This technique relies on (pseudo)
random number generators
A generator will give a number uniformly
distributed between 0 and 1
The RAND() function in Excel is a random
number
generator
We can therefore generate any
more or less complex situation
By repeating this simulation
thousands of times, we can estimate
the outcome probability function,
and deduce averages, medians,
standard deviations…
- A simple map is a model, because it is a simplified representation of buildings and streets
surrounding the main building of HEC Montreal
-Variables, also called decision variables, are mathematical
elements that can take values in specific sets. Usually, it is
the user who decides the value that each decision will take
in the model.
-Parameters are other mathematical elements whose values are
specified by the environment. We normally assume that the value
of each parameter is known with certainty. When this is not the
case, we will have to estimate this value. It will then be important to
analyze the sensitivity of the solution with respect to the possible
estimation errors of the parameters.
-Functions express relationships involving various parameters and variables in order to assess certain
quantities which will be subject to analysis.
- A variable must necessarily be a quantity
- The type of optimization problem in its general form is called a mathematical programming
problem:
The function f is called the objective function. The following inequalities
are called the constraints
- A special case in which we will focus is when the functions f and gi are
linear for i=1, 2,…, m, that is, they are under the form k1·x1 + k2·x2 + … +
kn·xn, where kj are known parameters and xj are the variables for j=1, 2, …,
n. These problems are referred to as linear programming problem
- This type of problem is usually solved using specialized software
- The blue rectangles indicate staffing needs, while red indicate the available personnel
- The trial and error method can yield a good solution to our problem, but will never guarantee
that this is the best solution. Using the Excel solver will allow us to guarantee the optimality of
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Document Summary
Analysis: 1) experience and intuition of the decision maker: to complement the intuition of the decision-maker, one can resort to a model -> abstraction -> modeling (formulation and data) -> model analysis (resolution and validation) -> interpretation. Benefits: easier to understand, often less expensive and deliver needed information faster, enable one to perform experiences that would be impossible to do in reality. Types of models > iconic (physical) models is a physical replica of a system, usually based on a smaller scale from the original. Iconic models may appear to scale in three dimensions, such as models of an airplane, car, bridge, or production line. > analog models: does not look like the real system but behaves like it. An analog model could be a physical model, but the shape of the model differs from that of the actual system. Some examples include organizational charts that depict structure, authority, and responsibility relationships.
