2-504-09 Lecture Notes - Lecture 4: Arm Cortex-M, Nuclear Energy Institute, Octane Rating
2-604-15 Lecture notes
1
3. Applications in finance, marketing, and production management
The aim of this chapter is to begin illustrating the diversity of possible
applications of mathematical programming, especially of linear programming.
Section 3.1 has an application in finance, Section 3.2 has an application in
marketing, and Section 3.3 has an application in production management. Other
applications in production management, in logistics and human resources will be
presented in the following sessions.
3.1 Applications in finance
Numerous decision support models are used in the field of finance, of course, in
financial firms (banks and insurance in particular), but also in large companies.
Some of these models may take the form of linear programs. In companies,
linear programming can, for example, be used to make investment choices (in
situations where you have to decide which projects to choose within a limited
budget) or to decide whether it is more profitable to manufacture a product than
buying it from a supplier. In financial companies, mathematical programming is
mainly used to guide the composition of investment portfolios.
In such a problem, a manager typically must choose between specific
investments (e.g., stocks and bonds) from a large number of possibilities. To do
so, he or she may seek to minimize the risk of the portfolio while maintaining a
minimum return on investment. Note that this problem can be solved through a
nonlinear decision support model. Our manager may also seek to maximize its
return on investment, taking into account risk using constraints that would
guarantee a certain diversification in the portfolio composition. This type of
problem is solved using linear programming.
The example we will now consider corresponds to a variant of the classical
portfolio selection problem, which we will model as a linear program.
The Alternative Bank of Canada case
The Alternative Bank of Canada (ABC) wishes to determine its new credit
policy with a limited budget of $13 million. It intends to distinguish the
following categories of credit: car loans, commercial loans, mortgages and
personal credits which yield the following interest rates:
find more resources at oneclass.com
find more resources at oneclass.com
2-604-15 Lecture notes
2
Categories
Car
Commercial
Mortgage
Personal
Rate
8 %
4 %
5 %
10 %
Table 3.1: Data for the case Alternative Bank of Canada
ABC wishes to allocate at least 40% of loans in commercial loans. To encourage
homeownership clients, ABC wishes to allocate at least 50% of its non-
commercial loans in mortgages. Finally, for diversification purposes, ABC does
not want personal credit to account for more than 60% of commercial loans and
mortgages.
What would be the best credit policy for ABC? Propose a mathematical model to
guide ABC in its decision making.
Step 1: Analysis
To build such a mathematical model, it is necessary to analyze ABC’s problem.
Which data should be considered?
• The maximum budget;
• the interest rates associated with each type of loan;
• the minimum percentage of commercial loans;
• the minimum percentage of mortgages;
• the maximum percentage of personal credits.
Which decisions should be taken?
ABC must determine how to allocate its funds in each category. Because the
bank has a limited budget in millions of dollars, we can more precisely state that
ABC must decide the allocated loan in millions of dollars for each category.
How can we assess the quality of the decision making?
Because ABC has specified interest rates associated with each type of loan (and
allocation constraints in order to diversify its portfolio), it is natural for the bank
to maximize its return on investment.
In which context should we make the decisions?
ABC must first meet its budget ($13 million). Then ABC imposes two types of
constraints. First, ABC wishes to have a minimum percentage of commercial
loans (40%) and mortgages (50% of non-commercial loans). Second, to diversify
its portfolio, it sets a maximum percentage on personal loans (60% of
commercial loans and mortgages).
find more resources at oneclass.com
find more resources at oneclass.com
2-604-15 Lecture notes
3
Step 2: Build a verbal model
To summarize our analysis, we can state the following problem:
1. We seek to determine the allocated loan in millions of dollars for each type
of credit;
2. in order to maximize return on investment;
3. while respecting: i) the budget, ii) the minimum allocation of commercial
loans, iii) the minimum allocation of mortgages, and iv) the maximum
allocation of personal credits.
Step 3: Build a mathematical model
The third and final step to design our model is to translate the verbal model in a
mathematical model.
To do this, we first select mathematical symbols, or variables, which reflect the
decisions. Since there are four types of loans, we select four variables (one for
each type), either:
• Ca, allocated car loans, in millions;
• Co, allocated commercial loans, in millions;
• M, allocated mortgages, in millions;
• P, allocated personal credits, in millions.
Decisions must be taken to maximize return on investment. The latter is
calculated from the specified interest rate for each loan. Thus, if ABC allocates
Ca million in car loans at 8%, the return on investment will be 0.08 × Ca million
for this type of loan. We argue the same for the other types of loans. The total
return on investment (in millions) is therefore calculated as: 0.08 Ca + 0.04 Co +
0.05 M + 0.1 P. We seek to maximize this amount. The objective function of the
model is therefore written mathematically as:
Max 0.08 Ca + 0.04 Co + 0.05 M + 0.1 P.
Then we have to consider the context in which decisions are taken. This
corresponds to the constraints that the decisions must respect. It must first meet
the budget by not allocating more credit than the 13 million available. This can
be expressed as follows:
Total allocated loans (in millions) ≤ 13.
The total allocated loans is calculated as the sum of allocated loans: Ca + Co +
M + P. The equation describing the budget constraint is then written as:
Ca + Co + M + P ≤ 13.
find more resources at oneclass.com
find more resources at oneclass.com
Document Summary
Lecture notes: applications in finance, marketing, and production management. The aim of this chapter is to begin illustrating the diversity of possible applications of mathematical programming, especially of linear programming. Section 3. 1 has an application in finance, section 3. 2 has an application in marketing, and section 3. 3 has an application in production management. Other applications in production management, in logistics and human resources will be presented in the following sessions. Numerous decision support models are used in the field of finance, of course, in financial firms (banks and insurance in particular), but also in large companies. Some of these models may take the form of linear programs. In financial companies, mathematical programming is mainly used to guide the composition of investment portfolios. In such a problem, a manager typically must choose between specific investments (e. g. , stocks and bonds) from a large number of possibilities.
