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MATH3161/MATH5165 Model Formulation

创建日期：2024-04-03 17:21:00

所属分类：统计学statistics

标签： MATH3161

Model Formulation

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MATH3161/MATH5165
Problem Sheet 1 – Model Formulation
1. Sydney Sounds Manufacturing produces speakers for hi-fi sets in two types, product
A and product B. Two types of processes are needed for their production. The first
process combines all machining operations, and the second consists of all assembly
operations. Each unit of product A requires 4 labor-hours of machining and 2 labor-
hours of assembly work, whereas each unit of product B requires 3 labor-hours of
machining and 0.5 labor-hours of assembly work. Manufacturing capacity available
during the coming production week is 2400 labor-hours, and the assembly work capacity
available for the same week is 750 labor-hours.
Previous sales experience indicates that product B sells at least as much as product A
and that there is already an order of 100 units for product A to be produced during the
next period. Furthermore, all products produced during the week can be sold during the
same week. Products A and B provide 7.00 and 5.00 profit per unit sold, respectively.
Management would like to know what quantities of A and B should be manufactured
during the next production week in order to maximize the total profit. The following
table summarizes the data.
Operation Product A Product B Capacity
Machining (labor-hours) 4 3 2400
Assembly (labor-hours) 2 0.5 750
Profit per unit ( ) 7 5
(a) Formulate this problem as a mathematical optimization problem.
(b) Sketch the feasible region, and find the vertices of the feasible region.
(c) *MATLAB: Use the MATLAB function linprog to find a solution.
Hint: In MATLAB type doc linprog.
2. The environmental Protection Agency (EPA) wants to restrict the amount of pollutants
added by a company to the river water. The concentrations of phenol and nitrogen in
the water are to be restricted to, respectively, P and N grams/ML (1ML = 1 mega-litre
= 106 litres) on a daily basis. The river has a flow of M ML/day. The company diverts
a portion of the river water, adds the pollutants, namely, phenol and nitrogen, to it,
and sends the water back to the river. The company has four possible ways to treat the
water it uses before returning it to the river. The characteristics of each treatment are
given in the following table.
Treatment 1 2 3 4
Phenol p1 p2 p3 p4
Nitrogen N1 N2 N3 N4
Cost/ML C1 C2 C3 C4
Table 1: Grams of pollutant/ML after treatment
Assuming that
the river is initially free of pollutants;
1
addition of pollutants does not a↵ect the amount of water flow;
the company has to process at least k ML/day of river water.
(a) Set up an optimization problem to solve for the amount of water to be processed
by each treatment so that total cost of treatment is minimized.
(b) How does the formulation change if EPA regulations apply not to total river con-
centration downstream from the plant, but rather to the concentration of e✏uent
from the plant?
3. An oil company has oil wells drilled at points A = (1, 1), B = (1, 1) and C = (0,3)
on a grid layout of a particular location. The scale of the grid is such that one grid unit
corresponds to one kilometre. The company must build pipelines from these wells to a
distribution depot. The cost of the pipelines from wells A and B is 1000/meter, but
only 500/meter from well C because of a lower flow rate from the well. For political
reasons the depot must lie on the same side of a state border as the town located at
the grid point (5,4). The state border is a straight line which passes through the
grid points (4,10) and (4, 6). There is also a circular lake 3 kilometres in diameter
centered at the grid point (2,1). Obviously the depot cannot be located in the lake.
Where should the depot be constructed so as to minimize the cost of building the
connecting pipelines?
Pose this as a mathematical optimization problem, and say as much as you can about
its structure.
4. * A Simplified Portfolio Selection Problem. A portfolio manager in charge of a
bank portfolio has 10 million to invest. The securities available for purchase, as well
as their respective quality ratings, maturities, and yields, are shown in the following
Table.
Bond Bond Quality scales Years to Yield to After-tax
name type maturity maturity yield
Moody’s Bank’s
A Municipal Aa 2 9 4.3% 4.3%
B Agency Aa 2 15 5.4% 2.7%
C Government Aaa 1 4 5.0% 2.5%
D Government Aaa 1 3 4.4% 2.2%
E Municipal Ba 5 2 4.5% 4.5%

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