LP Modeling
Examples discussed in class
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Example: Basketball Team Selection
Coach Jack is trying to choose the starting line-up for the basketball team. The team consists of
seven players who have been rated (on a scale of 1=poor to 3=excellent) according to their ballhandling, shooting, rebounding, and defensive abilities. The positions that each player is allowed to
play and the player’s abilities are listed in the following table:
The five-player starting line-up must satisfy the following restrictions:
1.
At least 4 members must be able to play guard, at least 2 members must be able to play forward,
and at least 1 member must be able to play centre.
2.
The average ball-handling, shooting, and rebounding level of the starting line-up must be at least 2.
3.
If player 3 starts, then player 6 cannot start.
4.
If player 1 starts, then player 4 and 5 must both start.
5.
Either player 2 or player 3, or both, must start.
•
Given these constraints, Coach Jack wants to maximize the total defensive ability of the starting
team. Formulate an IP that will help him choose his starting team.
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The General Structure of a Mathematical Model
Inputs
Uncontrollable
factors
(parameters)
Output
Independent
(decision)
variables
Mathematical
relationships
Dependent
variables
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A Simplified Model of a Manufacturing Situation
Mathematical relationship:
Decision variables:
x1, x2
(What quantities of
products 1 and 2 should
be produced?)
Maximize
revenue
Subject to:
x1 + x2  50
Objective
Constraint
Dependent variable:
R = 5x1 + 2x2
(total revenue)
Uncontrollable variables:
5, 2, and 50
(market prices, marketing limitation)
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Example 1: LP Formulation
• Papa Louis manufacturers wooden tables and chairs
for small kids. Each "table" built: Sells for $27 and uses
$10 worth of raw materials, increases Papa Louis’s
variable labor/overhead costs by $14. Requires 2 hours
of finishing labor and 1 hour of carpentry labor. Each
"chair" built: Sells for $21 and uses $9 worth of raw
materials, increase Papa Louis’s variable
labor/overhead costs by $10. Requires 1 hours of
finishing labor AND 1 hour of carpentry labor. Each
week Papa Louis can obtain only 100 finishing hours
and only 80 carpentry hours. Also demand for the
chairs is unlimited. However, at most 40 tables are
bought each week. Papa Louis wants to maximize
weekly profit (revenues - expenses).
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Example 2: LP Formulation
•
Production Planning: A manufacturing firm has discontinued production of a
certain unprofitable product line. This created considerable excess production
capacity. Management is considering devoting this excess capacity to one or more
of the three products: call them products 1, 2, and 3. The available capacity on the
machines that might limit output is summarized in the following table:
•
The number of machine hours required for each unit of the respective products is
given as:
Total Availability of resources
<500 hr/week
<350 hr/week
<150 hr/week
•
The sales department indicates that the sales potential for products 1, and 2
exceeds the maximum production rate and that the sales potential for product 3 is
20 units per week. The unit profit would be $ 30.0, $20, and $ 15, respectively, on
products 1, 2, and 3. Formulate the linear programming model for determining
how much of each product to produce in order to maximize profits.
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Example 3: Crude Petroleum
• Shell Petroleum runs a small refinery on the Texas coast. The refinery distills
crude petroleum from two sources, Saudi Arabia and Venezuela, into three
main products: gasoline, jet fuel, and lubricants. The two crudes differ in
chemical composition and thus yield different product mixes. Each barrel of
Saudi crude yields 0.3 barrel of gasoline, 0.4 barrel of jet fuel, and 0.2 barrel
of lubricants. On the other hand, each barrel of Venezuelan crude yields 0.4
barrel of gasoline but only 0.2 barrel of jet fuel and 0.3 barrel of lubricants.
The remaining 10 of each barrel is lost to refining.
• The crudes also differ in cost and availability, Shell can purchase up to 9000
barrels per day from Saudi Arabia at $20 per barrel. Up to 6000 barrels per
day of Venezuelan petroleum are also available at the lower cost of $15 per
barrel because of the shorter transportation distance. Shell's contracts with
independent distributors require it to produce 2000 barrels per day of
gasoline, 1500 barrels per day of jet fuel, and 500 barrels per day of
lubricants. How can these requirements be fulfilled most efficiently?
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Example 4: LP Formulation
•
Manpower Planning: Certain types of facilities operate seven days each week and
face the problem of allocating manpower during the week as staffing requirements
change as a function of the day of the week. Perhaps the most fundamental
staffing problem is the assignment of days off to full-time employees. In particular,
it is regularly the case that each employee is entitled to two consecutive days off
per week. If the number of employees required on each of the seven days of the
week is given, then the problem is to find the minimum workforce size which
allows these demands to be met and then to determine the days off for the people
in this work-force. To be specific, let us study the problem faced by Grand River
Transit Bus Company. The number of drivers required for each day of the week is
as follows:
a) How many drivers should be scheduled to start a five-days stint on each day of the
week? Formulate this problem as a linear program.
b) If daily pay is $50 per person on weekdays, $75 on Saturday, $90 on Sundays.
Modify the LP formulation so that the objective is now to minimize the weekly
payroll costs rather than to minimize the workforce size.
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Example 5: LP Formulation
• Target Shirt Company makes three varieties of shirts: Collegiate,
Traditional and European. These shirts are made from different
combinations of cotton and polyester. The cost per yard of
unblended cotton is $5 and for unblended polyester is $4. Target
can receive up to 4,000 yards of raw cotton and 3,000 yards of raw
polyester fabric weekly. The table shows below pertinent data
concerning the manufacture of the shirts. (Meet weekly contracts
or more; while not exceeding weekly demand).
• Formulate and solve a linear program that would give a
manufacturing policy for Target Shirt Company.
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Example 5: LP Formulation
Finco has the following investments available:
Investment A For each dollar invested at time 0, we receive $ 0.10 at
time 1 and $1.30 at time 2. Time 0= now; time 1 = one year from
now; and so on.)
Investment B For each dollar invested at time 1, we receive $1.60 at
time 2.
Investment C For each dollar invested at time 2, we receive $1.20 at
time 3.
At any time, leftover cash may be invested in T-bills, which pay 10%
per year. At time 0, we have $100. At most, $50 can be invested in
each of investments A, B, and C. Formulate an LP that can be used
to maximize Finco’s cash on hand at time 3.
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