A few examples of LP business applications

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A few examples of LP business applications
Marketing Advertising Media Mix Product
Location Retail Pricing Finance Currency
Trading Capital Budgeting Portfolio
Optimization Human Resources Healthcare Plan
Selection Workforce Planning Worker
Assignment Operations Logistical Supply
Chain Planning Production Planning Project
Planning Worker, Machine, Truck Scheduling
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Problem Statement Lets call the number of
Hudson notes we buy X and the number of
JFK notes Y Maximize Z 10X 18Y ST Ylt
100 Xlt 200 200X300Ylt55000
Decision Variables
Objective Function (Min or Max)
Constraints that solution is Subject To (ST)
One additional (often unstated) constraint called
non-negativity
A
X,Ygt 0 or X gt 0 Y gt 0
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Graphical Solution to the LP Formulation
Since we have two decision variables, we may
evaluate potential solutions to our problem
graphically. This process gets difficult with
three decision variables and impossible with 4 or
more!
Notes of JFK (y)
First, we note that our decision variables must
be non-negative. Further, our constraints may be
represented by linear functions.
Notes of Hudson (x)
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Graphical Solution to the LP Formulation
Constraint 1 y lt 100
100
Feasible Region
Notes of JFK (y)
Any point (x,y) on or below this constraint line
will satisfy constraint 1. All points above the
constraint line are infeasible.
Notes of Hudson (x)
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Graphical Solution to the LP Formulation
Constraint 2 x lt 200
Constraint 1 y lt 100
100
Notes of JFK (y)
Feasible Region
200
Notes of Hudson (x)
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Graphical Solution to the LP Formulation
183.3
Constraint 2 x lt 200
Constraint 1 y lt 100
100
Constraint 3 200x300ylt55000
Notes of JFK (y)
Feasible Region
275
200
Notes of Hudson (x)
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Graphical Solution to the LP Formulation
183.3
Constraint 2 x lt 200
Constraint 1 y lt 100
100
Constraint 3 200x300ylt55000
Notes of JFK (y)
The solution that maximizes Z must be on at least
one corner point of the feasible region
275
200
Notes of Hudson (x)
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Graphical Solution to the LP Formulation
183.3
Constraint 2 x lt 200
Constraint 1 y lt 100
100
Constraint 3 200x300ylt55000
Notes of JFK (y)
x 0 y 0 10(0)18(0)0
Buy Nothing
275
200
Notes of Hudson (x)
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Graphical Solution to the LP Formulation
183.3
Constraint 2 x lt 200
Constraint 1 y lt 100
100
Constraint 3 200x300ylt55000
Notes of JFK (y)
x 0 y
100 10(0)18(100)1800
Buy all the JFK you can but buy no Hudson
275
200
Notes of Hudson (x)
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Graphical Solution to the LP Formulation
183.3
Constraint 2 x lt 200
Constraint 1 y lt 100
100
Constraint 3 200x300ylt55000
Notes of JFK (y)
200x 300y 55000 - 300(
y 100) 200x
25000 x 125
10(125)18(100)3050
Buy all the JFK and spend the whole budget
275
200
Notes of Hudson (x)
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Graphical Solution to the LP Formulation
183.3
Constraint 2 x lt 200
Constraint 1 y lt 100
100
Constraint 3 200x300ylt55000
Notes of JFK (y)
200x 300y 55000 - 200( x
200) 300y
15000 y 50
10(200)18(50)2900
Buy all the Hudson and spend the whole budget
275
200
Notes of Hudson (x)
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Graphical Solution to the LP Formulation
183.3
Constraint 2 x lt 200
Constraint 1 y lt 100
100
Constraint 3 200x300ylt55000
Notes of JFK (y)
x 200 y
0 10(200)18(0)2000
Buy all the Hudson you can but buy no JFK
275
200
Notes of Hudson (x)
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WinQSB Solution
First Half of Combined Report
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WinQSB Solution
Decision variable names
Optimal value of each decision variable
The change in Z given a 1 unit increase in a
decision variable whose current value equals 0
The coefficient of each decision variable in the
objective function
The (solution value) (unit cost or profit)
The range of values within which the objective
function coefficients may change without changing
the current optimal values of the decision
varaiables
The optimal solution value of Z
Interpretation In the optimal solution, 125
notes of X (Hudson) and 100 notes of (JFK) are
purchased. If this plan is followed, the total
expected 6-month return will equal
3,050.00. If the expected 6-month return of X
(Hudson) increases beyond 12.00 per note, the
current solution will no longer be optimal.
If the expected 6-month return of Y (JFK)
decreases below 15.00 per note, the current
solution will no longer be optimal.
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WinQSB Solution
Second Half of Combined Report
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WinQSB Solution
Constraint Names
Quantity of each constraints RHS used by the
optimal solution
Form of constraint equality lt,, or gt
Quantity of each constraints RHS available
Quantity of each constraints RHS remaining after
solution RHS-LHS
The change in Z for a one unit change in the
constraints RHS
The range of RHS values within which the shadow
price provides the change in Z
Interpretation Constraint 1 (available notes
of JFK) and constraint 3 (budget) are fully
consumed. Constraint 2 (available notes of
Hudson) has not been fully consumed, 75 notes
will be available after solution is
implemented. Each additional note of JFK made
available and purchased (beyond the 100 currently
available) will increase the value of Z by
3.00. Likewise, notes of JFK made unavailable
(below the 100 currently available) will
decrease the value of Z by 3.00 each. This
3.00 change is only relevant if the total
number of shares of JFK available is between 50
and 183.3333 notes.
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Transportation Problem Solution using WinQSB
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Product Mix Example 1
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Product Mix Example 2
Smithco blends silicon and nitrogen to produce
two types of fertilizers. Fertilizer 1 must be
at least 40 nitrogen and sells for 7.00/lb.
Fertilizer 2 must be at least 70 silicon and
sells for 4.00/lb. Smithco can purchase up to
8,000 lb of nitrogen at 1.50/lb and up to 10,000
lb of silicon at 1.00/lb. Assuming that all
fertilizer produced can be sold, formulate an LP
model to maximize profits.
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Workforce Assignment Example
During each 4-hour period, the Smalltown Job Shop
requires the following number of on-duty
mechanics 12 midnight to 4 A.M. 4 4 to 8
A.M. 7 8 to 12 noon 6 12 noon to 4
P.M. 6 4 to 8 P.M. 5 8 to 12
midnight 4 Part a. Each mechanic works two
consecutive 4-hour shifts. Formulate an LP model
that can be used to minimize the number of
mechanics needed to meet Smalltowns daily
requirements. Part b. Assume that mechanics can
be scheduled for two consecutive 4-hour periods
and be paid at a rate of 6 per hour scheduled
or can be scheduled to work one 4- hour period at
a rate of 7 per hour.
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Warehousing, Facility Location, Facility Layout
Example
PJ Shops is planning to expand its facilities.
Currently they are planning for a new
manufacturing facility in Lyman. The shop they
have purchased for this expansion has 50,000
square feet of useable space. The budget for
the expansion project is 10,000,000. Management
is attempting to allocate this space among four
departments. Historical trends are shown in the
following table Profit (sqft) Invest
(sqft) Min. Size Grind 40.00 120.00 10,000 B
roach 30.00 240.00 20,000 Mill 25.00 150.00
5,000 Assemble 15.00 50.00
2,000 Additionally, the Grind department cannot
be more than 15 larger than the Broach
department. Formulate an LP model to allocate
space so that total profits are maximized.
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Transportation and Shipping Example
Water-Is-Us hauls water from its two reservoirs
to three communities. The manager wants to
determine how to supply these communities and
minimize the total number of miles traveled by
his fleet of trucks. The following table shows
the distances between reservoirs and communities
and the daily supply demand of water in
truckloads Reservoir 1 Reservoir
2 Demand Portland 25 35
150 Gorham 15 25
200 Scarborough 45 15
50 Supply 150 350 Formulate an LP
model to supply all demand with minimum total
travel.
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Scheduling Example
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