Koç University
OPSM 301 Operations Management
Class 11:
Linear Programming using Excel
Zeynep Aksin
zaksin@ku.edu.tr
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Announcement
Linear programming: Appendix A from our
book
Skip graphical solution, skip sensitivity
analysis for now
You can use examples done in class,
example A1, solved problem 1, Problem 3
as a study set (and all other problems if
you like)
Quiz 2 next Thursday
Office hours by Tas as previously
announced
Linear Programming (LP)
 Making decisions on the allocation of scarce
resources in environments of certainty using a
mathematical optimization approach
Example Applications
 OPSM: Product mix problem-how much of
each product should be produced given
resource constraints to maximize profits
 Finance: Construct a portfolio of securities
that maximizes return while keeping "risk"
below a predetermined level
 Marketing: Develop an advertising strategy
to maximize exposure of potential
customers while staying within a
predetermined budget
Conditions for Applicability of
Linear Programming
 Resources must be limited
 There must be an objective function
 There must be linearity in the constraints and in
the objective function
 Resources and products must be homogeneous
 Decision variables must be divisible and nonnegative
Components of Linear Programming
 A specified objective or a single goal, such as the
maximization of profit, minimization of machine idle
time etc.
 Decision variables represent choices available to
the decision maker in terms of amounts of either
inputs or outputs
 Constraints are limitations which restrict the
alternatives available to decision makers
Components of Linear Programming
 There are three types of constraints:
– (=<) An upper limit on the amount of some scarce
resource
– (>=) A lower bound that must be achieved in the
final solution
– (=) An exact specification of what a decision
variable should be equal to in the final solution
 Parameters are fixed and given values which
determine the relationships between the decision
variables of the problem
Example: Giapetto's Woodcarving
 Two types of toys are manufactured: soldiers and trains
 Soldiers:
– Sells for $27
– Uses raw materials worth $10
– Each soldier increases variable labor and overhead costs by $14
 Trains:
– Sells for $21
– Uses raw materials worth $9
– Each train increases variable labor and overhead costs by $10
Giapetto's Woodcarving
 Manufacture requires skilled labor of two types
– Carpentry
– Finishing
 Resource requirements by product
– Soldier: 1 hour of carpentry and 2 hours of finishing
– Train: 1 hour of carpentry and 1 hour of finishing
 Total resources available
– Unlimited raw materials
– 80 hours of carpentry
– 100 hours of finishing labor
Giapetto's Woodcarving
 Demand
– for trains is unlimited
– At most 40 soldiers can be sold each week
 Objective is to maximize weekly profit
 Formulate as a linear program (LP)
Towards the Mathematical Model:
Define (decision variables)
– x1 : number of soldiers produced each week
– x2 : number of trains produced each week
Objective function:
– maximize weekly profit = weekly profit from soldiers + weekly
profit from trains
Constraints:
– each week, no more than 100 hours of finishing time may be used
– each week, no more than 80 hours of carpentry time may be used
– each week, the number of soldiers produced should not exceed 40
because of limited demand
Giapetto's Woodcarving: The LP Model
max 3x1 + 2x2
subject to
2x1 + x2
 100
x1 + x2  80
x1
 40
x1
0
x2
0
 Where
(finishing hours)
(carpentry hours)
(demand for soldiers)
(nonnegativity constraint)
(nonnegativity constraint)
– x1 : number of soldiers produced each week
– x2 : number of trains produced each week
The Excel Model
soldiers
changing cells
profit
soldiers
finishing
carpenter
demand
trains
20
3
trains
2
1
Total(obj
ective)
60
2
180
used
1
1
capacity
100
80
20
100
80
40
Filled in by Excel Solver
Reading the variable information
 The optimal solution for Giapetto is to produce 20
soldiers and 60 trains per week, resulting in an optimal
profit of $180. (The maximum possible profit attainable is
$180, which can be achieved by producing 20 soldiers
and 60 trains)
Example 1:Product Mix Problem
P
Q
Sales price:90 $/unit
Max demand:100 units/week
D
10 min/un
Purchase
Part 5$/un
C
10 min/un
A
15 min/un.
C
5 min/un
B
15 min/un.
D
5 min/un
B
15 min/un
A
10 min/un.
Sales price: 100 $/unit
Max Demand:50 units/week
Products P and Q are produced using the given
process routing.
4 machines are used:A,B,C,D. (available for
2400 min/week)
The price and raw material costs are given.
Problem:
Formulate an LP to find the product mix that
maximizes weekly profit.
i.e. How many of each product should we
produce given the capacity and demand
constraints?
What is the bottleneck of this process?
RM1
20$/un
RM2
20$/un
RM3
20$/un
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Source: Paul Jensen
LP Formulation
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Decision variables:
– P:Amount of product P to produce per week
– Q:Amount of product Q to produce per week
Objective Function: Maximize Profit
– Max 45P+60 Q
Constraints:
Machine hours used should be less than or equal to 2400 minutes:
– A: 15 P + 10 Q <= 2400
– B: 15 P + 30 Q <= 2400
– C: 15 P + 5 Q <= 2400
– D: 10 P + 5 Q <= 2400
Production should not exceed demand:
– P<=100
– Q<=50
Non-negativity
–
–
P>=0,
Q>=0
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Solver Solution
P
changing cells
objective
Coefficients
Profit
Constraint
Coefficients:
Machine A
Machine B
Machine C
Machine D
Demand P
Demand Q
Q
100
30
45
6300
60
P
Q
15
15
15
10
1
0
L.H.S. Value
10
1800 <=
30
2400 <=
5
1650 <=
5
1150 <=
0
100 <=
1
30 <=
R.H.S.
2400
2400
2400
2400
100
50
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Example Problem
The Huntz Company purchases cucumbers and makes two kinds of
pickles: sweet and dill. The company policy is that at least 30%, but no
more than 60%, of the pickles be sweet. The demand for pickles is
SWEET:5000 jars + additional 3 jars for each $1 spent on
advertising
DILL:4000 jars + additional 5 jars for each $1 spent on advertising
 Sweet and dill pickles are advertised separately. The production costs
are:
SWEET:0.60 $/jar DILL:0.85 $/jar
and the selling prices are:
SWEET:1.45 $/jar DILL:1.75 $/jar
Huntz has $16,000 to spend on producing and advertising pickles.
Formulate an appropriate Linear Program.
Solution: LP Formulation
Xs: Number of Sweet pickle jars produced.
Xd: Number of Dill pickle jars produced.
As: Amount of advertisement done for Sweet Pickles in dollars
Ad: Amount of advertisement done for Dill Pickles in dollars
 Objective; maximize profits: (Revenue - Cost)
max[(1,45* Xs + 1,75* Xd)-( 0,6* Xs + 0,85* Xd + As + Ad)]
Subject to:
Demand Constraints:
Xs ≤ 5,000 + 3As
Xd ≤ 4,000 + 5Ad
Budget Constraint:
0,6* Xs + 0,85* Xd + As + Ad ≤ 16,000
Ratio Constraint:
Xs / (Xs + Xd) ≥ 0,3  0,7 Xs – 0,3 Xd ≥ 0
Xs / (Xs + Xd) ≤ 0,6  0,4 Xs – 0,6 Xd ≤ 0
Non-negativity Constraints:
Xs, Xd, As, Ad ≥0