opsm 301 operations management class 11: linear programming using excel koç university zeynep aksin
Post on 18-Jan-2018
226 Views
Preview:
DESCRIPTION
TRANSCRIPT
OPSM 301 Operations Management
Class 11:
Linear Programming using Excel
Koç University
Zeynep Aksinzaksin@ku.edu.tr
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 non-
negative
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 to2x1 + x2 100 (finishing hours)
x1 + x2 80 (carpentry hours)
x1 40 (demand for soldiers)
x1 0 (nonnegativity constraint)
x2 0 (nonnegativity constraint)
Where– x1 : number of soldiers produced each week– x2 : number of trains produced each week
The Excel Model
soldiers trainsTotal(objective)
changing cells 20 60
profit 3 2 180
soldiers trains used capacity
finishing 2 1 100 100 carpenter 1 1 80 80 demand 20 40
Filled in by Excel Solver
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)
Reading the variable information
C10 min/un
B15 min/un
C5 min/un
A15 min/un.
A10 min/un.
B15 min/un.
D10 min/un
D5 min/un
P QSales price:90 $/unitMax demand:100 units/week
Sales price: 100 $/unitMax Demand:50 units/week
RM120$/un
RM220$/un
RM320$/un
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?
Purchase Part 5$/un
Source: Paul Jensen
Example 1:Product Mix Problem
15
LP Formulation 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 16
Solver Solution
P Q changing cells 100 30
objective Coefficients 45 60
Profit 6300
Constraint Coefficients: P Q L.H.S. Value R.H.S.
Machine A 15 10 1800<= 2400 Machine B 15 30 2400<= 2400 Machine C 15 5 1650<= 2400 Machine D 10 5 1150<= 2400 Demand P 1 0 100<= 100 Demand Q 0 1 30<= 50
17
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 advertisingDILL: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.
Example Problem
Xs: Number of Sweet pickle jars produced.Xd: Number of Dill pickle jars produced.As: Amount of advertisement done for Sweet Pickles in dollarsAd: 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 + 3AsXd ≤ 4,000 + 5AdBudget Constraint:0,6* Xs + 0,85* Xd + As + Ad ≤ 16,000Ratio Constraint:Xs / (Xs + Xd) ≥ 0,3 0,7 Xs – 0,3 Xd ≥ 0Xs / (Xs + Xd) ≤ 0,6 0,4 Xs – 0,6 Xd ≤ 0
Non-negativity Constraints:Xs, Xd, As, Ad ≥0
Solution: LP Formulation
top related