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Unformatted text preview: Advance Decision
Making and Simulation
Module 1 --- Nonlinear Programming Topics Covered in OPR 330
Topics Nonlinear Programming Queuing Theory Markov Processes Dynamic Programming Simulation
Simulation Decision Analysis 1. Nonlinear Programming
1. Example of NLP application Resource allocation with nonlinear
Resource objective / constraints
objective 2. Queuing Theory
2. In such situations… Bank
Hospital Blood Bank
… Questions answered by queuing theory… What fraction of the time is each server idle?
What is the expected number of customers present in the
What is the expected time a customer spends in the queue?
What is the probability distribution of the number of customers
present in the queue?
… 3. Markov Processes
3. Markov Process analysis can be used to
Markov answer these questions…
answer a machine that is functioning in one period will
continue to function or break down in the next
a consumer purchasing brand A in one period
will purchase brand B in the next period.
… 4. Dynamic Programming
4. Example of DP problem… Knapsack problem A scout is packing for a trip and is considering fifteen
items. The fifteen items have different values and
weight which are provided. The maximum weight she
can carry is fifteen pounds. In addition, there must be
at least one food item and she may take either the
camera or the radio but not both. How should the
scout pack her bag? 5. Simulation
5. Examples of simulation problems: Risk Analysis Waiting Line Analysis 6. Decision Analysis
6. Examples of Decision Analysis: Decision Tree Nonlinear Programming (NLP)
Nonlinear OPR 320: Linear Programming (LP)
Maximize or minimize a linear function subject to
linear Real-world problems: often nonlinear objective
function and/or constraints
Optimization problems subject to nonlinearities
are called nonlinear programming problems
are LP Reminder
Product Labor (hr/unit) Clay (lb/unit) Profit ($/unit) Bowl 1 4 40 Mug 2 3 50 Availability 40 120 Objective: profit maximization
Constraint: limited resources Assumptions
• We can sell every unit produced.
• Each unit requires the same amount of labor.
• Each unit requires the same amount of clay. General Formulation
min imize f ( x)
subject to g ( x) ≥ 0 The objective function can be maximized or
The constraints can be equalities (=) or
inequalities (≤, ≥)
Unconstrained NLP: only objective function Example 1: Production Application
Armstrong Bike Co. produces two new lightweight Armstrong Bike Co. produces two new lightweight bicycle frames, the Flyer and the Razor, that are made from special aluminum and steel alloys. The cost to produce a Flyer frame is $100, and the cost to produce a Razor frame is $120. It is unsure whether Armstrong will sell all the frames it can produce. As the selling price of each frame model – Flyer and Razor increases, the quantity demanded for each model goes down. Part 1: Unconstrained NLP • Assume that the demand F for Flyer frames
and the demand R for Razor frames are given by: F = 750 – 5PF R = 400 – 2PR
where PF = the price of a Flyer frame •
• PR = the price of a Razor frame. Objective: maximize profit
No constraint Part 2: Constrained NLP Part 2: Constrained NLP A supplier can deliver a maximum of 500 pounds of the aluminum alloy and 420 pounds of the steel alloy weekly. The number of pounds of each alloy needed per frame is: Aluminum Alloy Steel Alloy Flyer 2 3 Razor 4 2
How many Flyer and Razor frames should
Armstrong produce each week? Graphical representation
Profit = $6,075.47
Contou r 50 100 150 200 250 300 x1 Example 2: Production & Advertising If Widgetco charges a price p for a product and spends $a on advertising, it can sell: 10000 + 5 a -100 p units of the product. If the product costs $10 per unit to produce, how can the company maximize profits? Local and Global Optima
Local and Global Optima
s A feasible solution is a local optimum if there are no other feasible solutions with a better objective function value in the immediate neighborhood. There could however be better solutions some distance away.
• For a maximization problem the local optimum corresponds to a local maximum.
• For a minimization problem the local optimum corresponds to a local minimum
A feasible solution is a global optimum if there are no other feasible points with a better objective function value in the feasible region.
A global optimum is also a local optimum. Multiple Local Optima
Multiple Local Optima
s s s s s Nonlinear optimization problems can have multiple local optimal solutions: find the best local optimum
Nonlinear problems with multiple local optima are very challenging
In these cases, the software can get “stuck” and terminate at a local optimum
There can be a severe penalty for finding a local optimum that is not a global optimum
Convex, concave functions are easier to optimize Example 3: Capital and Labor If K units of capital and L units of labor are used, a company can produce KL units of a manufactured good. Capital can be purchased at $4/unit, and labor can be purchased at $1/unit. A total of $8 is available to purchase capital and labor. How can the firm maximize the quantity of good that can be manufactured? Example 4: Facility Location
A firm must decide where to build a distribution center (DC) that will serve its sales centers (SC) in Amherst (a), Bowdoin (B), Colby (C) and Dartmouth (D). The table below shows the location in the xy coordinate plane of each SC and the number of shipments that must be made daily from the DC to each SC. The truck delivery cost is $1 per mile. Sales Centers
2 Number of Shipments
9 B 3 10 7 C 8 15 2 D 14 13 5 Where should the new DC be built in order to minimize the
transportation costs while carrying out the required
shipments? Example 4: Facility Location
Assume now that zoning laws impose limitations on the
possible location of the DC and assume that the DC must be
built in the region below.
built 11 y 5 10 19 13
x Example 5: Advertising
Part 1: Part 1:
QH Company advertises on soap operas and football games. Each soap opera ad costs $50,000, and each football ad costs $100,000. Giving all figures in millions of viewers, if S soap opera ads are bought, they will be seen by men and women. If F football ads are bought, they 20 S
will be seen by men and women. QH company 17 F
wants at least 40 million men and at least 60 million women 7F
to see its ads. Formulate an NLP that will minimize QH’s cost of reaching sufficient viewers. Example 5: Advertising Part 2: Suppose that the number of women reached by F football ads and S soap opera ads is 7 F + 20 S − 0.2 FS Why might this be a more realistic representation of the number of women viewers seeing QH’s ads? Example 6: Traveling Time
Each morning during rush hour, 10,000 people want to travel from New Jersey to New York City. To do so, they can take a train or they can drive. A train trip lasts 40 minutes. If x thousand people drive, it takes each person driving 20 + 5x minutes to make the trip. How many people need to travel by road to minimize the average travel time per person? Example 7: Forecasting adoption of a Example 7: Forecasting adoption of a new product Box office revenues and cumulative revenues in
$ millions for the doctor:
Week Revenue St Cumulative
Revenues Ct 1 0.10 0.10 2 3.00 3.10 3 5.20 8.30 4 7.00 15.30 5 5.25 20.55 6 4.90 25.45 7 3.00 28.45 8 2.40 30.85 9 1.90 32.75 10 1.30 34.05 11 0.80 34.85 12 0.60 35.45 ...
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- Spring '12