ESI 4313 Operations Research 2

1
ESI 4313Operations Research 2

• Nonlinear Programming Models
• Lecture 3

2
Example 2Warehouse location

• In OR1 we have looked at the warehouse (or
  facility) location problem.
• In particular, we formulated the problem of
  choosing a set of locations from a large set of
  candidate locations as a mixed-integer linear
  programming problem

3
Example 2 (contd.)Warehouse location

• Candidate locations are often found by solving
  location problems in the plane
• That is, location problems where we may locate a
  warehouse anywhere in some region

4
Example 2 (contd.)Warehouse location

• The Wareco Company wants to locate a new
  warehouse from which it will ship products to 4
  customers.
• The locations of the four customers and the of
  shipments per year are given by
• 1 (5,10) 200 shipments
• 2 (10,2) 150 shipments
• 3 (0,12) 200 shipments
• 4 (1,1) 300 shipments
• i (xi,yi) Di shipments (i 1,,4)

5
Example 2 (contd.)Warehouse location

• Suppose that the shipping costs per shipment are
  proportional to the distance traveled.
• Wareco now wants to find the warehouse location
  that minimizes the total shipment costs from the
  warehouse to the 4 customers.

6
Example 2 (contd.)Warehouse location

• Formulate this problem as an optimization problem
• How would/could/should you measure distances?
• Rectilinear distances (Manhattan metric)
• Euclidean distance

7
Example 2 (contd.)Warehouse location

• Decision variables
• x x-coordinate of warehouse
• y y-coordinate of warehouse
• Distance between warehouse and customer 1 at
  location (5,10)
• Manhattan
• Euclidean

8
Example 2 (contd.)Warehouse location

• Optimization problem
• Manhattan
• Euclidean

9
Example 3Fire station location

• Monroe county is trying to determine where to
  place its fire station.
• The centroid locations of the countys major
  towns are as follows
• (10,20) (60,20) (40,30) (80,60) (20,80)
• The county wants to build the fire station in a
  location that would allow the fire engine to
  respond to a fire in any of the five towns as
  quickly as possible.

10
Example 3 (contd.)Fire station location

• Formulate this problem as an optimization
  problem.
• The objective is not formulated very precisely
• How would/could/should you choose the objective
  in this case?
• Do you have sufficient data/information to
  formulate the optimization problem?
• Compare this situation to the warehouse location
  problem and the hazardous waste transportation
  problem

11
Example 4Newsboy problem

• Single period stochastic inventory model
• Joe is selling Christmas trees to (help) pay for
  his college tuition.
• He purchases trees for 10 each and sells them
  for 25 each.
• The number of trees he can sell during this
  Christmas season is unknown at the time that he
  must decide how many trees to purchase.
• He assumes that this number is uniformly
  distributed in the interval 10,100.
• How many trees should he purchase?

12
Example 4 (contd.)Newsboy problem

• Decision variable
• Q number of trees to purchase
• We will only consider values 10 ? Q ? 100 (why?)
• Objective
• Say Joe wants to maximize his expected profit
  revenue costs
• Costs 10Q
• Revenue 25 E( trees sold)
• Let the random variable D denote the (unknown!)
  number of trees that Joe can sell

13
Example 4 (contd.)Newsboy problem

• Then his revenue is
• 25Q if Q ? D
• 25D if Q gt D
• I.e., his revenue is 25 min(Q,D)
• His expected revenue is

14
Example 4 (contd.)Newsboy problem

• NLP formulation
• We can simplify the problem to

15
Example 4 (contd.)Newsboy problem

• Other applications
• Number of programs to be printed prior to a
  football game
• Number of newspapers a newsstand should order
  each day
• Etc.
• (In general seasonal items, i.e., items that
  loose their value after a certain date)

16
Example 5Advertising

• QH company advertises on soap operas and
  football games.
• Each soap opera ad costs 50,000
• Each football game ad costs 100,000
• QH wants at least 40 million men and at least 60
  million women to see its ads
• How many ads should QH purchase in each category?

17
Example 5 (contd.)Advertising

• Decision variables
• S number of soap opera ads
• F number of football game ads
• If S soap opera ads are bought, they will be seen
  by
• If F football game ads are bought, they will be
  seen by

18
Example 5 (contd.)Advertising

• Compare this model with a model that says that
  the number of men and women seeing a QH ad is
  linear in the number of ads S and F .
• Which one is more realistic?

19
Example 5 (contd.)Advertising

• Objective
• Constraints

20
Example 5 (contd.)Advertising

• Suppose now that the number of women (in
  millions) reached by F football ads and S soap
  opera ads is
• Why might this be a more realistic representation
  of the number of women viewers seeing QHs ads?

21
Nonlinear programming

• A general nonlinear programming problem (NLP) is
  written as
• x (x1,,xn) is the vector of decision variables
• f is the objective function
• we often write f (x )

22
Nonlinear programming

• gi are the constraint functions
• we often write gi (x )
• the corresponding (in)equalities are the
  constraints
• The set of points x satisfying all constraints is
  called the feasible region
• A point x that satisfies all constraints is
  called a feasible point
• A point that violates at least one constraint is
  called an infeasible point

23
Nonlinear programming

• A feasible point x with the property that
• is called an optimal solution to a maximization
  problem
• A feasible point x with the property that
• is called an optimal solution to a minimization
  problem