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ESI 4313 Operations Research 2. Nonlinear Programming Models Lecture 3. Example 2: Warehouse location. In OR1 we have looked at the warehouse (or facility) location problem.

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ESI 4313 Operations Research 2

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esi 4313 operations research 2

ESI 4313Operations Research 2

Nonlinear Programming Models

Lecture 3

example 2 warehouse location
Example 2:Warehouse 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
example 2 contd warehouse location
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
example 2 contd warehouse location4
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)
example 2 contd warehouse location5
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.
example 2 contd warehouse location6
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
example 2 contd warehouse location7
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:
example 2 contd warehouse location8
Example 2 (contd.):Warehouse location
  • Optimization problem
    • Manhattan:
    • Euclidean:
example 3 fire station location
Example 3:Fire station location
  • Monroe county is trying to determine where to place its fire station.
  • The centroid locations of the county’s 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.
example 3 contd fire station location
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
example 4 newsboy problem
Example 4:Newsboy 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?
example 4 contd newsboy problem
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 Ddenote the (unknown!) number of trees that Joe can sell
example 4 contd newsboy problem13
Example 4 (contd.):Newsboy problem
  • Then his revenue is:
    • 25Q if Q  D
    • 25Dif Q >D
  • I.e., his revenue is 25 min(Q,D)
  • His expected revenue is
example 4 contd newsboy problem14
Example 4 (contd.):Newsboy problem
  • NLP formulation:
  • We can simplify the problem to:
example 4 contd newsboy problem15
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)
example 5 advertising
Example 5:Advertising
  • Q&H company advertises on soap operas and football games.
    • Each soap opera ad costs $50,000
    • Each football game ad costs $100,000
  • Q&H wants at least 40 million men and at least 60 million women to see its ads
  • How many ads should Q&H purchase in each category?
example 5 contd advertising
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
example 5 contd advertising18
Example 5 (contd.):Advertising
  • Compare this model with a model that says that the number of men and women seeing a Q&H ad is linear in the number of ads S and F .
  • Which one is more realistic?
example 5 contd advertising19
Example 5 (contd.):Advertising
  • Objective:
  • Constraints:
example 5 contd advertising20
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 Q&H’s ads?
nonlinear programming
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 )
nonlinear programming22
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
nonlinear programming23
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