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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 l.jpg

ESI 4313Operations Research 2

Nonlinear Programming Models

Lecture 3

Example 2 warehouse location l.jpg
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 l.jpg
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 l.jpg
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 l.jpg
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 l.jpg
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 l.jpg
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 l.jpg
Example 2 (contd.):Warehouse location

  • Optimization problem

    • Manhattan:

    • Euclidean:

Example 3 fire station location l.jpg
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 l.jpg
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 l.jpg
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 l.jpg
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 l.jpg
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 l.jpg
Example 4 (contd.):Newsboy problem

  • NLP formulation:

  • We can simplify the problem to:

Example 4 contd newsboy problem15 l.jpg
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 l.jpg
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 l.jpg
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 l.jpg
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 l.jpg
Example 5 (contd.):Advertising

  • Objective:

  • Constraints:

Example 5 contd advertising20 l.jpg
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 l.jpg
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 l.jpg
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 l.jpg
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