ESI 4313 Operations Research 2

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# ESI 4313 Operations Research 2 - PowerPoint PPT Presentation

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 4313Operations Research 2

Nonlinear Programming Models

Lecture 3

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
• 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 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 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 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 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 location
• Optimization problem
• Manhattan:
• Euclidean:
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
• 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
• 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
• 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 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 problem
• NLP formulation:
• We can simplify the problem to:
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)
• 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?
• 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
• 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?
• Objective:
• Constraints:
• 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
• 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 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 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