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ESI 4313 Operations Research 2PowerPoint Presentation

ESI 4313 Operations Research 2

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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

- How would/could/should you measure distances?

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)

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

- 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.):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.):Advertising

- Objective:
- Constraints:

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

- 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

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