ESI 4313 Operations Research 2

1. ESI 4313Operations Research 2 Nonlinear Programming Models Lecture 3

2. 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

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 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.

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 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?

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 Ddenote the (unknown!) number of trees that Joe can sell

13. 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

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 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?

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 Q&H 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 Q&H’s 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