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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)**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