Jianlin Cheng, PhD Computer Science Department University of Missouri, Columbia Fall, 2013


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1 Jianlin Cheng, PhD Computer Science Department University of Missouri, Columbia Fall, 2013
2 Princeton s class notes on linear programming MIT s class notes on linear programming Xian Jiaotong University s class notes on linear programming Ohio University s class notes Rutgers University s class notes
3 Essen>al tool for op>mal alloca>on of scarce resource among a number of compe>ng ac>vi>es Powerful and general problem solving method  shortest path, max flow, min cost flow minimum spanning tree
4 Fast commercial solvers: CPLEX, OSL Ranked among most important scien>fic advances of 20 th century Also a general tool for aqacking NP hard op>miza>on problems Dominates world of industry  ex: Delta claims saving $100 million per year using LP
5 Agriculture: diet problem Computer science: data mining Electrical engineering: VLSI design Energy: blending petroleum products Environment: water quality management Finance: poryolio op>miza>on Logis>cs: supply chain management Management: hotel yield management Marke>ng: direct mail adver>sing manufacturing: produc>on line balancing Medicine: radioac>ve seed placement in cancer treatment Opera>on research: airline crew assignment Telecommunica>on: network design, internet rou>ng Sports: scheduling ACC basketball
6 An Example, IntuiFve Understanding of Linear Programming
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10 Profit can be considered as a series of linear lines having a certain profit. Scan the line through the feasible area.
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17 Another Example, FormulaFon, and Proof
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23 Slack Form min ax z = 4x s.t. 2x + 3y + x 0, y 5y w = 60 9, w 0. min s.t. x z = cx Ax x = b 0. rank( A) = m.
24 Optimal occurs at a vertex!!! z = 4 x + 5y 2x + 3y 60 Feasible domain
25 What s a vertex? A point x in a polyhadren Ω is called a if 1 x = ( y + z), y, z Ω 2 x = y = z. vertex
26 Fundamental Theorem Let Ω= { x Ax = b, x 0}. If min cx over x Ω has an optimal solution, then it can be found in vertices of Ω.
27 Proof. Consider an optimal solution x* with maximum number of zero components among all optimal solutions. We will show that x*is a vertex of Ω. By contradition, suppose x*is not, that is, there exist y, z Ω such that x* = ( y + z) / 2 and x*, y, z are distinct. Since cx* cy, cx* cz, and cx* = ( cy+cz) / 2, we must have cx* = cy = cz.this means that y and z are also optimal solutions. It follows that all feasible points on line x*+ α( yx*) are optimal solutions. However, Ω does infinite not contain any line.thus, the line must have a point x' not in Ω, that is, x' violates at least one constraint.
28 Z X* Y X X violates one constraint
29 Proof (cont s). Note that for any α, Thus, x' cannot violate constraint Ax = b. Moreover, for x* = 0, since x*= ( y +z i must have z =y =x of x*+ α( yx*) is equal to 0 for any α. This means that x' cannot violate constraint violate a constraint x i i i i A( x*+ α( yx*)) = b. ) / 2 and * = 0. Therefore, the ith component j i i 0. Hence, x' must 0 for some j with x 0, we *> 0. Now, we can easily find an optimal solution between x' and x*, which has one more zero  component than x*, a contradiction. x i y i,z i j
30 Z Y X* X X has fewer or equal number of 0 terms as X*. X has one nega>ve component, whose value in X* is posi>ve. As we move from X to X* into the feasible region, the component will become 0
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33 Polyhedron of simplex algorithm in 3D
34 Demo of MIT Linear Programming Solver: hqp://sourceforge.net/projects/lipside/
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40 Integer Programming Integer programming is a solution method for many discrete optimization problems Programming = Planning in this context Origins go back to military logistics in WWII (1940s). In a survey of Fortune 500 firms, 85% of those responding said that they had used linear or integer programming. Why is it so popular? Many different reallife situations can be modeled as integer programs (IPs). There are efficient algorithms to solve IPs.
41 Standard form of integer program (IP) maximize c 1 x 1 +c 2 x 2 + +c n x n (objective function) subject to a 11 x 1 +a 12 x 2 + +a 1n x n b 1 (functional constraints) a 21 x 1 +a 22 x 2 + +a 2n x n b 2. a m1 x 1 +a m2 x 2 + +a mn x n b m x 1, x 2,, x n Z + (set constraints)
42 Standard form of integer program (IP) In vector form: maximize cx (objective function) subject to Ax b (functional constraints) n x (set constraints) Z + Input for IP: 1 n vector c, m n matrice A, m 1 vector b. Output of IP: n 1 integer vector x. Note: More often, we will consider mixed integer programs (MIP), that is, some variables are integer, the others are continuous.
43 Example of Integer Program (Production PlanningFurniture Manufacturer) Technological data: Production of 1 table requires 5 ft pine, 2 ft oak, 3 hrs labor 1 chair requires 1 ft pine, 3 ft oak, 2 hrs labor 1 desk requires 9 ft pine, 4 ft oak, 5 hrs labor Capacities for 1 week: 1500 ft pine, 1000 ft oak, Market data: 20 employees (each works 40 hrs). profit Goal: Find a production schedule for 1 week that will maximize the profit. demand table $12/unit 40 chair $5/unit 130 desk $15/unit 30
44 Production PlanningFurniture Manufacturer: modeling the problem as integer program The goal can be achieved by making appropriate decisions. First define decision variables: Let x t be the number of tables to be produced; x c be the number of chairs to be produced; x d be the number of desks to be produced. (Always define decision variables properly!)
45 Production PlanningFurniture Manufacturer: modeling the problem as integer program Ø Objective is to maximize profit: max 12x t + 5x c + 15x d Ø Functional Constraints capacity constraints: pine: 5x t + 1x c + 9x d 1500 oak: 2x t + 3x c + 4x d 1000 labor: 3x t + 2x c + 5x d 800 market demand constraints: tables: x t 40 chairs: x c 130 desks: x d 30 Ø Set Constraints x t, x c, x d Z +
46 Solutions to integer programs A solution is an assignment of values to variables. A feasible solution is an assignment of values to variables such that all the constraints are satisfied. The objective function value of a solution is obtained by evaluating the objective function at the given point. An optimal solution (assuming maximization) is one whose objective function value is greater than or equal to that of all other feasible solutions. Integer program is NP complete There are efficient algorithms for finding the optimal solutions of an integer program based on LP relaxation.
47 Next: IP modeling techniques Modeling techniques: Ø Using binary variables Ø Restrictions on number of options Ø Contingent decisions Ø Variables with k possible values Applications: Ø Facility Location Problem Ø Knapsack Problem
48 Example of IP: Facility Location A company is thinking about building new facilities in LA and SF. Relevant data: capital needed 4. warehouse in SF $2M $4M Total capital available for investment: $10M Question: Which facilities should be built to maximize the total profit? expected profit 1. factory in LA $6M $9M 2. factory in SF $3M $5M 3. warehouse in LA $5M $6M
49 Example of IP: Facility Location Define decision variables (i = 1, 2, 3, 4): x i Then the total expected benefit: 9x 1 +5x 2 +6x 3 +4x the total capital needed: 6x 1 +3x 2 +5x 3 +2x 4 Ø Summarizing, the IP model is: max 9x 1 +5x 2 +6x 3 +4x 4 s.t. 6x 1 +3x 2 +5x 3 +2x 4 10 = not x 1, x 2, x 3, x 4 binary ( i.e., x i {0,1} ) if if facility i is built
50 Knapsack problem Any IP, which has only one constraint, is referred to as a knapsack problem. n items to be packed in a knapsack. The knapsack can hold up to W lb of items. Each item has weight w i lb and benefit b i. Goal: Pack the knapsack such that the total benefit is maximized.
51 IP model for Knapsack problem Define decision variables (i = 1,, n): 1 if item i is packed x i = n 0 if not Then the total benefit: b i x i n i= 1 the total weight: w i x i i= 1 Ø Summarizing, the IP model is: n max b i x i s.t. n i=1 i= 1 w x W x i binary (i = 1,, n) i i
52 Connection between the problems Note: The version of the facility location problem is a special case of the knapsack problem. Ø Important modeling skill: Suppose you know how to model Problem A 1,,A p ; You need to solve Problem B; Notice the similarities between Problems A i and B; Build a model for Problem B, using the model for Problem A i as a prototype.
53 The Facility Location Problem: adding new requirements Extra requirement: build at most one of the two warehouses. The corresponding constraint is: x 3 +x 4 1 Extra requirement: build at least one of the two factories. The corresponding constraint is: x 1 +x 2 1
54 Modeling Technique: Restrictions on the number of options Restrictions: At least p and at most q of the options can be chosen. The corresponding constraints are: n i=1 x i p n i=1 x i q
55 Modeling Technique: Contingent Decisions Back to the facility location problem. Requirement: Can t build a warehouse there is a factory in the city. The corresponding constraints are: x 3 x 1 (LA) x 4 x 2 (SF) Requirement: Can t select option 3 unless at least one of options 1 and 2 is selected. unless The constraint: x 3 x 1 + x 2 Requirement: Can t select option 4 unless at least two of options 1, 2 and 3 are selected. The constraint: 2x 4 x 1 + x 2 + x 3
56 Modeling Technique: Variables with k possible values Suppose variable y should take one of the values d 1, d 2,, d k. How to achieve that in the model? Introduce new decision variables. For i=1,,k, 1 if y takes value d i x i = 0 otherwise Then we need the following constraints. y k i= 1 k i= 1 x = d i = i x i 1 ( y can take only one value) ( y should take value d i if x i = 1)
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61 Rounding non integer solu/on values up to the nearest integer value can result in an infeasible solu/on. A feasible solu/on is ensured by rounding down non integer solu>on values but may result in a less than op>mal (sub op/mal) solu/on.
62 Integer Programming Example Graphical SoluFon of Machine Shop Model Maximize Z = $100x 1 + $150x 2 subject to: 8,000x 1 + 4,000x 2 $40,000 15x x ft 2 x 1, x 2 0 and integer Optimal Solution: Z = $1, x 1 = 2.22 presses x 2 = 5.55 lathes Figure 5.1 Feasible Solution Space with Integer Solution Points Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall
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68 Algorithms Use IP to solve the network flow problem Use IP to solve the min cut problem
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