MODELING (Integer Programming Examples)

Transcription

1 MODELING (Integer Programming Eamples)

2 IE 400 Principles of Engineering Management Integer Programming: Set 5

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7 Integer Programming: So far, we have considered problems under the following assumptions: i. Proportionality & Additivity ii. Divisibility iii. Certainty While many problems satisfy these assumptions, there are other problems in which we will need to either rela these assumptions. For eample, consider the following bus scheduling problem:

8 Integer Programming : Bus Scheduling: Periods Hours 0-4 AM 4-8 AM 8 AM 2 PM 2-4 PM 4 8 PM 8 PM-0AM Min. # of buses required Each bus starts to operate at the beginning of a period and operates for 8 consecutive hours and then receives 6 hours off. For eample, a bus operating from 4 AM to 2 PM must be off between 2 PM and 4 AM. Find the minimum # of busses to provide the required service.

9 Integer Programming : Bus Scheduling: Periods Hours 0-4 AM 4-8 AM 8 AM 2 PM 2-4 PM 4 8 PM 8 PM-0AM Min. # of buses required How do you define your decision variables here? a) i : # of busses operating in period i, i=,2,,6 or b) i : # of busses starting to operate in period i, i=,2,,6

10 Integer Programming : The alternative b makes modeling of this problem much easier. It is not possible to keep track of the total number of busses under alternative a. (WHY?) The model is as follows:

11 Integer Programming : The alternative b makes modeling of this problem much easier. It is not possible to keep track of the total number of busses under alternative a. (WHY?) The model is as follows: min s.t ³ ³ ³ ³ ³ ³ i ³ 0, i integer, i =,

12 Integer Programming : Note that each decision variable represents the number of busses. Clearly, it does not make sense to talk about 3.5 busses because busses are not divisible. Hence, we should impose the additional condition that each of,..., 6 can only take integer values. If some or all of the variables are restricted to be integer valued, we call it an integer program (IP). This last eample is hence an IP problem.

13 Work Scheduling: Burger Queen would like to determine the min # of kitchen employees. Each employee works si days a week and takes the seventh day off. Days Mon Tue Wed Thu Fri Sat Sun Min. # of workers How do you formulate this problem? The objective function? The constraints?

14 Work Scheduling: Burger Queen would like to determine the min # of kitchen employees. Each employee works si days a week and takes the seventh day off. Days Mon Tue Wed Thu Fri Sat Sun Min. # of workers i : # of workers who are not working on day i, i =,..., 7 (: Mon, 7: Sun)

15 Work Scheduling: Decision Variables: IP Model:

16 The Assignment Problem : There are n people and n jobs to carry out. Each person is assigned to carry out eactly one job. If person i is assigned to job j, then the cost is c ij. P P 2 J J 2 Find an assignment of n people to n jobs with minimum total cost. P n J n

17 The Assignment Problem : ij = ì í î, 0, person i isassigned to otherwise job j P P 2 J J 2 i =,2,!, n ; j =,2,!, n Such variables are called BINARY VARIABLES and offer great fleibility in modeling. They are ideal for YES/NO decisions. P n J n So how is the problem formulated?

18 The Assignment Problem : The problem is formulated as follows: min n n åå i= j= c ij ij s.t. n å j= ij =, i =,...,n (person i is assigned to eactly one job) n å i= ij =, j =,...,n (job j is assigned to eactly one person) ij Î{0,}, i =,...n; j =,... n This is still an eample of an IP problem since ij can only take one of the two integer values 0 and.

19 The 0- Knapsack Problem You are going on a trip. Your Suitcase has a capacity of b kg s. You have n different items that you can pack into your suitcase,..., n. Item i has a weight of a i and gives you a utility of c i. How should you pack your suitcase to maimize total utility?

20 The 0- Knapsack Problem Decision Variables: IP Model:

21 Workforce Scheduling Monthly demand for a shoe company Month A pair of shoes requires 4 hours of labor and $5 worth of raw materials 50 pairs of shoes in stock Monthly inventory cost is charged for shoes in stock at the end of each month at $30/pair Currently 3 workers are available Salary $500/month Each worker works 60 hours/month Hiring a worker costs $600 Firing a worker costs $ Demand

22 Workforce Scheduling Monthly demand for a shoe company Month A pair of shoes requires 4 hours of labor and $5 worth of raw materials 50 pairs of shoes in stock Monthly inventory cost is charged for shoes in stock at the end of each month at $30/pair Currently 3 workers are available How to meet demand with Salary $500/month minimum total cost? Each worker works 60 hours/month Hiring a worker costs $600 Firing a worker costs $ Demand

23 IP Model For Workforce Scheduling Decision Variables: LP Model: 23

24 The Cutting Stock Problem: A paper company manufactures and sells rolls of paper of fied width in 5 standard lengths: 5, 8, 2, 5 and 7 m. Demand for each size is given below. It produces 25 m length rolls and cuts orders from its stock of this size. What is the min # of rolls to be used to meet the total demand? Length (meters) Demand

25 The Cutting Stock Problem: First, identify the cutting patterns:

26 The Cutting Stock Problem: Then how do you define the decision variable? j : # of rolls cut according to pattern j.

27 The Cutting Stock Problem: Then how do you define the decision variable? j : # of rolls cut according to pattern j. What about the constraints? i.e. How do you write the constraint for the demand of 2 m rolls of paper? We look at the patterns that contain 2 m pieces, and then write the constraint based on j. Length (meters) Demand

28 The Cutting Stock Problem: The cutting patterns:

29 The Cutting Stock Problem: The cutting patterns: ³ Why do we have sign?

30 The Cutting Stock Problem: Then how do you define the decision variable? j : # of rolls cut according to pattern j. What about the constraints? i.e. What about demand for 5 m? Length (meters) Demand

31 The Cutting Stock Problem: The cutting patterns:

32 The Cutting Stock Problem: The cutting patterns: ³

33 The Cutting Stock Problem: The problem is formulated as follows: min s.t. j= ³ ³ ³ j ³ 0, j =,...,, integer. j å ³ ³ j

34 Project Selection A manager should choose from among 0 different projects: p j : profit if we invest in project j, j =,..., 0 c j : cost of project j, j =,..., 0 q: total budget available There are also some additional requirements: i. Projects 3 and 4 cannot be chosen together ii. Eactly 3 projects are to be selected iii. If project 2 is selected, then so is project iv. If project is selected, then project 3 should not be selected v. Either project or project 2 is chosen, but not both vi. Either both projects and 5 are chosen or neither are chosen vii. At least two and at most four projects should be chosen from the set {, 2, 7, 8, 9, 0} viii. Neither project 5 nor 6 can be chosen unless either 3 or 4 is chosen i. At least two of the investments, 2, 3 are chosen, or at most three of the investments 4, 5, 6, 7, 8 are chosen.. If both and 2 are chosen, then at least one of 9 and 0 should also be chosen.

35 Project Selection Under these requirements, the objective is to maimize the Profit. Decision Variables j ì, if project = í î 0, otherwise j =,...,0 j is chosen

36 Project Selection Objective function is: ma 0 å j= p j j

37 Project Selection Objective function is: ma å j= First the budget constraint: 0 p j j

38 Project Selection Objective function is: ma å j= First the budget constraint: 0 å j= 0 c j j p j q j

39 Project Selection Objective function is: ma å j= First the budget constraint: 0 å j= 0 p j c j j q i. Projects 3 and 4 cannot be chosen together: j

40 Project Selection Objective function is: ma å j= First the budget constraint: 0 å j= 0 p j c j j q i. Projects 3 and 4 cannot be chosen together: 3 4 j

41 Project Selection ii Eactly 3 projects are to be chosen: = 3 iii. If project 2 is chosen then so is project : 2

42 Project Selection iv If project is selected, then project 3 should not be selected - 3 v. Either project or project 2 is chosen, but not both 2 =

43 Project Selection vi. Either both projects and 5 are chosen or neither are chosen 0 or 5 5 = - = vii. At least two and at most four projects should be chosen from the set {, 2, 7, 8, 9, 0}: ³

44 Project Selection viii. Neither project 5 nor 6 can be chosen unless either 3 or 4 is chosen: Equivalently, = 0 then = 0 If = Another possibility, 5 6 2( 3 4)

45 Project Selection i. At least two of the investments, 2, 3 are chosen, or at most three of the investments 4, 5, 6, 7, 8 are chosen: It is more difficult to epress this requirement in mathematical terms due to or! 2 3 ³ 2 or ( I) ( II) 3 So we want to ensure that at least one of (I) and (II) is satisfied.

46 Either-Or Constraints The following situation commonly occurs in mathematical programming problems. We are given two constraints of the form, (, 2,... n ) 0 ( I) (,,... ) 0 ( II) f g 2 n

47 Either-Or Constraints The following situation commonly occurs in mathematical programming problems. We are given two constraints of the form, (, 2,... n ) 0 ( I) (,,... ) 0 ( II) f g 2 n We want to make sure that at least one of (I) and (II) is satisfied. These are often called either-or constraints.

48 Either-Or Constraints The following situation commonly occurs in mathematical programming problems. We are given two constraints of the form, (, 2,... n ) 0 ( I) (,,... ) 0 ( II) f g 2 n We want to make sure that at least one of (I) and (II) is satisfied. These are often called either-or constraints. Adding two new constraints as below to the formulation will ensure that at least one of (I) or (II) is satisfied, (, 2,... n ) ( I ) (, 2,... n ) ( - ) ( II ) y Î{ } f My g M y where 0,.

49 Either-Or Constraints M is a number chosen large enough to ensure that f (, 2,..., n ) M and g(, 2,..., n ) M are satisfied for all values of, 2,..., n that satisfy the other constraints in the problem.

50 Either-Or Constraints (, 2,... n ) ( I ) (, 2,... n ) ( - ) ( II ) y Î{ } f My g M y where 0,.

51 Either-Or Constraints (, 2,... n ) ( I ) (, 2,... n ) ( - ) ( II ) y Î{ } f My g M y where 0,. a) If y=0, then (I ) and (II ) becomes: f (, 2,..., n ) 0, and g (, 2,..., n ) M. Hence, this means that (I) must be satisfied, and possibly (II) is satisfied.

52 Either-Or Constraints (, 2,... n ) ( I ) (, 2,... n ) ( - ) ( II ) y Î{ } f My g M y where 0,.

53 Either-Or Constraints (, 2,... n ) ( I ) (, 2,... n ) ( - ) ( II ) y Î{ } f My g M y where 0,. b) If y=, then (I ) and (II ) becomes: f (, 2,..., n ) M, and g (, 2,..., n ) 0. Hence, this means that (II) must be satisfied, and possibly (I) is satisfied.

54 Either-Or Constraints f,,... My I ( ) ( ) 2 g,,... M - y II ( ) ( ) ( ) 2 where y Î 0,. n n { } Therefore whether y=0 or y=, (I ) and (II ) ensure that at least one of (I) and (II) is satisfied.. f,,... 0 I ( ) ( ) 2 g,,... 0 II ( ) ( ) 2 n n

55 Project Selection Now, let us return to our problem: i. At least two of the investments, 2, 3 are chosen, or at most three of the investments 4, 5, 6, 7, 8 are chosen: 2 3 ³ 2 or ( I) ( II) This is an eample of either-or constraint. Hence by defining the f(.) and g(.) functions, (I) and (II) becomes: ( ) g(.) f(.) 0 3

56 Project Selection Now, let us return to our problem: i. At least two of the investments, 2, 3 are chosen, or at most three of the investments 4, 5, 6, 7, 8 are chosen: 2 3 ³ 2 or ( I) ( II) Hence by defining a binary variable y, the formulation becomes: ( -3 4 yî{0,} ) M y M ( - y) M=2 works! We have to find M large enough to ensure that f(.) M and g(.) M for all values of,, n.

57 Project Selection Now, let us return to our problem: i. At least two of the investments, 2, 3 are chosen, or at most three of the investments 4, 5, 6, 7, 8 are chosen: 2 3 ³ 2 or ( I) ( II) Putting M=2 and rearranging the terms, we have: yî{0,} 3 6 ³ 2( - y) y

58 Project Selection. If both and 2 are chosen, then at least one of 9 and 0 should also be chosen. Both projects and 2 are chosen if 2 = 2. At least one of projects 9 and 0 is chosen if 9 0 Hence, we have 2 = 2 Þ 9 0 ³ So, how to epress this as a linear constraint?

59 If-Then Constraints The following situation commonly occurs in mathematical programming problems. We want to ensure that: If a constraint f(,, n )>0 is satisfied, then g(,, n ) 0 must be satisfied; while if a constraint f(,, n )>0 is not satisfied, then g(,, n ) 0 may or may not be satisfied.

60 If-Then Constraints In other words, we want to ensure that f(,, n )>0 implies g(,, n ) 0. Logically, this is equivalent to: f ( g(,, 2 or 2,...,,..., n n ) 0 ) 0 ( I) ( II)

61 If-Then Constraints f ( g(,, 2 or 2,...,,..., n n ) 0 ) 0 ( I) ( II) Now, these are either-or type constraints and we can handle them as before by defining a binary variable, say z: f ( g(,, 2 2,...,,..., ) ) where z Î{0,} n n M ( - Where as before M is a large enough number such that f(.) M and g(.) M holds for all values,, n. Mz z)

62 Project Selection Now, let us return to our problem:. If both and 2 are chosen, then at least one of 9 and 0 should also be chosen. 2 = 2 Þ 9 0 ³ We need to define f(.)>0 and g(.) 0 constraints: equivalently 2 > Þ 9 0 ³ 2 - > 0 Þ - ( 9 0) 0 f(.) g(.)

63 Project Selection. If both and 2 are chosen, then at least one of 9 and 0 should also be chosen. 2 = 2 Þ 9 0 ³ 2 - > 0 Þ - ( 9 0) 0 f(.) g(.) Now converting to either-or, we have: or - ( ) 0 (I) (II)

64 Project Selection To linearize the above constraints, define a binary variable, say z and a large enough value M: (II) 0 ) ( r (I) o M= works and after rearranging we have: {0,} ) M( ) ( Î z z Mz {0,} Î ³ z z z

65 Project Selection. If both and 2 are chosen, then at least one of 9 and 0 should also be chosen z Î{0,} ³ z z

66 Project Selection Finally our integer programming model with linear constraints becomes: ma 0 å j= 0 å p j j c j j j= 3 4 (i) = 2 q (iii) 3 (ii) = - 2 = 0 (iv) (v) (vi)

67 Project Selection Model cont. (vii) þ ý ü ³ (viii) þ ý ü (i) 2 5 y) 2( þ ý ü - - ³ y () þ ý ü ³ z z {0,} y,z {0,},...,, 0 2 Î Î

68 Modeling Fied Costs A set of potential warehouses:,...,m A set of clients :,..., n If warehouse i is used, then there is a fied cost of f i, i =,,m c ij : unit cost of transportation from warehouse i to client j, i =,...,m ; j =,...,n a i : capacity of warehouse i in terms of # of units, i =,,m d j : # of units demanded by client j, j =,,n Note: More than one warehouse can supply the client.

69 Modeling Fied Costs How to formulate the mathematical model in order to minimize total fied costs and transportation costs while meeting the demand and capacity constraints? Decision Variables: IP Model:

70 The Set Covering Problem You wish to decide where to locate hospitals in a city. Suppose that there are 5 different towns in the city. Town Potential Location Travel Time Travel time between each town and the nearest hospital should not be more than 0 minutes. What is the minimum number of hospitals?

71 The Set Covering Problem Decision Variables: IP Model:

72 Location Problems (Cover, Center, Median) Ankara municipality is providing service to 6 regions. The travel times (in minutes) between these regions, say t ij values are: R R2 R3 R4 R5 R6 R R R R R R

73 Location Problems (Cover, Center, Median) a) Cover: The municipality wants to build fire stations in some of these regions. Find the minimum number of fire stations required if the municipality wants to ensure that there is a fire station within 5 minute drive time from each region. (Similar to??) Decision Variables: IP Model:

74 Location Problems (Cover, Center, Median) b) Median: The municipality now wants to build 2 post offices. There is going to be a back and forth trip each day from each post office to each one of the regions assigned to this post office. Formulate a model which will minimize the total trip time. Decision Variables: IP Model:

75 Location Problems (Cover, Center, Median) c) What if additional to part (b), you have the restriction that a post office cannot provide service to more than 4 regions including itself? Add the following constraint to above model:

76 Location Problems (Cover, Center, Median) d) Center: The municipality wants to build 2 fire stations so that the time to reach the farthest region is kept at minimum. Decision Variables: IP Model:

77 Location Problems (Cover, Center, Median) e) Consider the median problem in part (b). Assume that in addition to the back and forth trips between the post offices and the assigned regions, there will be a trip between the two post offices. The municipality wants to locate the offices in such a way that the total trip time is minimized. Decision Variables: IP Model:

78 Traveling Salesman Problem (TSP): A sales person lives in city. He has to visit each of the cities 2,,n eactly once and return home. Let c ij be the travel time from city i to city j, i =,..., n; j =,..., n. What is the order in which she should make her tour to finish as quickly as possible? What should be the decision variables here?

79 Traveling Salesman Problem (TSP): A sales person lives in city. He has to visit each of the cities 2,,n eactly once and return home. Let c ij be the travel time from city i to city j, i =,..., n; j =,..., n. What is the order in which she should make her tour to finish as quickly as possible? What should be the decision variables here? ij ì = í î, if he goes directly from city i to city 0, otherwise i=,2, K, n ; j =,2, K, n j

80 Traveling Salesman Problem (TSP): Since each city has to be visited only once, the formulation is: min st.. n å j= n å i= n n i= j= =, i =,..., n ij =, j =,..., n ij = 0, i =,..., n ii { } ij ij Î 0,, " i, j =,..., n ij åå c

81 Traveling Salesman Problem (TSP): With this formulation, a feasible solution is: = = =, = = But this is a subtour! How can we eliminate subtours? We need to impose the constraints: åå iîs jïs ij ³ " S Í N, S ¹Æ

82 Traveling Salesman Problem (TSP): For eample, for n=5 and S={,2,3} this constraint implies: ³ The difficulty is that there are a total of 2 n such constraints. And that is a lot! For n = 00, 2 n How about the total number of valid tours? That is (n )! For n = 00, (n )! TSP is one of the hardest problems. But several efficient algorithms eist that can find good solutions.

83 Class Eercises (Blending problem revisited with binary variables): A company produces animal feed miture. The mi contains 2 active ingredients and a filler material. kg of feed mi must contain a minimum quantity of each of the following 4 nutrients: Nutrients A B C D Required amount (in grams) in kg of mi The ingredients have the following nutrient values and costs: A B C D Cost/kg Ingredient (gram/kg) $40 Ingredient2 (gram/kg) N/A $60

84 Class Eercises (Blending problem revisited with binary variables): a) What should be the amount of active ingredients and filler material in kg of feed mi in order to satisfy the nutrient requirements at minimum total cost? Decision Variables: Model (LP or IP?)

85 Class Eercises (Blending problem revisited with binary variables): b) Suppose now that we have the following additional limitations i. If we use any of ingredient 2, we incur a fied cost of $5 (in addition to the $60 unit cost). ii. It is enough to only satisfy 3 of the nutrition requirements rather than 4. Decision Variables (Additional to those for Part a) : Model: (LP or IP?)

86 Some Comments on Integer Programming : There are often multiple ways of modeling the same integer program. Solvers for integer programs are etremely sensitive to the formulation (not true for LPs). Not as easy to model. Not as easy to solve.