AM 121: Intro to Optimization Models and Methods Fall 2018

Transcription

1 AM 121: Intro to Optimization Models and Methods Fall 2018 Lecture 11: Integer programming Yiling Chen SEAS Lesson Plan Integer programs Examples: Packing, Covering, TSP problems Modeling approaches fixed costs penalizing a violated constraints alternate constraints conditional constraints n-fold constraints Jensen & Bard:

2 What s an IP? max c T x (IP) s.t. Ax b x 0 and integer max c T x s.t. Ax b x 2{0,1} n (Binary IP) max c T x + h T y (Mixed IP) s.t. Ax + Gy b x 0 y 0 and integer (or binary) Applications Scheduling problems e.g., crew and fleet scheduling (assign each crew and plane to a particular route) Procurement e.g., hospital system determining which suppliers to use for sourcing of medical/surgical equipment Electricity generation e.g., decide when to start-up plants, and what levels to run each plant at Kidney matching e.g., find swaps or chains of patient-donor pairs Facility location e.g., locate points of distribution during a disaster relief operation 2

3 Will IPs be easier or harder to solve? max 5x 1 + 8x 2 s.t. x 1 + x 2 6 5x 1 + 9x 2 45 x 1, x 2 0, integer cont round off nearest feas x x z infeas integer x optimal integer solution x * =(0,5), z * =40 optimal continuous solution x * =(9/4, 15/4), z * =41.25 isoprofit x 1 3

4 Modeling IPs Step 1: Define decision variables Step 2: Define constraints to capture the various elements of the problem introduce additional variables as necessary Step 3: Define the objective function Example: Project Selection Budget b to invest in projects. Project j 2 {1,,n} has cost a j and value c j. Let x j = 1 if project j selected, x j = 0 otherwise max j c j x j s.t. j a j x j b x j 2{0,1} (maximize value) (budget constraint) (no fractional projects) 4

5 Example: Facility Location Locate emergency response centers. Location j 2 {1,,n} has cost c j, and services regions R j R. Regions i 2{1,,m}. Goal: min total cost, but serve each region. a ij =1 if region i can be served by center j, with a ij =0 otherwise Let x j = 1 if center j selected, x j = 0 otherwise min j c j x j (min cost) s.t. j a ij x j 1, i 2R (cover each region) x j 2{0,1} (no fractional centers) Example: Traveling Salesperson A tour visits each of N={1,,n} cities once and returns to start. Cost c ij 0 to travel i to j. Goal: find tour that minimizes total cost. 1 4 A feasible tour in a seven-city TSP Examples: FedEx pick-up, robot placing modules on a circuit board, student visiting colleges. 5

6 Example: Traveling Salesperson How many solutions? Starting at city 1, there are n-1 choices For the next city, n-2 choices, (n-1)! feasible tours n log n n 0.5 n 2 2 n n! x x x x x x Example: Traveling Salesperson Proctor & Gamble ran a contest in The contest required solving a TSP on 33 cities

7 Groetschel (1977) found the optimal tour of 120 cities from what was then West Germany. 13 Applegate, Bixby, Chvátal, and Cook (2001) found the optimal tour of 15,112 cities in unified Germany. 14 7

8 Applegate, Bixby, Chvátal, Cook, and Helsgaun (2004) found the optimal tour of 24,978 cities in Sweden. This was at the time the largest solved TSP problem. 15 Example: Traveling Salesperson x ij = 1 if tour visits j immediately after i, x ij = 0 otherwise Objective: min i j c ij x ij j:j i x ij =1, 8 i leave i once i:i j x ij =1, 8 j enter j once 8

9 Example: Traveling Salesperson x ij = 1 if tour visits j immediately after i, x ij = 0 otherwise Objective: min i j c ij x ij j:j i x ij =1, 8 i leave i once i:i j x ij =1, 8 j enter j once 1 4 Not quite right: this allows infeasible subtours: Example: Traveling Salesperson Subtour elimination constraints Must not make more than S -1 trips between any strict subset S of cities: i S j S x ij S -1 for S N 2 2 A new problem: this requires #constraints exponential in n. 9

10 A Succinct TSP formulation New variables: t j for j 2{1,,n}, denote the position in sequence at which city j is visited (indexed by i or by j) Let t 1 =1 (wlog). Valid tour requires t j t i +1 if x ij =1 except for j= Capture this way: t j t i +1-n(1-x ij ) for all i and j (j 1) è x ij =0: t j t i -6 x ij =1: t j t i +1 2 doesn t work! No way to assign t values 7 Example: Radiation treatment Beams b 2B, pixels (i,j) 2 I, tumor T, critical C x b = power on beam b; d b ij relative intens b min ε T + ε C s.t. D ij +ε T γ L 8(i,j) 2T D ij -ε C γ U 8(i,j) 2C D ij = b 2 B d b ij x b 8(i,j) 2 I x b 0, D ij 0, ε T,ε C 0 10

11 Example: Radiation treatment Beams b 2B, pixels (i,j) 2 I, tumor T, critical C x b = power on beam b; d b ij relative intens b min ε T + ε C s.t. D ij +ε T γ L 8(i,j) 2T D ij -ε C γ U 8(i,j) 2C D ij = b 2 B d b ij x b 8(i,j) 2 I x b 0, D ij 0, ε T,ε C 0 What if each beam used also has cost w? Goal: min ε T + ε C + total cost of beams used Modified Radiation Formulation min ε T + ε C s.t. D ij +ε T γ L D ij -ε C γ U D ij = b B d b ij x 2 b 8(i,j) 2T 8(i,j) 2C 8(i,j) 2 I x b 0, D ij 0, ε T,ε C 0 11

12 Modified Radiation Formulation min ε T + ε C + b 2 B wα b s.t. D ij +ε T γ L 8(i,j) 2T D ij -ε C γ U 8(i,j) 2C D ij = b B d b ij x b (i,j) I x b Mα b (*) x b 0, D ij 0, ε T,ε C 0 α b 2{0,1} indicator variable Pick constant M so (*) is satisfied when α b =1. Need M max possible power. A big M J Modeling approaches 1. Fixed costs 2. Penalizing a Violated Constraint 3. Alternate constraints 4. Conditional constraints 5. n-fold constraints 12

13 1. Modeling Fixed costs c K Cost of K>0 if used, and additional cost c*x per unit of power. x 1. Modeling Fixed costs 0, if x b = 0 cost = K+cx b, if x b > 0 α b = indicator variable for when the fixed cost is incurred, so α b =1 when x b >0 and α b =0 when x=0 Can write cost = Kα b + cx b Constraints: x b Mα b x b 0, and α b 2{0,1} Need M to be larger than max possible x b when x b > 0, then α b =1. 13

14 2. Penalizing a Violated Constraint Penalize the objective if f(x) > b. the average power to critical region should not be more than 40% above γ U Let N C = #pixels in C Require: ij C D ij 1.4 N C γ U 2 Add: ij C D ij - Mα 1.4 N C γ U where α {0,1}. Penalize α in objective. Need M bigger than biggest possible difference Penalizing a Violated Constraint Penalize the objective if f(x) > b. Introduce an indicator variable α 2{0,1}. Write: f(x 1,,x n ) Mα b Define constant M to be larger than max x [f(x) b], so that (x, α = 1) feasible for any x. Penalize α in objective. 14

15 3. Modeling alternate constraints The power to critical region must be 20% or more below γ U or the power to tumor region must be 30% or more above γ L (or both). ij 2 C D ij M 1 α N C γ U ij 2 T D ij + M 2 α N T γ L α 1 + α 2 1 α 1, α 2 2 {0,1} (if violated, alpha1=1) (if violated, alpha2=1) (can t have both 1) 3. Modeling alternate constraints Need at least one of: f 1 (x) b (*) 1 f 2 (x) b 2 (**) Introduce indicator variables α 1, α 2 2{0,1} f 1 (x) M 1 α 1 b 1 (if * violated, need α 1 =1) f 2 (x) M 2 α 2 b 2 (if ** violated, need α 2 =1) α 1 + α 2 1 Big M constants: M 1 max x [f 1 (x)-b 1 ] M 2 max x [f 2 (x)-b 2 ] Can also use only one α by defining α 2 =1-α 1 15

16 4. Modeling conditional constraints If worst-case power to critical region is more than 10% above γ U then the average power to tumor should be at least 30% above γ L. (ε C > 0.1 γ U ) ) ( ij 2 T D ij 1.3 N T γ L ) (A) B) ε C M 1 α 1 0.1γ U ij 2 T D ij +M 2 α N T γ L α 1 +α 2 1 α 1,α 2 2{0,1} If A, need alpha1 = 1 If not B, need alpha2 = 1 precludes A and not B at same time for suitable big-m values of M 1 and M 2 4. Modeling conditional constraints f 1 (x) > b 1 ) f 2 (x) b 2 (A) B) ( A _ B) Because of this, can model as alternate constraints. At least one of: f 1 (x) b 1 f 2 (x) b 2 ) _ A B A B A A B F F T T T F T T T T T F F F F T T T F T 16

17 5. Modeling n-fold constraints Divide the critical region into 10 sections (C 1,,C 10 ). The average power must be less than γ U on 4 or more sections. Let N k = number of pixels in C k ij 2 Ck D ij N γ k U, 8 k 2 {1,,10} Solve as: ij 2 Ck D ij -M k α k N k γ U, 8 k 2 {1,,10} k (1-α k ) 4 α k 2 {0,1} 8 k 2{1,,10} at least four alpha s must be zero 5. Modeling n-fold constraints Require at least r of p constraints: f k (x) b k k 2{1,,p} Introduce indicator variables α k 2{0,1} Solve as: f k (x)-m k α k b k p k=1 (1-α k ) r with M k max x [f k (x)-b k ] at least r alpha s must be zero 17

18 Summary Integer programs are very flexible and can model many real-world problems. The key new component is 0/1 variables (and integer variables more generally). Lots of modeling tricks using indicator variables and big M constants. Next class: interactive modeling exercises! 18