Large-scale optimization and decomposition methods: outline. Column Generation and Cutting Plane methods: a unified view

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1 Large-scale optimization and decomposition methods: outline I Solution approaches for large-scaled problems: I Delayed column generation I Cutting plane methods (delayed constraint generation) 7 I Problems amenable to the above methods: I Cutting stock problem, etc. I Problems reformulated via decomposition methods I Benders decomposition I Dantzig-Wolfe decomposition 7 Not to be confused with the cutting stock problem, which is actually solved with a delayed column generation algorithm IOE 610: LP II, Fall 2013 Large-scale optimization Page 247 Column Generation and Cutting Plane methods: a unified view (P) min c T x (D) max b T p Ax = b p T A apple c T x 0 A 2< m n,wheren >> m. Typicaliteration: 1. Start with I {1,...,n} 2. Solve the Restricted/Relaxed Master Problem: P (ResMP) min j2i c jx j (RelMP) max b T p P j2i A jx j = b p T A j apple c T j j 2 I 0, j 2 I x j Let (x I, p) berespectiveoptimalsolutions. 3. Check if c j p T A j < 0 for some j 2 {1,...,n}. 4. If not, terminate: (x, p), where x j = 0 for j 62 I, are optimal for (P)/(D) 5. Otherwise I add one (or more) indices j (with at least one c j p T A j < 0) to I ; I if desired, remove one or more non-basic indices from I ; I return to step 1. IOE 610: LP II, Fall 2013 Overview Page 248

2 The pricing problem I If n is large, enumerating all columns of A in step 3 is impossible I Sometimes, structure of the columns of A is such that we can easily solve the pricing subproblem: Z? (p) = min (c j p T A j ) j2{1,...,n} without explicitly examining each column I Note: if Z? (p) 0, then the algorithm terminates; o/w solution to the pricing problem gives us an entering variable I The nature of the pricing problem depends on the setting; studied via examples I The Cutting Stock Problem is one of the most famous, and simple, examples IOE 610: LP II, Fall 2013 Overview Page 249 Cutting stock problem Introduction I A paper company has a supply of large rolls of paper, each of width W I Customer orders are for o i rolls of width w i, i =1,...,m (w i apple W ) IOE 610: LP II, Fall 2013 Cutting stock problem Page 250

3 Cutting stock problem Introduction I The demand can be met by slicing a large roll in a certain way, called a pattern. I For example, a large roll of width 100 can be cut into I 4 rolls each of width 25, or I 2 rolls each of width 35, with a waste of 30 I etc. I Goal: meet the demand with the lowest amount of waste IOE 610: LP II, Fall 2013 Cutting stock problem Page 251 Solution approach I: L.V. Kantorovich Formulation 1939 Russian, 1960 English L.V. Kantorovich, Mathematical Methods of Planning and Organising Production Management Science, 6, I K: Set of available rolls. I y k : 1 if roll k is used (cut), 0 otherwise, k 2 K. I x k i : number of times item i is cut on roll k, k 2 K, i =1,...,m. Model: min P Pk2K y k P k2k x i k o i for each item i =1,...,m, i w ixi k apple Wy k for each roll k 2 K, xi k 0, integer k 2 K, i =1,...,m y k binary k 2 K IOE 610: LP II, Fall 2013 Cutting stock problem Page 252

4 Solution approach I AMPL model The AMPL model for the problem: # # CUTTING STOCK USING KANTOROVICH'es MODEL # param roll_width > 0; param roll_number:=100; to be cut set WIDTHS; param orders {WIDTHS} > 0; # width of raw rolls # upper bound on the number of raw rolls # set of widths to be cut # number of each width to be cut var y{1..roll_number} binary; var x{widths,1..roll_number}>=0, integer; minimize Number: sum{k in 1..roll_number} y[k]; subj to Feas_Width{k in 1..roll_number}: sum {i in WIDTHS} i * x[i,k] <= roll_width*y[k]; subj to Demand{i in WIDTHS}: sum{k in 1..roll_number}x[i,k]>=orders[i]; option cplex_options 'mipdisplay=2 mipinterval=10000 timing=1'; IOE 610: LP II, Fall 2013 Cutting stock problem Page 253 Solution approach I Solution times 8 Experiment I: I o i : uniform, between 1 and 100 (rand(100)+1); I w i : uniform, between 1 and 30 (rand(30)+1); I Width of Roll, W = 3000; Rolls Items constr variables CPU (s) Warning: here and beyond, results of 5-year old runs IOE 610: LP II, Fall 2013 Cutting stock problem Page 254

5 Solution approach I Solution times Experiment II: Change width of the roll from 3000 to 150. Rolls Items constr variables CPU (s) out of memory The performance has deteriorated. Why? IOE 610: LP II, Fall 2013 Cutting stock problem Page 255 How good is the LP relaxation? P i w i o i W. Observation: z LP = The optimal solution of LP relaxation will satisfy: I Choose y k as small as possible. Therefore, for all k, w i xi k = Wy k I Choose x k i Objective value: y k = P i w ixi k W k2k k2k i as small as possible. Therefore, for all i = o i = i K k2k x k i w i W x k i = i w i W k2k x k i = i relaxation might be good if W is very large compared to w i, but not if their values are comparable. IOE 610: LP II, Fall 2013 Cutting stock problem Page 256 w i o i W

6 Solution approach I Adi cult instance Another example: W = 273 Quantity Ordered Order Width (inches) I LP relaxation: solved in less than 0.1s. Objective value I IP: found solution with objective value 230, then got stuck enumerating the branch-and-bound tree IOE 610: LP II, Fall 2013 Cutting stock problem Page 257 Solution approach II: Gilmore and Gomory Set covering formulation P.C. Gilmore and R.E. Gomory, A linear programming approach to the cutting-stock problem, Oper. Res., 8 (1961), pp Idea: Each way to cut the roll into items is a pattern I Pattern j characterized by a ij = number of times item i is cut in pattern j, i =1,...,m I For example, a large roll of width 100 can be cut into I 4 rolls each of width w i = 25 (pattern j, a ij = 4, all others 0) I 2 rolls each of width w k = 35 (pattern l, a kl = 2, all others 0) I x j = number of times pattern j is used IOE 610: LP II, Fall 2013 Cutting stock problem Page 258

7 Solution approach II Example Example: An instance: W = 100, m = 3. pattern w i o i Formulation: minimize P 6 j=1 x j x 1 x 2 x 3 x 4 x 5 x 6 RHS 4x 1 + 2x 2 + 2x 3 + 1x 4 + 0x 5 + 0x x 1 + 1x 2 + 0x 3 + 2x 4 + 1x 5 + 0x x 1 + 0x 2 + 1x 3 + 0x 4 + 1x 5 + 2x x j 0, integer, j =1,...,n. IOE 610: LP II, Fall 2013 Cutting stock problem Page 259 Solution approach II Formulation and relaxation Set-covering formulation: min P n j=1 x j P n j=1 a ijx j o i, i =1,...,m, x j 2 Z +, j =1,...,n min Linear relaxation (LP): P n j=1 x j P n j=1 a ijx j s i = o i, i =1,...m, x j 0, j =1,...,n s i 0, i =1,...,m I We will focus on solving the LP via column generation I Each variable corresponds to a di erent feasible pattern I Large-scale problem: n may be huge! I However, m is just the number of di erent orders IOE 610: LP II, Fall 2013 Cutting stock problem Page 260

8 Cutting stock problem Pricing subproblem I Let I {1,...,n}, andlet(x, p) solve the corresponding (ResMP) and (RelMP) I I should include all slacks, and enough patterns to ensure feasibility of (ResMP) I Pricing subproblem: Z? (p) = min (c j p T A j ) = min (1 j2{1,...,n} z is a feasible pattern pt z) I Characterizing feasible patterns: all vectors z 2 Z m + satisfying m w i z i apple W i=1 I So, the pricing problem can be formulated as the following IP: Z? P (p) =min 1 m P i=1 p iz i i w iz i apple W z i 2 Z + 8i IOE 610: LP II, Fall 2013 Cutting stock problem Page 261 Cutting stock problem Solving the pricing subproblem Knapsack problem I The pricing problem for the Cutting Stock Problem is equivalent to the following Knapsack problem: max P m Pi=1 iz i i w iz i apple W z i 2 Z + 8i I Computational results on random instances using the MIP solver from CPLE: n CPU (s) 1, , , I More specialized algorithm can be used to solve the Knapsack problem even more e ciently in practice. IOE 610: LP II, Fall 2013 Cutting stock problem Page 262

9 Cutting stock problem How good is the LP bound? Round Up Conjecture: v IP appledv LP e? Unfortunately, this is not true: I W = 273 w 1 = 18 o 1 = 233 w 2 = 91 o 2 = 310 w 3 = 21 o 3 = 122 w 4 = 136 o 4 = 157 w 5 = 51 o 5 = 120 I v LP (CG) = , v IP = 230. Modified Round Up Conjecture: v IP appledv LP e + 1? I This conjecture has not been answered. IOE 610: LP II, Fall 2013 Cutting stock problem Page 263 Cutting stock problem Getting integer solutions: Round-up heuristic I Let x j be the LP solution obtained from the column generation method possibly fractional. I Let x 0 j = dx j e integer. I P j a i,jx j o i implies P j a i,jx 0 j o i.sox 0 j defined in this way is a feasible integer solution. I How good is this heuristic? W = 273 w 1 = 18 o 1 = 233 w 2 = 91 o 2 = 310 w 3 = 21 o 3 = 122 w 4 = 136 o 4 = 157 w 5 = 51 o 5 = 120 I v LP (CG) = ; Round-Up produces a value of 231 IOE 610: LP II, Fall 2013 Cutting stock problem Page 264

10 Cutting stock problem Round-up heuristic Round-Up Fractional cut cut cut cut cut cut cut Can you give a bound on j x 0 j x j?? j IOE 610: LP II, Fall 2013 Cutting stock problem Page 265 Cutting stock problem Exact solution approach I For an exact solution, can apply B&B algorithm to the cutting stock problem I Every subproblem is an instance of the cutting stock problem with lower/upper bounds on some of the variables I LP relaxation of every subproblem can be solved by this column generation method IOE 610: LP II, Fall 2013 Cutting stock problem Page 266

11 Other applications of this column generation technique Vehicle routing Vehicle Routing with time window or other types of constraint: I A set of m customers. I Each customer must be served within certain time window. I Problem: Find a set of routes to serve all the customers, so that each customer will be visited by a vehicle within the stipulated time window. I Likely objectives could be: I Minimize the total number of vehicles I Minimize the total cost (fixed per-vehicle cost, plus travel costs) Note the set covering nature of this problem. IOE 610: LP II, Fall 2013 Column generation in other problems Page 267 Other applications of this column generation technique Vehicle routing IOE 610: LP II, Fall 2013 Column generation in other problems Page 268

12 Other applications of this column generation technique Vehicle routing formulation I Variables: x j 2 {0, 1} binary, representing whether the jth route is taken by one of the trucks I Each variable/column represents a feasible route, i.e., a subset of customers, and the order in which they are visited, so that each visit occurs within appropriate time windows I Parameters: a ij = 1 if customer i is included in route j; 0 otherwise I Formulation ( minimize number of vehicles version): min P j x j P j a ijx j 1, i =1,...,m x j 2 {0, 1} 8j IOE 610: LP II, Fall 2013 Column generation in other problems Page 269 Other applications of this column generation technique Vehicle routing Column generation and pricing subproblem I LP relaxation is hard to solve due to enormous number of variables use column generation I Solve LP relaxation with a small subset of columns/variables I Obtain dual variables p i for each demand point (can be interpreted as marginal profits of the customers) I Reduced cost c j =1 a ij p i =1 i i:route j visits i I Pricing subproblem: find a route that: I satisfies feasibility constraints (meets the time window constraints); I total profits accrued by serving the demand points on the route is maximized a variant of the Traveling Salesman problem. IOE 610: LP II, Fall 2013 Column generation in other problems Page 270 p i