इस सेक्शन में, असाइनमेंट से जुड़ी एक ऐसी समस्या के बारे में बताया गया है जिसमें सिर्फ़ कुछ लोगों को कर्मचारियों के ग्रुप को टास्क असाइन किए जा सकते हैं. उदाहरण में, बारह कर्मचारियों, नंबर 0 से 11 तक. अनुमति वाले ग्रुप, यहां दिए गए वर्कर जोड़ें के कॉम्बिनेशन हैं.
group1 = [[2, 3], # Subgroups of workers 0 - 3 [1, 3], [1, 2], [0, 1], [0, 2]]group2 = [[6, 7], # Subgroups of workers 4 - 7 [5, 7], [5, 6], [4, 5], [4, 7]]
group3 = [[10, 11], # Subgroups of workers 8 - 11 [9, 11], [9, 10], [8, 10], [8, 11]]
अनुमति वाले ग्रुप में, वर्कर के तीन जोड़े का कोई भी कॉम्बिनेशन हो सकता है.
group1, group2, और group3 में से हर एक का इस्तेमाल करना होगा.
उदाहरण के लिए, [2, 3], [6, 7], और [10, 11] नतीजों को जोड़ने की अनुमति है
ग्रुप [2, 3, 6, 7, 10, 11].
चूंकि तीन सेट में से प्रत्येक में पांच तत्व होते हैं, इसलिए कुल अनुमत संख्या
ग्रुप 5 * 5 * 5 = 125 है.
ध्यान दें कि कर्मचारियों का एक समूह समस्या का समाधान हो सकता है, अगर वह अनुमति वाले ग्रुप में से किसी एक को चुनें. दूसरे शब्दों में, संभव सेट में वे पॉइंट जिनके लिए कोई एक शर्त पूरी की जाती है. यह कन्वेक्स न होने वाली समस्या का उदाहरण है. इसके उलट, एमआईपी का उदाहरण, पहले, एक कॉम्प्लेक्स समस्या थी: किसी पॉइंट को संभव बनाने के लिए, सभी कंस्ट्रेंट संतुष्ट होना चाहिए.
इस तरह की सामान्य समस्याओं के लिए, CP-SAT सॉल्वर आम तौर पर इससे तेज़ होता है एक MIP सॉल्वर. इन सेक्शन में CP-SAT सॉल्वर का इस्तेमाल करके, समस्या का हल बताया गया है और एमआईपी सॉल्वर की मदद से हल करें. साथ ही, दोनों सॉल्वर के समाधान समय की तुलना करें.
CP-SAT समाधान
सबसे पहले, हम CP-SAT सॉल्वर की मदद से, समस्या को हल करने के बारे में बताएंगे.
लाइब्रेरी इंपोर्ट करें
यहां दिया गया कोड ज़रूरी लाइब्रेरी इंपोर्ट करता है.
Python
from ortools.sat.python import cp_model
C++
#include <stdlib.h> #include <cstdint> #include <numeric> #include <vector> #include "absl/strings/str_format.h" #include "absl/types/span.h" #include "ortools/base/logging.h" #include "ortools/sat/cp_model.h" #include "ortools/sat/cp_model.pb.h" #include "ortools/sat/cp_model_solver.h"
Java
import com.google.ortools.Loader; import com.google.ortools.sat.CpModel; import com.google.ortools.sat.CpSolver; import com.google.ortools.sat.CpSolverStatus; import com.google.ortools.sat.IntVar; import com.google.ortools.sat.LinearExpr; import com.google.ortools.sat.LinearExprBuilder; import com.google.ortools.sat.Literal; import java.util.ArrayList; import java.util.List; import java.util.stream.IntStream;
C#
using System; using System.Collections.Generic; using System.Linq; using Google.OrTools.Sat;
डेटा तय करना
यहां दिया गया कोड, प्रोग्राम के लिए डेटा बनाता है.
Python
costs = [
[90, 76, 75, 70, 50, 74],
[35, 85, 55, 65, 48, 101],
[125, 95, 90, 105, 59, 120],
[45, 110, 95, 115, 104, 83],
[60, 105, 80, 75, 59, 62],
[45, 65, 110, 95, 47, 31],
[38, 51, 107, 41, 69, 99],
[47, 85, 57, 71, 92, 77],
[39, 63, 97, 49, 118, 56],
[47, 101, 71, 60, 88, 109],
[17, 39, 103, 64, 61, 92],
[101, 45, 83, 59, 92, 27],
]
num_workers = len(costs)
num_tasks = len(costs[0])C++
const std::vector<std::vector<int>> costs = {{ {{90, 76, 75, 70, 50, 74}}, {{35, 85, 55, 65, 48, 101}}, {{125, 95, 90, 105, 59, 120}}, {{45, 110, 95, 115, 104, 83}}, {{60, 105, 80, 75, 59, 62}}, {{45, 65, 110, 95, 47, 31}}, {{38, 51, 107, 41, 69, 99}}, {{47, 85, 57, 71, 92, 77}}, {{39, 63, 97, 49, 118, 56}}, {{47, 101, 71, 60, 88, 109}}, {{17, 39, 103, 64, 61, 92}}, {{101, 45, 83, 59, 92, 27}}, }}; const int num_workers = static_cast<int>(costs.size()); std::vector<int> all_workers(num_workers); std::iota(all_workers.begin(), all_workers.end(), 0); const int num_tasks = static_cast<int>(costs[0].size()); std::vector<int> all_tasks(num_tasks); std::iota(all_tasks.begin(), all_tasks.end(), 0);
Java
int[][] costs = {
{90, 76, 75, 70, 50, 74},
{35, 85, 55, 65, 48, 101},
{125, 95, 90, 105, 59, 120},
{45, 110, 95, 115, 104, 83},
{60, 105, 80, 75, 59, 62},
{45, 65, 110, 95, 47, 31},
{38, 51, 107, 41, 69, 99},
{47, 85, 57, 71, 92, 77},
{39, 63, 97, 49, 118, 56},
{47, 101, 71, 60, 88, 109},
{17, 39, 103, 64, 61, 92},
{101, 45, 83, 59, 92, 27},
};
final int numWorkers = costs.length;
final int numTasks = costs[0].length;
final int[] allWorkers = IntStream.range(0, numWorkers).toArray();
final int[] allTasks = IntStream.range(0, numTasks).toArray();C#
int[,] costs = {
{ 90, 76, 75, 70, 50, 74 }, { 35, 85, 55, 65, 48, 101 }, { 125, 95, 90, 105, 59, 120 },
{ 45, 110, 95, 115, 104, 83 }, { 60, 105, 80, 75, 59, 62 }, { 45, 65, 110, 95, 47, 31 },
{ 38, 51, 107, 41, 69, 99 }, { 47, 85, 57, 71, 92, 77 }, { 39, 63, 97, 49, 118, 56 },
{ 47, 101, 71, 60, 88, 109 }, { 17, 39, 103, 64, 61, 92 }, { 101, 45, 83, 59, 92, 27 },
};
int numWorkers = costs.GetLength(0);
int numTasks = costs.GetLength(1);
int[] allWorkers = Enumerable.Range(0, numWorkers).ToArray();
int[] allTasks = Enumerable.Range(0, numTasks).ToArray();अनुमति वाले ग्रुप बनाएं
CP-SAT सॉल्वर के लिए, कर्मचारियों के अनुमति वाले ग्रुप तय करने के लिए, बाइनरी बनाएं
अरे जो बताता है कि कौनसे वर्कर किसी ग्रुप से जुड़े हैं. उदाहरण के लिए, group1 के लिए
(कर्मचारियों 0 - 3) का बाइनरी वेक्टर [0, 0, 1, 1]
कर्मचारी 2 और 3.
ये सरणियां, कर्मचारियों के अनुमति वाले ग्रुप के बारे में बताती हैं.
Python
group1 = [
[0, 0, 1, 1], # Workers 2, 3
[0, 1, 0, 1], # Workers 1, 3
[0, 1, 1, 0], # Workers 1, 2
[1, 1, 0, 0], # Workers 0, 1
[1, 0, 1, 0], # Workers 0, 2
]
group2 = [
[0, 0, 1, 1], # Workers 6, 7
[0, 1, 0, 1], # Workers 5, 7
[0, 1, 1, 0], # Workers 5, 6
[1, 1, 0, 0], # Workers 4, 5
[1, 0, 0, 1], # Workers 4, 7
]
group3 = [
[0, 0, 1, 1], # Workers 10, 11
[0, 1, 0, 1], # Workers 9, 11
[0, 1, 1, 0], # Workers 9, 10
[1, 0, 1, 0], # Workers 8, 10
[1, 0, 0, 1], # Workers 8, 11
]C++
const std::vector<std::vector<int64_t>> group1 = {{ {{0, 0, 1, 1}}, // Workers 2, 3 {{0, 1, 0, 1}}, // Workers 1, 3 {{0, 1, 1, 0}}, // Workers 1, 2 {{1, 1, 0, 0}}, // Workers 0, 1 {{1, 0, 1, 0}}, // Workers 0, 2 }}; const std::vector<std::vector<int64_t>> group2 = {{ {{0, 0, 1, 1}}, // Workers 6, 7 {{0, 1, 0, 1}}, // Workers 5, 7 {{0, 1, 1, 0}}, // Workers 5, 6 {{1, 1, 0, 0}}, // Workers 4, 5 {{1, 0, 0, 1}}, // Workers 4, 7 }}; const std::vector<std::vector<int64_t>> group3 = {{ {{0, 0, 1, 1}}, // Workers 10, 11 {{0, 1, 0, 1}}, // Workers 9, 11 {{0, 1, 1, 0}}, // Workers 9, 10 {{1, 0, 1, 0}}, // Workers 8, 10 {{1, 0, 0, 1}}, // Workers 8, 11 }};
Java
int[][] group1 = {
{0, 0, 1, 1}, // Workers 2, 3
{0, 1, 0, 1}, // Workers 1, 3
{0, 1, 1, 0}, // Workers 1, 2
{1, 1, 0, 0}, // Workers 0, 1
{1, 0, 1, 0}, // Workers 0, 2
};
int[][] group2 = {
{0, 0, 1, 1}, // Workers 6, 7
{0, 1, 0, 1}, // Workers 5, 7
{0, 1, 1, 0}, // Workers 5, 6
{1, 1, 0, 0}, // Workers 4, 5
{1, 0, 0, 1}, // Workers 4, 7
};
int[][] group3 = {
{0, 0, 1, 1}, // Workers 10, 11
{0, 1, 0, 1}, // Workers 9, 11
{0, 1, 1, 0}, // Workers 9, 10
{1, 0, 1, 0}, // Workers 8, 10
{1, 0, 0, 1}, // Workers 8, 11
};C#
long[,] group1 = {
{ 0, 0, 1, 1 }, // Workers 2, 3
{ 0, 1, 0, 1 }, // Workers 1, 3
{ 0, 1, 1, 0 }, // Workers 1, 2
{ 1, 1, 0, 0 }, // Workers 0, 1
{ 1, 0, 1, 0 }, // Workers 0, 2
};
long[,] group2 = {
{ 0, 0, 1, 1 }, // Workers 6, 7
{ 0, 1, 0, 1 }, // Workers 5, 7
{ 0, 1, 1, 0 }, // Workers 5, 6
{ 1, 1, 0, 0 }, // Workers 4, 5
{ 1, 0, 0, 1 }, // Workers 4, 7
};
long[,] group3 = {
{ 0, 0, 1, 1 }, // Workers 10, 11
{ 0, 1, 0, 1 }, // Workers 9, 11
{ 0, 1, 1, 0 }, // Workers 9, 10
{ 1, 0, 1, 0 }, // Workers 8, 10
{ 1, 0, 0, 1 }, // Workers 8, 11
};CP-SAT के लिए, इन वेक्टर के सभी 125 कॉम्बिनेशन बनाना ज़रूरी नहीं है
लूप में चलाना है. CP-SAT सॉल्वर एक तरीका उपलब्ध कराता है, AllowedAssignments,
इसकी मदद से, अनुमति वाले ग्रुप के लिए अलग-अलग सीमाएं तय की जा सकती हैं
कर्मचारियों के तीन सेट (0 - 3, 4 - 7, और 8 - 11) में से हर एक के लिए.
यह इस तरह से काम करता है:
Python
# Create variables for each worker, indicating whether they work on some task. work = {} for worker in range(num_workers): work[worker] = model.new_bool_var(f"work[{worker}]") for worker in range(num_workers): for task in range(num_tasks): model.add(work[worker] == sum(x[worker, task] for task in range(num_tasks))) # Define the allowed groups of worders model.add_allowed_assignments([work[0], work[1], work[2], work[3]], group1) model.add_allowed_assignments([work[4], work[5], work[6], work[7]], group2) model.add_allowed_assignments([work[8], work[9], work[10], work[11]], group3)
C++
// Create variables for each worker, indicating whether they work on some // task. std::vector<IntVar> work(num_workers); for (int worker : all_workers) { work[worker] = IntVar( cp_model.NewBoolVar().WithName(absl::StrFormat("work[%d]", worker))); } for (int worker : all_workers) { LinearExpr task_sum; for (int task : all_tasks) { task_sum += x[worker][task]; } cp_model.AddEquality(work[worker], task_sum); } // Define the allowed groups of worders auto table1 = cp_model.AddAllowedAssignments({work[0], work[1], work[2], work[3]}); for (const auto& t : group1) { table1.AddTuple(t); } auto table2 = cp_model.AddAllowedAssignments({work[4], work[5], work[6], work[7]}); for (const auto& t : group2) { table2.AddTuple(t); } auto table3 = cp_model.AddAllowedAssignments({work[8], work[9], work[10], work[11]}); for (const auto& t : group3) { table3.AddTuple(t); }
Java
// Create variables for each worker, indicating whether they work on some task. IntVar[] work = new IntVar[numWorkers]; for (int worker : allWorkers) { work[worker] = model.newBoolVar("work[" + worker + "]"); } for (int worker : allWorkers) { LinearExprBuilder expr = LinearExpr.newBuilder(); for (int task : allTasks) { expr.add(x[worker][task]); } model.addEquality(work[worker], expr); } // Define the allowed groups of worders model.addAllowedAssignments(new IntVar[] {work[0], work[1], work[2], work[3]}) .addTuples(group1); model.addAllowedAssignments(new IntVar[] {work[4], work[5], work[6], work[7]}) .addTuples(group2); model.addAllowedAssignments(new IntVar[] {work[8], work[9], work[10], work[11]}) .addTuples(group3);
C#
// Create variables for each worker, indicating whether they work on some task. BoolVar[] work = new BoolVar[numWorkers]; foreach (int worker in allWorkers) { work[worker] = model.NewBoolVar($"work[{worker}]"); } foreach (int worker in allWorkers) { List<ILiteral> tasks = new List<ILiteral>(); foreach (int task in allTasks) { tasks.Add(x[worker, task]); } model.Add(work[worker] == LinearExpr.Sum(tasks)); } // Define the allowed groups of worders model.AddAllowedAssignments(new IntVar[] { work[0], work[1], work[2], work[3] }).AddTuples(group1); model.AddAllowedAssignments(new IntVar[] { work[4], work[5], work[6], work[7] }).AddTuples(group2); model.AddAllowedAssignments(new IntVar[] { work[8], work[9], work[10], work[11] }).AddTuples(group3);
वैरिएबल work[i], 0-1 वैरिएबल होते हैं, जो काम की स्थिति दिखाते हैं या
हर काम के लिए ज़िम्मेदार है. इसका मतलब है कि अगर वर्कर i को किसी टास्क में असाइन किया गया है, तो work[i] 1 के बराबर है, और
0 नहीं. लाइन
solver.Add(solver.AllowedAssignments([work[0], work[1], work[2], work[3]], group1)) अभी तक किसी भी व्यक्ति ने चेक इन नहीं किया है
यह कंस्ट्रेंट तय करता है कि वर्कर 0 - 3 का वर्क स्टेटस, इनमें से किसी एक से मैच होना चाहिए
group1 में पैटर्न. अगले पेज पर, कोड के बारे में पूरी जानकारी देखी जा सकती है
सेक्शन में जाएं.
मॉडल बनाना
यह कोड मॉडल बनाता है.
Python
model = cp_model.CpModel()
C++
CpModelBuilder cp_model;
Java
CpModel model = new CpModel();
C#
CpModel model = new CpModel();
वैरिएबल बनाना
यह कोड, समस्या के लिए वैरिएबल का कलेक्शन बनाता है.
Python
x = {} for worker in range(num_workers): for task in range(num_tasks): x[worker, task] = model.new_bool_var(f"x[{worker},{task}]")
C++
// x[i][j] is an array of Boolean variables. x[i][j] is true // if worker i is assigned to task j. std::vector<std::vector<BoolVar>> x(num_workers, std::vector<BoolVar>(num_tasks)); for (int worker : all_workers) { for (int task : all_tasks) { x[worker][task] = cp_model.NewBoolVar().WithName( absl::StrFormat("x[%d,%d]", worker, task)); } }
Java
Literal[][] x = new Literal[numWorkers][numTasks]; for (int worker : allWorkers) { for (int task : allTasks) { x[worker][task] = model.newBoolVar("x[" + worker + "," + task + "]"); } }
C#
BoolVar[,] x = new BoolVar[numWorkers, numTasks];
// Variables in a 1-dim array.
foreach (int worker in allWorkers)
{
foreach (int task in allTasks)
{
x[worker, task] = model.NewBoolVar($"x[{worker},{task}]");
}
}कंस्ट्रेंट जोड़ें
यह कोड, प्रोग्राम के लिए सीमाएं बनाता है.
Python
# Each worker is assigned to at most one task. for worker in range(num_workers): model.add_at_most_one(x[worker, task] for task in range(num_tasks)) # Each task is assigned to exactly one worker. for task in range(num_tasks): model.add_exactly_one(x[worker, task] for worker in range(num_workers))
C++
// Each worker is assigned to at most one task. for (int worker : all_workers) { cp_model.AddAtMostOne(x[worker]); } // Each task is assigned to exactly one worker. for (int task : all_tasks) { std::vector<BoolVar> tasks; for (int worker : all_workers) { tasks.push_back(x[worker][task]); } cp_model.AddExactlyOne(tasks); }
Java
// Each worker is assigned to at most one task. for (int worker : allWorkers) { List<Literal> tasks = new ArrayList<>(); for (int task : allTasks) { tasks.add(x[worker][task]); } model.addAtMostOne(tasks); } // Each task is assigned to exactly one worker. for (int task : allTasks) { List<Literal> workers = new ArrayList<>(); for (int worker : allWorkers) { workers.add(x[worker][task]); } model.addExactlyOne(workers); }
C#
// Each worker is assigned to at most one task. foreach (int worker in allWorkers) { List<ILiteral> tasks = new List<ILiteral>(); foreach (int task in allTasks) { tasks.Add(x[worker, task]); } model.AddAtMostOne(tasks); } // Each task is assigned to exactly one worker. foreach (int task in allTasks) { List<ILiteral> workers = new List<ILiteral>(); foreach (int worker in allWorkers) { workers.Add(x[worker, task]); } model.AddExactlyOne(workers); }
मकसद तय करना
यह कोड, मकसद फ़ंक्शन बनाता है.
Python
objective_terms = [] for worker in range(num_workers): for task in range(num_tasks): objective_terms.append(costs[worker][task] * x[worker, task]) model.minimize(sum(objective_terms))
C++
LinearExpr total_cost; for (int worker : all_workers) { for (int task : all_tasks) { total_cost += x[worker][task] * costs[worker][task]; } } cp_model.Minimize(total_cost);
Java
LinearExprBuilder obj = LinearExpr.newBuilder(); for (int worker : allWorkers) { for (int task : allTasks) { obj.addTerm(x[worker][task], costs[worker][task]); } } model.minimize(obj);
C#
LinearExprBuilder obj = LinearExpr.NewBuilder();
foreach (int worker in allWorkers)
{
foreach (int task in allTasks)
{
obj.AddTerm(x[worker, task], costs[worker, task]);
}
}
model.Minimize(obj);सॉल्वर आज़माएं
यहां दिया गया कोड, सॉल्वर को शुरू करता है और नतीजे दिखाता है.
Python
solver = cp_model.CpSolver() status = solver.solve(model)
C++
const CpSolverResponse response = Solve(cp_model.Build());
Java
CpSolver solver = new CpSolver(); CpSolverStatus status = solver.solve(model);
C#
CpSolver solver = new CpSolver(); CpSolverStatus status = solver.Solve(model); Console.WriteLine($"Solve status: {status}");
नतीजे दिखाएं
अब हम समाधान को प्रिंट कर सकते हैं.
Python
if status == cp_model.OPTIMAL or status == cp_model.FEASIBLE: print(f"Total cost = {solver.objective_value}\n") for worker in range(num_workers): for task in range(num_tasks): if solver.boolean_value(x[worker, task]): print( f"Worker {worker} assigned to task {task}." + f" Cost = {costs[worker][task]}" ) else: print("No solution found.")
C++
if (response.status() == CpSolverStatus::INFEASIBLE) { LOG(FATAL) << "No solution found."; } LOG(INFO) << "Total cost: " << response.objective_value(); LOG(INFO); for (int worker : all_workers) { for (int task : all_tasks) { if (SolutionBooleanValue(response, x[worker][task])) { LOG(INFO) << "Worker " << worker << " assigned to task " << task << ". Cost: " << costs[worker][task]; } } }
Java
// Check that the problem has a feasible solution. if (status == CpSolverStatus.OPTIMAL || status == CpSolverStatus.FEASIBLE) { System.out.println("Total cost: " + solver.objectiveValue() + "\n"); for (int worker : allWorkers) { for (int task : allTasks) { if (solver.booleanValue(x[worker][task])) { System.out.println("Worker " + worker + " assigned to task " + task + ". Cost: " + costs[worker][task]); } } } } else { System.err.println("No solution found."); }
C#
// Check that the problem has a feasible solution. if (status == CpSolverStatus.Optimal || status == CpSolverStatus.Feasible) { Console.WriteLine($"Total cost: {solver.ObjectiveValue}\n"); foreach (int worker in allWorkers) { foreach (int task in allTasks) { if (solver.Value(x[worker, task]) > 0.5) { Console.WriteLine($"Worker {worker} assigned to task {task}. " + $"Cost: {costs[worker, task]}"); } } } } else { Console.WriteLine("No solution found."); }
ये रहे प्रोग्राम का आउटपुट.
Minimum cost = 239 Worker 0 assigned to task 4 Cost = 50 Worker 1 assigned to task 2 Cost = 55 Worker 5 assigned to task 5 Cost = 31 Worker 6 assigned to task 3 Cost = 41 Worker 10 assigned to task 0 Cost = 17 Worker 11 assigned to task 1 Cost = 45 Time = 0.0113 seconds
पूरा प्रोग्राम
यह रहा पूरा प्रोग्राम.
Python
"""Solves an assignment problem for given group of workers.""" from ortools.sat.python import cp_model def main() -> None: # Data costs = [ [90, 76, 75, 70, 50, 74], [35, 85, 55, 65, 48, 101], [125, 95, 90, 105, 59, 120], [45, 110, 95, 115, 104, 83], [60, 105, 80, 75, 59, 62], [45, 65, 110, 95, 47, 31], [38, 51, 107, 41, 69, 99], [47, 85, 57, 71, 92, 77], [39, 63, 97, 49, 118, 56], [47, 101, 71, 60, 88, 109], [17, 39, 103, 64, 61, 92], [101, 45, 83, 59, 92, 27], ] num_workers = len(costs) num_tasks = len(costs[0]) # Allowed groups of workers: group1 = [ [0, 0, 1, 1], # Workers 2, 3 [0, 1, 0, 1], # Workers 1, 3 [0, 1, 1, 0], # Workers 1, 2 [1, 1, 0, 0], # Workers 0, 1 [1, 0, 1, 0], # Workers 0, 2 ] group2 = [ [0, 0, 1, 1], # Workers 6, 7 [0, 1, 0, 1], # Workers 5, 7 [0, 1, 1, 0], # Workers 5, 6 [1, 1, 0, 0], # Workers 4, 5 [1, 0, 0, 1], # Workers 4, 7 ] group3 = [ [0, 0, 1, 1], # Workers 10, 11 [0, 1, 0, 1], # Workers 9, 11 [0, 1, 1, 0], # Workers 9, 10 [1, 0, 1, 0], # Workers 8, 10 [1, 0, 0, 1], # Workers 8, 11 ] # Model model = cp_model.CpModel() # Variables x = {} for worker in range(num_workers): for task in range(num_tasks): x[worker, task] = model.new_bool_var(f"x[{worker},{task}]") # Constraints # Each worker is assigned to at most one task. for worker in range(num_workers): model.add_at_most_one(x[worker, task] for task in range(num_tasks)) # Each task is assigned to exactly one worker. for task in range(num_tasks): model.add_exactly_one(x[worker, task] for worker in range(num_workers)) # Create variables for each worker, indicating whether they work on some task. work = {} for worker in range(num_workers): work[worker] = model.new_bool_var(f"work[{worker}]") for worker in range(num_workers): for task in range(num_tasks): model.add(work[worker] == sum(x[worker, task] for task in range(num_tasks))) # Define the allowed groups of worders model.add_allowed_assignments([work[0], work[1], work[2], work[3]], group1) model.add_allowed_assignments([work[4], work[5], work[6], work[7]], group2) model.add_allowed_assignments([work[8], work[9], work[10], work[11]], group3) # Objective objective_terms = [] for worker in range(num_workers): for task in range(num_tasks): objective_terms.append(costs[worker][task] * x[worker, task]) model.minimize(sum(objective_terms)) # Solve solver = cp_model.CpSolver() status = solver.solve(model) # Print solution. if status == cp_model.OPTIMAL or status == cp_model.FEASIBLE: print(f"Total cost = {solver.objective_value}\n") for worker in range(num_workers): for task in range(num_tasks): if solver.boolean_value(x[worker, task]): print( f"Worker {worker} assigned to task {task}." + f" Cost = {costs[worker][task]}" ) else: print("No solution found.") if __name__ == "__main__": main()
C++
// Solve assignment problem for given group of workers. #include <stdlib.h> #include <cstdint> #include <numeric> #include <vector> #include "absl/strings/str_format.h" #include "absl/types/span.h" #include "ortools/base/logging.h" #include "ortools/sat/cp_model.h" #include "ortools/sat/cp_model.pb.h" #include "ortools/sat/cp_model_solver.h" namespace operations_research { namespace sat { void AssignmentGroups() { // Data const std::vector<std::vector<int>> costs = {{ {{90, 76, 75, 70, 50, 74}}, {{35, 85, 55, 65, 48, 101}}, {{125, 95, 90, 105, 59, 120}}, {{45, 110, 95, 115, 104, 83}}, {{60, 105, 80, 75, 59, 62}}, {{45, 65, 110, 95, 47, 31}}, {{38, 51, 107, 41, 69, 99}}, {{47, 85, 57, 71, 92, 77}}, {{39, 63, 97, 49, 118, 56}}, {{47, 101, 71, 60, 88, 109}}, {{17, 39, 103, 64, 61, 92}}, {{101, 45, 83, 59, 92, 27}}, }}; const int num_workers = static_cast<int>(costs.size()); std::vector<int> all_workers(num_workers); std::iota(all_workers.begin(), all_workers.end(), 0); const int num_tasks = static_cast<int>(costs[0].size()); std::vector<int> all_tasks(num_tasks); std::iota(all_tasks.begin(), all_tasks.end(), 0); // Allowed groups of workers: const std::vector<std::vector<int64_t>> group1 = {{ {{0, 0, 1, 1}}, // Workers 2, 3 {{0, 1, 0, 1}}, // Workers 1, 3 {{0, 1, 1, 0}}, // Workers 1, 2 {{1, 1, 0, 0}}, // Workers 0, 1 {{1, 0, 1, 0}}, // Workers 0, 2 }}; const std::vector<std::vector<int64_t>> group2 = {{ {{0, 0, 1, 1}}, // Workers 6, 7 {{0, 1, 0, 1}}, // Workers 5, 7 {{0, 1, 1, 0}}, // Workers 5, 6 {{1, 1, 0, 0}}, // Workers 4, 5 {{1, 0, 0, 1}}, // Workers 4, 7 }}; const std::vector<std::vector<int64_t>> group3 = {{ {{0, 0, 1, 1}}, // Workers 10, 11 {{0, 1, 0, 1}}, // Workers 9, 11 {{0, 1, 1, 0}}, // Workers 9, 10 {{1, 0, 1, 0}}, // Workers 8, 10 {{1, 0, 0, 1}}, // Workers 8, 11 }}; // Model CpModelBuilder cp_model; // Variables // x[i][j] is an array of Boolean variables. x[i][j] is true // if worker i is assigned to task j. std::vector<std::vector<BoolVar>> x(num_workers, std::vector<BoolVar>(num_tasks)); for (int worker : all_workers) { for (int task : all_tasks) { x[worker][task] = cp_model.NewBoolVar().WithName( absl::StrFormat("x[%d,%d]", worker, task)); } } // Constraints // Each worker is assigned to at most one task. for (int worker : all_workers) { cp_model.AddAtMostOne(x[worker]); } // Each task is assigned to exactly one worker. for (int task : all_tasks) { std::vector<BoolVar> tasks; for (int worker : all_workers) { tasks.push_back(x[worker][task]); } cp_model.AddExactlyOne(tasks); } // Create variables for each worker, indicating whether they work on some // task. std::vector<IntVar> work(num_workers); for (int worker : all_workers) { work[worker] = IntVar( cp_model.NewBoolVar().WithName(absl::StrFormat("work[%d]", worker))); } for (int worker : all_workers) { LinearExpr task_sum; for (int task : all_tasks) { task_sum += x[worker][task]; } cp_model.AddEquality(work[worker], task_sum); } // Define the allowed groups of worders auto table1 = cp_model.AddAllowedAssignments({work[0], work[1], work[2], work[3]}); for (const auto& t : group1) { table1.AddTuple(t); } auto table2 = cp_model.AddAllowedAssignments({work[4], work[5], work[6], work[7]}); for (const auto& t : group2) { table2.AddTuple(t); } auto table3 = cp_model.AddAllowedAssignments({work[8], work[9], work[10], work[11]}); for (const auto& t : group3) { table3.AddTuple(t); } // Objective LinearExpr total_cost; for (int worker : all_workers) { for (int task : all_tasks) { total_cost += x[worker][task] * costs[worker][task]; } } cp_model.Minimize(total_cost); // Solve const CpSolverResponse response = Solve(cp_model.Build()); // Print solution. if (response.status() == CpSolverStatus::INFEASIBLE) { LOG(FATAL) << "No solution found."; } LOG(INFO) << "Total cost: " << response.objective_value(); LOG(INFO); for (int worker : all_workers) { for (int task : all_tasks) { if (SolutionBooleanValue(response, x[worker][task])) { LOG(INFO) << "Worker " << worker << " assigned to task " << task << ". Cost: " << costs[worker][task]; } } } } } // namespace sat } // namespace operations_research int main(int argc, char** argv) { operations_research::sat::AssignmentGroups(); return EXIT_SUCCESS; }
Java
// CP-SAT example that solves an assignment problem. package com.google.ortools.sat.samples; import com.google.ortools.Loader; import com.google.ortools.sat.CpModel; import com.google.ortools.sat.CpSolver; import com.google.ortools.sat.CpSolverStatus; import com.google.ortools.sat.IntVar; import com.google.ortools.sat.LinearExpr; import com.google.ortools.sat.LinearExprBuilder; import com.google.ortools.sat.Literal; import java.util.ArrayList; import java.util.List; import java.util.stream.IntStream; /** Assignment problem. */ public class AssignmentGroupsSat { public static void main(String[] args) { Loader.loadNativeLibraries(); // Data int[][] costs = { {90, 76, 75, 70, 50, 74}, {35, 85, 55, 65, 48, 101}, {125, 95, 90, 105, 59, 120}, {45, 110, 95, 115, 104, 83}, {60, 105, 80, 75, 59, 62}, {45, 65, 110, 95, 47, 31}, {38, 51, 107, 41, 69, 99}, {47, 85, 57, 71, 92, 77}, {39, 63, 97, 49, 118, 56}, {47, 101, 71, 60, 88, 109}, {17, 39, 103, 64, 61, 92}, {101, 45, 83, 59, 92, 27}, }; final int numWorkers = costs.length; final int numTasks = costs[0].length; final int[] allWorkers = IntStream.range(0, numWorkers).toArray(); final int[] allTasks = IntStream.range(0, numTasks).toArray(); // Allowed groups of workers: int[][] group1 = { {0, 0, 1, 1}, // Workers 2, 3 {0, 1, 0, 1}, // Workers 1, 3 {0, 1, 1, 0}, // Workers 1, 2 {1, 1, 0, 0}, // Workers 0, 1 {1, 0, 1, 0}, // Workers 0, 2 }; int[][] group2 = { {0, 0, 1, 1}, // Workers 6, 7 {0, 1, 0, 1}, // Workers 5, 7 {0, 1, 1, 0}, // Workers 5, 6 {1, 1, 0, 0}, // Workers 4, 5 {1, 0, 0, 1}, // Workers 4, 7 }; int[][] group3 = { {0, 0, 1, 1}, // Workers 10, 11 {0, 1, 0, 1}, // Workers 9, 11 {0, 1, 1, 0}, // Workers 9, 10 {1, 0, 1, 0}, // Workers 8, 10 {1, 0, 0, 1}, // Workers 8, 11 }; // Model CpModel model = new CpModel(); // Variables Literal[][] x = new Literal[numWorkers][numTasks]; for (int worker : allWorkers) { for (int task : allTasks) { x[worker][task] = model.newBoolVar("x[" + worker + "," + task + "]"); } } // Constraints // Each worker is assigned to at most one task. for (int worker : allWorkers) { List<Literal> tasks = new ArrayList<>(); for (int task : allTasks) { tasks.add(x[worker][task]); } model.addAtMostOne(tasks); } // Each task is assigned to exactly one worker. for (int task : allTasks) { List<Literal> workers = new ArrayList<>(); for (int worker : allWorkers) { workers.add(x[worker][task]); } model.addExactlyOne(workers); } // Create variables for each worker, indicating whether they work on some task. IntVar[] work = new IntVar[numWorkers]; for (int worker : allWorkers) { work[worker] = model.newBoolVar("work[" + worker + "]"); } for (int worker : allWorkers) { LinearExprBuilder expr = LinearExpr.newBuilder(); for (int task : allTasks) { expr.add(x[worker][task]); } model.addEquality(work[worker], expr); } // Define the allowed groups of worders model.addAllowedAssignments(new IntVar[] {work[0], work[1], work[2], work[3]}) .addTuples(group1); model.addAllowedAssignments(new IntVar[] {work[4], work[5], work[6], work[7]}) .addTuples(group2); model.addAllowedAssignments(new IntVar[] {work[8], work[9], work[10], work[11]}) .addTuples(group3); // Objective LinearExprBuilder obj = LinearExpr.newBuilder(); for (int worker : allWorkers) { for (int task : allTasks) { obj.addTerm(x[worker][task], costs[worker][task]); } } model.minimize(obj); // Solve CpSolver solver = new CpSolver(); CpSolverStatus status = solver.solve(model); // Print solution. // Check that the problem has a feasible solution. if (status == CpSolverStatus.OPTIMAL || status == CpSolverStatus.FEASIBLE) { System.out.println("Total cost: " + solver.objectiveValue() + "\n"); for (int worker : allWorkers) { for (int task : allTasks) { if (solver.booleanValue(x[worker][task])) { System.out.println("Worker " + worker + " assigned to task " + task + ". Cost: " + costs[worker][task]); } } } } else { System.err.println("No solution found."); } } private AssignmentGroupsSat() {} }
C#
using System; using System.Collections.Generic; using System.Linq; using Google.OrTools.Sat; public class AssignmentGroupsSat { public static void Main(String[] args) { // Data. int[,] costs = { { 90, 76, 75, 70, 50, 74 }, { 35, 85, 55, 65, 48, 101 }, { 125, 95, 90, 105, 59, 120 }, { 45, 110, 95, 115, 104, 83 }, { 60, 105, 80, 75, 59, 62 }, { 45, 65, 110, 95, 47, 31 }, { 38, 51, 107, 41, 69, 99 }, { 47, 85, 57, 71, 92, 77 }, { 39, 63, 97, 49, 118, 56 }, { 47, 101, 71, 60, 88, 109 }, { 17, 39, 103, 64, 61, 92 }, { 101, 45, 83, 59, 92, 27 }, }; int numWorkers = costs.GetLength(0); int numTasks = costs.GetLength(1); int[] allWorkers = Enumerable.Range(0, numWorkers).ToArray(); int[] allTasks = Enumerable.Range(0, numTasks).ToArray(); // Allowed groups of workers: long[,] group1 = { { 0, 0, 1, 1 }, // Workers 2, 3 { 0, 1, 0, 1 }, // Workers 1, 3 { 0, 1, 1, 0 }, // Workers 1, 2 { 1, 1, 0, 0 }, // Workers 0, 1 { 1, 0, 1, 0 }, // Workers 0, 2 }; long[,] group2 = { { 0, 0, 1, 1 }, // Workers 6, 7 { 0, 1, 0, 1 }, // Workers 5, 7 { 0, 1, 1, 0 }, // Workers 5, 6 { 1, 1, 0, 0 }, // Workers 4, 5 { 1, 0, 0, 1 }, // Workers 4, 7 }; long[,] group3 = { { 0, 0, 1, 1 }, // Workers 10, 11 { 0, 1, 0, 1 }, // Workers 9, 11 { 0, 1, 1, 0 }, // Workers 9, 10 { 1, 0, 1, 0 }, // Workers 8, 10 { 1, 0, 0, 1 }, // Workers 8, 11 }; // Model. CpModel model = new CpModel(); // Variables. BoolVar[,] x = new BoolVar[numWorkers, numTasks]; // Variables in a 1-dim array. foreach (int worker in allWorkers) { foreach (int task in allTasks) { x[worker, task] = model.NewBoolVar($"x[{worker},{task}]"); } } // Constraints // Each worker is assigned to at most one task. foreach (int worker in allWorkers) { List<ILiteral> tasks = new List<ILiteral>(); foreach (int task in allTasks) { tasks.Add(x[worker, task]); } model.AddAtMostOne(tasks); } // Each task is assigned to exactly one worker. foreach (int task in allTasks) { List<ILiteral> workers = new List<ILiteral>(); foreach (int worker in allWorkers) { workers.Add(x[worker, task]); } model.AddExactlyOne(workers); } // Create variables for each worker, indicating whether they work on some task. BoolVar[] work = new BoolVar[numWorkers]; foreach (int worker in allWorkers) { work[worker] = model.NewBoolVar($"work[{worker}]"); } foreach (int worker in allWorkers) { List<ILiteral> tasks = new List<ILiteral>(); foreach (int task in allTasks) { tasks.Add(x[worker, task]); } model.Add(work[worker] == LinearExpr.Sum(tasks)); } // Define the allowed groups of worders model.AddAllowedAssignments(new IntVar[] { work[0], work[1], work[2], work[3] }).AddTuples(group1); model.AddAllowedAssignments(new IntVar[] { work[4], work[5], work[6], work[7] }).AddTuples(group2); model.AddAllowedAssignments(new IntVar[] { work[8], work[9], work[10], work[11] }).AddTuples(group3); // Objective LinearExprBuilder obj = LinearExpr.NewBuilder(); foreach (int worker in allWorkers) { foreach (int task in allTasks) { obj.AddTerm(x[worker, task], costs[worker, task]); } } model.Minimize(obj); // Solve CpSolver solver = new CpSolver(); CpSolverStatus status = solver.Solve(model); Console.WriteLine($"Solve status: {status}"); // Print solution. // Check that the problem has a feasible solution. if (status == CpSolverStatus.Optimal || status == CpSolverStatus.Feasible) { Console.WriteLine($"Total cost: {solver.ObjectiveValue}\n"); foreach (int worker in allWorkers) { foreach (int task in allTasks) { if (solver.Value(x[worker, task]) > 0.5) { Console.WriteLine($"Worker {worker} assigned to task {task}. " + $"Cost: {costs[worker, task]}"); } } } } else { Console.WriteLine("No solution found."); } Console.WriteLine("Statistics"); Console.WriteLine($" - conflicts : {solver.NumConflicts()}"); Console.WriteLine($" - branches : {solver.NumBranches()}"); Console.WriteLine($" - wall time : {solver.WallTime()}s"); } }
एमआईपी सलूशन
इसके बाद, हम MIP सॉल्वर का इस्तेमाल करके, समस्या के समाधान के बारे में बताते हैं.
लाइब्रेरी इंपोर्ट करें
यहां दिया गया कोड ज़रूरी लाइब्रेरी इंपोर्ट करता है.
Python
from ortools.linear_solver import pywraplp
C++
#include <cstdint> #include <memory> #include <numeric> #include <utility> #include <vector> #include "absl/strings/str_format.h" #include "ortools/base/logging.h" #include "ortools/linear_solver/linear_solver.h"
Java
import com.google.ortools.Loader; import com.google.ortools.linearsolver.MPConstraint; import com.google.ortools.linearsolver.MPObjective; import com.google.ortools.linearsolver.MPSolver; import com.google.ortools.linearsolver.MPVariable; import java.util.stream.IntStream;
C#
using System; using System.Collections.Generic; using System.Linq; using Google.OrTools.LinearSolver;
डेटा तय करना
यहां दिया गया कोड, प्रोग्राम के लिए डेटा बनाता है.
Python
costs = [
[90, 76, 75, 70, 50, 74],
[35, 85, 55, 65, 48, 101],
[125, 95, 90, 105, 59, 120],
[45, 110, 95, 115, 104, 83],
[60, 105, 80, 75, 59, 62],
[45, 65, 110, 95, 47, 31],
[38, 51, 107, 41, 69, 99],
[47, 85, 57, 71, 92, 77],
[39, 63, 97, 49, 118, 56],
[47, 101, 71, 60, 88, 109],
[17, 39, 103, 64, 61, 92],
[101, 45, 83, 59, 92, 27],
]
num_workers = len(costs)
num_tasks = len(costs[0])C++
const std::vector<std::vector<int64_t>> costs = {{ {{90, 76, 75, 70, 50, 74}}, {{35, 85, 55, 65, 48, 101}}, {{125, 95, 90, 105, 59, 120}}, {{45, 110, 95, 115, 104, 83}}, {{60, 105, 80, 75, 59, 62}}, {{45, 65, 110, 95, 47, 31}}, {{38, 51, 107, 41, 69, 99}}, {{47, 85, 57, 71, 92, 77}}, {{39, 63, 97, 49, 118, 56}}, {{47, 101, 71, 60, 88, 109}}, {{17, 39, 103, 64, 61, 92}}, {{101, 45, 83, 59, 92, 27}}, }}; const int num_workers = costs.size(); std::vector<int> all_workers(num_workers); std::iota(all_workers.begin(), all_workers.end(), 0); const int num_tasks = costs[0].size(); std::vector<int> all_tasks(num_tasks); std::iota(all_tasks.begin(), all_tasks.end(), 0);
Java
double[][] costs = {
{90, 76, 75, 70, 50, 74},
{35, 85, 55, 65, 48, 101},
{125, 95, 90, 105, 59, 120},
{45, 110, 95, 115, 104, 83},
{60, 105, 80, 75, 59, 62},
{45, 65, 110, 95, 47, 31},
{38, 51, 107, 41, 69, 99},
{47, 85, 57, 71, 92, 77},
{39, 63, 97, 49, 118, 56},
{47, 101, 71, 60, 88, 109},
{17, 39, 103, 64, 61, 92},
{101, 45, 83, 59, 92, 27},
};
int numWorkers = costs.length;
int numTasks = costs[0].length;
final int[] allWorkers = IntStream.range(0, numWorkers).toArray();
final int[] allTasks = IntStream.range(0, numTasks).toArray();C#
int[,] costs = {
{ 90, 76, 75, 70, 50, 74 }, { 35, 85, 55, 65, 48, 101 }, { 125, 95, 90, 105, 59, 120 },
{ 45, 110, 95, 115, 104, 83 }, { 60, 105, 80, 75, 59, 62 }, { 45, 65, 110, 95, 47, 31 },
{ 38, 51, 107, 41, 69, 99 }, { 47, 85, 57, 71, 92, 77 }, { 39, 63, 97, 49, 118, 56 },
{ 47, 101, 71, 60, 88, 109 }, { 17, 39, 103, 64, 61, 92 }, { 101, 45, 83, 59, 92, 27 },
};
int numWorkers = costs.GetLength(0);
int numTasks = costs.GetLength(1);
int[] allWorkers = Enumerable.Range(0, numWorkers).ToArray();
int[] allTasks = Enumerable.Range(0, numTasks).ToArray();अनुमति वाले ग्रुप बनाएं
यह कोड, तीन सेट को लूप में रखकर, ऐसे ग्रुप बनाता है जिन्हें अनुमति मिली है में सबग्रुप के नाम को शामिल करें.
Python
group1 = [ # Subgroups of workers 0 - 3
[2, 3],
[1, 3],
[1, 2],
[0, 1],
[0, 2],
]
group2 = [ # Subgroups of workers 4 - 7
[6, 7],
[5, 7],
[5, 6],
[4, 5],
[4, 7],
]
group3 = [ # Subgroups of workers 8 - 11
[10, 11],
[9, 11],
[9, 10],
[8, 10],
[8, 11],
]C++
using WorkerIndex = int; using Binome = std::pair<WorkerIndex, WorkerIndex>; using AllowedBinomes = std::vector<Binome>; const AllowedBinomes group1 = {{ // group of worker 0-3 {2, 3}, {1, 3}, {1, 2}, {0, 1}, {0, 2}, }}; const AllowedBinomes group2 = {{ // group of worker 4-7 {6, 7}, {5, 7}, {5, 6}, {4, 5}, {4, 7}, }}; const AllowedBinomes group3 = {{ // group of worker 8-11 {10, 11}, {9, 11}, {9, 10}, {8, 10}, {8, 11}, }};
Java
int[][] group1 = {
// group of worker 0-3
{2, 3},
{1, 3},
{1, 2},
{0, 1},
{0, 2},
};
int[][] group2 = {
// group of worker 4-7
{6, 7},
{5, 7},
{5, 6},
{4, 5},
{4, 7},
};
int[][] group3 = {
// group of worker 8-11
{10, 11},
{9, 11},
{9, 10},
{8, 10},
{8, 11},
};C#
int[,] group1 = {
// group of worker 0-3
{ 2, 3 }, { 1, 3 }, { 1, 2 }, { 0, 1 }, { 0, 2 },
};
int[,] group2 = {
// group of worker 4-7
{ 6, 7 }, { 5, 7 }, { 5, 6 }, { 4, 5 }, { 4, 7 },
};
int[,] group3 = {
// group of worker 8-11
{ 10, 11 }, { 9, 11 }, { 9, 10 }, { 8, 10 }, { 8, 11 },
};सॉल्वर के बारे में बताओ
इस कोड से सॉल्वर बनता है.
Python
# Create the mip solver with the SCIP backend. solver = pywraplp.Solver.CreateSolver("SCIP") if not solver: return
C++
// Create the mip solver with the SCIP backend. std::unique_ptr<MPSolver> solver(MPSolver::CreateSolver("SCIP")); if (!solver) { LOG(WARNING) << "SCIP solver unavailable."; return; }
Java
// Create the linear solver with the SCIP backend. MPSolver solver = MPSolver.createSolver("SCIP"); if (solver == null) { System.out.println("Could not create solver SCIP"); return; }
C#
Solver solver = Solver.CreateSolver("SCIP"); if (solver is null) { return; }
वैरिएबल बनाना
यह कोड, समस्या के लिए वैरिएबल का कलेक्शन बनाता है.
Python
# x[worker, task] is an array of 0-1 variables, which will be 1 # if the worker is assigned to the task. x = {} for worker in range(num_workers): for task in range(num_tasks): x[worker, task] = solver.BoolVar(f"x[{worker},{task}]")
C++
// x[i][j] is an array of 0-1 variables, which will be 1 // if worker i is assigned to task j. std::vector<std::vector<const MPVariable*>> x( num_workers, std::vector<const MPVariable*>(num_tasks)); for (int worker : all_workers) { for (int task : all_tasks) { x[worker][task] = solver->MakeBoolVar(absl::StrFormat("x[%d,%d]", worker, task)); } }
Java
// x[i][j] is an array of 0-1 variables, which will be 1 // if worker i is assigned to task j. MPVariable[][] x = new MPVariable[numWorkers][numTasks]; for (int worker : allWorkers) { for (int task : allTasks) { x[worker][task] = solver.makeBoolVar("x[" + worker + "," + task + "]"); } }
C#
// x[i, j] is an array of 0-1 variables, which will be 1 // if worker i is assigned to task j. Variable[,] x = new Variable[numWorkers, numTasks]; foreach (int worker in allWorkers) { foreach (int task in allTasks) { x[worker, task] = solver.MakeBoolVar($"x[{worker},{task}]"); } }
कंस्ट्रेंट जोड़ें
यह कोड, प्रोग्राम के लिए सीमाएं बनाता है.
Python
# The total size of the tasks each worker takes on is at most total_size_max. for worker in range(num_workers): solver.Add(solver.Sum([x[worker, task] for task in range(num_tasks)]) <= 1) # Each task is assigned to exactly one worker. for task in range(num_tasks): solver.Add(solver.Sum([x[worker, task] for worker in range(num_workers)]) == 1)
C++
// Each worker is assigned to at most one task. for (int worker : all_workers) { LinearExpr worker_sum; for (int task : all_tasks) { worker_sum += x[worker][task]; } solver->MakeRowConstraint(worker_sum <= 1.0); } // Each task is assigned to exactly one worker. for (int task : all_tasks) { LinearExpr task_sum; for (int worker : all_workers) { task_sum += x[worker][task]; } solver->MakeRowConstraint(task_sum == 1.0); }
Java
// Each worker is assigned to at most one task. for (int worker : allWorkers) { MPConstraint constraint = solver.makeConstraint(0, 1, ""); for (int task : allTasks) { constraint.setCoefficient(x[worker][task], 1); } } // Each task is assigned to exactly one worker. for (int task : allTasks) { MPConstraint constraint = solver.makeConstraint(1, 1, ""); for (int worker : allWorkers) { constraint.setCoefficient(x[worker][task], 1); } }
C#
// Each worker is assigned to at most one task. foreach (int worker in allWorkers) { Constraint constraint = solver.MakeConstraint(0, 1, ""); foreach (int task in allTasks) { constraint.SetCoefficient(x[worker, task], 1); } } // Each task is assigned to exactly one worker. foreach (int task in allTasks) { Constraint constraint = solver.MakeConstraint(1, 1, ""); foreach (int worker in allWorkers) { constraint.SetCoefficient(x[worker, task], 1); } }
मकसद तय करना
यह कोड, मकसद फ़ंक्शन बनाता है.
Python
objective_terms = [] for worker in range(num_workers): for task in range(num_tasks): objective_terms.append(costs[worker][task] * x[worker, task]) solver.Minimize(solver.Sum(objective_terms))
C++
MPObjective* const objective = solver->MutableObjective(); for (int worker : all_workers) { for (int task : all_tasks) { objective->SetCoefficient(x[worker][task], costs[worker][task]); } } objective->SetMinimization();
Java
MPObjective objective = solver.objective(); for (int worker : allWorkers) { for (int task : allTasks) { objective.setCoefficient(x[worker][task], costs[worker][task]); } } objective.setMinimization();
C#
Objective objective = solver.Objective();
foreach (int worker in allWorkers)
{
foreach (int task in allTasks)
{
objective.SetCoefficient(x[worker, task], costs[worker, task]);
}
}
objective.SetMinimization();सॉल्वर आज़माएं
यहां दिया गया कोड, सॉल्वर को शुरू करता है और नतीजे दिखाता है.
Python
print(f"Solving with {solver.SolverVersion()}")
status = solver.Solve()C++
const MPSolver::ResultStatus result_status = solver->Solve();
Java
MPSolver.ResultStatus resultStatus = solver.solve();
C#
Solver.ResultStatus resultStatus = solver.Solve();
नतीजे दिखाएं
अब हम समाधान को प्रिंट कर सकते हैं.
Python
if status == pywraplp.Solver.OPTIMAL or status == pywraplp.Solver.FEASIBLE: print(f"Total cost = {solver.Objective().Value()}\n") for worker in range(num_workers): for task in range(num_tasks): if x[worker, task].solution_value() > 0.5: print( f"Worker {worker} assigned to task {task}." + f" Cost: {costs[worker][task]}" ) else: print("No solution found.")
C++
// Check that the problem has a feasible solution. if (result_status != MPSolver::OPTIMAL && result_status != MPSolver::FEASIBLE) { LOG(FATAL) << "No solution found."; } LOG(INFO) << "Total cost = " << objective->Value() << "\n\n"; for (int worker : all_workers) { for (int task : all_tasks) { // Test if x[i][j] is 0 or 1 (with tolerance for floating point // arithmetic). if (x[worker][task]->solution_value() > 0.5) { LOG(INFO) << "Worker " << worker << " assigned to task " << task << ". Cost: " << costs[worker][task]; } } }
Java
// Check that the problem has a feasible solution. if (resultStatus == MPSolver.ResultStatus.OPTIMAL || resultStatus == MPSolver.ResultStatus.FEASIBLE) { System.out.println("Total cost: " + objective.value() + "\n"); for (int worker : allWorkers) { for (int task : allTasks) { // Test if x[i][j] is 0 or 1 (with tolerance for floating point // arithmetic). if (x[worker][task].solutionValue() > 0.5) { System.out.println("Worker " + worker + " assigned to task " + task + ". Cost: " + costs[worker][task]); } } } } else { System.err.println("No solution found."); }
C#
// Check that the problem has a feasible solution. if (resultStatus == Solver.ResultStatus.OPTIMAL || resultStatus == Solver.ResultStatus.FEASIBLE) { Console.WriteLine($"Total cost: {solver.Objective().Value()}\n"); foreach (int worker in allWorkers) { foreach (int task in allTasks) { // Test if x[i, j] is 0 or 1 (with tolerance for floating point // arithmetic). if (x[worker, task].SolutionValue() > 0.5) { Console.WriteLine($"Worker {worker} assigned to task {task}. Cost: {costs[worker, task]}"); } } } } else { Console.WriteLine("No solution found."); }
यहां प्रोग्राम का आउटपुट दिया गया है:
Minimum cost = 239.0 Worker 0 assigned to task 4 Cost = 50 Worker 1 assigned to task 2 Cost = 55 Worker 5 assigned to task 5 Cost = 31 Worker 6 assigned to task 3 Cost = 41 Worker 10 assigned to task 0 Cost = 17 Worker 11 assigned to task 1 Cost = 45 Time = 0.3281 seconds
पूरा प्रोग्राम
यह रहा पूरा प्रोग्राम.
Python
"""Solve assignment problem for given group of workers.""" from ortools.linear_solver import pywraplp def main(): # Data costs = [ [90, 76, 75, 70, 50, 74], [35, 85, 55, 65, 48, 101], [125, 95, 90, 105, 59, 120], [45, 110, 95, 115, 104, 83], [60, 105, 80, 75, 59, 62], [45, 65, 110, 95, 47, 31], [38, 51, 107, 41, 69, 99], [47, 85, 57, 71, 92, 77], [39, 63, 97, 49, 118, 56], [47, 101, 71, 60, 88, 109], [17, 39, 103, 64, 61, 92], [101, 45, 83, 59, 92, 27], ] num_workers = len(costs) num_tasks = len(costs[0]) # Allowed groups of workers: group1 = [ # Subgroups of workers 0 - 3 [2, 3], [1, 3], [1, 2], [0, 1], [0, 2], ] group2 = [ # Subgroups of workers 4 - 7 [6, 7], [5, 7], [5, 6], [4, 5], [4, 7], ] group3 = [ # Subgroups of workers 8 - 11 [10, 11], [9, 11], [9, 10], [8, 10], [8, 11], ] # Solver. # Create the mip solver with the SCIP backend. solver = pywraplp.Solver.CreateSolver("SCIP") if not solver: return # Variables # x[worker, task] is an array of 0-1 variables, which will be 1 # if the worker is assigned to the task. x = {} for worker in range(num_workers): for task in range(num_tasks): x[worker, task] = solver.BoolVar(f"x[{worker},{task}]") # Constraints # The total size of the tasks each worker takes on is at most total_size_max. for worker in range(num_workers): solver.Add(solver.Sum([x[worker, task] for task in range(num_tasks)]) <= 1) # Each task is assigned to exactly one worker. for task in range(num_tasks): solver.Add(solver.Sum([x[worker, task] for worker in range(num_workers)]) == 1) # Create variables for each worker, indicating whether they work on some task. work = {} for worker in range(num_workers): work[worker] = solver.BoolVar(f"work[{worker}]") for worker in range(num_workers): solver.Add( work[worker] == solver.Sum([x[worker, task] for task in range(num_tasks)]) ) # Group1 constraint_g1 = solver.Constraint(1, 1) for index, _ in enumerate(group1): # a*b can be transformed into 0 <= a + b - 2*p <= 1 with p in [0,1] # p is True if a AND b, False otherwise constraint = solver.Constraint(0, 1) constraint.SetCoefficient(work[group1[index][0]], 1) constraint.SetCoefficient(work[group1[index][1]], 1) p = solver.BoolVar(f"g1_p{index}") constraint.SetCoefficient(p, -2) constraint_g1.SetCoefficient(p, 1) # Group2 constraint_g2 = solver.Constraint(1, 1) for index, _ in enumerate(group2): # a*b can be transformed into 0 <= a + b - 2*p <= 1 with p in [0,1] # p is True if a AND b, False otherwise constraint = solver.Constraint(0, 1) constraint.SetCoefficient(work[group2[index][0]], 1) constraint.SetCoefficient(work[group2[index][1]], 1) p = solver.BoolVar(f"g2_p{index}") constraint.SetCoefficient(p, -2) constraint_g2.SetCoefficient(p, 1) # Group3 constraint_g3 = solver.Constraint(1, 1) for index, _ in enumerate(group3): # a*b can be transformed into 0 <= a + b - 2*p <= 1 with p in [0,1] # p is True if a AND b, False otherwise constraint = solver.Constraint(0, 1) constraint.SetCoefficient(work[group3[index][0]], 1) constraint.SetCoefficient(work[group3[index][1]], 1) p = solver.BoolVar(f"g3_p{index}") constraint.SetCoefficient(p, -2) constraint_g3.SetCoefficient(p, 1) # Objective objective_terms = [] for worker in range(num_workers): for task in range(num_tasks): objective_terms.append(costs[worker][task] * x[worker, task]) solver.Minimize(solver.Sum(objective_terms)) # Solve print(f"Solving with {solver.SolverVersion()}") status = solver.Solve() # Print solution. if status == pywraplp.Solver.OPTIMAL or status == pywraplp.Solver.FEASIBLE: print(f"Total cost = {solver.Objective().Value()}\n") for worker in range(num_workers): for task in range(num_tasks): if x[worker, task].solution_value() > 0.5: print( f"Worker {worker} assigned to task {task}." + f" Cost: {costs[worker][task]}" ) else: print("No solution found.") if __name__ == "__main__": main()
C++
// Solve a simple assignment problem. #include <cstdint> #include <memory> #include <numeric> #include <utility> #include <vector> #include "absl/strings/str_format.h" #include "ortools/base/logging.h" #include "ortools/linear_solver/linear_solver.h" namespace operations_research { void AssignmentTeamsMip() { // Data const std::vector<std::vector<int64_t>> costs = {{ {{90, 76, 75, 70, 50, 74}}, {{35, 85, 55, 65, 48, 101}}, {{125, 95, 90, 105, 59, 120}}, {{45, 110, 95, 115, 104, 83}}, {{60, 105, 80, 75, 59, 62}}, {{45, 65, 110, 95, 47, 31}}, {{38, 51, 107, 41, 69, 99}}, {{47, 85, 57, 71, 92, 77}}, {{39, 63, 97, 49, 118, 56}}, {{47, 101, 71, 60, 88, 109}}, {{17, 39, 103, 64, 61, 92}}, {{101, 45, 83, 59, 92, 27}}, }}; const int num_workers = costs.size(); std::vector<int> all_workers(num_workers); std::iota(all_workers.begin(), all_workers.end(), 0); const int num_tasks = costs[0].size(); std::vector<int> all_tasks(num_tasks); std::iota(all_tasks.begin(), all_tasks.end(), 0); // Allowed groups of workers: using WorkerIndex = int; using Binome = std::pair<WorkerIndex, WorkerIndex>; using AllowedBinomes = std::vector<Binome>; const AllowedBinomes group1 = {{ // group of worker 0-3 {2, 3}, {1, 3}, {1, 2}, {0, 1}, {0, 2}, }}; const AllowedBinomes group2 = {{ // group of worker 4-7 {6, 7}, {5, 7}, {5, 6}, {4, 5}, {4, 7}, }}; const AllowedBinomes group3 = {{ // group of worker 8-11 {10, 11}, {9, 11}, {9, 10}, {8, 10}, {8, 11}, }}; // Solver // Create the mip solver with the SCIP backend. std::unique_ptr<MPSolver> solver(MPSolver::CreateSolver("SCIP")); if (!solver) { LOG(WARNING) << "SCIP solver unavailable."; return; } // Variables // x[i][j] is an array of 0-1 variables, which will be 1 // if worker i is assigned to task j. std::vector<std::vector<const MPVariable*>> x( num_workers, std::vector<const MPVariable*>(num_tasks)); for (int worker : all_workers) { for (int task : all_tasks) { x[worker][task] = solver->MakeBoolVar(absl::StrFormat("x[%d,%d]", worker, task)); } } // Constraints // Each worker is assigned to at most one task. for (int worker : all_workers) { LinearExpr worker_sum; for (int task : all_tasks) { worker_sum += x[worker][task]; } solver->MakeRowConstraint(worker_sum <= 1.0); } // Each task is assigned to exactly one worker. for (int task : all_tasks) { LinearExpr task_sum; for (int worker : all_workers) { task_sum += x[worker][task]; } solver->MakeRowConstraint(task_sum == 1.0); } // Create variables for each worker, indicating whether they work on some // task. std::vector<const MPVariable*> work(num_workers); for (int worker : all_workers) { work[worker] = solver->MakeBoolVar(absl::StrFormat("work[%d]", worker)); } for (int worker : all_workers) { LinearExpr task_sum; for (int task : all_tasks) { task_sum += x[worker][task]; } solver->MakeRowConstraint(work[worker] == task_sum); } // Group1 { MPConstraint* g1 = solver->MakeRowConstraint(1, 1); for (int i = 0; i < group1.size(); ++i) { // a*b can be transformed into 0 <= a + b - 2*p <= 1 with p in [0,1] // p is true if a AND b, false otherwise MPConstraint* tmp = solver->MakeRowConstraint(0, 1); tmp->SetCoefficient(work[group1[i].first], 1); tmp->SetCoefficient(work[group1[i].second], 1); MPVariable* p = solver->MakeBoolVar(absl::StrFormat("g1_p%d", i)); tmp->SetCoefficient(p, -2); g1->SetCoefficient(p, 1); } } // Group2 { MPConstraint* g2 = solver->MakeRowConstraint(1, 1); for (int i = 0; i < group2.size(); ++i) { // a*b can be transformed into 0 <= a + b - 2*p <= 1 with p in [0,1] // p is true if a AND b, false otherwise MPConstraint* tmp = solver->MakeRowConstraint(0, 1); tmp->SetCoefficient(work[group2[i].first], 1); tmp->SetCoefficient(work[group2[i].second], 1); MPVariable* p = solver->MakeBoolVar(absl::StrFormat("g2_p%d", i)); tmp->SetCoefficient(p, -2); g2->SetCoefficient(p, 1); } } // Group3 { MPConstraint* g3 = solver->MakeRowConstraint(1, 1); for (int i = 0; i < group3.size(); ++i) { // a*b can be transformed into 0 <= a + b - 2*p <= 1 with p in [0,1] // p is true if a AND b, false otherwise MPConstraint* tmp = solver->MakeRowConstraint(0, 1); tmp->SetCoefficient(work[group3[i].first], 1); tmp->SetCoefficient(work[group3[i].second], 1); MPVariable* p = solver->MakeBoolVar(absl::StrFormat("g3_p%d", i)); tmp->SetCoefficient(p, -2); g3->SetCoefficient(p, 1); } } // Objective. MPObjective* const objective = solver->MutableObjective(); for (int worker : all_workers) { for (int task : all_tasks) { objective->SetCoefficient(x[worker][task], costs[worker][task]); } } objective->SetMinimization(); // Solve const MPSolver::ResultStatus result_status = solver->Solve(); // Print solution. // Check that the problem has a feasible solution. if (result_status != MPSolver::OPTIMAL && result_status != MPSolver::FEASIBLE) { LOG(FATAL) << "No solution found."; } LOG(INFO) << "Total cost = " << objective->Value() << "\n\n"; for (int worker : all_workers) { for (int task : all_tasks) { // Test if x[i][j] is 0 or 1 (with tolerance for floating point // arithmetic). if (x[worker][task]->solution_value() > 0.5) { LOG(INFO) << "Worker " << worker << " assigned to task " << task << ". Cost: " << costs[worker][task]; } } } } } // namespace operations_research int main(int argc, char** argv) { operations_research::AssignmentTeamsMip(); return EXIT_SUCCESS; }
Java
package com.google.ortools.linearsolver.samples; import com.google.ortools.Loader; import com.google.ortools.linearsolver.MPConstraint; import com.google.ortools.linearsolver.MPObjective; import com.google.ortools.linearsolver.MPSolver; import com.google.ortools.linearsolver.MPVariable; import java.util.stream.IntStream; /** MIP example that solves an assignment problem. */ public class AssignmentGroupsMip { public static void main(String[] args) { Loader.loadNativeLibraries(); // Data double[][] costs = { {90, 76, 75, 70, 50, 74}, {35, 85, 55, 65, 48, 101}, {125, 95, 90, 105, 59, 120}, {45, 110, 95, 115, 104, 83}, {60, 105, 80, 75, 59, 62}, {45, 65, 110, 95, 47, 31}, {38, 51, 107, 41, 69, 99}, {47, 85, 57, 71, 92, 77}, {39, 63, 97, 49, 118, 56}, {47, 101, 71, 60, 88, 109}, {17, 39, 103, 64, 61, 92}, {101, 45, 83, 59, 92, 27}, }; int numWorkers = costs.length; int numTasks = costs[0].length; final int[] allWorkers = IntStream.range(0, numWorkers).toArray(); final int[] allTasks = IntStream.range(0, numTasks).toArray(); // Allowed groups of workers: int[][] group1 = { // group of worker 0-3 {2, 3}, {1, 3}, {1, 2}, {0, 1}, {0, 2}, }; int[][] group2 = { // group of worker 4-7 {6, 7}, {5, 7}, {5, 6}, {4, 5}, {4, 7}, }; int[][] group3 = { // group of worker 8-11 {10, 11}, {9, 11}, {9, 10}, {8, 10}, {8, 11}, }; // Solver // Create the linear solver with the SCIP backend. MPSolver solver = MPSolver.createSolver("SCIP"); if (solver == null) { System.out.println("Could not create solver SCIP"); return; } // Variables // x[i][j] is an array of 0-1 variables, which will be 1 // if worker i is assigned to task j. MPVariable[][] x = new MPVariable[numWorkers][numTasks]; for (int worker : allWorkers) { for (int task : allTasks) { x[worker][task] = solver.makeBoolVar("x[" + worker + "," + task + "]"); } } // Constraints // Each worker is assigned to at most one task. for (int worker : allWorkers) { MPConstraint constraint = solver.makeConstraint(0, 1, ""); for (int task : allTasks) { constraint.setCoefficient(x[worker][task], 1); } } // Each task is assigned to exactly one worker. for (int task : allTasks) { MPConstraint constraint = solver.makeConstraint(1, 1, ""); for (int worker : allWorkers) { constraint.setCoefficient(x[worker][task], 1); } } // Create variables for each worker, indicating whether they work on some task. MPVariable[] work = new MPVariable[numWorkers]; for (int worker : allWorkers) { work[worker] = solver.makeBoolVar("work[" + worker + "]"); } for (int worker : allWorkers) { // MPVariable[] vars = new MPVariable[numTasks]; MPConstraint constraint = solver.makeConstraint(0, 0, ""); for (int task : allTasks) { // vars[task] = x[worker][task]; constraint.setCoefficient(x[worker][task], 1); } // solver.addEquality(work[worker], LinearExpr.sum(vars)); constraint.setCoefficient(work[worker], -1); } // Group1 MPConstraint constraintG1 = solver.makeConstraint(1, 1, ""); for (int i = 0; i < group1.length; ++i) { // a*b can be transformed into 0 <= a + b - 2*p <= 1 with p in [0,1] // p is True if a AND b, False otherwise MPConstraint constraint = solver.makeConstraint(0, 1, ""); constraint.setCoefficient(work[group1[i][0]], 1); constraint.setCoefficient(work[group1[i][1]], 1); MPVariable p = solver.makeBoolVar("g1_p" + i); constraint.setCoefficient(p, -2); constraintG1.setCoefficient(p, 1); } // Group2 MPConstraint constraintG2 = solver.makeConstraint(1, 1, ""); for (int i = 0; i < group2.length; ++i) { // a*b can be transformed into 0 <= a + b - 2*p <= 1 with p in [0,1] // p is True if a AND b, False otherwise MPConstraint constraint = solver.makeConstraint(0, 1, ""); constraint.setCoefficient(work[group2[i][0]], 1); constraint.setCoefficient(work[group2[i][1]], 1); MPVariable p = solver.makeBoolVar("g2_p" + i); constraint.setCoefficient(p, -2); constraintG2.setCoefficient(p, 1); } // Group3 MPConstraint constraintG3 = solver.makeConstraint(1, 1, ""); for (int i = 0; i < group3.length; ++i) { // a*b can be transformed into 0 <= a + b - 2*p <= 1 with p in [0,1] // p is True if a AND b, False otherwise MPConstraint constraint = solver.makeConstraint(0, 1, ""); constraint.setCoefficient(work[group3[i][0]], 1); constraint.setCoefficient(work[group3[i][1]], 1); MPVariable p = solver.makeBoolVar("g3_p" + i); constraint.setCoefficient(p, -2); constraintG3.setCoefficient(p, 1); } // Objective MPObjective objective = solver.objective(); for (int worker : allWorkers) { for (int task : allTasks) { objective.setCoefficient(x[worker][task], costs[worker][task]); } } objective.setMinimization(); // Solve MPSolver.ResultStatus resultStatus = solver.solve(); // Print solution. // Check that the problem has a feasible solution. if (resultStatus == MPSolver.ResultStatus.OPTIMAL || resultStatus == MPSolver.ResultStatus.FEASIBLE) { System.out.println("Total cost: " + objective.value() + "\n"); for (int worker : allWorkers) { for (int task : allTasks) { // Test if x[i][j] is 0 or 1 (with tolerance for floating point // arithmetic). if (x[worker][task].solutionValue() > 0.5) { System.out.println("Worker " + worker + " assigned to task " + task + ". Cost: " + costs[worker][task]); } } } } else { System.err.println("No solution found."); } } private AssignmentGroupsMip() {} }
C#
using System; using System.Collections.Generic; using System.Linq; using Google.OrTools.LinearSolver; public class AssignmentGroupsMip { static void Main() { // Data. int[,] costs = { { 90, 76, 75, 70, 50, 74 }, { 35, 85, 55, 65, 48, 101 }, { 125, 95, 90, 105, 59, 120 }, { 45, 110, 95, 115, 104, 83 }, { 60, 105, 80, 75, 59, 62 }, { 45, 65, 110, 95, 47, 31 }, { 38, 51, 107, 41, 69, 99 }, { 47, 85, 57, 71, 92, 77 }, { 39, 63, 97, 49, 118, 56 }, { 47, 101, 71, 60, 88, 109 }, { 17, 39, 103, 64, 61, 92 }, { 101, 45, 83, 59, 92, 27 }, }; int numWorkers = costs.GetLength(0); int numTasks = costs.GetLength(1); int[] allWorkers = Enumerable.Range(0, numWorkers).ToArray(); int[] allTasks = Enumerable.Range(0, numTasks).ToArray(); // Allowed groups of workers: int[,] group1 = { // group of worker 0-3 { 2, 3 }, { 1, 3 }, { 1, 2 }, { 0, 1 }, { 0, 2 }, }; int[,] group2 = { // group of worker 4-7 { 6, 7 }, { 5, 7 }, { 5, 6 }, { 4, 5 }, { 4, 7 }, }; int[,] group3 = { // group of worker 8-11 { 10, 11 }, { 9, 11 }, { 9, 10 }, { 8, 10 }, { 8, 11 }, }; // Solver. Solver solver = Solver.CreateSolver("SCIP"); if (solver is null) { return; } // Variables. // x[i, j] is an array of 0-1 variables, which will be 1 // if worker i is assigned to task j. Variable[,] x = new Variable[numWorkers, numTasks]; foreach (int worker in allWorkers) { foreach (int task in allTasks) { x[worker, task] = solver.MakeBoolVar($"x[{worker},{task}]"); } } // Constraints // Each worker is assigned to at most one task. foreach (int worker in allWorkers) { Constraint constraint = solver.MakeConstraint(0, 1, ""); foreach (int task in allTasks) { constraint.SetCoefficient(x[worker, task], 1); } } // Each task is assigned to exactly one worker. foreach (int task in allTasks) { Constraint constraint = solver.MakeConstraint(1, 1, ""); foreach (int worker in allWorkers) { constraint.SetCoefficient(x[worker, task], 1); } } // Create variables for each worker, indicating whether they work on some task. Variable[] work = new Variable[numWorkers]; foreach (int worker in allWorkers) { work[worker] = solver.MakeBoolVar($"work[{worker}]"); } foreach (int worker in allWorkers) { Variable[] vars = new Variable[numTasks]; foreach (int task in allTasks) { vars[task] = x[worker, task]; } solver.Add(work[worker] == LinearExprArrayHelper.Sum(vars)); } // Group1 Constraint constraint_g1 = solver.MakeConstraint(1, 1, ""); for (int i = 0; i < group1.GetLength(0); ++i) { // a*b can be transformed into 0 <= a + b - 2*p <= 1 with p in [0,1] // p is True if a AND b, False otherwise Constraint constraint = solver.MakeConstraint(0, 1, ""); constraint.SetCoefficient(work[group1[i, 0]], 1); constraint.SetCoefficient(work[group1[i, 1]], 1); Variable p = solver.MakeBoolVar($"g1_p{i}"); constraint.SetCoefficient(p, -2); constraint_g1.SetCoefficient(p, 1); } // Group2 Constraint constraint_g2 = solver.MakeConstraint(1, 1, ""); for (int i = 0; i < group2.GetLength(0); ++i) { // a*b can be transformed into 0 <= a + b - 2*p <= 1 with p in [0,1] // p is True if a AND b, False otherwise Constraint constraint = solver.MakeConstraint(0, 1, ""); constraint.SetCoefficient(work[group2[i, 0]], 1); constraint.SetCoefficient(work[group2[i, 1]], 1); Variable p = solver.MakeBoolVar($"g2_p{i}"); constraint.SetCoefficient(p, -2); constraint_g2.SetCoefficient(p, 1); } // Group3 Constraint constraint_g3 = solver.MakeConstraint(1, 1, ""); for (int i = 0; i < group3.GetLength(0); ++i) { // a*b can be transformed into 0 <= a + b - 2*p <= 1 with p in [0,1] // p is True if a AND b, False otherwise Constraint constraint = solver.MakeConstraint(0, 1, ""); constraint.SetCoefficient(work[group3[i, 0]], 1); constraint.SetCoefficient(work[group3[i, 1]], 1); Variable p = solver.MakeBoolVar($"g3_p{i}"); constraint.SetCoefficient(p, -2); constraint_g3.SetCoefficient(p, 1); } // Objective Objective objective = solver.Objective(); foreach (int worker in allWorkers) { foreach (int task in allTasks) { objective.SetCoefficient(x[worker, task], costs[worker, task]); } } objective.SetMinimization(); // Solve Solver.ResultStatus resultStatus = solver.Solve(); // Print solution. // Check that the problem has a feasible solution. if (resultStatus == Solver.ResultStatus.OPTIMAL || resultStatus == Solver.ResultStatus.FEASIBLE) { Console.WriteLine($"Total cost: {solver.Objective().Value()}\n"); foreach (int worker in allWorkers) { foreach (int task in allTasks) { // Test if x[i, j] is 0 or 1 (with tolerance for floating point // arithmetic). if (x[worker, task].SolutionValue() > 0.5) { Console.WriteLine($"Worker {worker} assigned to task {task}. Cost: {costs[worker, task]}"); } } } } else { Console.WriteLine("No solution found."); } } }
समाधान का समय
दोनों सॉल्वर के लिए, हल का समय इस तरह है:
- CP-SAT: 0.0113 सेकंड
- MIP: 0.3281 सेकंड
इस समस्या के लिए CP-SAT, MIP की तुलना में काफ़ी तेज़ है.