Avella et al. (2006) [Avella, P., D'Auria, B., Salerno, S. (2006). A LP-based heuristic for a time-constrained routing problem. European Journal of Operational Research 173:120-124] investigate a time-constrained routing (TCR) problem. The core of the proposed solution approach is a large-scale linear program (LP) that grows both row- and column-wise when new variables are introduced. Thus, a column-and-row generation algorithm is proposed to solve this LP optimally, and an optimality condition is presented to terminate the column-and-row generation algorithm. We demonstrate that this optimality condition is incorrect and may lead to a suboptimal solution at termination. We identify the source of this error and discuss how the generic column-and-row generation algorithm proposed by Muter et al. (2010) may be applied to this TCR problem in order to solve the proposed large-scale LP correctly

Birbil, S. Ilker

Birbil, Ş. İlker

Bulbul, Kerem

Bülbül, Kerem

Muter, İbrahim

Muter, Ibrahim

Sahin, Guvenc

Şahin, Güvenç

English

Sabanci University Research Database

SABANCI UNIVERSITYOrhanlı-Tuzla, 34956 Istanbul, TurkeyPhone: +90 (216) 483-9500Fax: +90 (216) 483-9550http://www.sabanciuniv.eduhttp://algopt.sabanciuniv.edu/projectsNovember 19, 2010A Note on “A LP-based Heuristic for a Time-Constrained Routing Problem”I˙brahim Muter, S¸. I˙lker Birbil, Kerem Bu¨lbu¨l, and Gu¨venc¸ S¸ahinSabancı University, Manufacturing Systems and Industrial Engineering, Orhanlı-Tuzla, 34956 Istanbul, Turkeyimuter@su.sabanciuniv.edu, sibirbil@sabanciuniv.edu, bulbul@sabanciuniv.edu, guvencs@sabanciuniv.eduAbstract: Avella et al. (2006) [Avella, P., D’Auria, B., Salerno, S. (2006). A LP-based heuristic for atime-constrained routing problem. European Journal of Operational Research 173:120-124] investigate a time-constrained routing (TCR) problem. The core of the proposed solution approach is a large-scale linear program(LP) that grows both row- and column-wise when new variables are introduced. Thus, a column-and-row gen-eration algorithm is proposed to solve this LP optimally, and an optimality condition is presented to terminatethe column-and-row generation algorithm. We demonstrate that this optimality condition is incorrect and maylead to a suboptimal solution at termination. We identify the source of this error and discuss how the genericcolumn-and-row generation algorithm proposed by Muter et al. (2010) may be applied to this TCR problem inorder to solve the proposed large-scale LP correctly.Keywords: Time-constrained routing; large-scale optimization; linear programming; column generation; column-and-row generation; column-dependent-rows.1. Introduction. Avella et al. (2006) study a time-constrained routing problem motivated by anapplication that schedules the visit of a tourist to a given geographical area. The problem is to send thetourist on one tour during each day in the vacation period by maximizing her total satisfaction whileensuring that each attraction is visited no more than once. The authors refer to this problem as theIntelligent Tourist Problem (ITP).ITP is formulated as a set packing problem with side constraints. The set of sites that may be visitedby a tourist in a vacation period T is denoted by V , and J represents the set of daily tours that originatefrom and terminate at the same location. Each tour is a sequence of sites to be visited on the same day,provided that it satisfies the time-windows restrictions of the tourist and the other feasibility criteria.The total satisfaction of the tourist from participating in tour j is given by rj , and the binary variableyj is set to one, if tour j is incorporated into the itinerary of the tourist. If tour j is performed on day t,the binary variable xjt takes the value one. The overall mathematical model for ITP then becomesmaximize∑j∈Jrjyj , (1)subject to∑j∈D(i)yj ≤ 1, i ∈ V, (2)yj −∑t∈S(j)xjt = 0, j ∈ J, (3)∑j∈F (t)xjt = 1, t ∈ T, (4)12 Muter, Birbil, Bu¨lbu¨l, and S¸ahin: A Note on the TCR ProblemSabancı University, c©November 19, 2010yj ∈ {0, 1}, j ∈ J, (5)xjt ∈ {0, 1}, j ∈ J, T ∈ T, (6)where D(i) ⊂ J denotes the subset of the tours containing site i, S(j) ⊂ T represents the set of dayson which tour j can be performed, and F (t) ⊂ J denotes the subset of the tours allowed on day t. Byconstraints (2), at most one tour in the selected itinerary includes site i. Constraints (3) impose thata tour to be included in the itinerary is assigned to one of the days in T on which the tour is allowed.We also require that exactly one tour is selected on each day of the vacation period as prescribed byconstraints (4). Finally, the objective function (1) maximizes the total satisfaction of the tourist over thevacation period T .Avella et al. (2006) solve ITP heuristically in two steps. In the first step, the LP relaxation of (1)-(6)is solved by a column-and-row generation approach due to a potentially huge number of tours. To thisend, a large number of possible tours is enumerated and added to J . A subset J¯ ⊂ J of these toursis selected to form the restricted master problem (RMP) for the column-and-row generation procedure.At each iteration, a set of new tours j ∈ L ⊆ (J \ J¯) is introduced to the RMP. For each j ∈ L, thisimplies adding xjt, t ∈ S(j), and the associated linking constraint yj −∑t∈S(j) xjt = 0 to the RMP.The column-and-row generation procedure terminates when the condition stated below in Theorem 1.1is satisfied. Observe that the authors evaluate this condition for each tour in j ∈ (J \ J¯) following eachoptimization of the RMP, where J is known explicitly. Furthermore, in the computational experiments| L |= 1; that is, the tour that violates the condition in Theorem 1.1 to the largest extent is added to theRMP. In the second step of the proposed solution approach, (1)-(6) is solved by a commercial solver overthe set of tours currently present in the RMP to obtain an integer feasible solution for ITP.Now, let pii, i ∈ V , γj , j ∈ J , and λt, t ∈ T , denote the dual variables associated with the constraints(2)-(4), respectively. The following theorem, given in Avella et al. (2006) without a proof, defines thestopping condition for the column-and-row generation algorithm applied to ITP by the authors.Theorem 1.1 The solution of the current RMP is optimal for the LP relaxation of (1)-(6) if r¯j =rj −∑i:j∈D(i) pii −∑t:j∈F (t) λt ≤ 0, for each j ∈ J .The statement of the theorem above corrects two typos in the original Theorem 3.1 given byAvella et al. (2006), where the termination condition appears as r¯j = rj−∑i:j∈D(i) pii−∑t:j∈F (t) λt ≥ 0,for each t ∈ T and j ∈ D(t). Next, we demonstrate that the stopping condition in Theorem 1.1 is incor-rect and may lead to a suboptimal solution when the column-and-row generation algorithm proposed byAvella et al. (2006) terminates. Then, we discuss how the generic column-and-row generation algorithmproposed by Muter et al. (2010) may be applied here to solve the proposed large-scale LP correctly.The following is the dual of the linear programming relaxation of (1)-(6):minimize∑i∈Vpii +∑t∈Tλt, (7)subject to∑i:j∈D(i)pii + γj ≥ rj , j ∈ J, (8)− γj + λt ≥ 0, j ∈ J, t ∈ S(j), (9)pii ≥ 0, i ∈ V. (10)Muter, Birbil, Bu¨lbu¨l, and S¸ahin: A Note on the TCR ProblemSabancı University, c©November 19, 2010 3Given the optimal dual variables associated with the current RMP, the resulting pricing subproblem tobe solved becomesζyx = maxj∈J\J¯ c¯j , (11)where c¯j = rj −∑i:j∈D(i) pii − γj denotes the reduced cost of tour j. If the optimal objective functionvalue of this subproblem is positive with c¯j > 0, the variables yj , xjt, t ∈ S(j), and the primal constraintyj −∑t∈S(j) xjt = 0 should be added to the RMP. Otherwise, the optimal solution of the current RMPis declared as optimal for the LP relaxation of (1)-(6), and the algorithm terminates. The challengehere is that the value of the dual variable γj , j ∈ J \ J¯ is unknown because the corresponding constraintis currently absent from the RMP. Hence, in order to design an optimal column-and-row generationalgorithm for ITP, we must devise a method that anticipates the correct value of γj , j ∈ J \ J¯ tobe incorporated into the pricing subproblem. According to Theorem 1.1,∑t:j∈F (t) λt =∑t∈S(j) λt isconsidered as an appropriate estimate for γj , j ∈ J \ J¯ . In the sequel, we first show that this is incorrect,and then illustrate how to determine the correct value of γj , j ∈ J\J¯ based on the generic column-and-rowgeneration framework given by Muter et al. (2010).Consider an iteration of the column-and-row-generation algorithm, where the current RMP is solvedto optimality and the associated optimal dual solution is represented by pii, i ∈ V , γj , j ∈ J¯ , and λt, t ∈ T .Suppose that yj′ , j′ ∈ J \ J¯ is to be priced out, where r¯j′ = rj′ −∑i:j′∈D(i) pii −∑t:j′∈F (t) λt = rj′ −∑i:j′∈D(i) pii −∑t∈S(j′) λt ≤ 0. Observe for the currently unknown dual variable γj′ that the dualconstraints (9) implyγj′ ≤ mint∈S(j′)λt, (12)and if | S(j′) |> 1 and maxt∈S(j′) λt > 0, then we may haveγj′ ≤ mint∈S(j′)λt <∑t∈S(j′)λt. (13)Clearly, this may lead tor¯j′ = rj′ −∑i:j′∈D(i)pii −∑t∈S(j′)λt ≤ 0 < rj′ −∑i:j′∈D(i)pii − γj′ = c¯j′ . (14)Thus, we conclude that while r¯j ≤ 0 for all j ∈ J as required by Theorem 1.1 due to Avella et al.(2006), there may exist a tour j′ with c¯j′ > 0. In other words,∑t∈S(j′) λt is not an appropriate estimateof the missing dual variable γj′ , and the column-and-row generation algorithm of Avella et al. (2006)may terminate prematurely, while there exists a tour j′ with a positive reduced cost. On the other hand,if λt ≤ 0, t ∈ S(j′), then ∑t∈S(j′)λt < γj′ ≤ mint∈S(j′)λt (15)may hold, and this may result inc¯j′ = rj′ −∑i:j′∈D(i)pii − γj′ ≤ 0 < rj′ −∑i:j′∈D(i)pii −∑t∈S(j′)λt = r¯j′ . (16)That is, a tour j′ with a nonpositive reduced cost may violate the condition in Theorem 1.1 and it isunnecessarily added to the RMP.Using the column-and-row generation algorithm given in Muter et al. (2010), the pricing subproblembecomes a two-stage problem:ζyx = maxj∈J\J¯rj −∑i:j∈D(i)pii − αj , (17)4 Muter, Birbil, Bu¨lbu¨l, and S¸ahin: A Note on the TCR ProblemSabancı University, c©November 19, 2010whereαj =minimize γj , (18)subject to − γj + λt ≥ 0, t ∈ S(j), (19)at least one constraint in (19) is tight. (20)To calculate the correct reduced cost of yj , we solve the problem (18)-(20) separately for each j ∈ J \ J¯ .Constraints (19) in the problem (18)-(20) impose that the dual constraints (9) corresponding to thevariables xjt, t ∈ S(j), induced by any j ∈ J \ J¯ are not violated. Consequently, if the optimal objectiveof (17) is ζyx = c¯j > 0, then the RMP is augmented with yj and xjt, t ∈ S(j), along with one newlinking constraint of type (3), and yj is the natural candidate to enter the basis at the next iteration ofthe column-and-row generation algorithm. In order to warm-start the primal simplex algorithm to re-optimize the RMP, the optimal basis from the previous iteration must be augmented with a new variable.This new basic variable xjt is provided by the optimal solution of (18)-(20) for yj and corresponds toa tight constraint in (19). Thus, the complementary slackness condition for the new basic variable issatisfied.We observe that the problem (18)-(20) is unbounded without the restriction (20) for all j ∈ J \ J¯ .Therefore, for each j ∈ J \J¯ , the solution of (18)-(20) is trivial and αj = γj = mint∈S(j) λt. Consequently,the correct termination condition for the column-row-generation algorithm that solves the LP relaxationof (1)-(6) optimally is formalized in the theorem below.Theorem 1.2 Let pii, i ∈ V , γj , j ∈ J¯ , and λt, t ∈ T be the optimal dual solution corresponding tothe optimal basis B of the current RMP. The primal solution associated with B is optimal for the LPrelaxation of (1)-(6) if rj −∑i:j∈D(i) pii −mint∈S(j) λt ≤ 0 for every j ∈ J \ J¯ .The proof of this theorem follows from the analysis of the column-and-row generation algorithm developedin Muter et al. (2010).To support these observations, we generated a random instance of ITP. The column-and-row generationalgorithm is first applied to the LP relaxation of this instance using the stopping condition of Avella et al.(2006) as stated in Theorem 1.1. We observe that this stopping condition leads to a premature terminationwith a suboptimal solution. Then, we employ the stopping condition of Theorem 1.2 and determinethe true optimal solution. These results confirm also empirically that the stopping condition given byAvella et al. (2006) is not correct1.ReferencesAvella, P., D’Auria, B., and Salerno, S. (2006). A LP-based heuristic for a time-constrained routingproblem. European Journal of Operational Research, 173:120–124.Muter, I., Birbil, S. I., and Bulbul, K. (2010). Simultaneous column-and-row generation for large-scale linear programs with column-dependent-rows. Technical Report SU FENS 2010/0004, SabancıUniversity, Istanbul, Turkey. http://research.sabanciuniv.edu/14197/.1This counterexample is provided at http://people.sabanciuniv.edu/sibirbil/ejorsupp/.

A note on "A LP-based heuristic for a time-constrained routing problem"

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A note on "A LP-based heuristic for a time-constrained routing problem"

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