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Routing problems: constraints and optimality*
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IFAC-PapersOnLine 49-12 (2016) 640–644 Routing problems: constraints and Routing problems: constraints and Routing problems: constraints and ⋆ Routing problems: constraints and optimality ⋆ Routing problems: constraints and optimality ⋆ ⋆ optimality ⋆ optimality optimality Alexander G. Chentsov ∗∗ Pavel A. Chentsov ∗∗ ∗∗
Alexander Chentsov ∗∗∗ ∗ Pavel A. Chentsov ∗∗ ∗∗∗∗ Alexander A. G. Petunin Sesekin∗∗ ∗ Pavel A. N. Alexander G. Chentsov Chentsov ∗∗∗ Alexander ∗∗∗∗ ∗ ∗∗ Alexander G. Chentsov A. Chentsov Alexander A. Petunin Alexander N. Sesekin ∗ Pavel ∗∗ Alexander G. Chentsov Pavel A. Chentsov ∗∗∗ ∗∗∗∗ Alexander G. Chentsov Pavel A. Chentsov ∗∗∗ Alexander ∗∗∗∗ Alexander A. Petunin Sesekin ∗∗∗ Alexander N. ∗∗∗∗ Alexander A. Petunin N. Sesekin ∗∗∗ ∗∗∗∗ A. Petunin Alexander N. Sesekin ∗ Alexander Alexander A. Petunin Alexander N. Sesekin Ural Federal University, Russia. Institute of Mathematics and ∗ Ural Federal University, Russia. Institute of Mathematics and ∗ UB RAS, Russia, (Tel: +7-343-375-34-57; e-mail: ∗Mechanics Ural Federal University, Russia. Institute of Mathematics and ∗Mechanics Ural Federal University, Russia. Institute of Mathematics and UB RAS, Russia, (Tel: +7-343-375-34-57; e-mail: ∗ Ural University, Russia. Institute of and Ural Federal Federal University, Russia. Institute of Mathematics Mathematics and [email protected]). Mechanics UB RAS, Russia, (Tel: +7-343-375-34-57; e-mail: Mechanics UB RAS, Russia, (Tel: +7-343-375-34-57; e-mail: [email protected]). ∗∗Mechanics UB RAS, Russia, (Tel: +7-343-375-34-57; e-mail: Mechanics UB RAS, Russia, (Tel: +7-343-375-34-57; e-mail: Ural Federal University, Russia. Institute of Mathematics and [email protected]). ∗∗ [email protected]). Ural Federal University, Russia. Institute of Mathematics and [email protected]). ∗∗Mechanics [email protected]). UB RAS, Russia, (Tel: +7-343-362-81-62; e-mail: ∗∗ Ural Federal University, Russia. Institute of Mathematics and ∗∗ Ural Federal Russia. of Mathematics and UBUniversity, RAS,[email protected]). Russia, (Tel:Institute +7-343-362-81-62; e-mail: ∗∗Mechanics Ural Federal University, Russia. Institute of Mathematics and Ural Federal University, Russia. Institute of Mathematics and Mechanics UB RAS, Russia, (Tel: +7-343-362-81-62; e-mail: UB RAS, Russia, (Tel: +7-343-362-81-62; e-mail: [email protected]). ∗∗∗ Mechanics Mechanics UB RAS, Russia, (Tel: +7-343-362-81-62; e-mail: Mechanics UB RAS, Russia, (Tel: +7-343-362-81-62; e-mail: Ural Federal University, Russia, (Tel: +7-343-375-97-14; e-mail: [email protected]). ∗∗∗ [email protected]). Russia, (Tel: +7-343-375-97-14; e-mail: [email protected]). ∗∗∗ Ural Federal University, [email protected]). [email protected]). ∗∗∗ Ural Federal University, Russia, (Tel: +7-343-375-97-14; e-mail: ∗∗∗ Ural Federal University, Russia, (Tel: +7-343-375-97-14; e-mail: [email protected]). ∗∗∗ Ural Federal University, ∗∗∗∗ Russia, (Tel: +7-343-375-97-14; e-mail: Russia, (Tel: +7-343-375-97-14; e-mail: Ural Federal Federal University, University, Russia, (Tel: +7-343-375-41-40; e-mail: [email protected]). ∗∗∗∗Ural [email protected]). Ural Federal University, Russia, (Tel: +7-343-375-41-40; e-mail: [email protected]). ∗∗∗∗ [email protected]). [email protected]). ∗∗∗∗ Ural Federal University, Russia, (Tel: +7-343-375-41-40; e-mail: ∗∗∗∗ Federal University, Russia, (Tel: +7-343-375-41-40; e-mail: [email protected]). ∗∗∗∗ Ural Ural Russia, Ural Federal Federal University, University, Russia, (Tel: (Tel: +7-343-375-41-40; +7-343-375-41-40; e-mail: e-mail: [email protected]). [email protected]). [email protected]). [email protected]). Abstract: We consider the issues of routing under constraints and formulate a mathematical Abstract: consider the issues The of routing constraints formulate aconstraints. mathematical problem of We visiting megalopolises. order under of visits is subjectand to precedence In Abstract: We consider the issues of routing under constraints and formulate aa mathematical Abstract: We consider the issues of routing under constraints and formulate mathematical problem of visiting megalopolises. The order of visits is subject to precedence constraints. Ina Abstract: We consider the issues of routing under constraints and formulate a mathematical Abstract: We consider the issues of on routing under constraints and formulate aconstraints. mathematical addition, the cost functions depend the set of pending tasks. The quality criterion is problem of visiting megalopolises. The order of visits is subject to precedence In problem ofthe visiting megalopolises. The of visits is subject to precedence addition, cost functions depend onorder the set of pending tasks. The qualityconstraints. criterion is In problem visiting megalopolises. The order of is to precedence constraints. In problem ofthe visiting megalopolises. The order of visits is subject subject to the precedence constraints. Inaa variety ofof the additive criterion. The problem is visits established within dynamic programming addition, cost functions depend on the set of pending tasks. The quality criterion is addition, the cost functions depend on the set of pending tasks. The quality criterion is a variety of the additive criterion. The problem is established within the dynamic programming addition, the cost functions depend on of pending tasks. The quality criterion addition, thehowever, cost functions depend on the the set set ofimplemented pending within tasks. The practical quality programming criterion is ofaa framework, heuristic is proposed and to solve problemsis variety additive The problem is the variety of of the the additiveaa criterion. criterion. The problemand is established established within the dynamic dynamic programming framework, however, heuristic is proposed implemented to solve practical problems of variety of the additive criterion. The problem is established within the dynamic programming variety of the additivea criterion. Theproposed problemand is established within the dynamic programming large dimensionality. framework, however, framework, however, a heuristic heuristic is is proposed and implemented implemented to to solve solve practical practical problems problems of of large dimensionality. framework, however, framework, however, a a heuristic heuristic is is proposed proposed and and implemented implemented to to solve solve practical practical problems problems of of large dimensionality. large dimensionality. © 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. large large dimensionality. dimensionality. 1. INTRODUCTION large dimension. For those, we have to resort to heuristic 1. INTRODUCTION large dimension. For those, we have to resort to heuristic algorithms. 1. large dimension. 1. INTRODUCTION INTRODUCTION large dimension. For For those, those, we we have have to to resort resort to to heuristic heuristic algorithms. 1. INTRODUCTION large 1. of INTRODUCTION large dimension. dimension. For For those, those, we we have have to to resort resort to to heuristic heuristic There is a number distinct applications where rout- algorithms. algorithms. There is a number of distinct applications whereatomic rout- algorithms. algorithms. ing problems arise (e.g. transportation problems, There is a number of distinct applications where rout2. PROBLEM STATEMENT AND GENERAL There is aa number number of distinct distinct applications whereatomic routing problems arise (e.g. transportation problems, There is of applications rout2. PROBLEM STATEMENT GENERAL There is a number of distinct applications where routpower generation, toolpath optimization forwhere CNC cuting problems arise (e.g. transportation problems, atomic NOTATION AND 2. PROBLEM STATEMENT AND GENERAL ing problems arise (e.g. transportation problems, atomic power generation, toolpath optimization for CNC cut2. PROBLEM STATEMENT AND GENERAL ing problems arise (e.g. transportation problems, atomic NOTATION ing problems arise (e.g. transportation problems, atomic 2. PROBLEM STATEMENT AND GENERAL ting machines). In many problems, there is a need to power generation, toolpath optimization for CNC cut2. PROBLEM STATEMENT AND GENERAL NOTATION powermachines). generation, toolpath optimization for CNC cutting In many problems, there is a need to NOTATION power generation, toolpath optimization for CNC cutpowermachines). generation, toolpath optimization for CNC cutNOTATION order, or sequence, the operations. Often, the sequencing ting In many problems, there is a need to NOTATION Let us fix an arbitrary nonempty set X, which will serve ting machines). In many problems, there is a need to order, or sequence, the operations. Often, the ting machines). many problems, there is aait need to us ground fix an arbitrary nonempty setcorresponding X, which willinitial serve ting machines). Invarious many problems, there is sequencing need to Let is complicated byIn constraints. At times, is more 0 order, or sequence, the operations. Often, the sequencing as the set. Let x ∈ X be the order, or sequence, the operations. Often, the sequencing Let us fix an arbitrary nonempty set X, which will serve is complicated by various constraints. At times, it is more 0 order, or the operations. Often, the sequencing sequencing Let us fix an arbitrary nonempty set X, which will serve as the ground set. Let x ∈ X be the corresponding initial order, or sequence, sequence, the operations. Often, the important to satisfy all the constraints than to achieve Let us fix an arbitrary nonempty set X, which will serve is complicated by various constraints. At times, it is more 0 Let us fix an arbitrary nonempty set X, which will serve state, or the base (depot) of our process. Let N be a is complicated by various constraints. At times, it is more 0 as the ground set. Let x ∈ X be the corresponding initial important to satisfy all the constraints than to achieve is complicated by various constraints. At times, it is more 0 as the ground set. Let x ∈ X be the corresponding initial state, or the base (depot) of our process. Let N be a is complicated by various constraints. At times, it is more 0 an optimum for the quality criterion—the typical case in as the ground set. Let x ∈ X be the corresponding initial important to satisfy all the constraints than to achieve as the ground set. Let x ∈ X be the corresponding initial natural number for which N ≥ 2. Fix nonempty finite important tofor satisfy all the thecriterion—the constraints than than to achieve achieve state, or or the the base base (depot) (depot) of of our our process. process. Let Let N N be be a a an optimum the quality typical case in important to satisfy all constraints to state, natural number for which N ≥ 2. Fix nonempty finite important tofor satisfy all thecriterion—the constraints than to achieve real-life engineering. state, or the base (depot) of our process. Let N be an optimum the quality typical case in state,Mor the (depot) of ≥ our2. process. Let N finite be a a ..., MNbase for which an optimum optimum for the the quality quality criterion—the criterion—the typical typical case case in in sets natural for which N Fix nonempty 1 ,number real-life engineering. an for natural number for which N ≥ 2. Fix nonempty finite sets M1 ,number ..., MN for which an optimum for the quality criterion—the typical case in natural for which N ≥ 2. Fix nonempty finite real-life engineering. natural number for which N ≥ 2. Fix nonempty finite real-life engineering. sets M , ..., M for which We consider an optimization problem for the sake of 1 N Mwhich real-life engineering. for 1 ⊂ X, ..., MN ⊂ X. real-life engineering. sets M M111 ,,, ..., ..., M MN for which We consider an optimization problem formore the effective. sake of sets N Mwhich sets M ..., M 1 ⊂ X, ..., MN ⊂ X. N for making the search for admissibleproblem solutionsfor We consider an optimization the sake of ⊂ X, ..., M ⊂ X.Assume M 1 as We regard these sets megalopolises. We consider an optimization problem for the sake of ⊂ X, ..., MN ⊂ M making the search for admissible solutions more effective. We consider an optimization problem for the sake of ⊂ ..., ⊂ X. X. M We this consider an employ optimization problem formore thevariant sake of We regard these sets N megalopolises. To end,search we a pretty complicated ⊂ X, X, ..., M M0N M0 111 as making the search for admissible admissible solutions effective. N ⊂ X.Assume making the for solutions more effective. We regard these sets as megalopolises. Assume To this end, we employ a pretty complicated variant of ∈ / M , ..., x ∈ / M ; x making the search for admissible solutions more effective. We regard these sets as megalopolises. 1 N Assume 0 0 making the search for admissible solutions more effective. dynamic programming (DP) supplemented with special To this end, we employ aa pretty complicated variant of We regard these sets as megalopolises. Assume ∈ / M , ..., x ∈ / M ; x We regard these sets as megalopolises. Assume To this end, we employ pretty complicated variant of 1 N 0 0 dynamic programming (DP) supplemented with special To this end, we employ a pretty complicated variant of 0M 0are ∈ / M , ..., x ∈ / M ;; x To this end, we employ a pretty complicated variant of 1 N constructions which decrease the computational complex, ..., M pairwise disjoint. The moreover, the sets 0 0 dynamic programming (DP) supplemented with special ∈ / M , ..., x ∈ / M x 1 N 1 N 0 0 dynamic programming (DP) supplemented with special ∈ M , ..., ...,Nx x are ∈ M ; x M∈ constructions which decrease computational complex1 N ..., pairwise disjoint. and, The moreover, the must setsx //1 ,M ∈ //a M dynamic programming (DP) supplemented with special 1 ,M N; dynamic programming (DP) the supplemented with special ity though the use precedence constraints. It is natural to megalopolises be visited in certain sequence constructions which decrease the computational complex, ..., M are pairwise The moreover, the sets M 1 , ..., MN are pairwise disjoint. constructions which decrease the computational complexdisjoint. The moreover, the sets M ity though the use precedence constraints. It is natural to 1 , visited N in megalopolises must be a certain sequence and, constructions which decrease the computational complex..., M are pairwise disjoint. The moreover, the sets M 1 N constructions which decrease the computational complex, ..., M are pairwise disjoint. The moreover, the sets M use DP in local way, which is implemented as insertions (or in each of them, a certain job has to be completed, ity though the use precedence constraints. It is natural to 1 N megalopolises must be visited in a certain sequence and, ity though the use precedence constraints. It is natural to megalopolises must be visited in a certain sequence and, use DP in local way, which is implemented as insertions (or in each of them, a certain job has to be completed, ity though the use precedence constraints. It is natural to megalopolises must be visited in a certain sequence and, ity though the use precedence constraints. It is natural to megalopolises must be visited in a certain sequence and, “incuts”) of moderate dimension. Basic constructions of which results into a kind of interior movement within use DP in local way, which is implemented as insertions (or in each of them, a certain job has to be completed, use DP DP in in local local way, which which is implemented implemented as insertions insertions (or (or in each of them, aa certain has to be completed, “incuts”) of moderate dimension. Basic(2008a, constructions of which results intoWe kind of job interior movement within use way, is as in each of to be completed, use DP in local way, which isinimplemented as insertions (or in each of them, them, certain job has to be completed, this approach are reflected Chentsov 2014a,b). megalopolis. consider thehas arrival point and the “incuts”) of dimension. Basic constructions constructions of which results into aaaa certain kind of job interior movement within “incuts”) of moderate moderate dimension. Basic of each which results into kind of interior movement within this approach are reflected in Chentsov (2008a, 2014a,b). each megalopolis. We consider the arrival point and the “incuts”) of moderate dimension. Basic constructions of which results into a kind of interior movement within “incuts”) of moderate dimension. Basic(2008a, constructions of departure which results into a kind of interior movement within Of course, the above-mentioned mathematical problem has point. The pairs these points are evaluated this approach are reflected in Chentsov 2014a,b). each megalopolis. We consider the arrival point and the thiscourse, approach are reflected in Chentsov (2008a, 2014a,b). each megalopolis. We consider the arrival point and the Of the above-mentioned mathematical problem has departure point. The pairs of these points are evaluated this approach are reflected in Chentsov (2008a, 2014a,b). each megalopolis. We consider the arrival point and the thiscourse, approach are reflected in Chentsov (2008a, 2014a,b). each megalopolis. We consider the arrival point and the many applications. by the corresponding cost functions. Of the above-mentioned mathematical problem has departure point. The pairs of these points are evaluated Of course, the above-mentioned mathematical problem has departure point. The pairs of these points are evaluated many applications. by the corresponding cost functions. Of course, the above-mentioned mathematical problem has departure point. The pairs of these points are evaluated Of course, the above-mentioned mathematical problem has departure point. The pairs of these points are evaluated many applications. by the corresponding cost functions. 0 many applications. by the corresponding cost functions. It is important that the DP-based procedure we employ Next, exterior permutation (from x0 to megalopolises many applications. the corresponding cost many applications. by theeach corresponding cost functions. functions. It is important that the DP-based procedure constrained we employ by Next, each exterior permutation (from x0 to megalopolises provides for optimal solutions of precedence and between megalopolises) is evaluated. Finally, for terIt is important that the DP-based procedure we employ 0 to Next, each exterior permutation (from x megalopolises It is is important important that the the DP-based procedure constrained we employ employ Next, 0 each exterior permutation (from x megalopolises provides for optimal solutions of precedence and between megalopolises) is evaluated. Finally, for ter0 to It that DP-based procedure we Next, each exterior permutation (from x to megalopolises It is important that the DP-based procedure we employ Next, each exterior permutation (from x to megalopolises problems where the cost functions depend on the set minal state, the corresponding evaluation is defined. We provides for optimal solutions of precedence constrained and between megalopolises) is evaluated. Finally, for terprovides for for optimal solutions of precedence precedence constrained and between megalopolises) is evaluated. Finally, for terproblems where the cost functions depend on the set minal state, the corresponding evaluation is defined. We provides optimal solutions of constrained and between megalopolises) evaluated. for provides for optimal solutions of precedence and between megalopolises) isthe evaluated. Finally, for terterof pending tasks. This is its main feature asconstrained farthe as the consider the the casecorresponding when all is costs areFinally, summed, the problems where the cost functions depend on set minal state, the corresponding evaluation is defined. We problems where the cost functions depend on the set minal state, evaluation is defined. We of pending tasks. This is its main feature as far as the consider the case when all the costs are summed, the problems where the cost functions depend on the set minal state, the corresponding evaluation is defined. We problems where the cost functions depend on the set minal state, the corresponding evaluation defined. We DP structure is concerned (the optional dependence of additive case. However, the optimization is complicated of pending tasks. This is its main feature as far as the consider the case when all the costs are summed, the of pending pending tasks. This is its main feature as far as as the the consider the case when all the costs are summed, the DP structure is concerned (the optional dependence of additive case. However, the optimization is complicated of tasks. This is its main feature as far consider the case when all the costs are summed, the of pending tasks. This is its main feature as far as the consider the case when all the costs are summed, the cost functions on the set of pending tasks being its most by the constraints, which restrict both the routes defined DP structure is concerned (the optional dependence of additive case. However, the optimization is complicated DP structure is concerned (the optional dependence of additive case. However, the optimization is complicated cost functions on the set of pending tasks being its most by the constraints, which restrict both the routes defined DP structure is concerned (the optional dependence of additive case. However, the optimization is complicated DP structure is concerned (the optional dependence of additive case. However, the optimization is complicated essential and innovative part). The main issue in using DP as permutations of the megalopolises’ indices and the cost functions on the set of pending tasks being its most by the constraints, which restrict both the routes defined cost functions functions on the of tasks being its by the constraints, restrict both the routes defined essential andof innovative The(DP main using DP as permutations ofwhich themotions. megalopolises’ indices the cost on the set setpart). of pending pending tasks being its most most by constraints, restrict both routes defined cost on the set of pending tasks its most by the constraints, which restrictSpecifically, both the the routes defined is thefunctions deficit computer memory is issue notbeing ain polynomial of specific we and consider essential and innovative part). The main issue in using as the permutations ofwhich the megalopolises’ megalopolises’ indices and the essential andof innovative part). The(DP main issue in using DP DP trajectories as permutations of the indices and the is the deficit computer memory is not a polynomial trajectories of specific motions. Specifically, we consider essential and innovative part). The main issue in using DP as permutations of the megalopolises’ indices and the essential andwhich innovative The main using DP asprocess permutations of the themotions. megalopolises’ and the procedure), makespart). its use infeasible foraain problems of atrajectories defined by following scheme:indices is the deficit of computer memory (DP is issue not polynomial of specific Specifically, we consider is the deficit of computer memory (DP is not polynomial trajectories of specific motions. Specifically, we consider procedure), which makes its use infeasible for problems of atrajectories process defined by the following scheme: is the deficit of computer memory (DP is not a polynomial trajectories of specific motions. Specifically, we consider is the deficit of computer memory (DP is not a polynomial of specific motions. Specifically, we consider procedure), which makes its use infeasible for problems of a process defined by the following scheme: 0 ⋆ procedure), which makesbyits its use use infeasible for ofproblems problems of defined by the x0 −→ (x1,1 ∈ ❀ x1,2 ∈scheme: Mα(1) ) −→ ... This work was supported 211 Government the Russian α(1)following procedure), which makes infeasible for of aaa process process defined byM the following scheme: procedure), which makesbyitsAct use infeasible for ofproblems of ⋆ process defined by the (1) x0...−→ (x(x M ❀❀ x1,2x ∈scheme: M∈α(1) ) −→), ... This workcontract was supported Act 211 Government the Russian 1,1 ∈ α(1)following ⋆ Federation, N 02.A03.21.0006 ∈ M Mα(N −→ 0 −→ (1) x (x ∈ M ❀ x ∈ )) −→ N,1 N,2 This work was supported by Act 211 Government of the Russian α(N ) ) ... 1,1 1,2 ⋆ α(1) ❀ x1,2 ∈ M α(1) 0 x −→ (x ∈ M M −→ ... This work was supported by Act 211 Government of the Russian 1,1 0 Federation, contract N 02.A03.21.0006 ⋆ α(1) α(1) ∈ M ❀ x ∈ M ), ... −→ (x (1) N,1 N,2 x (x ∈ ❀ x ∈ )) −→ ⋆ α(N α(N ) ... This was by 1,1 α(1) (1) x ...−→ −→ (x(x ∈M M ❀)) ❀ x1,2 ∈M M∈α(1) −→ ... This work workcontract was supported supported by Act Act 211 211 Government Government of of the the Russian Russian 1,1N,1 1,2xN,2 Federation, N 02.A03.21.0006 α(1) α(1) ∈ M M ), −→ (1) α(N α(N ) Federation, contract N 02.A03.21.0006 ∈ M ❀ x ∈ M ), ... −→ (x (1) N,1 N,2 α(N ) α(N ) Federation, contract N 02.A03.21.0006 ∈ M ❀ x ∈ M ), ... −→ (x N,1 N,2 Federation, contract N 02.A03.21.0006 α(N ) α(N ) ∈ M ❀ x ∈ M ), ... −→ (x N,1 N,2 α(N ) α(N ) Copyright © 2016 IFAC 640 Copyright © 2016, 2016 IFAC 640Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2016 IFAC 640 Copyright 2016 IFAC 640 Peer review© of International Federation of Automatic Control. Copyright ©under 2016 responsibility IFAC 640 Copyright © 2016 IFAC 640 10.1016/j.ifacol.2016.07.756
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where α is a route, or indices’ permutation. Constraints are imposed on α, x1 = (x1,1 , x1,2 ), ..., xN −1 = (xN −1,1 , xN −1,2 ), and xN = (xN,1 , xN,2 ) (here, it is supposed that N > 2). In (1), the straight arrows denote the exterior movements and the sinuous arrows denote the movements connected with interior jobs. As mentioned above, all steps in (1) are evaluated through the corresponding cost functions. The sum of their costs defines the additive criterion. Now, let us recall the constraints. We note only the most important. The choice of the route α in (1) can be constrained by precedence constraints. Namely, there may be a nonempty set of ordered sender-receiver pairs. The route α must order the megalopolises so as to make sure that each sender megalopolis appears before its corresponding receiver. These are the route constraints. Now for trajectory constraints, which restrict x1 , ..., xN in (1). Previously, we remarked that, in many cases, this trajectory must be realized in the form x1 ∈ Mα(1) , ..., xN ∈ Mα(N ) , where Mj ⊂ Mj × Mj for all j ∈ 1, N . Now, we note that Mj defines all possible variants of conducting the interior jobs connected with the megalopolis Mj , which are denoted by sinuous arrows in (1). In this connection, see the following example: Example. Consider the problem of routing the tool in sheet cutting CNC machines. The tool arrives at the contour to cut it; it must arrive at a given penetration point near the equidistant of the contour, and it is the equidistant that is to be cut to create a desired contour— roughly speaking, the cut is made such that there is some space between the desired contour and the cut. After the contour is cut, the tool, while still active, must be driven to the corresponding shut-off point. It is important to observe precedence constraints: each interior (child) contour of a feature must be cut out before the exterior (parent) contour. There are also other constraints, for example, the sheet rigidity must be maintained at all times (a new cut may only be done in the rigidity zone), and the sheet must remain in one piece (i.e., not cut in two, etc.), which may be formalized as the dependence of the cost on the set of pending tasks. Thus there appears a kind of memory, or sequence dependence. Our principal aim in constructing the route and trajectory is to satisfy all those constraints. The quality criterion is the total amount of time required to cut all the features. To provide for satisfaction of certain constraints, we may make the cost of “prohibited” actions effectively infinite; thus optimization would contribute to constraint satisfaction. We propose a method for constructing a route subject to certain constraints that is related to DP. However, the mentioned constraints complicate the construction of the DP procedure.
641
△
and, for p ∈ N0 and q ∈ N0 , p, q = {i ∈ N0 | (p ≤ i)&(i ≤ q)} (of course, p, q = ∅ under q < p). For a nonempty set H, denote by R+ [H] set of all nonnegative real-valued △
functions defined on H. Let Mj = {pr2 (z) : z ∈ Mj } for j ∈ 1, N (pr1 (h) and pr2 (h) denote the first and second elements of an arbitrary ordered pair h). Let N N △ △ X = {x0 } ( Mi ), X = {x0 } ( Mj ), i=1
i=1
X ⊂ X ⊂ X. In addition, X and X are nonempty finite sets. Consider the possible variants of precedence constraints. Fix the set K with the property K ⊂ 1, N × 1, N . For every nonempty set K0 , K0 ⊂ K, assume (2) ∃z0 ∈ K0 : pr1 (z0 ) �= pr2 (z) ∀z ∈ K0 ; see (Chentsov , 2008a, Condition 2.2.1) (condition (2) is fulfilled in many practically interesting cases; see (Chentsov , 2008a, ch.2)). Then, △
A = {α ∈ P | ∀z ∈ K ∀t1 ∈ 1, N ∀t2 ∈ 1, N ((α(t1 ) = pr1 (z))&(α(t2 ) = pr2 (z))) ⇒ (t1 < t2 )} = (3) = {α ∈ P | α−1 (pr1 (z)) < α−1 (pr2 (z)) ∀z ∈ K} �= ∅, where P is the set of all permutations of 1, N and, for every β ∈ P, the symbol β −1 denotes the inverse of the permutation β. Thus, an admissible (in view of precedence constraints) route exists. ˜ we denote the set of all tuples (zi ) By Z i∈0,N : 0, N −→ X × X and △ ˜ | (z0 = (x0 , x0 ))&(zt ∈ Mα(t) Zα = {(zi )i∈0,N ∈ Z (4) ∀t ∈ 1, N )} �= ∅ ∀α ∈ A. In (4), the trajectories coordinated with a route are introduced. Then, △
˜ | (zi ) D = {(α, (zi )i∈0,N ) ∈ A × Z i∈0,N ∈ Zα } �= ∅ is the set of all admissible solutions. Cost functions. In the following, we consider an “additive” routing problem similar to Chentsov (2014a,b) (a variant of bottleneck problem was considered in Chentsov (2008b, 2015) and in other articles). Now, we introduce the costs of movements, interior jobs, and the terminal state. Denote by N the family of all nonempty subsets of 1, N . Fix c ∈ R+ [X × X × N], c1 ∈ R+ [X × X × N],..., cN ∈ R+ [X × X × N] and f ∈ R+ [X ]. Mathematical setting of problem. If α ∈ A and (zt )i∈0,N ∈ Zα , then △
Cα [(zt )t∈0,N ] =
N
[c(pr2 (zt−1 ), pr1 (zt ),
t=1
{α(j) : j ∈ t, N }) + cα(t) (zt , {αj : j ∈ t, N })]+ +f (pr2 (zN ))
3. EXTREMAL PROBLEM (GENERAL REMARKS) In this section, we describe the mathematical setting of the routing problem with constraints. We follow the informative setting of the previous section. In addition, we △ △ assume that N = {1; 2; ...}, N0 = {0} N = {0; 1; 2; ...}, 641
(5)
(recall that, for three nonempty sets A, B and C, A × B × C = (A×B)×C; in particular, X ×X ×N = (X ×X )×N). Then, our basic problem is formulated as (6) Cα [(zt )t∈0,N ] −→ min, (α, (zt )t∈0,N ) ∈ D.
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4. ALGORITHM ON FUNCTIONAL LEVEL To solve (6), in Chentsov (2008a, 2014a,b) (and in other articles), an economic variant of DP was constructed. On this basis, algorithms were developed and implemented on PC. Of course, these algorithms can be applied for problems of moderate dimensions. But it is also possible to employ them in construction of optimized insertions and iterated procedures making use of such insertions; see Chentsov (2014c); Petunin (2014). In connection with the problem concerning sheet cutting on the CNC machines, we note Petunin (2009); Frolovskij (2005). Some questions on tool path route optmization for CNC cutting machines were considered in Hoeft (1997); Dewil (2011, 2014); Xie (2009); Yang (2010). However, unlike the proposed mathematical model, this works do not take into account the dependence of cost functions on the list of completed jobs, which is important for technological restrictions compliance (Petunin (2015)). General questions of solution of traveling salesman problem (TSP) are considered in Melamed (1989a,b,c); Gutin (2002), and many other publications. In connection with the application of DP for solution of TSP, we note Bellman (1958); Held (1962). Let us briefly sketch the procedure. To this end, we introduce the operator I of (Chentsov , 2008a, Ch.2), I : N −→ N. Namely, for K ∈ N, let
Thus we obtain the required sets Dj , j ∈ 0, N . For a more detailed construction, refer to (Chentsov , 2008a, §4.9). If s ∈ 1, N , (x, K) ∈ Ds , j ∈ I(K), and z ∈ Mj , then (pr2 (z), K \ {j}) ∈ Ds−1 . We note that D0 �= ∅, D1 �= ∅,..., DN �= ∅. The Layers of the Bellman function. We are going to describe the following procedure of construction of the realvalued functions: v0 −→ v1 −→ ... −→ vN . Let v0 ∈ R+ [D0 ] and let △ ˜ v0 (x, ∅) = f (x) ∀x ∈ M.
Let s ∈ 1, N and suppose vs−1 is already constructed. Then, vs ∈ R+ [Ds ] is defined by the rule △
vs (x, K) = min min [c(x, pr1 (z), K)+
(7)
j∈I(K) z∈Mj
+cj (z, K) + vs−1 (pr2 (z), K \ {j})] ∀(x, K) ∈ Ds . Thus we obtain the recurrent procedure v0 −→ v1 −→ △ ... −→ vN . In addition, V = vN (x0 , 1, N ) ∈ [0, ∞[ is realized. Moreover, V is the global extremum for our routing problem (6); namely, V = min Cα [z] = min (α,z)∈D
min
α∈A (zt )t∈0,N ∈Zα
Cα [(zt )t∈0,N ]. (8)
Optimal solution. From (7) and (8), we obtain
△
I(K) = K \ {pr2 (z) : z ∈ Ξ[K]},
V =
△
where Ξ[K] = {z ∈ K | (pr1 (z) ∈ K)&(pr2 (z) ∈ K)}.
△
let Gs = {K ∈ G | s = |K|}, where, for a nonempty finite set F, its cardinality is denoted by |F|. Of course, GN = {1, N } (the singleton containing the set 1, N ). Let
Gs−1 = {K \ {t} : K ∈ Gs , t ∈ I(K)} ∀s ∈ 2, N . We obtain a natural step-by-step procedure: GN −→ GN −1 −→ ... −→ G1 (if N > 2, obviously). Layers of position space. An ordered pair (x, K), where x ∈ X and K ∈ N {∅}, is considered a position. We construct sets D0 , D1 , ..., DN in the state space. Suppose ˜ is the union of all sets Mj , j ∈ 1, N \ K1 . Then, that M △ △ ˜ and DN = D0 = {(x, ∅) : x ∈ M} {(x0 , 1, N )} (a 0 singleton containing the ordered pair (x , 1, N )). Consider the construction procedure for the set Ds , where s ∈ 1, N − 1. For K ∈ Gs , let △ △ ˆ s [K] = Js (K) = {j ∈ 1, N \ K | {j} K ∈ Gs+1 }, M △ = Mj , j∈Js (K)
(9)
△
Let z0 = (x0 , x0 ). Using (9), we choose η1 ∈ I(1, N ) and z(1) ∈ Mη1 for which V = c(x0 , pr1 (z(1) ), 1, N ) + cη1 (z(1) , 1, N )+ +vN −1 (pr2 (z(1) ), 1, N \ {η1 }).
△
K1 = {pr1 (z) : z ∈ K}; then, G1 = {{t} : t ∈ 1, N \ K1 }. Finally,
min [c(x0 , pr1 (z), 1, N )+
+cj (z, 1, N ) + vN −1 (pr2 (N ), 1, N \ {j})].
△
Admissible task lists. Set G = {K ∈ N | ∀z ∈ K (pr1 (z) ∈ K) ⇒ (pr2 (z) ∈ K)}; moreover, for s ∈ 1, N ,
min
j∈I(1,N ) z∈Mj
(10)
Then, (pr2 (z(1) ), 1, N \ {η1 }) ∈ DN −1 and, by (7),
=
vN −1 (pr2 (z(1) ), 1, N \ {η1 }) = min min [c(pr2 (z(1) ), pr1 (z), 1, N \ {η1 })+
j∈I(1,N \{η1 }) z∈Mj
+cj (z, 1, N \ {η1 }) + vN −2 (pr2 (z), 1, N \ {η1 ; j})]. We choose η2 ∈ I(1, N \ {η1 }) and z(2) ∈ Mη2 such that vN −1 (pr2 (z(1) ), 1, N \ {η1 }) = = c(pr2 (z(1) ), pr1 (z(2) ), 1, N \ {η1 })+ +cη2 (z(2) , 1, N \ {η1 })+ +vN −2 (pr2 (z(2) ), 1, N \ {η1 , η2 }) ∈ DN −2 .
(11)
From (10) and (11), we obtain V = c(x0 , pr1 (z(1) ), 1, N )+ +c(pr2 (z ), pr1 (z(2) ), 1, N \ {η1 }) + cη1 (z(1) , 1, N )+ +cη2 (z(2) , 1, N \ {η1 }) + vN −2 (pr2 (z(2) ), 1, N \ {η1 ; η2 }). (1)
△ ˆ s [K]}. Finally, let and let Ds [K] = {(x, K) : x ∈ M △ Ds = Ds [K].
This procedure should be continued until the whole list of tasks is exhausted. As a result, we obtain the route
K∈Gs
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Fig. 1. Sample 1. Dynamic programming solution. △
η = (ηj )j∈1,N ∈ A and trajectory (z(j) )j∈0,N ∈ Zη for which Cη [(z(j) )j∈0,N ] = V. Thus, (η, (z(j) )j∈0,N ) ∈ D is the required optimal solution (a more detailed presentation of this scheme is contained in Chentsov (2014a)). 5. COMPUTATIONAL EXPERIMENT Calculations were made on the computer with the Intel i7-2630QM processor, 8GB memory, and the Windows 7 (64-bit) operating system. For small parts, these numbers could be reduced. There is a much data on the coordinates of points and the resulting route; we omit these due to space constraints. All external cost functions are distances between points. Internal cost functions depend on the sets of pending tasks. Example 1. Number of megalopolises N = 27. Dynamic programming method. The calculation took 1 hour 22 min. 41 sec. The value obtained was 89.07. We again note that DP may reasonably be applied to the problems with moderate values of N (N ≈ 31). For problems of larger dimensions, it is possible to use DP to construction local insertions; this possibility was studied in Chentsov A.A., Chentsov A.G. 2 (2014) and Petunin A.A., Chentsov A.G. and Chentsov P.A. (2014). On this basis, we can construct iterative procedures. 6. HEURISTIC ALGORITHM Now, we consider a variant of heuristic algorithm for solving “big” constrained routing problems. To account for some natural constraints, we use cost functions with dependence on the set of pending tasks. However, it is quite hard to implement such cost functions from the computational perspective (the complexity is especially pronounced for cost functions with dependence on the set of pending tasks; however, such cost functions are important for complying with the above-mentioned constraints). The proposed algorithm does not require a construction of “full” cost functions. The corresponding calculations of the required values of cost functions are done as a specific trajectory of a process develops. Using the fragments of 643
Fig. 2. Sample 2. Heuristic solution. these functions, we implement the local greedy choice of the next task. This idea is conceptually similar to the feedback control: calculations are realized for a fully developed position. This is an original approach to solution of large-scale problems, for which DP is computationally infeasible. Moreover, in the given algorithm, system improving corrections are provided; in addition, the corresponding exclusions on permutations are used. Heuristic algorithm consists of several steps. It creates a route by adding the contours step by step. Every next contour to be added to the route is obtained as the one that provides the minimum cost (internal cost functions depend on the preceding part of the route) and satisfies precedence constraints. Example 2. Number of megalopolises N = 112. Heuristic method. The calculation took 1 min. 2 sec. The value obtained was 383.98. For Example 1, the heuristic method obtained the value 89.35. It is close to the DP result. It is the reason why the heuristic algorithm may be used for problems of large dimension. 7. CONCLUSION The considered class of problems has essential singularities that were not considered previously. The principal singularity of this type is connected with the possible dependence of cost functions on the set of pending tasks, which might be necessary to satisfy all the necessary constraints arising in real-life problems and, in particular, in sheet cutting problems for CNC machines. Such means of accounting for the mentioned constraints is a new contribution to the field. It is important that, in this case, the cost functions depend on the set of pending tasks. The authors know of only two works concerned with solving a routing problem where the costs depend on the set of pending tasks, Alkaya (2015); Leon (1996). However, in these papers, there was only considered a very particular case of the TSP with a complicated cost matrix, for which heuristic algorithms were proposed; optimal solution was not attempted. It is not possible to apply those algorithms to the problem our paper is concerned with (we deal
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with a greater number of constraints and, in particular, the precedence constraints). A particular obstacle is the real-time computation of complicated cost functions. The problem considered in the paper is much more complex, and much more specialized algorithms are necessary to solve it. Such an algorithm was presented in Section 6. We note that a DP-based procedure can be applied to testing the heuristics for this problem on sample instances of limited dimension. DP and heuristic procedures ought to be considered together as we did in this paper. 8. ACKNOWLEDGMENTS This work was supported by Act 211 Government of the Russian Federation, contract N 02.A03.21.0006. REFERENCES Ali Fuat Alkaya, Ekrem Duman Combining and solving sequence dependent traveling salesman and quadratic assignment problems in PCB assembly Discrete Applied Mathematics, 10 September 2015, Vol. 192, p. 2-16. Bellman R. On a Routing Problem Quart. Appl. Math, 16, 1958, P.87-90. Chentsov A.G. Ekstremal’nye zadachi marshrutizatsii i raspredeleniya zadaniy: voprosy teorii. M.Izhevsk: NITs ”Regulyarnaya i khaoticheskaya dinamika”, Izhevskiy institut komp’yuternykh issledovaniy, 2008, 240 s. Chentsov A.A., Chentsov A.G. Extremal Bottleneck Routing Problem with Constraints in the Form of Precedence Conditions. Proceedings of the Steklov Institute of Mathematics (Supplementary issues), 2008, vol. 263, no.2 p.23-36. Chentsov A.G. Problem of successive megalopolis traversal with the precedence conditions. Automation and Remote Control, April 2014, Volume 75, Issue 4, p. 728744. Chentsov A.A., Chentsov A.G., Chentsov P.A. Elements of dynamic programming in extremal routing problems. Automation and Remote Control, Volume 75, Issue 3, March 2014, p. 537-550. Chentsov A.A., Chentsov A.G. Zadacha posledovatel’nogo obhoda megapolisov. Vestn. Tambov. un-ta. Ser. Estestv. i tehn. nauki, 2014, T. 19, vyp. 2, S.454-475. Chentsov A.G., Salii Ya.V. A Model of ”Nonadditive” Routing Problem Where the Costs Depend on the Set of Pending Tasks. Bulletin SUSU MMCS, Mathematical Modelling Programming & Computer Software, 2015, vol.8, no.1. p. 24-44. Dewil R., Vansteenwegen P., Cattrysse D. Cutting Path Optimization using Tabu Search. Key Engineering Materials, 473, 2011. p. 739-748. Dewil R., Vansteenwegen P., Cattrysse D. Construction heuristics for generating tool paths for laser cutters. International Journal of Production Research, Mar. 2014, p. 1-20. Frolovskij V.D. Avtomatizacija proektirovanija upravljajushhih programm teplovoj rezki metalla na oborudovanii s ChPU Informacionnye tehnologii v proektirovanii i proizvodstve. N4. M. 2005. S. 63-66. Gutin G., Punnen A. The Traveling Salesman Problem and Its Variations. Berlin: Springer, 2002, 850 p. 644
Held M., Karp R.M. A Dynamic Programming Approach to Sequencing Problems Journal of the Society for Industrial and Applied Mathematics, 1962, N10 (1), P. 196-210. Hoeft J., Palekar U.S. Heuristics for the plate-cutting traveling salesman problem. IIE Transactions, 29, 1997, p. 719-731. V.Jorge Leon, Brett A.Peters Replanning and Analysis of Partial Setup Strategies in Printed Circuit Board Assembly Systems The International Journal of Flexible Manufacturing Systems, 1996, N8, p. 389-412. Melamed I.I., Sergeev S.I., Sigal I.Kh. The traveling salesman problem. Issues in theory Automation and Remote Control, 1989, 50:9, P. 1147-1173. Melamed I.I., Sergeev S.I., Sigal I.Kh. The traveling salesman’s problem. Exact methods. Automation and Remote Control, 1989, 50:10, P. 1303-1324. Melamed I.I., Sergeev S.I., Sigal I.Kh. The traveling salesman problem. Approximate algorithms. Automation and Remote Control, 1989, 50:11, P. 1459-1479. Petunin A.A. O nekotoryh strategijah formirovanija marshruta instrumenta pri razrabotke upravljajushhih programm dlja mashin termicheskoj rezki metalla. Vestnik UGATU. 2009, T.13, N.2, p.280-286. Petunin A.A., Chentsov A.G., Chentsov P.A. Local dynamic programming incuts in routing problems with restrictions. Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2014, N2, p. 56-75. Petunin A.A. Modelling of Tool Path for the CNC Sheet Cutting Machines. AIP Conference Proceedings. 41st International Conference Applications of Mathematics in Engineering and Economics (AMEE’15), 2015, V.1690, p. 060002(1)-060002(7). Xie S.Q., et al. Optimal process planning for compound laser cutting and punch using Genetic Algorithms. International Journal of Mechatronics and Manufacturing Systems, 2009, 2 (1/2), p.20-38. Yang W.B., et al. An Effective Algorithm for ToolPath Airtime Optimization during Leather Cutting. Advanced Materials Research, 2010, 102-104, 373-377.
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IFAC-PapersOnLine 49-12 (2016) 640–644 Routing problems: constraints and Routing problems: constraints and Routing problems: constraints and ⋆ Routing problems: constraints and optimality ⋆ Routing problems: constraints and optimality ⋆ ⋆ optimality ⋆ optimality optimality Alexander G. Chentsov ∗∗ Pavel A. Chentsov ∗∗ ∗∗
Alexander Chentsov ∗∗∗ ∗ Pavel A. Chentsov ∗∗ ∗∗∗∗ Alexander A. G. Petunin Sesekin∗∗ ∗ Pavel A. N. Alexander G. Chentsov Chentsov ∗∗∗ Alexander ∗∗∗∗ ∗ ∗∗ Alexander G. Chentsov A. Chentsov Alexander A. Petunin Alexander N. Sesekin ∗ Pavel ∗∗ Alexander G. Chentsov Pavel A. Chentsov ∗∗∗ ∗∗∗∗ Alexander G. Chentsov Pavel A. Chentsov ∗∗∗ Alexander ∗∗∗∗ Alexander A. Petunin Sesekin ∗∗∗ Alexander N. ∗∗∗∗ Alexander A. Petunin N. Sesekin ∗∗∗ ∗∗∗∗ A. Petunin Alexander N. Sesekin ∗ Alexander Alexander A. Petunin Alexander N. Sesekin Ural Federal University, Russia. Institute of Mathematics and ∗ Ural Federal University, Russia. Institute of Mathematics and ∗ UB RAS, Russia, (Tel: +7-343-375-34-57; e-mail: ∗Mechanics Ural Federal University, Russia. Institute of Mathematics and ∗Mechanics Ural Federal University, Russia. Institute of Mathematics and UB RAS, Russia, (Tel: +7-343-375-34-57; e-mail: ∗ Ural University, Russia. Institute of and Ural Federal Federal University, Russia. Institute of Mathematics Mathematics and [email protected]). Mechanics UB RAS, Russia, (Tel: +7-343-375-34-57; e-mail: Mechanics UB RAS, Russia, (Tel: +7-343-375-34-57; e-mail: [email protected]). ∗∗Mechanics UB RAS, Russia, (Tel: +7-343-375-34-57; e-mail: Mechanics UB RAS, Russia, (Tel: +7-343-375-34-57; e-mail: Ural Federal University, Russia. Institute of Mathematics and [email protected]). ∗∗ [email protected]). Ural Federal University, Russia. Institute of Mathematics and [email protected]). ∗∗Mechanics [email protected]). UB RAS, Russia, (Tel: +7-343-362-81-62; e-mail: ∗∗ Ural Federal University, Russia. Institute of Mathematics and ∗∗ Ural Federal Russia. of Mathematics and UBUniversity, RAS,[email protected]). Russia, (Tel:Institute +7-343-362-81-62; e-mail: ∗∗Mechanics Ural Federal University, Russia. Institute of Mathematics and Ural Federal University, Russia. Institute of Mathematics and Mechanics UB RAS, Russia, (Tel: +7-343-362-81-62; e-mail: UB RAS, Russia, (Tel: +7-343-362-81-62; e-mail: [email protected]). ∗∗∗ Mechanics Mechanics UB RAS, Russia, (Tel: +7-343-362-81-62; e-mail: Mechanics UB RAS, Russia, (Tel: +7-343-362-81-62; e-mail: Ural Federal University, Russia, (Tel: +7-343-375-97-14; e-mail: [email protected]). ∗∗∗ [email protected]). Russia, (Tel: +7-343-375-97-14; e-mail: [email protected]). ∗∗∗ Ural Federal University, [email protected]). [email protected]). ∗∗∗ Ural Federal University, Russia, (Tel: +7-343-375-97-14; e-mail: ∗∗∗ Ural Federal University, Russia, (Tel: +7-343-375-97-14; e-mail: [email protected]). ∗∗∗ Ural Federal University, ∗∗∗∗ Russia, (Tel: +7-343-375-97-14; e-mail: Russia, (Tel: +7-343-375-97-14; e-mail: Ural Federal Federal University, University, Russia, (Tel: +7-343-375-41-40; e-mail: [email protected]). ∗∗∗∗Ural [email protected]). Ural Federal University, Russia, (Tel: +7-343-375-41-40; e-mail: [email protected]). ∗∗∗∗ [email protected]). [email protected]). ∗∗∗∗ Ural Federal University, Russia, (Tel: +7-343-375-41-40; e-mail: ∗∗∗∗ Federal University, Russia, (Tel: +7-343-375-41-40; e-mail: [email protected]). ∗∗∗∗ Ural Ural Russia, Ural Federal Federal University, University, Russia, (Tel: (Tel: +7-343-375-41-40; +7-343-375-41-40; e-mail: e-mail: [email protected]). [email protected]). [email protected]). [email protected]). Abstract: We consider the issues of routing under constraints and formulate a mathematical Abstract: consider the issues The of routing constraints formulate aconstraints. mathematical problem of We visiting megalopolises. order under of visits is subjectand to precedence In Abstract: We consider the issues of routing under constraints and formulate aa mathematical Abstract: We consider the issues of routing under constraints and formulate mathematical problem of visiting megalopolises. The order of visits is subject to precedence constraints. Ina Abstract: We consider the issues of routing under constraints and formulate a mathematical Abstract: We consider the issues of on routing under constraints and formulate aconstraints. mathematical addition, the cost functions depend the set of pending tasks. The quality criterion is problem of visiting megalopolises. The order of visits is subject to precedence In problem ofthe visiting megalopolises. The of visits is subject to precedence addition, cost functions depend onorder the set of pending tasks. The qualityconstraints. criterion is In problem visiting megalopolises. The order of is to precedence constraints. In problem ofthe visiting megalopolises. The order of visits is subject subject to the precedence constraints. Inaa variety ofof the additive criterion. The problem is visits established within dynamic programming addition, cost functions depend on the set of pending tasks. The quality criterion is addition, the cost functions depend on the set of pending tasks. The quality criterion is a variety of the additive criterion. The problem is established within the dynamic programming addition, the cost functions depend on of pending tasks. The quality criterion addition, thehowever, cost functions depend on the the set set ofimplemented pending within tasks. The practical quality programming criterion is ofaa framework, heuristic is proposed and to solve problemsis variety additive The problem is the variety of of the the additiveaa criterion. criterion. The problemand is established established within the dynamic dynamic programming framework, however, heuristic is proposed implemented to solve practical problems of variety of the additive criterion. The problem is established within the dynamic programming variety of the additivea criterion. Theproposed problemand is established within the dynamic programming large dimensionality. framework, however, framework, however, a heuristic heuristic is is proposed and implemented implemented to to solve solve practical practical problems problems of of large dimensionality. framework, however, framework, however, a a heuristic heuristic is is proposed proposed and and implemented implemented to to solve solve practical practical problems problems of of large dimensionality. large dimensionality. © 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. large large dimensionality. dimensionality. 1. INTRODUCTION large dimension. For those, we have to resort to heuristic 1. INTRODUCTION large dimension. For those, we have to resort to heuristic algorithms. 1. large dimension. 1. INTRODUCTION INTRODUCTION large dimension. For For those, those, we we have have to to resort resort to to heuristic heuristic algorithms. 1. INTRODUCTION large 1. of INTRODUCTION large dimension. dimension. For For those, those, we we have have to to resort resort to to heuristic heuristic There is a number distinct applications where rout- algorithms. algorithms. There is a number of distinct applications whereatomic rout- algorithms. algorithms. ing problems arise (e.g. transportation problems, There is a number of distinct applications where rout2. PROBLEM STATEMENT AND GENERAL There is aa number number of distinct distinct applications whereatomic routing problems arise (e.g. transportation problems, There is of applications rout2. PROBLEM STATEMENT GENERAL There is a number of distinct applications where routpower generation, toolpath optimization forwhere CNC cuting problems arise (e.g. transportation problems, atomic NOTATION AND 2. PROBLEM STATEMENT AND GENERAL ing problems arise (e.g. transportation problems, atomic power generation, toolpath optimization for CNC cut2. PROBLEM STATEMENT AND GENERAL ing problems arise (e.g. transportation problems, atomic NOTATION ing problems arise (e.g. transportation problems, atomic 2. PROBLEM STATEMENT AND GENERAL ting machines). In many problems, there is a need to power generation, toolpath optimization for CNC cut2. PROBLEM STATEMENT AND GENERAL NOTATION powermachines). generation, toolpath optimization for CNC cutting In many problems, there is a need to NOTATION power generation, toolpath optimization for CNC cutpowermachines). generation, toolpath optimization for CNC cutNOTATION order, or sequence, the operations. Often, the sequencing ting In many problems, there is a need to NOTATION Let us fix an arbitrary nonempty set X, which will serve ting machines). In many problems, there is a need to order, or sequence, the operations. Often, the ting machines). many problems, there is aait need to us ground fix an arbitrary nonempty setcorresponding X, which willinitial serve ting machines). Invarious many problems, there is sequencing need to Let is complicated byIn constraints. At times, is more 0 order, or sequence, the operations. Often, the sequencing as the set. Let x ∈ X be the order, or sequence, the operations. Often, the sequencing Let us fix an arbitrary nonempty set X, which will serve is complicated by various constraints. At times, it is more 0 order, or the operations. Often, the sequencing sequencing Let us fix an arbitrary nonempty set X, which will serve as the ground set. Let x ∈ X be the corresponding initial order, or sequence, sequence, the operations. Often, the important to satisfy all the constraints than to achieve Let us fix an arbitrary nonempty set X, which will serve is complicated by various constraints. At times, it is more 0 Let us fix an arbitrary nonempty set X, which will serve state, or the base (depot) of our process. Let N be a is complicated by various constraints. At times, it is more 0 as the ground set. Let x ∈ X be the corresponding initial important to satisfy all the constraints than to achieve is complicated by various constraints. At times, it is more 0 as the ground set. Let x ∈ X be the corresponding initial state, or the base (depot) of our process. Let N be a is complicated by various constraints. At times, it is more 0 an optimum for the quality criterion—the typical case in as the ground set. Let x ∈ X be the corresponding initial important to satisfy all the constraints than to achieve as the ground set. Let x ∈ X be the corresponding initial natural number for which N ≥ 2. Fix nonempty finite important tofor satisfy all the thecriterion—the constraints than than to achieve achieve state, or or the the base base (depot) (depot) of of our our process. process. Let Let N N be be a a an optimum the quality typical case in important to satisfy all constraints to state, natural number for which N ≥ 2. Fix nonempty finite important tofor satisfy all thecriterion—the constraints than to achieve real-life engineering. state, or the base (depot) of our process. Let N be an optimum the quality typical case in state,Mor the (depot) of ≥ our2. process. Let N finite be a a ..., MNbase for which an optimum optimum for the the quality quality criterion—the criterion—the typical typical case case in in sets natural for which N Fix nonempty 1 ,number real-life engineering. an for natural number for which N ≥ 2. Fix nonempty finite sets M1 ,number ..., MN for which an optimum for the quality criterion—the typical case in natural for which N ≥ 2. Fix nonempty finite real-life engineering. natural number for which N ≥ 2. Fix nonempty finite real-life engineering. sets M , ..., M for which We consider an optimization problem for the sake of 1 N Mwhich real-life engineering. for 1 ⊂ X, ..., MN ⊂ X. real-life engineering. sets M M111 ,,, ..., ..., M MN for which We consider an optimization problem formore the effective. sake of sets N Mwhich sets M ..., M 1 ⊂ X, ..., MN ⊂ X. N for making the search for admissibleproblem solutionsfor We consider an optimization the sake of ⊂ X, ..., M ⊂ X.Assume M 1 as We regard these sets megalopolises. We consider an optimization problem for the sake of ⊂ X, ..., MN ⊂ M making the search for admissible solutions more effective. We consider an optimization problem for the sake of ⊂ ..., ⊂ X. X. M We this consider an employ optimization problem formore thevariant sake of We regard these sets N megalopolises. To end,search we a pretty complicated ⊂ X, X, ..., M M0N M0 111 as making the search for admissible admissible solutions effective. N ⊂ X.Assume making the for solutions more effective. We regard these sets as megalopolises. Assume To this end, we employ a pretty complicated variant of ∈ / M , ..., x ∈ / M ; x making the search for admissible solutions more effective. We regard these sets as megalopolises. 1 N Assume 0 0 making the search for admissible solutions more effective. dynamic programming (DP) supplemented with special To this end, we employ aa pretty complicated variant of We regard these sets as megalopolises. Assume ∈ / M , ..., x ∈ / M ; x We regard these sets as megalopolises. Assume To this end, we employ pretty complicated variant of 1 N 0 0 dynamic programming (DP) supplemented with special To this end, we employ a pretty complicated variant of 0M 0are ∈ / M , ..., x ∈ / M ;; x To this end, we employ a pretty complicated variant of 1 N constructions which decrease the computational complex, ..., M pairwise disjoint. The moreover, the sets 0 0 dynamic programming (DP) supplemented with special ∈ / M , ..., x ∈ / M x 1 N 1 N 0 0 dynamic programming (DP) supplemented with special ∈ M , ..., ...,Nx x are ∈ M ; x M∈ constructions which decrease computational complex1 N ..., pairwise disjoint. and, The moreover, the must setsx //1 ,M ∈ //a M dynamic programming (DP) supplemented with special 1 ,M N; dynamic programming (DP) the supplemented with special ity though the use precedence constraints. It is natural to megalopolises be visited in certain sequence constructions which decrease the computational complex, ..., M are pairwise The moreover, the sets M 1 , ..., MN are pairwise disjoint. constructions which decrease the computational complexdisjoint. The moreover, the sets M ity though the use precedence constraints. It is natural to 1 , visited N in megalopolises must be a certain sequence and, constructions which decrease the computational complex..., M are pairwise disjoint. The moreover, the sets M 1 N constructions which decrease the computational complex, ..., M are pairwise disjoint. The moreover, the sets M use DP in local way, which is implemented as insertions (or in each of them, a certain job has to be completed, ity though the use precedence constraints. It is natural to 1 N megalopolises must be visited in a certain sequence and, ity though the use precedence constraints. It is natural to megalopolises must be visited in a certain sequence and, use DP in local way, which is implemented as insertions (or in each of them, a certain job has to be completed, ity though the use precedence constraints. It is natural to megalopolises must be visited in a certain sequence and, ity though the use precedence constraints. It is natural to megalopolises must be visited in a certain sequence and, “incuts”) of moderate dimension. Basic constructions of which results into a kind of interior movement within use DP in local way, which is implemented as insertions (or in each of them, a certain job has to be completed, use DP DP in in local local way, which which is implemented implemented as insertions insertions (or (or in each of them, aa certain has to be completed, “incuts”) of moderate dimension. Basic(2008a, constructions of which results intoWe kind of job interior movement within use way, is as in each of to be completed, use DP in local way, which isinimplemented as insertions (or in each of them, them, certain job has to be completed, this approach are reflected Chentsov 2014a,b). megalopolis. consider thehas arrival point and the “incuts”) of dimension. Basic constructions constructions of which results into aaaa certain kind of job interior movement within “incuts”) of moderate moderate dimension. Basic of each which results into kind of interior movement within this approach are reflected in Chentsov (2008a, 2014a,b). each megalopolis. We consider the arrival point and the “incuts”) of moderate dimension. Basic constructions of which results into a kind of interior movement within “incuts”) of moderate dimension. Basic(2008a, constructions of departure which results into a kind of interior movement within Of course, the above-mentioned mathematical problem has point. The pairs these points are evaluated this approach are reflected in Chentsov 2014a,b). each megalopolis. We consider the arrival point and the thiscourse, approach are reflected in Chentsov (2008a, 2014a,b). each megalopolis. We consider the arrival point and the Of the above-mentioned mathematical problem has departure point. The pairs of these points are evaluated this approach are reflected in Chentsov (2008a, 2014a,b). each megalopolis. We consider the arrival point and the thiscourse, approach are reflected in Chentsov (2008a, 2014a,b). each megalopolis. We consider the arrival point and the many applications. by the corresponding cost functions. Of the above-mentioned mathematical problem has departure point. The pairs of these points are evaluated Of course, the above-mentioned mathematical problem has departure point. The pairs of these points are evaluated many applications. by the corresponding cost functions. Of course, the above-mentioned mathematical problem has departure point. The pairs of these points are evaluated Of course, the above-mentioned mathematical problem has departure point. The pairs of these points are evaluated many applications. by the corresponding cost functions. 0 many applications. by the corresponding cost functions. It is important that the DP-based procedure we employ Next, exterior permutation (from x0 to megalopolises many applications. the corresponding cost many applications. by theeach corresponding cost functions. functions. It is important that the DP-based procedure constrained we employ by Next, each exterior permutation (from x0 to megalopolises provides for optimal solutions of precedence and between megalopolises) is evaluated. Finally, for terIt is important that the DP-based procedure we employ 0 to Next, each exterior permutation (from x megalopolises It is is important important that the the DP-based procedure constrained we employ employ Next, 0 each exterior permutation (from x megalopolises provides for optimal solutions of precedence and between megalopolises) is evaluated. Finally, for ter0 to It that DP-based procedure we Next, each exterior permutation (from x to megalopolises It is important that the DP-based procedure we employ Next, each exterior permutation (from x to megalopolises problems where the cost functions depend on the set minal state, the corresponding evaluation is defined. We provides for optimal solutions of precedence constrained and between megalopolises) is evaluated. Finally, for terprovides for for optimal solutions of precedence precedence constrained and between megalopolises) is evaluated. Finally, for terproblems where the cost functions depend on the set minal state, the corresponding evaluation is defined. We provides optimal solutions of constrained and between megalopolises) evaluated. for provides for optimal solutions of precedence and between megalopolises) isthe evaluated. Finally, for terterof pending tasks. This is its main feature asconstrained farthe as the consider the the casecorresponding when all is costs areFinally, summed, the problems where the cost functions depend on set minal state, the corresponding evaluation is defined. We problems where the cost functions depend on the set minal state, evaluation is defined. We of pending tasks. This is its main feature as far as the consider the case when all the costs are summed, the problems where the cost functions depend on the set minal state, the corresponding evaluation is defined. We problems where the cost functions depend on the set minal state, the corresponding evaluation defined. We DP structure is concerned (the optional dependence of additive case. However, the optimization is complicated of pending tasks. This is its main feature as far as the consider the case when all the costs are summed, the of pending pending tasks. This is its main feature as far as as the the consider the case when all the costs are summed, the DP structure is concerned (the optional dependence of additive case. However, the optimization is complicated of tasks. This is its main feature as far consider the case when all the costs are summed, the of pending tasks. This is its main feature as far as the consider the case when all the costs are summed, the cost functions on the set of pending tasks being its most by the constraints, which restrict both the routes defined DP structure is concerned (the optional dependence of additive case. However, the optimization is complicated DP structure is concerned (the optional dependence of additive case. However, the optimization is complicated cost functions on the set of pending tasks being its most by the constraints, which restrict both the routes defined DP structure is concerned (the optional dependence of additive case. However, the optimization is complicated DP structure is concerned (the optional dependence of additive case. However, the optimization is complicated essential and innovative part). The main issue in using DP as permutations of the megalopolises’ indices and the cost functions on the set of pending tasks being its most by the constraints, which restrict both the routes defined cost functions functions on the of tasks being its by the constraints, restrict both the routes defined essential andof innovative The(DP main using DP as permutations ofwhich themotions. megalopolises’ indices the cost on the set setpart). of pending pending tasks being its most most by constraints, restrict both routes defined cost on the set of pending tasks its most by the constraints, which restrictSpecifically, both the the routes defined is thefunctions deficit computer memory is issue notbeing ain polynomial of specific we and consider essential and innovative part). The main issue in using as the permutations ofwhich the megalopolises’ megalopolises’ indices and the essential andof innovative part). The(DP main issue in using DP DP trajectories as permutations of the indices and the is the deficit computer memory is not a polynomial trajectories of specific motions. Specifically, we consider essential and innovative part). The main issue in using DP as permutations of the megalopolises’ indices and the essential andwhich innovative The main using DP asprocess permutations of the themotions. megalopolises’ and the procedure), makespart). its use infeasible foraain problems of atrajectories defined by following scheme:indices is the deficit of computer memory (DP is issue not polynomial of specific Specifically, we consider is the deficit of computer memory (DP is not polynomial trajectories of specific motions. Specifically, we consider procedure), which makes its use infeasible for problems of atrajectories process defined by the following scheme: is the deficit of computer memory (DP is not a polynomial trajectories of specific motions. Specifically, we consider is the deficit of computer memory (DP is not a polynomial of specific motions. Specifically, we consider procedure), which makes its use infeasible for problems of a process defined by the following scheme: 0 ⋆ procedure), which makesbyits its use use infeasible for ofproblems problems of defined by the x0 −→ (x1,1 ∈ ❀ x1,2 ∈scheme: Mα(1) ) −→ ... This work was supported 211 Government the Russian α(1)following procedure), which makes infeasible for of aaa process process defined byM the following scheme: procedure), which makesbyitsAct use infeasible for ofproblems of ⋆ process defined by the (1) x0...−→ (x(x M ❀❀ x1,2x ∈scheme: M∈α(1) ) −→), ... This workcontract was supported Act 211 Government the Russian 1,1 ∈ α(1)following ⋆ Federation, N 02.A03.21.0006 ∈ M Mα(N −→ 0 −→ (1) x (x ∈ M ❀ x ∈ )) −→ N,1 N,2 This work was supported by Act 211 Government of the Russian α(N ) ) ... 1,1 1,2 ⋆ α(1) ❀ x1,2 ∈ M α(1) 0 x −→ (x ∈ M M −→ ... This work was supported by Act 211 Government of the Russian 1,1 0 Federation, contract N 02.A03.21.0006 ⋆ α(1) α(1) ∈ M ❀ x ∈ M ), ... −→ (x (1) N,1 N,2 x (x ∈ ❀ x ∈ )) −→ ⋆ α(N α(N ) ... This was by 1,1 α(1) (1) x ...−→ −→ (x(x ∈M M ❀)) ❀ x1,2 ∈M M∈α(1) −→ ... This work workcontract was supported supported by Act Act 211 211 Government Government of of the the Russian Russian 1,1N,1 1,2xN,2 Federation, N 02.A03.21.0006 α(1) α(1) ∈ M M ), −→ (1) α(N α(N ) Federation, contract N 02.A03.21.0006 ∈ M ❀ x ∈ M ), ... −→ (x (1) N,1 N,2 α(N ) α(N ) Federation, contract N 02.A03.21.0006 ∈ M ❀ x ∈ M ), ... −→ (x N,1 N,2 Federation, contract N 02.A03.21.0006 α(N ) α(N ) ∈ M ❀ x ∈ M ), ... −→ (x N,1 N,2 α(N ) α(N ) Copyright © 2016 IFAC 640 Copyright © 2016, 2016 IFAC 640Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2016 IFAC 640 Copyright 2016 IFAC 640 Peer review© of International Federation of Automatic Control. Copyright ©under 2016 responsibility IFAC 640 Copyright © 2016 IFAC 640 10.1016/j.ifacol.2016.07.756
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where α is a route, or indices’ permutation. Constraints are imposed on α, x1 = (x1,1 , x1,2 ), ..., xN −1 = (xN −1,1 , xN −1,2 ), and xN = (xN,1 , xN,2 ) (here, it is supposed that N > 2). In (1), the straight arrows denote the exterior movements and the sinuous arrows denote the movements connected with interior jobs. As mentioned above, all steps in (1) are evaluated through the corresponding cost functions. The sum of their costs defines the additive criterion. Now, let us recall the constraints. We note only the most important. The choice of the route α in (1) can be constrained by precedence constraints. Namely, there may be a nonempty set of ordered sender-receiver pairs. The route α must order the megalopolises so as to make sure that each sender megalopolis appears before its corresponding receiver. These are the route constraints. Now for trajectory constraints, which restrict x1 , ..., xN in (1). Previously, we remarked that, in many cases, this trajectory must be realized in the form x1 ∈ Mα(1) , ..., xN ∈ Mα(N ) , where Mj ⊂ Mj × Mj for all j ∈ 1, N . Now, we note that Mj defines all possible variants of conducting the interior jobs connected with the megalopolis Mj , which are denoted by sinuous arrows in (1). In this connection, see the following example: Example. Consider the problem of routing the tool in sheet cutting CNC machines. The tool arrives at the contour to cut it; it must arrive at a given penetration point near the equidistant of the contour, and it is the equidistant that is to be cut to create a desired contour— roughly speaking, the cut is made such that there is some space between the desired contour and the cut. After the contour is cut, the tool, while still active, must be driven to the corresponding shut-off point. It is important to observe precedence constraints: each interior (child) contour of a feature must be cut out before the exterior (parent) contour. There are also other constraints, for example, the sheet rigidity must be maintained at all times (a new cut may only be done in the rigidity zone), and the sheet must remain in one piece (i.e., not cut in two, etc.), which may be formalized as the dependence of the cost on the set of pending tasks. Thus there appears a kind of memory, or sequence dependence. Our principal aim in constructing the route and trajectory is to satisfy all those constraints. The quality criterion is the total amount of time required to cut all the features. To provide for satisfaction of certain constraints, we may make the cost of “prohibited” actions effectively infinite; thus optimization would contribute to constraint satisfaction. We propose a method for constructing a route subject to certain constraints that is related to DP. However, the mentioned constraints complicate the construction of the DP procedure.
641
△
and, for p ∈ N0 and q ∈ N0 , p, q = {i ∈ N0 | (p ≤ i)&(i ≤ q)} (of course, p, q = ∅ under q < p). For a nonempty set H, denote by R+ [H] set of all nonnegative real-valued △
functions defined on H. Let Mj = {pr2 (z) : z ∈ Mj } for j ∈ 1, N (pr1 (h) and pr2 (h) denote the first and second elements of an arbitrary ordered pair h). Let N N △ △ X = {x0 } ( Mi ), X = {x0 } ( Mj ), i=1
i=1
X ⊂ X ⊂ X. In addition, X and X are nonempty finite sets. Consider the possible variants of precedence constraints. Fix the set K with the property K ⊂ 1, N × 1, N . For every nonempty set K0 , K0 ⊂ K, assume (2) ∃z0 ∈ K0 : pr1 (z0 ) �= pr2 (z) ∀z ∈ K0 ; see (Chentsov , 2008a, Condition 2.2.1) (condition (2) is fulfilled in many practically interesting cases; see (Chentsov , 2008a, ch.2)). Then, △
A = {α ∈ P | ∀z ∈ K ∀t1 ∈ 1, N ∀t2 ∈ 1, N ((α(t1 ) = pr1 (z))&(α(t2 ) = pr2 (z))) ⇒ (t1 < t2 )} = (3) = {α ∈ P | α−1 (pr1 (z)) < α−1 (pr2 (z)) ∀z ∈ K} �= ∅, where P is the set of all permutations of 1, N and, for every β ∈ P, the symbol β −1 denotes the inverse of the permutation β. Thus, an admissible (in view of precedence constraints) route exists. ˜ we denote the set of all tuples (zi ) By Z i∈0,N : 0, N −→ X × X and △ ˜ | (z0 = (x0 , x0 ))&(zt ∈ Mα(t) Zα = {(zi )i∈0,N ∈ Z (4) ∀t ∈ 1, N )} �= ∅ ∀α ∈ A. In (4), the trajectories coordinated with a route are introduced. Then, △
˜ | (zi ) D = {(α, (zi )i∈0,N ) ∈ A × Z i∈0,N ∈ Zα } �= ∅ is the set of all admissible solutions. Cost functions. In the following, we consider an “additive” routing problem similar to Chentsov (2014a,b) (a variant of bottleneck problem was considered in Chentsov (2008b, 2015) and in other articles). Now, we introduce the costs of movements, interior jobs, and the terminal state. Denote by N the family of all nonempty subsets of 1, N . Fix c ∈ R+ [X × X × N], c1 ∈ R+ [X × X × N],..., cN ∈ R+ [X × X × N] and f ∈ R+ [X ]. Mathematical setting of problem. If α ∈ A and (zt )i∈0,N ∈ Zα , then △
Cα [(zt )t∈0,N ] =
N
[c(pr2 (zt−1 ), pr1 (zt ),
t=1
{α(j) : j ∈ t, N }) + cα(t) (zt , {αj : j ∈ t, N })]+ +f (pr2 (zN ))
3. EXTREMAL PROBLEM (GENERAL REMARKS) In this section, we describe the mathematical setting of the routing problem with constraints. We follow the informative setting of the previous section. In addition, we △ △ assume that N = {1; 2; ...}, N0 = {0} N = {0; 1; 2; ...}, 641
(5)
(recall that, for three nonempty sets A, B and C, A × B × C = (A×B)×C; in particular, X ×X ×N = (X ×X )×N). Then, our basic problem is formulated as (6) Cα [(zt )t∈0,N ] −→ min, (α, (zt )t∈0,N ) ∈ D.
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4. ALGORITHM ON FUNCTIONAL LEVEL To solve (6), in Chentsov (2008a, 2014a,b) (and in other articles), an economic variant of DP was constructed. On this basis, algorithms were developed and implemented on PC. Of course, these algorithms can be applied for problems of moderate dimensions. But it is also possible to employ them in construction of optimized insertions and iterated procedures making use of such insertions; see Chentsov (2014c); Petunin (2014). In connection with the problem concerning sheet cutting on the CNC machines, we note Petunin (2009); Frolovskij (2005). Some questions on tool path route optmization for CNC cutting machines were considered in Hoeft (1997); Dewil (2011, 2014); Xie (2009); Yang (2010). However, unlike the proposed mathematical model, this works do not take into account the dependence of cost functions on the list of completed jobs, which is important for technological restrictions compliance (Petunin (2015)). General questions of solution of traveling salesman problem (TSP) are considered in Melamed (1989a,b,c); Gutin (2002), and many other publications. In connection with the application of DP for solution of TSP, we note Bellman (1958); Held (1962). Let us briefly sketch the procedure. To this end, we introduce the operator I of (Chentsov , 2008a, Ch.2), I : N −→ N. Namely, for K ∈ N, let
Thus we obtain the required sets Dj , j ∈ 0, N . For a more detailed construction, refer to (Chentsov , 2008a, §4.9). If s ∈ 1, N , (x, K) ∈ Ds , j ∈ I(K), and z ∈ Mj , then (pr2 (z), K \ {j}) ∈ Ds−1 . We note that D0 �= ∅, D1 �= ∅,..., DN �= ∅. The Layers of the Bellman function. We are going to describe the following procedure of construction of the realvalued functions: v0 −→ v1 −→ ... −→ vN . Let v0 ∈ R+ [D0 ] and let △ ˜ v0 (x, ∅) = f (x) ∀x ∈ M.
Let s ∈ 1, N and suppose vs−1 is already constructed. Then, vs ∈ R+ [Ds ] is defined by the rule △
vs (x, K) = min min [c(x, pr1 (z), K)+
(7)
j∈I(K) z∈Mj
+cj (z, K) + vs−1 (pr2 (z), K \ {j})] ∀(x, K) ∈ Ds . Thus we obtain the recurrent procedure v0 −→ v1 −→ △ ... −→ vN . In addition, V = vN (x0 , 1, N ) ∈ [0, ∞[ is realized. Moreover, V is the global extremum for our routing problem (6); namely, V = min Cα [z] = min (α,z)∈D
min
α∈A (zt )t∈0,N ∈Zα
Cα [(zt )t∈0,N ]. (8)
Optimal solution. From (7) and (8), we obtain
△
I(K) = K \ {pr2 (z) : z ∈ Ξ[K]},
V =
△
where Ξ[K] = {z ∈ K | (pr1 (z) ∈ K)&(pr2 (z) ∈ K)}.
△
let Gs = {K ∈ G | s = |K|}, where, for a nonempty finite set F, its cardinality is denoted by |F|. Of course, GN = {1, N } (the singleton containing the set 1, N ). Let
Gs−1 = {K \ {t} : K ∈ Gs , t ∈ I(K)} ∀s ∈ 2, N . We obtain a natural step-by-step procedure: GN −→ GN −1 −→ ... −→ G1 (if N > 2, obviously). Layers of position space. An ordered pair (x, K), where x ∈ X and K ∈ N {∅}, is considered a position. We construct sets D0 , D1 , ..., DN in the state space. Suppose ˜ is the union of all sets Mj , j ∈ 1, N \ K1 . Then, that M △ △ ˜ and DN = D0 = {(x, ∅) : x ∈ M} {(x0 , 1, N )} (a 0 singleton containing the ordered pair (x , 1, N )). Consider the construction procedure for the set Ds , where s ∈ 1, N − 1. For K ∈ Gs , let △ △ ˆ s [K] = Js (K) = {j ∈ 1, N \ K | {j} K ∈ Gs+1 }, M △ = Mj , j∈Js (K)
(9)
△
Let z0 = (x0 , x0 ). Using (9), we choose η1 ∈ I(1, N ) and z(1) ∈ Mη1 for which V = c(x0 , pr1 (z(1) ), 1, N ) + cη1 (z(1) , 1, N )+ +vN −1 (pr2 (z(1) ), 1, N \ {η1 }).
△
K1 = {pr1 (z) : z ∈ K}; then, G1 = {{t} : t ∈ 1, N \ K1 }. Finally,
min [c(x0 , pr1 (z), 1, N )+
+cj (z, 1, N ) + vN −1 (pr2 (N ), 1, N \ {j})].
△
Admissible task lists. Set G = {K ∈ N | ∀z ∈ K (pr1 (z) ∈ K) ⇒ (pr2 (z) ∈ K)}; moreover, for s ∈ 1, N ,
min
j∈I(1,N ) z∈Mj
(10)
Then, (pr2 (z(1) ), 1, N \ {η1 }) ∈ DN −1 and, by (7),
=
vN −1 (pr2 (z(1) ), 1, N \ {η1 }) = min min [c(pr2 (z(1) ), pr1 (z), 1, N \ {η1 })+
j∈I(1,N \{η1 }) z∈Mj
+cj (z, 1, N \ {η1 }) + vN −2 (pr2 (z), 1, N \ {η1 ; j})]. We choose η2 ∈ I(1, N \ {η1 }) and z(2) ∈ Mη2 such that vN −1 (pr2 (z(1) ), 1, N \ {η1 }) = = c(pr2 (z(1) ), pr1 (z(2) ), 1, N \ {η1 })+ +cη2 (z(2) , 1, N \ {η1 })+ +vN −2 (pr2 (z(2) ), 1, N \ {η1 , η2 }) ∈ DN −2 .
(11)
From (10) and (11), we obtain V = c(x0 , pr1 (z(1) ), 1, N )+ +c(pr2 (z ), pr1 (z(2) ), 1, N \ {η1 }) + cη1 (z(1) , 1, N )+ +cη2 (z(2) , 1, N \ {η1 }) + vN −2 (pr2 (z(2) ), 1, N \ {η1 ; η2 }). (1)
△ ˆ s [K]}. Finally, let and let Ds [K] = {(x, K) : x ∈ M △ Ds = Ds [K].
This procedure should be continued until the whole list of tasks is exhausted. As a result, we obtain the route
K∈Gs
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Fig. 1. Sample 1. Dynamic programming solution. △
η = (ηj )j∈1,N ∈ A and trajectory (z(j) )j∈0,N ∈ Zη for which Cη [(z(j) )j∈0,N ] = V. Thus, (η, (z(j) )j∈0,N ) ∈ D is the required optimal solution (a more detailed presentation of this scheme is contained in Chentsov (2014a)). 5. COMPUTATIONAL EXPERIMENT Calculations were made on the computer with the Intel i7-2630QM processor, 8GB memory, and the Windows 7 (64-bit) operating system. For small parts, these numbers could be reduced. There is a much data on the coordinates of points and the resulting route; we omit these due to space constraints. All external cost functions are distances between points. Internal cost functions depend on the sets of pending tasks. Example 1. Number of megalopolises N = 27. Dynamic programming method. The calculation took 1 hour 22 min. 41 sec. The value obtained was 89.07. We again note that DP may reasonably be applied to the problems with moderate values of N (N ≈ 31). For problems of larger dimensions, it is possible to use DP to construction local insertions; this possibility was studied in Chentsov A.A., Chentsov A.G. 2 (2014) and Petunin A.A., Chentsov A.G. and Chentsov P.A. (2014). On this basis, we can construct iterative procedures. 6. HEURISTIC ALGORITHM Now, we consider a variant of heuristic algorithm for solving “big” constrained routing problems. To account for some natural constraints, we use cost functions with dependence on the set of pending tasks. However, it is quite hard to implement such cost functions from the computational perspective (the complexity is especially pronounced for cost functions with dependence on the set of pending tasks; however, such cost functions are important for complying with the above-mentioned constraints). The proposed algorithm does not require a construction of “full” cost functions. The corresponding calculations of the required values of cost functions are done as a specific trajectory of a process develops. Using the fragments of 643
Fig. 2. Sample 2. Heuristic solution. these functions, we implement the local greedy choice of the next task. This idea is conceptually similar to the feedback control: calculations are realized for a fully developed position. This is an original approach to solution of large-scale problems, for which DP is computationally infeasible. Moreover, in the given algorithm, system improving corrections are provided; in addition, the corresponding exclusions on permutations are used. Heuristic algorithm consists of several steps. It creates a route by adding the contours step by step. Every next contour to be added to the route is obtained as the one that provides the minimum cost (internal cost functions depend on the preceding part of the route) and satisfies precedence constraints. Example 2. Number of megalopolises N = 112. Heuristic method. The calculation took 1 min. 2 sec. The value obtained was 383.98. For Example 1, the heuristic method obtained the value 89.35. It is close to the DP result. It is the reason why the heuristic algorithm may be used for problems of large dimension. 7. CONCLUSION The considered class of problems has essential singularities that were not considered previously. The principal singularity of this type is connected with the possible dependence of cost functions on the set of pending tasks, which might be necessary to satisfy all the necessary constraints arising in real-life problems and, in particular, in sheet cutting problems for CNC machines. Such means of accounting for the mentioned constraints is a new contribution to the field. It is important that, in this case, the cost functions depend on the set of pending tasks. The authors know of only two works concerned with solving a routing problem where the costs depend on the set of pending tasks, Alkaya (2015); Leon (1996). However, in these papers, there was only considered a very particular case of the TSP with a complicated cost matrix, for which heuristic algorithms were proposed; optimal solution was not attempted. It is not possible to apply those algorithms to the problem our paper is concerned with (we deal
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