Perencanaan Drainase Jalan Pahlawan Dan Jalan Sriwijaya, Semarang

Banjir adalah aliran air di permukaan tanah yang relatif tinggi dan tidak dapat ditampung oleh saluran drainase atau sungai, sehingga melimpah ke kanan dan ke kiri dari saluran drainase serta menimbulkan genangan/aliran dalam jumlah yang melebihi normal dan mengakibatkan kerugian pada manusia. Jalan Pahlawan dan Jalan Sriwijaya Kota Semarang merupakan kawasan pusat kegiatan dan pelayanan di Kota Semarang. Pada saat hujan terjadi banjir atau genangan di jalan tersebut. Hal ini disebabkan karena saluran drainase berkurang kapasitasnya akibat sedimentasi, serta kurangnya saluran inlet yang beroperasi maksimal. Oleh karena itu diperlukan analisis hidrologi untuk mencari debit banjir rencana pada saluran drainase sehingga didapat dimensi optimal untuk saluran drainase dan jumlah serta dimensi saluran inlet yang dibutuhkan. Evaluasi kapasitas eksisting dengan debit rencana yang diperoleh dari aplikasi InfoSWMM untuk mengetahui daerah mana yang berpotensi terjadi genangan. Hasil evaluasi menunjukkan ada beberapa saluran yang tidak memenuhi syarat dari segi kapasitas dan kemiringan dasar saluran, serta kurangnya jumlah saluran inlet. Sehingga perlu dilakukan perencanaan dimensi saluran baru dan kemiringan saluran baru. Perencanaan kemiringan saluran dilakukan dengan bentuk saluran tetap. Perencanaan gorong-gorong menggunakan gorong-gorong U-Ditch. Perencanaan inlet menggunakan dua jenis inlet yaitu kerb inlet A2nh dengan dimensi 13/16 x 30 x 50 cm dan gutter inlet dengan dimensi gril saluran 49 x 35 x 2 cm

A branch-and-cut method for the bi-objective bi-dimensional knapsack problem

International audienceMulti-objective multi-dimensional knapsack problems (pOmDKP) are widely used to represent practical problems as capital budgeting or allocating processors. It aims to select a subset of n items such that the sum of weight of the selected items does not exceed the capacity on any of the m dimensions, while maximizing p objective functions. Each item has a weight on each dimension and a profit for each objective function. This problem is known for being particularly difficult as soon as the number of dimensions exceeds one, even in its single-objective version.There are many published papers focusing on the exact solution of multi-objective single-dimensional knapsack. The solutions methods are often two-phases methods. The second phase is either a branch-and-bound method (as in [1] for the bi-objective case or in [2] for the three-objective case), either a dynamic programming method [3], or a dedicated ranking method [2].Only a few works have studied exactly the multi-objective multi-dimensional case. Concerning the single-objective multi-dimensional knapsack problem, many works have investigated cutting inequalities to speed-up the computation of solution [4].In this work we are interested in the exact solution of the bi-objective bi-dimensional knapsack problem (2O2DKP), using a branch-and-cut method. A branch-and-cut method is a combination of a cutting plane method and a branch-and-bound method. According to its name, one of the main component of a branch-and-bound method aims at computing bounds of the problem. Convex relaxation has been a key component for successful bi-objective branch-and-bound algorithm (see for example [5]). It defines indeed a tight upper bound set, which can be computed easily if the single-objective version of the problem can be solved in (pseudo-)polynomial time. However, this is not the case for 2O2DKP. On the contrary, the linear relaxation remains relatively easy to compute, but the resulting bound set is less tight, which makes more difficult the exploration of nodes and leads to larger search-trees. To improve the quality of the upper bound set based on linear relaxation, we introduce cover inequalities at each node of the branch-and-bound method, turning it to a branch-and-cut method. Cover inequalities consist of cuts defined for single-objective binary problems [6]. A cover is a set of objects such that the sum of the weights associated to these objects exceeds the capacity. In [6], the authors remark that computing all possible cover inequalities would be time-consuming and even impossible to implement. Instead, they consider the optimal solution of the linear relaxation and solve a smaller binary problem to find a cover inequality that is violated. In the bi-objective context, the linear relaxation is described by a set of extreme points, which are associated to efficient solutions. Moreover, each of these efficient solutions may be fractional and have a different subset of fractional variables. The generation of cover inequalities is therefore more complex, particularly to get a good tradeoff between quality of the improved upper bound set defined and computational time. This leads to numerous strategies to generate cover inequalities. This presentation will describe the mechanisms used in the multi-objective branch-and-cut method that we have developed (separation procedure, bound sets, generation of cover inequalities...). These strategies have been then experimentally validated. [1] Visée, M., Teghem, J., Pirlot, M., Ulungu, E. L., March 1998. Two-phases method and branch and bound procedures to solve the bi–objective knapsack problem. Journal of Global Optimization 12, 139–155. [2] Jorge, J., May 2010. Nouvelles propositions pour la résolution exacte du sac à dos multi-objectif unidimensionnel en variables binaires. Thèse, Université de Nantes.[3] Delort, C., Spanjaard, O., 2010. Using bound sets in multiobjective optimization: Application to the biobjective binary knapsack problem. In: Festa, P. (Ed.), SEA. Vol 6049 of Lecture Notes in Computer Science. Springer, 253-265.[4] Osorio, M. A., Glover, F., Hammer, P., 2002. Cutting and surrogate constraint analysis for improved multidimensional knapsack solutions. Annals of Operations Research 117 (1-4), 71–93.[5] Sourd F. and Spanjaard O., 2008. A multi-objective branch-and bound framework: Application to the biobjective spanning tree problem. INFORMS Journal on Computing, 20:472-484.[6] Crowder, H., Johnson, E. L., Padberg, M. W., 1983. Solving large-scale zero-one linear programming problems. Operations Research 31 (5), 803–834

The binary knapsack problem with qualitative levels

A variant of the classical knapsack problem is considered in which each item is associated with an integer weight and a qualitative level. We define a dominance relation over the feasible subsets of the given item set and show that this relation defines a preorder. We propose a dynamic programming algorithm to compute the entire set of non-dominated rank cardinality vectors and we state two greedy algorithms, which efficiently compute a single efficient solution

Approximating the nondominated set of an MOLP by approximately solving its dual problem

The geometric duality theory of Heyde and Lohne (2006) defines a dual to a multiple objective linear programme (MOLP). In objective space, the primal problem can be solved by Benson’s outer approximation method (Benson, 1998a,b) while the dual problem can be solved by a dual variant of Benson’s algorithm (Ehrgott et al., 2007). Duality theory then assures that it is possible to find the nondominated set of the primal MOLP by solving its dual. In this paper, we propose an algorithm to solve the dual MOLP approximately but within specified tolerance. This approximate solution set can be used to calculate an approximation of the nondominated set of the primal. We show that this set is an ε-nondominated set of the original primal MOLP and provide numerical evidence that this approach can be faster than solving the primal MOLP approximately

An approximation algorithm for convex multi-objective programming problems

Multi-objective optimization, Convex optimization, Approximation algorithm, ε-nondominated point,