Browsing by Subject "Linear programming."

Now showing 1 - 8 of 8

Open Access
Aggregate production planning with linear programming: a case study
(1991) Alperat, Haluk
The aim o f A g g r e g a t e P r o d u c t io n P la n n in g i s t o b a la n c e th e e x p e c te d denuind and s u p p ly f o r a company. I f demand i s g iv e n , p ro d u c t io n r a t e s , w o rk fo r c e s i z e s , and in v e n t o r y l e v e l s a r e d e te rm in ed f o r e v e r y p e r io d in th e p la n n in g h o r iz o n in o rd e r t o m in im ize th e t o t a l p r o d u c t io n c o s t . In t h i s s tu d y . A g g r e g a t e P r o d u c t io n P la n n in g i s a p p l i e d t o a sm a ll company th a t p ro d u c e s s p a r e p a r t s f o r T o fa ^ and F o rd f a c t o r i e s . L in e a r Progranun ing i s one ap p ro a ch t o s o l v in g th e A g g r e g a t e P ro d u c t io n P la n n in g p ro b lem w ith l i n e a r c o s t fu n c t io n s . T h is s o p h i s t i c a t e d t e c h n iq u e , wh ich i s v e r y s im p le t o a p p ly , y i e l d s a c c u r a t e r e s u l t s in a r e l a t i v e l y s h o r t p r o c e s s in g tim e . In s o l v in g t h i s p ro b lem , th e r e q u i r e d d a t a f o r th e company a r e c o l l e c t e d and p r o c e s s e d . The c o s t fu n c t io n s a r e fou n d t o be l i n e a r wh ich e n a b le s th e u se o f LP. F i n a l l y , s e n s i t i v i t y a n a ly s i s i s p e r fo rm ed t o f i n d th e a l lo w e d ra n g e s f o r demand f o r e c a s t s a s w e l l a s r e g u l a r and o v e r t im e w o rk in g h o u rs .
Open Access
Algorithms for linear and convex feasibility problems: A brief study of iterative projection, localization and subgradient methods
(1998) Özaktaş, Hakan
Several algorithms for the feasibility problem are investigated. For linear systems, a number of different block projections approaches have been implemented and compared. The parallel algorithm of Yang and Murty is observed to be much slower than its sequential counterpart. Modification of the step size has allowed us to obtain a much better algorithm, exhibiting considerable speedup when compared to the sequential algorithm. For the convex feasibility problem an approach combining rectangular cutting planes and subgradients is developed. Theoretical convergence results are established for both ca^es. Two broad classes of image recovery problems are formulated as linear feasibility problems and successfully solved with the algorithms developed.
Open Access
Decomposing linear programs for parallel solution
(1996) Pınar, Ali
Many current research efforts are based on better exploitation of sparsity— common in most large scaled problems—for computational efEciency. This work proposes different methods for permuting sparse matrices to block angular form with specified number of equal sized blocks for efficient parallelism. The problem has applications in linear programming, where there is a lot of work on the solution of problems with existing block angular structure. However, these works depend on the existing block angular structure of the matrix, and hence suffer from unscalability. We propose two hypergraph models for decomposition, and these models reduce the problem to the well-known hypergraph partitioning problem. We also propose a graph model, which reduces the problem to the graph partitioning by node separator problem. We were able to decompose very large problems, the results are quite attractive both in terms solution quality and running times.
Open Access
Parallelization of an interior point algorithm for linear programming
(1994) Simitçi, Hüseyin
In this study, we present the parallelization of Mehrotra’s predictor-corrector interior point algorithm, which is a Karmarkar-type optimization method for linear programming. Computation types needed by the algorithm are identified and parallel algorithms for each type are presented. The repeated solution of large symmetric sets of linear equations, which constitutes the major computational effort in Karmarkar-type algorithms, is studied in detail. Several forward and backward solution algorithms are tested, and buffered backward solution algorithm is developed. Heurustic bin-packing algorithms are used to schedule sparse matrix-vector product and factorization operations. Algorithms having the best performance results are used to implement a system to solve linear programs in parallel on multicomputers. Design considerations and implementation details of the system are discussed, and performance results are presented from a number of real problems.
Open Access
Pricing and optimal exercise of perpetual American options with linear programming
(2010) Bozkaya, Efe Burak
An American option is the right but not the obligation to purchase or sell an underlying equity at any time up to a predetermined expiration date for a predetermined amount. A perpetual American option differs from a plain American option in that it does not expire. In this study, we solve the optimal stopping problem of a perpetual American option with methods from the linear programming literature. Under the assumption that the underlying’s price follows a discrete time and discrete state Markov process, we formulate the problem with an infinite dimensional linear program using the excessive and majorant properties of the value function. This formulation allows us to solve complementary slackness conditions efficiently, revealing an optimal stopping strategy which highlights the set of stock-prices for which the option should be exercised. Under two different stock-price movement scenarios (simple and geometric random walks), we show that the optimal strategy is to exercise the option when the stock-price hits a special critical value. The analysis also reveals that such a critical value exists only for some special cases under the geometric random walk, dependent on a combination of state-transition probabilities and the economic discount factor. We further demonstrate that the method is useful for determining the optimal stopping time for combinations of plain vanilla options, by solving the same problem for spread and strangle positions under simple random walks.
Open Access
Row generation techniques for approximate solution of linear programming problems
(2010) Paç, A. Burak
In this study, row generation techniques are applied on general linear programming problems with a very large number of constraints with respect to the problem dimension. A lower bound is obtained for the change in the objective value caused by the generation of a specific row. To achieve row selection that results in a large shift in the feasible region and the objective value at each row generation iteration, the lower bound is used in the comparison of row generation candidates. For a warm-start to the solution procedure, an effective selection of the subset of constraints that constitutes the initial LP is considered. Several strategies are discussed to form such a small subset of constraints so as to obtain an initial solution close to the feasible region of the original LP. Approximation schemes are designed and compared to make possible the termination of row generation at a solution in the proximity of an optimal solution of the input LP. The row generation algorithm presented in this study, which is enhanced with a warm-start strategy and an approximation scheme is implemented and tested for computation time and the number of rows generated. Two efficient primal simplex method variants are used for benchmarking computation times, and the row generation algorithm appears to perform better than at least one of them especially when number of constraints is large.
Open Access
Solution of feasibility problems via non-smooth optimization
(1990) Ouveysi, Iradj
In this study we present a penalty function approach for linear feasibility problems. Our attempt is to find an eiL· coive algorithm based on an exterior method. Any given feasibility (for a set of linear inequalities) problem, is first transformed into an unconstrained minimization of a penalty function, and then the problem is reduced to minimizing a convex, non-smooth, quadratic function. Due to non-differentiability of the penalty function, the gradient type methods can not be applied directly, so a modified nonlinear programming technique will be used in order to overcome the difficulties of the break points. In this research we present a new algorithm for minimizing this non-smooth penalty function. By dropping the nonnegativity constraints and using conjugate gradient method we compute a maximum set of conjugate directions and then we perform line searches on these directions in order to minimize our penalty function. Whenever the optimality criteria is not satisfied and the improvements in all directions are not enough, we calculate the new set of conjugate directions by conjugate Gram Schmit process, but one of the directions is the element of sub differential at the present point.
Open Access
Valid inequalities for the problem of optimizing a nonseparable piecewise linear function on 0-1 variables
(2011) Aksüt, Ziyaattin Hüsrev
In many procurement and transportation applications, the unit prices depend on the amount purchased or transported. This results in piecewise linear cost functions. Our aim is to study the structure that arises due to a piecewise linear objective function and to propose valid inequalities that can be used to solve large procurement and transportation problems. We consider the problem of optimizing a nonseparable piecewise linear function on 0-1 variables. We linearize this problem using a multiple-choice model and investigate properties of facet defining inequalities. We propose valid inequalities and lifting results.