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Production planning is an important activity in manufacturing industries. The main goal of production planning is to minimize the cost under the condition that the customer requirement in terms of quality, quantity, and time is satisfied. An important player (human) is with little attention in traditional production planning. This thesis studied production planning with consideration of human factor, especially human work injuries as a result of performing a repetitive operation for a certain period of time in production systems. Production planning in this thesis only takes the minimization of total production cost as its goal. A linear programming technique was employed to incorporate the cost of work injury into the total production cost model. The LINDOTM software was used to solve the linear production planning model and to analyze the solution. Finally, the benefits of the production planning, which considers work injury, were discussed. Several conclusions can be drawn from this study: (1) the traditional production planning model, which only takes the material costs and labor costs into account, cannot deal with the cost related to work injury; (2) the work injury cost could be significant in those manual-intensive assembly systems, especially with high production rates; (3) the careful design of the worker’s postures can significantly reduce the work injury cost and thus the total cost of production. The significant contributions of this thesis are: (1) the development of a mathematical model for the total production cost including the work injury cost and (2) the finding that the work injury cost may be a significant portion in the total cost of production in the assembly system that has intensive manual works.

Recent advances in linear programming solution methodology have focused on interior point algorithms. These are powerful new methods, achieving significant reductions in computer time for large LPs and solving problems significantly larger than previously possible. This dissertation describes the implementation of interior point algorithms. It focuses on applications of direct sparse matrix methods to sparse symmetric positive definite systems of linear equations on scalar computers and vector supercomputers. The most computationally intensive step in each iteration of any interior point algorithm is the numerical factorization of a sparse symmetric positive definite matrix. In large problems or relatively dense problems, 80-90% or more of computational time is spent in this step. This study concentrates on solution methods for such linear systems. It is based on modifications and extensions of graph theory applied to sparse matrices. The row and column permutation of a sparse symmetric positive definite matrix dramatically affects the performance of solution algorithms. Various reordering methods are considered to find the best ordering for various numerical factorization methods and computer architectures. It is assumed that the reordering method will follow the fill-preserving rule, i.e., not allow additional fill-ins beyond that provided by the initial ordering. To follow this rule, a modular approach is used. In this approach, the matrix is first permuted by using any minimum degree heuristic, and then the permuted matrix is again reordered according to a specific reordering objective. Results of different reordering methods are described. There are several ways to compute the Cholesky factor of a symmetric positive definite matrix. A column Cholesky algorithm is a popular method for dense and sparse matrix factorization on serial and parallel computers. Applying this algorithm to a sparse matrix requires the use of sparse vector operations. Graph theory is applied to reduce sparse vector computations. A second and relatively new algorithm is the multifrontal algorithm. This method uses dense operations for sparse matrix computation at the expense of some data manipulation. The performance of the column Cholesky and multifrontal algorithms in the numerical factorization of a sparse symmetric positive definite matrix on an IBM 3090 vector supercomputer is described.

The basis handling procedures of the simplex method are formulated in terms of a "double basis". That is, the basis is factored as (DIAGRAM OMITTED...PLEASE SEE DAI) where ‘B, the pseudobasis matrix, is the basis matrix at the last refactorization. P and Q are permutation matrices. Forward and backward transformations and update are presented for each of two implementations of the double-basis method. The first implementation utilizes an explicit G⁻¹ matrix. The second uses a sparse LU factorization of G. Both are based on Marsten's modularized XMP package, in which standard simplex method routines are replaced by corresponding double-basis method routines. XMP and the LU double-basis method implementation employ Reid's LA05 routines for handling sparse linear programming bases. All calculations are done without reference to the H matrix. Therefore, the update is restricted to G, which has dimension limited by the refactorization frequency, and P and Q, which are held as lists. This can lead to a saving in storage space and updating time. The cost is that time for transformations will be about double. Computational comparisons of storage and speed performance are made with the standard simplex method on problems of up to 1480 constraints. It is found that, generally, the double-basis method performs best on larger, denser problems. Density seems to be the more important factor, and the problems with large nonzero growth between refactorizations are the better ones for the double-basis method. Storage saving in the basis inverse representation versus the standard method is as high as 36%, whereas the double-basis run times are 1.2 or more times as great.

The Defense Language Institute (DLI) offers 23 beginning language courses and in 2004 began to provide a smaller class size for these courses. Restrictions on when classes can begin and a limited number of instructors prevent all students from being trained in a smaller class. This thesis develops integer linear programs (ILPs) that generate schedules for all student classes and maximize the number of smaller class starts for a given number of instructors. Secondary scheduling goals include avoiding weekly changes to instructor levels and scheduling preferences such as the number of classes to start simultaneously. The ILPs solve in less than one minute and offer a significant improvement in the number of students that may be trained in the smaller class size. Computational results using real data for the Arabic, Chinese-Mandarin, and Persian-Farsi courses verify the ILPs find feasible multiyear schedules that incorporate the DLI's scheduling preferences while exceeding the DLI's published schedule results. For example, the ILPs find schedules for Arabic that train 8%, 34% and 76% of students in the smaller class in 2006, 2007, and 2008, whereas DLI's manual schedules at best can train 8%, 7% and 64%.

This thesis considered the potential benefits of employing linear programming in cheese manufacturing plants as a decision tool for management. Its potential has been enhanced by the recent approval of acid orange 12 as a chemical for testing the percent protein in milk; therefore, a practical test is now available for monitoring protein as well as milk fat in milk manufacturing and fluid milk plants. Seven models, each one differing only in the milk fat and protein percentages or means of standardizing the cheese milk, were manipulated individually and simultaneously to test the managerial benefits of linear programming under various plant and market conditions. Each model consisted of five cheese activities or variables, two butter activities, three powder activities, and a selling activity for each product produced. The maximum price that could be paid the farm producer per hundred-weight of milk and the minimum wholesale rice per pound of manufactured product, to cover variable costs were determined for each variety of cheese and composition of milk. There was a definite interaction between each of the activities. This caused the cost to produce a Pound of cheese to vary according to the alternative uses for milk, cream, skim milk, and whey. When the simulated plant was being utilized at or near full capacity and the cheese milk was standardized with non fat dry milk powder, total cheese yield increased as did total profits. When the plant was not being utilized to full capacity, profits were higher by not standardizing.

Call number: LD2668 .T4 1957 U57 / Master of Science

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This report presents a qualitative case study, which explored how a Grade 11 mathematics teacher in a multilingual classroom used the learners’ home languages in order to support their understanding of concepts in Linear Programming. The study involved one teacher together with his Grade 11 learners and was carried out in a township school located in the Eastrand, Johannesburg. Data was collected through lesson observations of five consecutive lessons and a reflective interview with the teacher. The situated-sociocultural perspectives guided the study. The analysis shows that the teacher used learners’ home languages deliberately; in mathematics tasks, for asking questions, to re-voice learners’ contributions, for encouraging learners’ participation in mathematical discourses and practices, and for probing learners’ thinking. In general, the use of learners’ home languages enhanced learners’ understanding of Linear Programming concepts. The study also highlights the complexities of translating mathematics tasks from English to learners’ home languages.