Оптимальный раскрой промышленных материалов

                      М о с к о в с к и й

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                        т е х н и к у м

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                                 на тему:

                 @'ОПТИМАЛТНЫЙ РАСКРОЙ ПРОМЫШЛЕННЫХ МАТЕРИАЛОВ'

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          @'МОДЕЛИРОВАНИЕ ПРОИЗВОДСТВЕННЫХ И ЭКОНОМИЧЕСКИХ ПРОЦЕССОВ'

        Работу выполнил:                             Работу проверил:

        ученик группы П-406                          преподаватель

        Горбатов Р.С.                                Капустина Р.Н.

                                    1995 г.

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       │                         СОДЕРЖАНИЕ:                       │

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       │  @СОДЕРЖАНИЕ                                               @СТРАНИЦА   │

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       │  ВВЕДЕНИЕ...................................................   2     │

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       │  1. ЭКОНОМИЧЕСКАЯ ПОСТАНОВКА ЗАДАЧИ.........................   3     │

       │                                                                      │

       │  2. МАТЕМАТИЧЕСКАЯ ПОСТАНОВКА ЗАДАЧИ.                                │

       │     ПОСТРОЕНИЕ МАТЕМАТИЧЕСКОЙ МОДЕЛИ........................   4     │

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       │  3. ВЫБОР МЕТОДА РЕАЛИЗАЦИИ МОДЕЛИ.                                  │

       │     ОБОСНОВАНИЕ ВЫБОРА......................................   5     │

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       │  4. СХЕМА АЛГОРИТМА И ЕЕ ОПИСАНИЕ........................... 6 - 10  │

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       │  5. КРАТКАЯ ХАРАКТЕРИСТИКА ЭВМ И ПРОГРАММНОГО ОБЕСПЕЧЕНИЯ...   11    │

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       │  6. КРАТКАЯ ХАРАКТЕРИСТИКА ВЫБРАННГО ЯЗЫКА ПРОГРАММИРОВАНИЯ.   12    │

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       │  7. РЕШЕНИЕ ЗАДАЧИ-ТЕСТА ДЛЯ НАПИСАНИЯ И ОТЛАДКИ ПРОГРАММЫ..13 - 14  │

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       │  8. АНАЛИЗ ПОЛУЧЕННЫХ РЕЗУЛЬТАТОВ...........................   15    │

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       │  9. ИНСТРУКЦИЯ ПОЛЬЗОВАТЕЛЮ.................................16 - 17  │

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       │  СПИСОК ЛИТЕРАТУРЫ..........................................   18    │

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       │  ЗАКЛЮЧЕНИЕ.ВЫВОДЫ ПО РАБОТЕ................................   19    │

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       │  ПРОГРАММА.ОПИСАНИЕ ПРОГРАММЫ...............................   20    │

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       │  ПРИЛОЖЕНИЕ 1...............................................21 - 26  │

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       │  ПРИЛОЖЕНИЕ 2...............................................27 - 30  │

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       │                                 - 2 -                                │

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       │                           ВВЕДЕНИЕ.                         │

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       │   В настоящее время новейшие достижения математики и современной     │

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       │ вычислительной техники находят все более широкое применение в  эко-  │

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       │ номических исследованиях в планировании. Накоплен достаточный  опыт  │

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       │ постановки и решения экономических задач с помощью математических    │

       │                                                                      │

       │ методов. Особенно успешно развиваются методы оптимального планиро-   │

       │                                                                      │

       │ вания.                                                               │

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       │   В промышленном производстве применяется большое количество  мате-  │

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       │ риалов,которые подвергаются разрезке на штучные заготовки.В про-     │

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       │ цессе раскроя неизбежны отходы из-за некратности размеров заготовки  │

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       │ размерам исходного материала.На промышленных предприятиях исполь-    │

       │                                                                      │

       │ зуются различные методы борьбы с потерями из-за отходов.Наиболее     │

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       │ рациональным считается метод проведения совместных раскроев.@Сов-     │

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       │ @местный раскроя означает разрезку единицы материала на комплект      │

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       │ разных деталей.                                                      │

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       │   Идея совместного раскроя состоит в следующем.Известны размеры      │

       │                                                                      │

       │ заготовок и размер исходного материала.На основании этого разра-     │

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       │ батываются варианты раскроя единицы исходного материала с различ-    │

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       │ ным составом заготовок и различной величиной отходов.Поскольк у      │

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       │ варианты раскроя разрабатываются для единицы исходного материала,    │

       │                                                                      │

       │ в них не учитывается требуемое количество заготовок.Поэтому на       │

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       │ основании этих вариантов строится модель линейного программирова-    │

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       │ ния,где в качестве переменных берется количество исходного матери-   │

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       │ ала,раскраиваемого по каждому варианту.Так как модель строится       │

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       │ на основании вариантов раскроя,она названа @вариантная модель         │

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       │ @оптимального раскроя.С помощью данной модели можно определить ,      │

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       │ какое количество исходного материала и по каким вариантам нужно      │

       │                                                                      │

       │ раскраивать,чтобы получить требуемое количество заготовок с мини-    │

       │                                                                      │

       │ мальными отходами.Этот набор вариантов будет оптимальным.            │

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       │   В данном курсовом проекте будет рассмотрено решение экономичес-    │

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       │ кой задачи на оптимальный раскроя материалов универсальным методом   │

       │                                                                      │

       │ линейного программирования @Симплекс-методом.                         │

       │                                                                      │

       └──────────────────────────────────────────────────────────────────────┘

       ┌──────────────────────────────────────────────────────────────────────┐

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       │                                 - 3 -                                │

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       │1.ЭКОНОМИЧЕСКАЯ ПОСТАНОВКА ЗАДАЧИ.  │

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       │                    ОПТИМАЛЬНЫЙ РАСКРОЙ МАТЕРИАЛОВ.                   │

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       │   В соответствии с производственными заданиями заготовительный       │

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       │ цех должен нарезать из стальных прутков длиною 11,0 м следующее      │

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       │ количество заготовок:                                                │

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       │                                                                      │

       │    Длиною по:     1,6 м - 480 штук.                                  │

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       │                   1,3 м - 760 штук.                                  │

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       │                   3,6 м - 180 штук.                                  │

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       │                                                                      │

       │   Требуется: 1) Составить план раскроя прутков , обеспечивающий      │

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       │                 минимальное количество отходов.                      │

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       │              2) Определить абсолютную величину отходов и коэф -      │

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       │                 фициент использования металла.                       │

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       │   Предварительно , перед решением задачи , необходимо составить      │

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       │ таблицу возможных вариантов раскроя поступающих прутков данной       │

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       │ партии. После решения задачи сделать проверку полученных резуль-     │

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       │ татов.                                                               │

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       └──────────────────────────────────────────────────────────────────────┘

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       │                                 - 4 -                                │

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       │2.МАТЕМАТИЧЕСКАЯ ПОСТАНОВКА ЗАДАЧИ.│

       │  ПОСТРОЕНИЕ МАТЕМАТИЧЕСКОЙ МОДЕЛИ.│

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       │    Для решения данной задачи введем следующие обозначения:           │

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       │    m ( i=1,2,...,m ) - виды заготовок.                               │

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       │    n ( j=1,2,...,n ) - способы раскроя.                              │

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       │    Bi - план по заготовкам "i"-того вида.                            │

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       │    bij - количество заготовок "i"-того вида ,                        │

       │          полученные "j"-тым способом раскроя.                        │

       │                                                                      │

       │    Xj - количество единиц (штук) исходного материала,                │

       │         которое следует раскраивать по "j"-тому способу.             │

       │                                                                      │

       │    Cj - количество отходов при "j"-том способе раскроя.              │

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       │    При решении задачи надо учитывать следующие формулы:              │

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       │    СИСТЕМА ОГРАНИЧЕНИЙ:                                              │

       │                                                                      │

       │        n                                                             │

       │       [1]                                                               │

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       │  1)   [1]/    bij * Xj Є Bj   i=(1,2,...,m)                             │

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       │       j=1                                                            │

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       │  2)   Xj Є 0                                                         │

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       │    ЦЕЛЕВАЯ ФУНКЦИЯ:                                                  │

       │           n                                                          │

       │          [1]                                                            │

       │          \                                                           │

       │  3)  F = [1]/    Cj * Xj ^#& min                                          │

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       │          j=1                                                         │

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       │        Составим таблицу возможных вариантов раскроя прутков:         │

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       │                                            Таблица 1.                │

       │              ┌───────┬───────────────────────┬──────┐                │

       │              │Заготов│    Способы раскроя    │ План │                │

       │              │  ки   ├───┬───┬───┬───┬───┬───┤      │                │

       │              │       │ 1 │ 2 │ 3 │ 4 │ 5 │ 6 │      │                │

       │              ├───────┼───┼───┼───┼───┼───┼───┼──────┤                │

       │              │  1,6  │ 6 │ 5 │ 2 │ - │ 2 │ 2 │ 480  │                │

       │              │       │   │   │   │   │   │   │      │                │

       │              │  1,3  │ 1 │ 2 │ 6 │ 5 │ 3 │ - │ 760  │                │

       │              │       │   │   │   │   │   │   │      │                │

       │              │  3,6  │ - │ - │ - │ 1 │ 1 │ 2 │ 180  │                │

       │              ├───────┼───┼───┼───┼───┼───┼───┼──────┤                │

       │              │Отходы │0,1│0,4│ 0 │0,9│0,3│0,6│      │                │

       │              └───────┴───┴───┴───┴───┴───┴───┴──────┘                │

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       │        Система уравнений будет строится по данной таблице.           │

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       │                                 - 5 -                                │

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       │ 3. ВЫБОР МЕТОДА РЕАЛИЗАЦИИ МОДЕЛИ. │

       │                  ОБОСНОВАНИЕ ВЫБОРА.              │

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       │   Данная задача была решена @Симплекс-методом, т.к. указанный метод   │

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       │ является универсальным методом для решения задач линейного програм-  │

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       │ мирования.                                                           │

       │                                                                      │

       │   Известно, что оптимальные решения задачи линейного программирования│

       │                                                                      │

       │ связаны с угловыми точками многогранника решений. Угловых точек может│

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       │ быть много, если есть много ограничений. Количество угловых точек    │

       │                                                                      │

       │ соответствует количеству базисных решений. Для каждого базисного ре- │

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       │ шения однозначно определяется значение целевой функции. Найти опти-  │

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       │ мальное решение (оптимальный план), беспорядочно перебирая все базис-│

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       │ ные решения, в поисках такого, которое приносит целевой функции экс- │

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       │ тремальное значение, весьма затруднительно.                          │

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       │   В связи с этим необходим такой переход от одного базисного решения │

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       │ к другому, в результате которого новое решение приносило бы, в невы- │

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       │ рожденной задачи на максимум, большее значение целевой функции, а в  │

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       │ невырожденной задаче на минимум - меньшее. Такой процесс решения     │

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       │ задачи реализует Симплекс-метод. Процесс решения задачи продолжается │

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       │ до получения оптимального плана либо до установления факта отсутст-  │

       │                                                                      │

       │ вия решения задачи. Переход от одного базисного решения к другому    │

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       │ называется @итерацией Симплекс-метода.                                │

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       ┌──────────────────────────────────────────────────────────────────────┐

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       │                                 - 6 -                                │

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       │   4.СХЕМА АЛГОРИТМА И ЕЕ ОПИСАНИЕ.   │

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       ┌──────────────────────────────────────────────────────────────────────┐

       │                                                                      │

       │                                 - 7 -                                │

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       │                                                                      │

       │                                 - 8 -                                │

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       ┌──────────────────────────────────────────────────────────────────────┐

       │                                                                      │

       │                                 - 9 -                                │

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       └──────────────────────────────────────────────────────────────────────┘

       ┌──────────────────────────────────────────────────────────────────────┐

       │                                                                      │

       │                                 - 10 -                               │

       │                                                                      │

       │                                                                      │

       │   В данной блок-схеме блоки означают следующее:                      │

       │                                                                      │

       │                                                                      │

       │ 1: Начало алгоритма.                                                 │

       │                                                                      │

       │ 2,3,4,5: Ввод данных о системе уравнений.                            │

       │                                                                      │

       │ 6: Определение общего размера матрицы.                               │

       │                                                                      │

       │ 7: Ввод коэффициентов при Х, стоящих в целевой функции.              │

       │                                                                      │

       │ 8: Ввод свободных членов для каждого уравнения                       │

       │                                                                      │

       │ 9: Ввод коэффициентов при Х и Y, стоящие в каждом уравнении.         │

       │                                                                      │

       │10: YV- отключить признак конца подсчета,                             │

       │    itr - счетчик количества итераций.                                │

       │11: Если YV=True (признак конца подсчета), то выполнить вывод         │

       │    конечного результата (Блок 31), иначе продолжить решение.         │

       │12: Сделать копию индексной строки.                                   │

       │                                                                      │

       │13: Вызов функции testY (эта функция проверяет наличие искуственных   │

       │    переменных в массиве).                                            │

       │14: Если testY=True (массив содержит искуственные переменные),        │

       │    вызвать процедуру indY (Блок 15), иначе вызвать процедуру indX.   │

       │15: Вызов процедуры indY. Эта процедура ведет подсчет индексной       │

       │    строки с учетом искуственных переменных.                          │

       │16: Вызов процедуры indX. Эта процедура ведет подсчет индексной строки│

       │    в том случае, если искуственные переменные выведены из матрицы.   │

       │17: Вызов функции test0. Эта функция проверяет наличие положительных  │

       │    элементов в индексной строке.                                     │

       │18: Если test0=false (в индексной строке содержатся положительные     │

       │    элементы), выдать промежуточные результаты на экран (Блок 19),    │

       │    иначе завершить решение задачи (Блок 30).                         │

       │19: Вывод промежуточного результата на экран.                         │

       │                                                                      │

       │20: Процедура MaxSt выделяет ключевой столбец.                        │

       │                                                                      │

       │21: Процедура Str выделяет ключевую строку и находит разрешающий      │

       │    элемент в матрице.                                                │

       │22: Выводится базис ключевой строки и ключевого столбца.              │

       │                                                                      │

       │23: Сделать копию основного массива (a) и столбца (H).                │

       │                                                                      │

       │24: Заполняется строка введеного базиса путем деления соответствующих │

       │    элементов выведенной строки на разрешающий элемент.               │

       │25: Заполнить новую матрицу по заданной формуле.                      │

       │                                                                      │

       │26: Вычисление столбца H.                                             │

       │                                                                      │

       │27: Отключить признак конца подсчета.                                 │

       │                                                                      │

       │28: Получение новой матрицы, столбца Н и индексной строки.            │

       │                                                                      │

       │29: Определение следующей итерации.                                   │

       │                                                                      │

       │30: Определение завершения решения задачи. (Конец подсчета итерации)  │

       │                                                                      │

       │31: Вывод конечного результата решения задачи на экран.               │

       │                                                                      │

       │32: Конец алгоритма.                                                  │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       └──────────────────────────────────────────────────────────────────────┘

       ┌──────────────────────────────────────────────────────────────────────┐

       │                                                                      │

       │                                 - 11 -                               │

       │                                                                      │

       │ 5. КРАТКАЯ ХАРАКТЕРИСТИКА ЭВМ И     │

       │        ПРОГРАММНОГО ОБЕСПЕЧЕНИЯ.            │

       │                                                                      │

       │                                                                      │

       │    Слово 'компьютер' означает 'вычислитель', т.е. устройство для     │

       │                                                                      │

       │ вычисления. В 1981г. фирмой 'IBM Corporation' был выпущен первый     │

       │                                                                      │

       │ персональный компьютер IBM PC, на базе 16-разрядного процессора      │

       │                                                                      │

       │ Intel-8088. Персональный компьютер состоит из нескольких основных    │

       │                                                                      │

       │ частей: устройства ввода (клавиатура), устройство вывода (монитор),  │

       │                                                                      │

       │ центральный процессор, выполняющий функции управления и счетный      │

       │                                                                      │

       │ процесс, внутреняя память ОЗУ и ПЗУ, различные устройства для работы │

       │                                                                      │

       │ с внешней памятью и шины данных, соединяющей все устройства воедино  │

       │                                                                      │

       │ и служащей для передачи данных между устройствами. Персональный ком- │

       │                                                                      │

       │ пьютер типа IBM собирается по принципу открытой архитектуры, т.е.    │

       │                                                                      │

       │ владелец компьютера может постепенно докупать дополнительные устрой- │

       │                                                                      │

       │ ства (модем, сканер, CD-ROM) и без проблем устанавливать их.         │

       │                                                                      │

       │   Есть много параметров, которыми характеризуется персональный       │

       │                                                                      │

       │ компьютер (тип монитора, количество оперативной памяти, емкость винт-│

       │                                                                      │

       │ честера), но главные параметры - тип процессора и тактовая частота.  │

       │                                                                      │

       │ Данная программа не требует высоких характеристик вычислительной     │

       │                                                                      │

       │ системы, она писалась на компьютере типа IBM PC со следующими харак- │

       │                                                                      │

       │ теристиками: процессор Intel-80286, тактовая частота 16 Мгц, опера-  │

       │                                                                      │

       │ тивная память 1 Мб, монитор типа VGA, свободное место на винтчестере │

       │                                                                      │

       │ для программы 14 Кб. Эту программу так же можно запустить и на дру-  │

       │                                                                      │

       │ гой вычислительной системе, с более низкими характеристиками.        │

       │                                                                      │

       │   Операционная система - программа, которая загружается сразу после  │

       │                                                                      │

       │ включения компьютера. Она осуществляет диалог с пользователем, управ-│

       │                                                                      │

       │ ление компьютером, его ресурсами, запускает другие программы на вы-  │

       │                                                                      │

       │ полнение. ОС обеспечивает пользователю и прикладным программам удоб- │

       │                                                                      │

       │ ный способ общения с устройствами компьютера. Для данной программы   │

       │                                                                      │

       │ не требуется специального программного обеспечения. Программный мини-│

       │                                                                      │

       │ мум: Операционная система MS-DOS 3.0 или выше и резидентная програм- │

       │                                                                      │

       │ ма 'экранный руссификатор', которая позволяет выводить на экран      │

       │                                                                      │

       │ буквы русского алфавита.                                             │

       │                                                                      │

       │                                                                      │

       └──────────────────────────────────────────────────────────────────────┘

       ┌──────────────────────────────────────────────────────────────────────┐

       │                                                                      │

       │                                 - 12 -                               │

       │                                                                      │

       │          6. КРАТКАЯ ХАРАКТЕРИСТИКА         │

       │                ЯЗЫКА ПРОГРАММИРОВАНИЯ.        │

       │                                                                      │

       │                                                                      │

       │   Программы для первых компьютеров приходилось писать на машинном    │

       │                                                                      │

       │ языке, т.е. в кодах, непосредственно воспринимаемых компьютером.     │

       │                                                                      │

       │ Это было очень тяжелой и кропотливой работой, поэтому в начале 50-х  │

       │                                                                      │

       │ годов были разработаны так называемые @языки низкого уровня, которые  │

       │                                                                      │

       │ позволяли писать программы не в машинных кодах, а с использованием   │

       │                                                                      │

       │ мнемонических обозначений. К таким языкам относится язык ASSEMBLER,  │

       │                                                                      │

       │ однако и этот язык слишком сложен: программист должен очень хорошо   │

       │                                                                      │

       │ знать архитектуру вычислительной машины. С развитием вычислительной  │

       │                                                                      │

       │ техники появились @языки высокого уровня. Программа на таком языке    │

       │                                                                      │

       │ состоит из последовательности команд и операторов, понятных пользова-│

       │                                                                      │

       │ телю. К таким языкам относится язык программирования @Turbo PASCAL.   │

       │                                                                      │

       │   Язык Паскаль впервые появился на машинах III поколения. В середине │

       │                                                                      │

       │ 80-х годов фирма Borland выпустила язык Turbo PASCAL для персональных│

       │                                                                      │

       │ компьютеров, который обладал удобной интерактивной оболочкой. Язык   │

       │                                                                      │

       │ сразу же завоевал огромную популярность. Объясняется это сочетанием  │

       │                                                                      │

       │ двух безусловных его достоинств: исключительной простотой и естест-  │

       │                                                                      │

       │ венностью языка и великолепными сервисными возможностями диалоговой  │

       │                                                                      │

       │ среды программирования. Последняя версия Турбо Паскаля 7.0 представ- │

       │                                                                      │

       │ ляет собой мощную, гибкую, удобную и почти универсальную систему     │

       │                                                                      │

       │ программирования с удобной интерактивной оболочкой, с большим коли-  │

       │                                                                      │

       │ чеством сервисных возможностей. Сам язык программирования сильно     │

       │                                                                      │

       │ изменен и доработан, по сравнению с первой версией.                  │

       │                                                                      │

       │   С помощью Турбо Паскаля можно создавать любые программы - от про-  │

       │                                                                      │

       │ грамм, предназначенных для решения простейших вычислительных задач,  │

       │                                                                      │

       │ до сложных современных систем управления базами данных и операционных│

       │                                                                      │

       │ систем.                                                              │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       └──────────────────────────────────────────────────────────────────────┘

       ┌──────────────────────────────────────────────────────────────────────┐

       │                                                                      │

       │                                 - 13 -                               │

       │                                                                      │

       │       7.РЕШЕНИЕ ЗАДАЧИ-ТЕСТА             │

       │                                                                      │

       │                                                                      │

       │   По таблице 1 строим систему линейных уравнений :                   │

       │                                                                      │

       │  ┌─ 6X1 + 5X2 + 2X3       + 2X5 + 2X6 Є 480                          │

       │  │                                                                   │

       │ <    X1 + 2X2 + 6X3 + 5X4 + 3X5       Є 760       Xj Є 0             │

       │  │                                                                   │

       │  └─                    X4 +  X5 + 2X6 Є 180                          │

       │                                                                      │

       │  Функция, которую будем исследовать на минимум, имеет вид :          │

       │                                                                      │

       │  Fmin = 0,1X1 + 0,4X2 + 0X3 + 0,9X4 + 0,3X6                          │

       │                                                                      │

       │                                                                      │

       │  Приведем систему уравнений к кононическому виду:                    │

       │                                                                      │

       │  ┌─ 6X1 + 5X2 + 2X3       + 2X5 + 2X6 - X7       + Y1       =  480   │

       │  │                                                                   │

       │ <    X1 + 2X2 + 6X3 + 5X4 + 3X5          - X8       + Y2    =  760   │

       │  │                                                                   │

       │  └─                    X4 +  X5 + 2X6       - X9       + Y3 =  180   │

       │                                                                      │

       │  Функция примет следующий вид :                                      │

       │                                                                      │

       │  Fmin = 0,1X1 + 0,4X2 + 0X3 + 0,9X4 + 0,3X6 + 0X7 + 0X8 + 0X9 +      │

       │                                                                      │

       │         + MY1 + MY2 + MY3                                         │

       │                                                                      │

       │  Решим эту систему уравнений универсальным Симплекс-методом.         │

       │                                                                      │

       │               (Решение смотри в таблице 2).                          │

       │                                                                      │

       │    На 5-й итерации получено оптимальное решение, т.к. в индексной    │

       │                                                                      │

       │  строке отсутствуют положительные элементы. Чтобы получить 480 заго- │

       │                                                                      │

       │  товок по 1,6 м, 760 по 1,3 м, 180 по 3,6 м нужно разрезать 150 пру- │

       │                                                                      │

       │  тков по 11 м на 2 по 1,6 м и 6 по 1,3 м каждый и 90 прутков на 2 по │

       │                                                                      │

       │  1,6 м и 2 по 3,6 м каждый. При этом получается перевыполнение плана │

       │                                                                      │

       │  по изготовлению заготовок по 1,3 м на 140 шт.                       │

       │                                                                      │

       │         Для проверки подставим полученные Х в таблицу 1.             │

       │                                                                      │

       │        2 * 150 = 300  (по 1,6)      2 * 90 = 180 (по 1,6)            │

       │                                                                      │

       │        6 * 150 = 900  (по 1,3)      2 * 90 = 180 (по 3,6)            │

       │                                                                      │

       │  Действительно, план по всем заготовкам выполняется. Решение верное. │

       │                                                                      │

       │  Wобщ. - общий объем раскраиваемого материала.                       │

       │                                                                      │

       │  Wзат. - общий объем затраченного материала.                         │

       │                                                                      │

       │  Ки.м. - коэффициент использования материала.                        │

       │                                                                      │

       │     (Решение смотри на следующей странице, ниже таблицы 2)           │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       └──────────────────────────────────────────────────────────────────────┘

       ┌──────────────────────────────────────────────────────────────────────┐

       │                                                                      │

       │                                 - 14 -                               │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                            Таблица 2.│

       │┌───┬────┬─────┬────┬────┬────┬────┬────┬────┬────┬────┬────┬──┬──┬──┐│

       ││   │    │     │ 0,1│ 0,4│ 0  │ 0,9│ 0,3│ 0,6│ 0  │ 0  │ 0  │M │M │M ││

       ││ C │ B  │  Н  ├────┼────┼────┼────┼────┼────┼────┼────┼────┼──┼──┼──┤│

       ││   │    │     │ X1 │ X2 │ X3 │ X4 │ X5 │ X6 │ X7 │ X8 │ X9 │Y1│Y2│Y3││

       │├───┼────┼─────┼────┼────┼────┼────┼────┼────┼────┼────┼────┼──┼──┼──┤│

       ││ M │ Y1 │ 480 │ 6  │ 5  │ 2  │ 0  │ 2  │ 2  │ -1 │ 0  │ 0  │1 │0 │0 ││

       ││ M │ Y2 │ 760 │ 1  │ 2  │ 6  │ 5  │ 3  │ 0  │ 0  │ -1 │ 0  │0 │1 │0 ││

       ││ M │ Y3 │ 180 │ 0  │ 0  │ 0  │ 1  │ 1  │ 2  │ 0  │ 0  │ -1 │0 │0 │1 ││

       │├───┼────┼─────┼────┼────┼────┼────┼────┼────┼────┼────┼────┼──┼──┼──┤│

       ││ ! │Fmin│1420 │ 7  │ 7  │ 8  │ 6  │ 6  │ 4  │ -1 │ -1 │ -1 │0 │0 │0 ││

       │├───┼────┼─────┼────┼────┼────┼────┼────┼────┼────┼────┼────┼──┼──┼──┤│

       ││ M │ Y1 │     │    │    │ 0  │    │ 1  │ 2  │ -1 │    │ 0  │1 │  │0 ││

       ││ 0 │ X3 │     │    │    │ 1  │    │0,5 │ 0  │ 0  │    │ 0  │0 │  │0 ││

       ││ M │ Y3 │ 180 │ 0  │ 0  │ 0  │ 1  │ 1  │ 2  │ 0  │ 0  │ -1 │0 │  │1 ││

       │├───┼────┼─────┼────┼────┼────┼────┼────┼────┼────┼────┼────┼──┼──┼──┤│

       ││ ! │Fmin│     │    │    │ 0  │    │ 2  │ 4  │ -1 │    │ -1 │0 │  │0 ││

       │├───┼────┼─────┼────┼────┼────┼────┼────┼────┼────┼────┼────┼──┼──┼──┤│

       ││0,1│ X1 │     │ 1  │    │ 0  │    │    │    │    │    │ 0  │  │  │0 ││

       ││ 0 │ X3 │     │ 0  │    │ 1  │    │    │    │    │    │ 0  │  │  │0 ││

       ││ M │ Y3 │ 180 │ 0  │ 0  │ 0  │ 1  │ 1  │ 2  │ 0  │ 0  │ -1 │  │  │1 ││

       │├───┼────┼─────┼────┼────┼────┼────┼────┼────┼────┼────┼────┼──┼──┼──┤│

       ││ ! │Fmin│ 180 │ 0  │ 0  │ 0  │ 1  │ 1  │ 2  │ 0  │ 0  │ -1 │  │  │0 ││

       │├───┼────┼─────┼────┼────┼────┼────┼────┼────┼────┼────┼────┼──┼──┼──┤│

       ││0,1│ X1 │     │ 1  │    │ 0  │    │    │    │    │    │ 0  │  │  │  ││

       ││ 0 │ X3 │     │ 0  │    │ 1  │    │    │    │    │    │ 0  │  │  │  ││

       ││0,6│ X6 │ 90  │ 0  │ 0  │ 0  │0,5 │0,5 │ 1  │ 0  │ 0  │-0,5│  │  │  ││

       │├───┼────┼─────┼────┼────┼────┼────┼────┼────┼────┼────┼────┼──┼──┼──┤│

       ││   │Fmin│     │ 0  │    │ 0  │    │    │    │    │    │    │  │  │  ││

       │├───┼────┼─────┼────┼────┼────┼────┼────┼────┼────┼────┼────┼──┼──┼──┤│

       ││ 0 │ X8 │     │    │    │ 0  │    │    │ 0  │    │ 1  │ 3  │  │  │  ││

       ││ 0 │ X3 │     │ 3  │    │ 1  │-0,5│    │ 0  │    │ 0  │    │  │  │  ││

       ││0,6│ X6 │ 90  │ 0  │ 0  │ 0  │0,5 │0,5 │ 1  │ 0  │ 0  │-0,5│  │  │  ││

       │├───┼────┼─────┼────┼────┼────┼────┼────┼────┼────┼────┼────┼──┼──┼──┤│

       ││   │Fmin│ 54  │-0,1│-0,4│ 0  │-0,6│ 0  │ 0  │ 0  │ 0  │-0,3│  │  │  ││

       │└───┴────┴─────┴────┴────┴────┴────┴────┴────┴────┴────┴────┴──┴──┴──┘│

       │                                                                      │

       │                                                                      │

       │ Wобщ. = 11 * 149,981 + 11 * 90 = 2639,791                            │

       │                                                                      │

       │ Wзат. = ( 2 * 149,981 + 2 * 90 )* 1,6 + 6 * 149,981 * 1,3 +          │

       │                                                                      │

       │         + 2 * 90 * 3,6 = 2585,791                                    │

       │                                                                      │

       │ Ки.м. = Wзат. / Wобщ. = 0,9795 ў 0,98                                │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       └──────────────────────────────────────────────────────────────────────┘

       ┌──────────────────────────────────────────────────────────────────────┐

       │                                                                      │

       │                                 - 15 -                               │

       │                                                                      │

       │   8.АНАЛИЗ ПОЛУЧЕННЫХ РЕЗУЛЬТАТОВ.   │

       │                                                                      │

       │                                                                      │

       │    В результате решения задачи-теста и прогона программы были полу-  │

       │                                                                      │

       │ чены следующие результаты:                                           │

       │                                                                      │

       │   Значение базиса:                                                   │

       │                                                                      │

       │    Fmin = 54                         X3 ў 150 ─┐ количество исходного│

       │                                                ├ материала для 3 и 6 │

       │    X8 ў 140 - перевыполнение плана   X6 = 90  ─┘ способов раскроя    │

       │                                                                      │

       │   Значение индексной строки:                                         │

       │                                                                      │

       │    X1 = -0,1                         X6 = 0                          │

       │                                                                      │

       │    X2 = -0,4                         X7 = 0                          │

       │                                                                      │

       │    X3 = 0                            X8 = 0                          │

       │                                                                      │

       │    X4 = -0,6                         X9 = -0,3                       │

       │                                                                      │

       │    X5 = 0                                                            │

       │                                                                      │

       │    Проанализировав данные результаты получим следующее экономическое │

       │                                                                      │

       │ обоснование данной задачи:                                           │

       │                                                                      │

       │                  @Экономическое обоснование задачи.                   │

       │                                                                      │

       │    На 5-той итерации получено оптимальное решение, т.к. в индексной  │

       │                                                                      │

       │ строке отсутствуют положительные элементы. Чтобы получить 480 заго-  │

       │                                                                      │

       │ товок по 1,6 м, 760 по 1,3 м, 180 по 3,6 м нужно разрезать 150 пру-  │

       │                                                                      │

       │ тков (11 м) на 2 по 1,6 м и 6 по 1,3 м каждый и 90 прутков на 2 по   │

       │                                                                      │

       │ 1,6 м и 2 по 3,6 м каждый. При этом получается перевыполнение плана  │

       │                                                                      │

       │ по изготовлению заготовок по 1,3 м на 140 штук.                      │

       │                                                                      │

       │     Для проверки подставим полученные результаты в таблицу 1.        │

       │                                                                      │

       │        2 * 150 (по 1,6) + 2 * 90 (по 1,6) = 480                     │

       │                                                                      │

       │        6 * 150 (по 1,3) = 900 - перевыполнение плана на 140 штук     │

       │                                                                      │

       │        2 * 90  (по 3,6) = 180                                        │

       │                                                                      │

       │  Действительно, план по всем заготовкам выполняется. Решение верное. │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       └──────────────────────────────────────────────────────────────────────┘

       ┌──────────────────────────────────────────────────────────────────────┐

       │                                                                      │

       │                                 - 16 -                               │

       │                                                                      │

       │         9. ИНСТРУКЦИЯ ПОЛЬЗОВАТЕЛЮ.       │

       │                                                                      │

       │                                                                      │

       │    Данная программа очень проста в управлении, и не требует специ-   │

       │                                                                      │

       │  альных знаний от пользователя.                                      │

       │                                                                      │

       │    После включения ПЭВМ и начальной загрузки операционной системы    │

       │                                                                      │

       │  MS-DOS, следует загрузить экранный руссификатор (если он еще не     │

       │                                                                      │

       │  загружен), иначе пользователь не сможет прочесть подсказки при      │

       │                                                                      │

       │  вводе данных, просмотре промежуточных и конечного результатов.      │

       │                                                                      │

       │  Запустить на выполнение файл KURS.EXE .                             │

       │                                                                      │

       │    На экране появится табличка с информацией о программе, об авто-   │

       │                                                                      │

       │  рах и название метода. Следует нажать любую клавишу. Дальше про-    │

       │                                                                      │

       │  грамма попросит пользователя ввести количество уравнений в системе, │

       │                                                                      │

       │  количество основных, дополнительных и искуственных переменных.      │

       │                                                                      │

       │  Следует ввести последовательно ЦЕЛЫЕ числа, заканчивая ввод клави-  │

       │                                                                      │

       │  шей Enter ( ─┘ ). Дальше, по запросу программы, следует ввести по- │

       │                                                                      │

       │  следовательно коэффициенты, стоящие при Х в целевой функции. Дальше │

       │                                                                      │

       │  программа попросит ввести свободные члены ко всем уравнениям и коэф-│

       │                                                                      │

       │  фициенты при X и Y, стоящие в каждом уравнении. Ввод производить    │

       │                                                                      │

       │  последовательно, заканчивать ввод следует клавишей Enter.           │

       │                                                                      │

       │    После ввода данных программа начнет оптимизировать функцию, вве-  │

       │                                                                      │

       │  денную пользователем, автоматически, выдавая на экран после каждой  │

       │                                                                      │

       │  итерации промежуточные данные. На экран будет выдаватся следующее:  │

       │                                                                      │

       │  номер итерации, значение функции, индексная строка для всех Х, пре- │

       │                                                                      │

       │  дупреждение "Функция НЕ минимизирована" и запрос "Продолжим (Y/N)?".│

       │                                                                      │

       │  Если данные введены правильно и промежуточные результаты схожи с    │

       │                                                                      │

       │  теоретическими, следует ответить клавишами 'Y' + '─┘'. Если поль-  │

       │                                                                      │

       │  зователь хочет прервать вычислительный процесс, следует ответить    │

       │                                                                      │

       │  так: 'N' + '─┘'. (Вообще программу можно прервать в любом месте    │

       │                                                                      │

       │  комбинацией клавиш 'Ctrl' + 'Break'). Вычислительный процесс будет  │

       │                                                                      │

       │  продолжатся до тех пор, пока не будет получено оптимальное решение. │

       │                                                                      │

       │  В этом случае будет подан звуковой сигнал и на экране отобразится   │

       │                                                                      │

       │  информация: на какой итерации получено оптимальное решение, значение│

       │                                                                      │

       │  функции, значение индексной строки и приглашение "Нажмите любую     │

       │                                                                      │

       └──────────────────────────────────────────────────────────────────────┘

       ┌──────────────────────────────────────────────────────────────────────┐

       │                                                                      │

       │                                 - 17 -                               │

       │                                                                      │

       │                                                                      │

       │  клавишу!". После нажатия любой клавиши программа закончит свою      │

       │                                                                      │

       │  работу и вернет пользователя в MS-DOS.                              │

       │                                                                      │

       │    Данная программа не является универсальной. Программа оптимизиру- │

       │                                                                      │

       │  ет функции только на минимум. Функция и система уравнений должны    │

       │                                                                      │

       │  быть приведены к кононическому виду.                                │

       │                                                                      │

       │    Данные для решения задачи удобнее всего вводить прямо с таблицы,  │

       │                                                                      │

       │  которую используют при решении системы уравнений Симплекс-методом   │

       │                                                                      │

       │  ручным способом.                                                    │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       └──────────────────────────────────────────────────────────────────────┘

       ┌──────────────────────────────────────────────────────────────────────┐

       │                                                                      │

       │                                 - 18 -                               │

       │                                                                      │

       │                СПИСОК ЛИТЕРАТУРЫ:                  │

       │                                                                      │

       │                                                                      │

       │ Малик Г.С.       Основы экономики и математические методы            │

       │                                                                      │

       │                  в планировании.                                     │

       │                                                                      │

       │                  Москва ; "Высшая школа" ; 1988г.                    │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │ Кузнецов Ю.Н.                                                        │

       │                  Математическое программирование                     │

       │ Кузубов В.И.                                                         │

       │                  Москва ; "Высшая школа" ; 1980г.                    │

       │ Волощенко А.Б.                                                       │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │ Фигурнов В.Э.    IBM PC для пользователя                             │

       │                                                                      │

       │                  Москва ; "Финансы и статистика" ; 1994г.            │

       │                                                                      │

       │                                                                      │

       │                                                                      │

       │ Фаронов В.В.     Основы Турбо Паскаля  6.0                           │

       │                                                                      │

       │                  Москва ; "МВТУ-ФЕСТО ДИДАКТИК" ; 1992г.             │

       │                                                                      │

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       ┌──────────────────────────────────────────────────────────────────────┐

       │                                                                      │

       │                                 - 19 -                               │

       │                                                                      │

       │      ЗАКЛЮЧЕНИЕ. ВЫВОДЫ ПО РАБОТЕ.   │

       │                                                                      │

       │                                                                      │

       │    Для решения данной задачи линейного программирования на тему      │

       │                                                                      │

       │ "оптимальный раскрой промышленных материалов" был использован        │

       │                                                                      │

       │ Симплекс-метод. Решение задачи помогло более глубоко и основательно  │

       │                                                                      │

       │ изучить и укрепить на практике все тонкости и моменты этого метода.  │

       │                                                                      │

       │    Симплекс-метод действительно является универсальным методом для   │

       │                                                                      │

       │ решения любой задачи линейного программирования. При ходе решения    │

       │                                                                      │

       │ заданной задачи была разработана универсальная программа для решения │

       │                                                                      │

       │ любой задачи при определении оптимального плана на минимум.          │

       │                                                                      │

       │    Разработка программы помогла более подробно изучить работу опера- │

       │                                                                      │

       │ торов алгоритмического языка Turbo-Pascal и особенности Симплекс-    │

       │                                                                      │

       │ метода.                                                              │

       │                                                                      │

       │    В результате прогона программы и решения задачи-теста получены    │

       │                                                                      │

       │ эдентичные результаты, следовательно программа составлена и отлажена │

       │                                                                      │

       │ правильно.                                                           │

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       └──────────────────────────────────────────────────────────────────────┘

       ┌──────────────────────────────────────────────────────────────────────┐

       │                                                                      │

       │                                 - 20 -                               │

       │                                                                      │

       │                 ОПИСАНИЕ ПРОГРАММЫ.               │

       │                                                                      │

       │                                                                      │

       │    Данная программа предназначена для решения задач линейного прог-  │

       │                                                                      │

       │ раммирования на минимум универсальным Симплекс-методом и состоит     │

       │                                                                      │

       │ из следующих основных частей:                                        │

       │                                                                      │

       │    @Основная программа. Обеспечивает ввод данных о системе уравнений  │

       │                                                                      │

       │ и ввод самой системы, оптимизация функции Симплекс-методом, вывод    │

       │                                                                      │

       │ промежуточного и конечного результатов.                              │

       │                                                                      │

       │    @Функция test0. Проверяет наличие положительных элементов в индекс-│

       │                                                                      │

       │ ной строке. Если такие элементы есть, возвращает в программу логиче- │

       │                                                                      │

       │ скую 'ЛОЖЬ', иначе 'ПРАВДА'.                                         │

       │                                                                      │

       │    @Функция testY. Проверяет наличие искуственных переменных в рабочем│

       │                                                                      │

       │ массиве. Если такие есть, возвращает 'ЛОЖЬ', иначе 'ПРАВДА'.         │

       │                                                                      │

       │    @Процедура indxY. Производит подсчет индексной строки в том случае,│

       │                                                                      │

       │ если искуственные переменные не выведены из базиса.                  │

       │                                                                      │

       │    @Процедура indxX. Производит подсчет индексной строки в том случае,│

       │                                                                      │

       │ если искуственные переменные выведены из базиса.                     │

       │                                                                      │

       │    @Процедура maxSt. Ищет в общем рабочем массиве максимальный (клю-  │

       │                                                                      │

       │ чевой) столбец и возвращает в основную программу его местоположение. │

       │                                                                      │

       │    @Процедура Str. Ищет в общем рабочем массиве ключевую строку и     │

       │                                                                      │

       │ возвращает в основную программу местоположение разрешающего элемента.│

       │                                                                      │

       │                                                                      │

       │ В листинге программы на языке PASCAL содержатся подробные коммента-  │

       │                                                                      │

       │ рии, которые описывают практически все действия, производимые прог-  │

       │                                                                      │

       │ раммой.                                                              │

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       └──────────────────────────────────────────────────────────────────────┘