Визначаємо мету розв’язування задачі.
З’ясовуємо, які величини в задачі є невідомими, і вводимо відповідні змінні.
І. Побудуємо математичну модель задачі.
РОЗВ’ЯЗУВАННЯ ЗАДАЧ ЗА ДОПОМОГОЮ СИМПЛЕКС-МЕТОДУ
Використовуючи графічний метод, знайти мінімум і максимум функції L при вказаних обмеженнях.
| 101. L = 3x1 + x2, x1 + 2x2 ≤ 14, -5x1 + 3x2 ≤15, x1 + 2x2 ≥ 8, x1, x2 ≥ 0. | 102. L = x1 + 4x2, 4x1 - 2x2 ≤ 12, -x1 + x2 ≤ 5, x1 + 2x2 ≥ 8, x1, x2 ≥ 0. | 103. L = -2x1 + x2, x1 - x2 ≤ 3, 3x1 - 4x2 ≥ -12, x1 ≤ 5, x1, x2 ≥ 0. | 104. L = -2x1 + x2, x1 + x2 ≤ 6, x1 ≤ 5, x2 ≤ 3, x1, x2 ≥ 0. |
| 105. L = 9x1 + x2, x1 - 5x2 ≤ 5, -x1 + 4x2 ≤ 4, x1 + x2 ≤ 8, x1, x2 ≥ 0. | 106. L = 4x1 + 3x2, x1 + x2 ≤ 6, 3x1 + 10x2 ≤ 26, x1 + 11x2 ≤ 20, x1, x2 ≥ 0. | 107. L = -2x1 - 5x2, 2x1 + x2 ≤ 8, -2x1 + 3x2 ≤ 6, 2x1 + 4x2 ≥ 8, x1, x2 ≥ 0. | 108. L = x1 + 3x2, 2x1 - x2 ≥ 4, x1 - x2 ≥ 1, 2x1 + x2 ≤ 6, x1, x2 ≥ 0. |
| 109. L = -2x1 + 5x2, x1 + x2 ≥ 3, x1 ≥ 2, x2 ≤ 8, x1, x2 ≥ 0. | 110. L = -3x1 + 6x2, 5x1 - 2x2 ≤ 4, x1 - 2x2 ≥ -4, x1 + x2 ≥ 4, x1, x2 ≥ 0. | 111. L = x1 + 2x2, 5x1 - 2x2 ≤ 4, x1 - 2x2 ≥ -4, x1 + x2 ≥ 4, x1, x2 ≥ 0. | 112. L = 2x1 + 5x2, -x1 + x2 ≤ 1, x1 - x2 ≤ 1, x1 + x2 ≤ 2, x1, x2 ≥ 0. |
| 113. L = 4x1 + 3x2, 2x1 + x2 ≤ 30, x1 + x2 ≤ 20, 3x1 + 2x2 ≥ 6, x1 ≤ 11, x2 ≤ 12, x1, x2 ≥ 0. | 114. L = x1 + x2, 3x1 + 4x2 ≤ 70, 4x1 + 3x2 ≤ 70, x1 + 2x2 ≥ 2, x1 ≤ 13, x2 ≤ 13, x1, x2 ≥ 0. | 115. L = 5x1 + 2x2, 2x1 + x2 ≤ 30, 5x1 + 3x2 ≤ 80, x1 + 3x2 ≥ 3, x1 ≤ 15, x2 ≤ 12, x1, x2 ≥ 0. | 116. L = 2x1 + x2, 3x1 + x2 ≤ 40, x1 + x2 ≤ 20, 2x1 + 3x2 ≥ 6, x1 ≤ 11, x2 ≤ 11, x1, x2 ≥ 0. |
| 117. L = 7x1 + 3x2, 3x1 + x2 ≤ 40, 3x1 + 2x2 ≤ 50, x1 + x2 ≥ 1, x1 ≤ 13, x2 ≤ 11, x1, x2 ≥ 0. | 118. L = 5x1 + 3x2, 5x1 + 2x2 ≤ 70, 5x1 + 4x2 ≤ 50, 3x1 + x2 ≥ 3, x1 ≤ 15, x2 ≤ 12, x1, x2 ≥ 0. | 119. L = x1 + x2, 3x1 + 5x2 ≤ 80, 5x1 + 3x2 ≤ 80, x1 + 3x2 ≥ 3, x1 ≤ 13, x2 ≤ 13, x1, x2 ≥ 0. | 120. L = 6x1 + 5x2, 3x1 + 2x2 ≤ 50, x1 + x2 ≤ 20, x1 + 2x2 ≥ 2, x1 ≤ 11, x2 ≤ 12, x1, x2 ≥ 0. |
| 121. L = 7x1 + 4x2, 2x1 + x2 ≤ 30, 7x1 + 5x2 ≤ 120, x1 + 2x2 ≥ 4, x1 ≤ 17, x2 ≤ 11, x1, x2 ≥ 0. | 122. L = 2x1 + x2, 3x1 + x2 ≤ 40, 2x1 + 3x2 ≤ 50, x1 + x2 ≥ 2, x1 ≤ 11, x2 ≤ 12, x1, x2 ≥ 0. | 123. L = 3x1 + x2, 4x1 + x2 ≤ 50, 3x1 + 2x2 ≤ 50, 3x1 + x2 ≥ 6, x1 ≤ 13, x2 ≤ 11, x1, x2 ≥ 0. | 124. L = 7x1 + 5x2, 6x1 + 5x2 ≤ 110, x1 + x2 ≤ 20, 2x1 + x2 ≥ 2, x1 ≤ 11, x2 ≤ 15, x1, x2 ≥ 0. |
| 125. L = 5x1 + 4x2, 5x1 + 3x2 ≤ 80, x1 + x2 ≤ 20, 2x1 + x2 ≥ 4, x1 ≤ 11, x2 ≤ 13, x1, x2 ≥ 0. | 126. L = 7x1 + 6x2, 2x1 + x2 ≤ 30, 4x1 + 5x2 ≤ 90, x1 + x2 ≥ 4, x1 ≤ 14, x2 ≤ 11, x1, x2 ≥ 0. | 127. L = x1 + x2, x1 + 2x2 ≤ 30, 2x1 + x2 ≤ 30, 3x1 + x2 ≥ 6, x1 ≤ 12, x2 ≤ 12, x1, x2 ≥ 0. | 128. L = 3x1 + 2x2, 3x1 + x2 ≤ 40, x1 + x2 ≤ 20, x1 + 4x2 ≥ 8, x1 ≤ 12, x2 ≤ 11, x1, x2 ≥ 0. |
| 129. L = x1 + 2x2, x1 + 3x2 ≤ 40, x1 + x2 ≤ 20, 4x1 + x2 ≥ 8, x1 ≤ 11, x2 ≤ 11, x1, x2 ≥ 0. | 130. L = 3x1 + 4x2, x1 + 2x2 ≤ 30, x1 + x2 ≤ 20, 2x1 + x2 ≥ 8, x1 ≤ 12, x2 ≤ 11, x1, x2 ≥ 0. | 131. L = 3x1 + 7x2, x1 + 3x2 ≤ 40, 2x1 + 3x2 ≤ 50, x1 + x2 ≥ 5, x1 ≤ 11, x2 ≤ 13, x1, x2 ≥ 0. | 132. L = 2x1 + 5x2, x1 + 2x2 ≤ 30, 3x1 + 5x2 ≤ 80, x1 + x2 ≥ 3, x1 ≤ 12, x2 ≤ 15, x1, x2 ≥ 0. |
| 133. L = x1 + 3x2, x1 + 4x2 ≤ 50, 2x1 + 3x2 ≤ 50, x1 + 2x2 ≥ 6, x1 ≤ 11, x2 ≤ 13, x1, x2 ≥ 0. | 134. L = 4x1 + 5x2, 3x1 + 5x2 ≤ 80, x1 + x2 ≤ 20, 3x1 + 2x2 ≥ 6, x1 ≤ 13, x2 ≤ 11, x1, x2 ≥ 0. | 135. L = 3x1 + 5x2, 2x1 + 5x2 ≤ 70, 4x1 + 5x2 ≤ 90, 3x1 + x2 ≥ 6, x1 ≤ 12, x2 ≤ 15, x1, x2 ≥ 0. | 136. L = 5x1 + 6x2, 2x1 + 3x2 ≤ 50, x1 + x2 ≤ 20, x1 + 2x2 ≥ 8, x1 ≤ 12, x2 ≤ 11, x1, x2 ≥ 0. |
| 137. L = 2x1 + 3x2, x1 + 3x2 ≤ 40, x1 + x2 ≤ 20, 2x1 + x2 ≥ 6, x1 ≤ 11, x2 ≤ 12, x1, x2 ≥ 0. | 138. L = 6x1 + 7x2, x1 + 2x2 ≤ 30, 5x1 + 4x2 ≤ 90, x1 + x2 ≥ 4, x1 ≤ 11, x2 ≤ 14, x1, x2 ≥ 0. | 139. L = 5x1 + 7x2, 5x1 + 6x2 ≤ 110, x1 + x2 ≤ 20, x1 + 3x2 ≥ 6, x1 ≤ 15, x2 ≤ 11, x1, x2 ≥ 0. | 140. L = x1 + x2, 2x1 + 3x2 ≤ 50, 3x1 + 2x2 ≤ 50, x1 + 4x2 ≥ 8, x1 ≤ 12, x2 ≤ 15, x1, x2 ≥ 0. |
| 141. L = 4x1 + 7x2, x1 + 2x2 ≤ 30, 5x1 + 7x2 ≤ 120, 2x1 + 3x2 ≥ 12, x1 ≤ 11, x2 ≤ 17, x1, x2 ≥ 0. | 142. L = x1 + 2x2, -x1 + 5x2 ≤ 19, x1 - x2 ≤ 1, 3x1 + x2 ≥ 7, x1, x2 ≥ 0. | 143. L = x1 + x2, x1 + x2 ≥ 1, -5x1 + x2 ≤ 0, -x1 + 5x2 ≥ 0, x1 + x2 ≤ 6, x1, x2 ≥ 0. | 144. L = x1 + x2, x1 + 3x2 ≤ 40, 3x1 + 2x2 ≤ 50, x1 + x2 ≥ 5, x1 ≤ 12, x2 ≤ 11, x1, x2 ≥ 0. |
| 145. L = x1 + x2, 5x1 - 2x2 ≤ 7, -x1 + x2 ≤ 5, x1 + x2 ≤ 6, x1, x2 ≥ 0. | 146. L = -2x1 + x2, 2x1 + x2 ≤ 8, x1 + 3x2 ≥ 6, 3x1 + x2 ≥ 3, x1, x2 ≥ 0. | 147. L = 2x1 - 4x2, 8x1 - 5x2 ≤ 16, x1 + 3x2 ≥ 2, 2x1 + 7x2 ≤ 9, x1, x2 ≥ 0. | 148. L = 2x1 - x2, x1 - x2 ≥ -3, 6x1 + 7x2 ≤ 42, 2x1 - 3x2 ≤ 6, x1, x2 ≥ 0. |
| 149. L = -2x1 + x2, 2x1 + x2 ≤ 8, x1 + x2 ≤ 6, -3x1 + 2x2 ≥ 3, x1, x2 ≥ 0. | 150. L = 3x1 + x2, 3x1 + 5x2 ≥ 15, 5x1 + 3x2 ≥ 15, x1 ≥ 1, x2 ≥ 1, x1, x2 ≥ 0. | 151. L = 4x1 + 6x2 + 6, 2x1 - 3x2 ≤ 12, -x1 + 2x2 ≤ 8, 3x1 + 2x2 ≤ 24, x1, x2 ≥ 0. | 152. L = 7x1 + 5x2, 7x1 + 5x2 ≥ 7, 7x1 – 5x2 ≥ 35, x1 - x2 ≤ 0, x1, x2 ≥ 0. |
| 153. L = 3x1 + 2x2 – 10, 2x1 - x2 ≥ 0, 2x1 + 3x2 ≥ 6, x1 - 2x2 ≤ 4, x1, x2 ≥ 0. | 154. L = 4x1 + 3x2 + 25, 4x1 + x2 ≥ 8, x1 - x2 ≤ 5, 7x1 + 10x2 ≤ 70, x1, x2 ≥ 0. | 155. L = x1 + 2x2 - 5, 5x1 - 2x2 ≤ 4, -x1 + 2x2 ≤ 4, x1 + x2 ≥ 4, x1, x2 ≥ 0. | 156. L = 7x1 + 5x2 + 20, 3x1 + 8x2 ≥ 24, 2x1 - x2 ≤ 6, -x1 + 2x2 ≤ 4, x1, x2 ≥ 0. |
| 157. L = -3x1 - 2x2 +15, x1 - 4x2 ≤ 0, -3x1 + x2 ≤ 3, 6x1 + 5x2 ≥ 30, x1, x2 ≥ 0. | 158. L = 2x1 + 5x2 + 12, 6x1 + 5x2 ≥ 30, 3x1 - 2x2 ≤ 12, -3x1 + 6x2 ≤ 12, x1, x2 ≥ 0. | 159. L = 6x1 + 4x2 + 20 2x1 + x2 ≥ 4, x1 + 2x2 ≥ 4, -x1 + x2 ≤ 5, x1, x2 ≥ 0. | 160. L = -7x1 - 4x2 + 15, 3x1 - 2x2 ≤ 12, 3x1 + 2x2 ≥ 6, -x1 + 2x2 ≥ 4, x1, x2 ≥ 0. |
| 161. L = x1 + x2 – 2, 5x1 - 2x2 ≤ 7, x1 - x2 ≥ -5, x1 + x2 ≤ 6, x1, x2 ≥ 0. | 162. L = 2x1 + 3x2 + 16, 2x1 - 5x2 ≤ 10, -2x1 + 5x2 ≤ 10, 2x1 + 3x2 ≥ 12, x1, x2 ≥ 0. | 163. L = 4x1 + 3x2 +5, x1 - x2 ≥ 0, x1 + 3x2 ≥ 6, 2x1 - 3x2 ≤ 21, x1, x2 ≥ 0. | 164. L = -x1 + 2x2 – 3, 5x1 - 2x2 ≤ 4, -x1 + 2x2 ≤ 4, x1 + x2 ≥ 4, x1, x2 ≥ 0. |
| 165. L = 8x1 + 2x2, x1 - 4x2 ≤ 4, -4x1 + x2 ≤ 4, x1 + x2 ≤ 6, x1, x2 ≥ 0. | 166. L = -2x1 + x2, 4x1 + 3x2 ≥ 22, 2x1 + 5x2 ≥ 22, x1 - x2 ≤ 2, x1, x2 ≥ 0. | 167. L = -x1 + 2x2, 5x1 + 2x2 ≤ 37, x1 - x2 ≤ -1, 2x1 + 5x2 ≥ 19, x1, x2 ≥ 0. | 168. L = -3x1 + 6x2, 5x1 - 2x2 ≥ 4, x1 - 2x2 ≤ -4, x1 + x2 ≥ 4, x1, x2 ≥ 0. |
| 169. L = x1 + x2, x1 + x2 ≥ 2, -5x1 + x2 ≤ 0, -x1 + 5x2 ≥ 0, x1 + x2 ≤ 5, x1, x2 ≥ 0. | 170. L = 3x1 + 5x2 + 20, 2x1 + 3x2 ≥ 11, x1 + x2 ≤ 5, x1 ≤ 2, x2 ≤ 4, x1, x2 ≥ 0. | 171. L = -5x1 + 3x2, 3x1 + 2x2 ≤ 18, -x1 + x2 ≤ 3, 5x1 + 2x2 ≥ 10, x1 - 2x2 ≤ 4, x1, x2 ≥ 0. | 172. L = 2x1 - 2x2, -x1 + 4x2 ≥ 10, 2x1 + 3x2 ≥ 13, 3x1 - x2 ≥ 14, x1 + x2 ≤ 20, x1, x2 ≥ 0. |
| 173. L = x1 + 2x2, 2x1 + 3x2 ≤ 23, 4x1 - x2 ≥ 11, x1 - 2x2 ≤ 1, x1, x2 ≥ 0. | 174. L = 3x1 + x2, 3x1 + 5x2 ≤ 15, x1 ≥ 1, x2 ≥ 1, x1, x2 ≥ 0. | 175. L = 7x1 - 4x2, 3x1 - 2x2 ≤ 15, 3x1 - 2x2 ≥ 6, -x1 + 2x2 ≤ 4, x1, x2 ≥ 0. | 176. L = -3x1 - 4x2 – 20, 3x1 + 4x2 ≥ 12, -4x1 + 3x2 ≤ 12, x1 - 2x2 ≤ 4, x1, x2 ≥ 0. |
| 177. L = -7x1 - 6x2 + 19, x1 + 2x2 ≤ 10, 2x1 + x2 ≤ 10, x1 + x2 ≥ 2, x1, x2 ≥ 0. | 178. L = 6x1 + 5x2 – 30, x1 + 2x2 ≥ 8, 2x1 + x2 ≥ 8, 2x1 - x2 ≥ 0, x1, x2 ≥ 0. | 179. L = 2x1 + 3x2 + 16, 5x1 - 2x2 ≤ 10, x1 + x2 ≤ 3, 4x1 + 3x2 ≥ 13, x1, x2 ≥ 0. | 180. L = 9x1 + 3x2 + 8, -5x1 + 4x2 ≤ 20, 5x1 - 4x2 ≤ 20, 8x1 + 7x2 ≤ 56, x1, x2 ≥ 0. |
| 181. L = 4x1 + 7x2, 5x1 + 3x2 ≥ 15, x1 + x2 ≤ 6, -5x1 + x2 ≤ 0, x1, x2 ≥ 0. | 182. L = 3x1 + 2x2 – 10, 2x1 - x2 ≥ 0, 2x1 + 3x2 ≥ 6, x1 - 2x2 ≤ 4, x1, x2 ≥ 0. | 183. L = 7x1 + 4x2 + 40, 9x1 - 5x2 ≤ 36, -x1 + 3x2 ≤ 9, 7x1 + 6x2 ≤ 42, x1, x2 ≥ 0. | 184. L = -7x1 - 4x2 + 15, 3x1 - 2x2 ≤ 12, 3x1 + 2x2 ≥ 6, -x1 + 2x2 ≤ 4, x1, x2 ≥ 0. |
| 185. L = 8x1 + 4x2 + 22, x1 - 2x2 ≤ 0, -x1 + x2 ≤ 8, 10x1 + 9x2 ≤ 90, x1, x2 ≥ 0. | 186. L = -5x1 - 2x2, 4x1 + 3x2 ≤ 24, x1 + x2 ≥ 2, x1 - 2x2 ≤ 2, x1, x2 ≥ 0. | 187. L = -x1 + 3x2 + 20, 2x1 + 3x2 ≥ 11, x1 + x2 ≤ 5, x1 ≤ 2, x2 ≤ 5, x1, x2 ≥ 0. | 188. L = x1 + 2x2, 2x1 + 3x2 ≥ 3, -x1 + 2x2 ≤ 1, 3x1 - x2 ≤ 6, x1, x2 ≥ 0. |
| 189. L = 4x1 + 2x2 – 20, x1 + 2x2 ≥ 5, 2x1 - x2 ≥ 0, x2 ≤ 5, x1, x2 ≥ 0. | 190. L = 3x1 + 2x2, x1 - 3x2 ≤ 3, -4x1 + 3x2 ≤ 24, x1 ≥ 6, x1, x2 ≥ 0. | 191. L = 7x1 + 3x2 + 30, 10x1 + 9x2 ≤ 90, -x1 + 2x2 ≤ 6, 6x1 + 5x2 ≤ 30, x1, x2 ≥ 0. | 192. L = 7x1 + 4x2 + 40, 9x1 - 5x2 ≤ 36, -x1 + 3x2 ≤ 9, 7x1 + 6x2 ≤ 42, x1, x2 ≥ 0. |
| 193. L = 4x1 + 6x2 + 20, 8x1 + 7x2 ≤ 56, 3x1 + 5x2 ≥ 15, 5x1 + 3x2 ≥ 15, x1, x2 ≥ 0. | 194. L = -3x1 + 2x2 + 10, x1 - x2 ≤ 10, x1 + x2 ≥ 2, -x1 + x2 ≤ 2, x1, x2 ≥ 0. | 195. L = -3x1 - 2x2 + 14, 2x1 + x2 ≥ 4, x1 + x2 ≤ 4, -x1 + 2x2 ≤ 6, x1, x2 ≥ 0. | 196. L = -4x1 - 3x2 + 15, 6x1 + 3x2 ≤ 24, -4x1 + x2 ≤ 12, 3x1 - x2 ≤ 12, x1, x2 ≥ 0. |
| 197. L = -4x1 - 3x2 + 8, 4x1 + 5x2 ≤ 20, 7x1 + 3x2 ≤ 21, 2x1 + x2 ≥ 2, x1, x2 ≥ 0. | 198. L = -3x1 + 6x2 + 20, 5x1 + 2x2 ≤ 20, x1 + 3x2 ≥ 6, x1 ≥ 2, x1, x2 ≥ 0. | 199. L = 3x1 + 2x2 + 10, x1 - x2 ≤ 10, x1 + x2 ≥ 2, -x1 + x2 ≤ 1, x1, x2 ≥ 0. | 200. L = 7x1 + 5x2 + 20, 3x1 + 8x2 ≥ 24, 2x1 - x2 ≤ 6, -x1 + 2x2 ≤ 4, x1, x2 ≥ 0. |
Приклад 3.
Нехай на підприємстві можна виробляти два види продукції Р1 і Р2, для чого необхідно використовувати три види сировини S1, S2, S3, запаси яких обмежені і відповідно дорівнюють 60, 65, 135. Відомі норми витрат кожного виду сировини (і) на виробництві кожного виду продукції (j) – aij та прибуток від реалізації кожного виду продукції. Треба визначити такий план випуску кількості продукції кожного виду, щоб загальний прибуток від її реалізації був максимальним. Наведені умови задачі наочно подамо у вигляді таблиці.
| Вид сировини | Вид продукції Р1 | Вид продукції Р2 | Запаси сировини |
| S1 | |||
| S2 | |||
| S3 | |||
| Прибуток |
Розв’язання:
В розглянутій задачі невідомими є кількість продукції видів Р1 і Р2. Отже, вводимо відповідні змінні х1 і х2.
В даному прикладі – це максимізація загального прибутку від реалізації.