油混合

Question

油混合

一家石油公司生产三种品牌的石油：普通，多级和最高。每种品牌的油由四种原油中的一种或多种组成，每种原油具有不同的润滑指数。有关原油库存的相关数据如下。

+-------------+-------------------+------------------+--------------------------+
| Crude Stock | Lubrication Index | Cost (€/barrell) | Supply per day (barrels) |
+-------------+-------------------+------------------+--------------------------+
| 1           | 20                | 7,10             | 1000                     |
+-------------+-------------------+------------------+--------------------------+
| 2           | 40                | 8,50             | 1100                     |
+-------------+-------------------+------------------+--------------------------+
| 3           | 30                | 7,70             | 1200                     |
+-------------+-------------------+------------------+--------------------------+
| 4           | 55                | 9,00             | 1100                     |
+-------------+-------------------+------------------+--------------------------+

每个品牌的油必须满足润滑指数和每个品牌的最低标准因此以不同的价格出售。有关三个品牌的石油的相关数据如下如下：

+------------+---------------------------+---------------+--------------+
| Brand      | Minimum Lubrication index | Selling price | Daily demand |
+------------+---------------------------+---------------+--------------+
| Regular    | 25                        | 8,50          | 2000         |
+------------+---------------------------+---------------+--------------+
| Multigrade | 35                        | 9,00          | 1500         |
+------------+---------------------------+---------------+--------------+
| Supreme    | 50                        | 10,00         | 750          |
+------------+---------------------------+---------------+--------------+

确定一天的最佳输出计划，假设生产可以是出售或以可忽略的成本存储。每日需求数据可能有其他解释。调查以下：
（a）每日需求代表潜在销售额。换句话说，该模型应包含需求上限（上限）。什么是最佳利润？
（b）日常要求是严格的义务。换句话说，模型应该包含精确满足的需求约束。什么是最佳利润？
（c）每日需求代表最低销售承诺，但所有产出都可以出售。换句话说，该模型应允许生产超过每日承诺。什么是最佳利润？

问题

我已经能够在Excel中构建以下模型并通过OpenSolver解决它，但我只能将常规油的混合物整合在一起。我正试图通过Kenneth R. Baker的“使用电子表格进行优化建模”一书来解决这个问题，但我仍然坚持这个练习。虽然我可以从另一个混合问题转移逻辑，但我不确定如何一次构建多个混合的模型。我把这个问题建模为不同原油库存成本的最小化问题。使用润滑指数数据，我将R-Lub指数的约束建立为线性约束。到目前为止，答案似乎适合普通石油。然而，使用这种方法，我不知道如何包括第二种多元油。

+--------------------+--------+--------+--------+--------+--+-------------+----+------+
| Decision Variables |        |        |        |        |  |             |    |      |
+--------------------+--------+--------+--------+--------+--+-------------+----+------+
|                    | C1     | C2     | C3     | C4     |  |             |    |      |
+--------------------+--------+--------+--------+--------+--+-------------+----+------+
| Inputs             | 1000   | 0      | 1000   | 0      |  |             |    |      |
+--------------------+--------+--------+--------+--------+--+-------------+----+------+
|                    |        |        |        |        |  |             |    |      |
+--------------------+--------+--------+--------+--------+--+-------------+----+------+
| Objective Function |        |        |        |        |  | Total       |    |      |
+--------------------+--------+--------+--------+--------+--+-------------+----+------+
| Cost               | 7,10 € | 8,50 € | 7,70 € | 9,00 € |  | 14.800,00 € |    |      |
+--------------------+--------+--------+--------+--------+--+-------------+----+------+
|                    |        |        |        |        |  |             |    |      |
+--------------------+--------+--------+--------+--------+--+-------------+----+------+
| Constraints        |        |        |        |        |  | LHS         |    | RHS  |
+--------------------+--------+--------+--------+--------+--+-------------+----+------+
| C1 supply          | 1      |        |        |        |  | 1000        | <= | 1000 |
+--------------------+--------+--------+--------+--------+--+-------------+----+------+
| C2 supply          |        | 1      |        |        |  | 0           | <= | 1100 |
+--------------------+--------+--------+--------+--------+--+-------------+----+------+
| C3 supply          |        |        | 1      |        |  | 1000        | <= | 1200 |
+--------------------+--------+--------+--------+--------+--+-------------+----+------+
| C4 supply          |        |        |        | 1      |  | 0           | <= | 1100 |
+--------------------+--------+--------+--------+--------+--+-------------+----+------+
| R- Lub Index       | -5     | 15     | 5      | 30     |  | 0           | >= | 0    |
+--------------------+--------+--------+--------+--------+--+-------------+----+------+
| R- Output          | 1      | 1      | 1      | 1      |  | 2000        | =  | 2000 |
+--------------------+--------+--------+--------+--------+--+-------------+----+------+
|                    |        |        |        |        |  |             |    |      |
+--------------------+--------+--------+--------+--------+--+-------------+----+------+
| Blending Data      |        |        |        |        |  |             |    |      |
+--------------------+--------+--------+--------+--------+--+-------------+----+------+
| R- Lub             | 20     | 40     | 30     | 55     |  | 25          | >= | 25   |
+--------------------+--------+--------+--------+--------+--+-------------+----+------+

以下是使用Excel公式的模型：

+--------------------+------+-----+------+----+--+----------------------------------------------------+----+------+
| Decision Variables |      |     |      |    |  |                                                    |    |      |
+--------------------+------+-----+------+----+--+----------------------------------------------------+----+------+
|                    | C1   | C2  | C3   | C4 |  |                                                    |    |      |
+--------------------+------+-----+------+----+--+----------------------------------------------------+----+------+
| Inputs             | 1000 | 0   | 1000 | 0  |  |                                                    |    |      |
+--------------------+------+-----+------+----+--+----------------------------------------------------+----+------+
|                    |      |     |      |    |  |                                                    |    |      |
+--------------------+------+-----+------+----+--+----------------------------------------------------+----+------+
| Objective Function |      |     |      |    |  | Total                                              |    |      |
+--------------------+------+-----+------+----+--+----------------------------------------------------+----+------+
| Cost               | 7,1  | 8,5 | 7,7  | 9  |  | =SUMMENPRODUKT(B5:E5;B8:E8)                        |    |      |
+--------------------+------+-----+------+----+--+----------------------------------------------------+----+------+
|                    |      |     |      |    |  |                                                    |    |      |
+--------------------+------+-----+------+----+--+----------------------------------------------------+----+------+
| Constraints        |      |     |      |    |  | LHS                                                |    | RHS  |
+--------------------+------+-----+------+----+--+----------------------------------------------------+----+------+
| C1 supply          | 1    |     |      |    |  | =SUMMENPRODUKT($B$5:$E$5;B11:E11)                  | <= | 1000 |
+--------------------+------+-----+------+----+--+----------------------------------------------------+----+------+
| C2 supply          |      | 1   |      |    |  | =SUMMENPRODUKT($B$5:$E$5;B12:E12)                  | <= | 1100 |
+--------------------+------+-----+------+----+--+----------------------------------------------------+----+------+
| C3 supply          |      |     | 1    |    |  | =SUMMENPRODUKT($B$5:$E$5;B13:E13)                  | <= | 1200 |
+--------------------+------+-----+------+----+--+----------------------------------------------------+----+------+
| C4 supply          |      |     |      | 1  |  | =SUMMENPRODUKT($B$5:$E$5;B14:E14)                  | <= | 1100 |
+--------------------+------+-----+------+----+--+----------------------------------------------------+----+------+
| R- Lub Index       | -5   | 15  | 5    | 30 |  | =SUMMENPRODUKT($B$5:$E$5;B15:E15)                  | >= | 0    |
+--------------------+------+-----+------+----+--+----------------------------------------------------+----+------+
| R- Output          | 1    | 1   | 1    | 1  |  | =SUMMENPRODUKT($B$5:$E$5;B16:E16)                  | =  | 2000 |
+--------------------+------+-----+------+----+--+----------------------------------------------------+----+------+
|                    |      |     |      |    |  |                                                    |    |      |
+--------------------+------+-----+------+----+--+----------------------------------------------------+----+------+
| Blending Data      |      |     |      |    |  |                                                    |    |      |
+--------------------+------+-----+------+----+--+----------------------------------------------------+----+------+
| R- Lub             | 20   | 40  | 30   | 55 |  | =SUMMENPRODUKT($B$5:$E$5;B19:E19)/SUMME($B$5:$E$5) | >= | 25   |
+--------------------+------+-----+------+----+--+----------------------------------------------------+----+------+

向正确的方向推动将是一个巨大的帮助。

Answer 1

我认为您希望您的目标是利润，我将其定义为销售额的总和 - 成本总和。

要包括所有混合物，请为每种混合物生成产量，润滑油指数，成本和价值的计算。对所用库存量，产量和润滑油指数应用约束，并优化利润。

我将模型放在一起如下......

列A到D是您提供的信息。
G2中的10：J5是每种混合物中使用的原料量的种子值。解决者会操纵这些。
K列包含产生的总产品量。根据您的调查（a），（b）和（c），这些将受到不同方式的约束。已填写=SUM(G3:J3)。
L列是产品的润滑油指数。如您所述，它是线性混合 - 对于混合问题通常不是这样。这些值将在Solver中受到约束。填充{=SUMPRODUCT(G3:J3,TRANSPOSE($B$2:$B$5))/$K3}。请注意，它是一个Control-Shift-Enter（CSE）公式，因为TRANSPOSE而需要。
M列是用于创建产品的库存成本。这用于利润计算。填充为{=SUMPRODUCT(G3:J3,TRANSPOSE($C$2:$C$5))}。这也是一个CSE公式。
N列是生产的产品的价值。这用于利润计算。已填写=K3*C8。
第7行是用于生成所有混合物的总库存量。这些值将在Solver中受到约束。它是=SUM(G3:G5)，填充在右侧。
利润计算为=SUM(N3:N5)-SUM(M3:M5)。

下面是解算器对话框的快照...

它做了以下......

目标是最大化利润。
它将通过操纵进入每种混合物的库存量来实现。
前四个约束（$G$7 through $J$7）确保不会违反可用库存量。
接下来的三个约束（$K$3 through $K$5）适用于案例（a） - 不超过产品而不是需求。
最后三个约束（$L$3 through $L$5）确保润滑油指数符合最低规格。
未显示 - 我为GRG Nonlinear选择了选项并选择了“Use Multistart”并取消选择“Require of Bounds on Variables”。

以下是案例（a）的结果......

对于情况（b），将列K上的约束更改为“=”而不是“＆lt; =”。以下是结果......

对于情况（c），将列K上的约束更改为“＆gt; =”。以下是结果......

Answer 2

我想我想出了一个解决方案，但我不确定这是否正确。

| Decision Variables |         |        |        |        |             |        |        |        |        |        |        |        |   |                                |    |      |
|--------------------|---------|--------|--------|--------|-------------|--------|--------|--------|--------|--------|--------|--------|---|--------------------------------|----|------|
|                    | C1R     | C1M    | C1S    | C2R    | C2M         | C2S    | C3R    | C3M    | C3S    | C4R    | C4M    | C4S    |   |                                |    |      |
| Inputs             | 1000    | 0      | 0      | 800    | 0           | 300    | 0      | 1200   | 0      | 200    | 300    | 600    |   |                                |    |      |
|                    |         |        |        |        |             |        |        |        |        |        |        |        |   |                                |    |      |
| Objective Function |         |        |        |        |             |        |        |        |        |        |        |        |   | Total Profit (Selling  - Cost) |    |      |
| Cost               | 7,10 €  | 7,10 € | 7,10 € | 8,50 € | 8,50 €      | 8,50 € | 7,70 € | 7,70 € | 7,70 € | 9,00 € | 9,00 € | 9,00 € |   | 3.910,00 €                     |    |      |
|                    |         |        |        |        |             |        |        |        |        |        |        |        |   |                                |    |      |
| Constraints        |         |        |        |        |             |        |        |        |        |        |        |        |   | LHS                            |    | RHS  |
| Regular            | -5      |        |        | 15     |             |        | 5      |        |        | 30     |        |        |   | 13000                          | >= | 0    |
| Multi              |         | -15    |        |        | 5           |        |        | -5     |        |        | 20     |        |   | 0                              | >= | 0    |
| Supreme            |         |        | -30    |        |             | -10    |        |        | -20    |        |        | 5      |   | 0                              | >= | 0    |
| C1 Supply          | 1       | 1      | 1      |        |             |        |        |        |        |        |        |        |   | 1000                           | <= | 1000 |
| C2 Supply          |         |        |        | 1      | 1           | 1      |        |        |        |        |        |        |   | 1100                           | <= | 1100 |
| C3 Supply          |         |        |        |        |             |        | 1      | 1      | 1      |        |        |        |   | 1200                           | <= | 1200 |
| C4 Supply          |         |        |        |        |             |        |        |        |        | 1      | 1      | 1      |   | 1100                           | <= | 1100 |
| Regular Demand     | 1       |        |        | 1      |             |        | 1      |        |        | 1      |        |        |   | 2000                           | >= | 2000 |
| Multi Demand       |         | 1      |        |        | 1           |        |        | 1      |        |        | 1      |        |   | 1500                           | >= | 1500 |
| Supreme Demand     |         |        | 1      |        |             | 1      |        |        | 1      |        |        | 1      |   | 900                            | >= | 750  |
|                    |         |        |        |        |             |        |        |        |        |        |        |        |   |                                |    |      |
|                    |         |        |        |        |             |        |        |        |        |        |        |        |   |                                |    |      |
| Selling            |         |        |        |        |             |        |        |        |        |        |        |        |   |                                |    |      |
| Regular            | 8,50 €  | x      | 2000   | =      | 17.000,00 € |        |        |        |        |        |        |        |   |                                |    |      |
| Multi              | 9,00 €  | x      | 1500   | =      | 13.500,00 € |        |        |        |        |        |        |        |   |                                |    |      |
| Supreme            | 10,00 € | x      | 900    | =      | 9.000,00 €  |        |        |        |        |        |        |        |   |                                |    |      |
|                    |         |        |        |        | 39.500,00 € |        |        |        |        |        |        |        |   |                                |    |      |

混合模型：石油生产

油混合

问题

2 个答案: