本次英国代写主要为数理统计的限时测试

1(i)使用两阶段方法来找到以下直线的最佳解决方案:
耳朵编程问题:
最大z = −2×1 − 4×2
服从x1,x2≥0且
x1 + x2≤5
3×1 + 2×2≥6。
明确说明您的最终最佳解决方案。
提示:不计算预处理步骤,您只需要一个单纯形
阶段1的迭代。(15分)
(ii)使用对偶单纯形法找到问题的最佳解决方案
(i)部分所述。 (10分)

2 CompanyMmix四种成分和一个装填器一起生产营养粉。这
粉末包含三种营养素:A,B和C。营养素含量(单位:克/每克
公斤的原料)和成本(单位:磅/千克的成分-
ent)的每种成分都在此表中给出:
A(克/千克)B(克/千克)C(克/千克)成本(英镑/千克)
成分1 1 8 4 4
成分2 2 1 5 6
成分3 7 4 1 8
成分4 2 6 2 5
已知以下信息:
•根据行销法规,如果M公司要声明
粉末中含有一种营养素,那么该营养素的含量必须不少于
比最小值。对于不同的营养素,最小值是不同的
ents。一公斤粉末中的营养素最小值为5克
ent A,B为7克,C为4克。
•CompanyMintend声称该粉末含有至少两种营养素。
他们没有选择哪个。
•M公司计划混合5000公斤粉末。
•如果使用成分2或成分4,则固定安装成本为300英镑。
•两种成分的化学特性均满足以下条件:
使用3和3,则不能使用成分2。
•CompanyM从P公司购买成分1和2。
出于种种考虑,P公司不接受小订单。因此,com-
三色堇M可以单独购买不少于100公斤的成分1,
或单独不少于120公斤的成分2,或不少于290公斤,
两种成分的克合计。
•淀粉用作填充物。它的营养成分和成本可以忽略不计。
对于应使用多少没有单独的限制。
令x1,x2,x3和x4为1中四种成分的量(以千克为单位)
公斤的粉末。制定混合整数线性规划问题
从中可以找到最佳解决方案,以最大程度地降低总成本。注意:做
不要试图找到问题的数值解决方案;您可能需要介绍
更多变量。 (25分)
1(I) shǐyòng liǎng jiēduàn fāngfǎ lái zhǎodào yǐx

1 (i) Use the two-phase method to find the optimal solution for the following lin-
ear programming problem:
max z = −2×1 − 4×2
subject to x1, x2 ≥ 0 and
x1 + x2 ≤ 5,
3×1 + 2×2 ≥ 6.
State clearly your final optimal solution.
Hint: not counting the preprocessing step, you need only one simplex
iteration in phase 1. (15 marks)
(ii) Use the dual simplex method to find the optimal solution for the problem
stated in Part (i). (10 marks)

2 CompanyMmixes four ingredients and a filler to produce a nutritional powder. The
powder contains three nutrients: A, B and C. The nutrient contents (unit: grams per
kilogram of the ingredient) and the costs (unit: pounds per kilogram of the ingredi-
ent) of each ingredient are given in this table:
A (g/kg) B (g/kg) C (g/kg) Cost (£/kg)
Ingredient 1 1 8 4 4
Ingredient 2 2 1 5 6
Ingredient 3 7 4 1 8
Ingredient 4 2 6 2 5
The following information is known:
• According to marketing regulations, if company M wants to claim that the
powder contains a nutrient, then the amount of that nutrient must be no less
than a minimum value. The minimum value is different for different nutri-
ents. In one kilogram of the powder, the minimum value is 5 grams for nutri-
ent A, 7 grams for B, and 4 grams for C.
• CompanyMintends to claim that the powder contains at least two nutrients.
They have no preference for which ones.
• Company M plans to mix 5000 kilograms of the powder.
• A fixed set-up cost of £300 is incurred if either ingredient 2 or 4 are used.
• The chemical properties of the ingredients are such that, if both ingredients 1
and 3 are used, then ingredient 2 cannot be used.
• CompanyMpurchases ingredient 1 and 2 from company P. Due to profitabil-
ity considerations, company P does not accept small orders. Therefore, com-
pany M can purchase either no less than 100 kilograms of ingredient 1 alone,
or no less than 120 kilograms of ingredient 2 alone, or no less than 290 kilo-
grams of the two ingredients combined.
• Starch is used as the filler. Its nutritional contents and cost can be neglected.
There is no separate constraint on how much it should be used.
Let x1, x2, x3, and x4 be the amounts (in kilograms) of the four ingredients in 1
kilogram of the powder. Formulate the mixed integer linear programming problem
from which one can find the optimal solution to minimise the total cost. Note: do
NOT try to find the numerical solution of the problem; youmay need to introduce
more variables. (25 marks)