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线性规划模型

线性规划是起源最早， 使用最为广泛的数学规划模型，

$$\begin{array}{rlr}\underset{x\in {\mathbb{R}}^n}{\min }& \sum _{i=1}^n{c}_i{x}_i& \\ \text{s.t.}& l_j\le \sum _{i=1}^n{a}_{ij}{x}_i\le u_j& j=1\dots m\\ & l_i\le x_i\le u_i& i=1+m\dots n.\end{array}$$

例子

using JuMP; import HiGHS; import DataFrames
model = Model(HiGHS.Optimizer)
# 变量
@variable(model, x >= 0); @variable(model, 0 <= y <= 3);@variable(model, z <= 1)  
# 目标函数
@objective(model, Min, 12x + 20y - z)
# 约束
@constraint(model, c1, 6x + 8y >= 100)
@constraint(model, c2, 7x + 12y >= 120)
@constraint(model, c3, x + y <= 20)
# 打印模型
print(model)

$$\begin{array}{rl}\min & 12x+20y-z\\ \mathrm{Subject}\mathrm{to}& 6x+8y\ge 100.0\\ & 7x+12y\ge 120.0\\ & x+y\le 20.0\\ & x\ge 0.0\\ & y\ge 0.0\\ & y\le 3.0\\ & z\le 1.0\end{array}$$
​

#  求解
optimize!(model)
solution_summary(model; verbose = true)  # 汇总主要信息

线性规划灵敏性分析

report = lp_sensitivity_report(model)
function variable_report(xi)return (name = name(xi),   # 变量名lower_bound = has_lower_bound(xi) ? lower_bound(xi) : -Inf,  #下界value = value(xi),  # 数值解upper_bound = has_upper_bound(xi) ? upper_bound(xi) : Inf, # 上界reduced_cost = reduced_cost(xi),  # 降低成本, 等价于有效变量的影子价格obj_coefficient = coefficient(objective_function(model), xi),  # 目标函数的线性系数# 目标函数系数的变化范围, 保持基底最优性  allowed_decrease = report[xi][1], # 允许减少allowed_increase = report[xi][2], # 允许增长)
end
variable_report(x)
variable_df =  DataFrames.DataFrame(variable_report(xi) for xi in all_variables(model))

name	lower_bound	value	upper_bound	reduced_cost	obj_coefficient	allowed_decrease
String	Float64	Float64	Float64	Float64	Float64	Float64
1	x	0.0	15.0	Inf	0.0	12.0
2	y	0.0	1.25	3.0	0.0	20.0
3	z	-Inf	1.0	1.0	-1.0	-1.0

约束集合灵敏性

function constraint_report(ci)return (name = name(ci),  # 变量名value = value(ci),  # 约束左侧值rhs = normalized_rhs(ci),  # 约束右侧值slack = normalized_rhs(ci) - value(ci),shadow_price = shadow_price(ci), # 影子价格allowed_decrease = report[ci][1], # 基底最优性 allowed_increase = report[ci][2], # 基底最优性 )
end
c1_report = constraint_report(c1)
constraint_df = DataFrames.DataFrame(constraint_report(ci) for (F, S) in list_of_constraint_types(model) for ci in all_constraints(model, F, S) if F == AffExpr
)

name	value	rhs	slack	shadow_price	allowed_decrease	allowed_increase
String	Float64	Float64	Float64	Float64	Float64	Float64
1	c1	100.0	100.0	0.0	-0.25	-4.0
2	c2	120.0	120.0	0.0	-1.5	-3.33333
3	c3	16.25	20.0	3.75	0.0	-3.75

分析问题

basic = filter(row -> iszero(row.reduced_cost), variable_df)  # 过滤等号成立的变量

name	lower_bound	value	upper_bound	reduced_cost	obj_coefficient	allowed_decrease
String	Float64	Float64	Float64	Float64	Float64	Float64
1	x	0.0	15.0	Inf	0.0	12.0
2	y	0.0	1.25	3.0	0.0	20.0

non_basic = filter(row -> !iszero(row.reduced_cost), variable_df)  # 过滤等号不成立的变量
binding = filter(row -> iszero(row.slack), constraint_df)  # 过滤等号成立的约束条件
binding2 = filter(row -> !iszero(row.shadow_price), constraint_df)#  过滤等号不成立的约束条件