基于JuMP与GLPK的员工调度优化代码报错排查

DevOps 小满

2026-7-4

JuMP+GLPK员工调度优化问题错误修复

问题背景

使用JuMP与GLPK求解员工调度优化问题，目标是确定满足工作量需求的最优员工数量，代码基于数学模型编写，但运行时出现类型转换错误，无法得到结果。

出错代码

using JuMP
using GLPK

# Constants and parameters
Wjk = [[19, 19, 14]  [16, 16, 11]  [22, 22, 16]  [22, 22, 16]  [22, 22, 16]  [22, 22, 16]  [22, 22, 16]]   # Workload requirement
Ajk = 8  # Some constant value for Ajk
M = 395  # Total number of shifts

# Create a JuMP model
model = Model(GLPK.Optimizer)

# Decision variables for shift allocation
@variable(model, x[1:N, 1:3, 1:7] >= 0, Int)  # Decision variable for shift allocation


# Objective function
@objective(model, Min, sum(x[i, j, k] for i in 1:N, j in 1:3, k in 1:7))


# Objective function
#@objective(model, Min, sum(x[i, j, k] for i in 1:N for j in 1:3 for k in 1:7))


# Constraints
for i in 1:N
    for k in 1:7
        @constraint(model, sum(x[i, j, k] for j in 1:3) <= 2)  # Constraints (8) - (10)
        @constraint(model, sum(x[i, j, k]) + sum(x[i, 2, mod(k + 1, 7) + 1] for j in 2:3) <= 2)  # Constraint (9)
        @constraint(model, sum(x[i, 3, k]) + sum(x[i, 1, mod(k + 1, 7) + 1]) +
                    sum(x[i, 2, (mod(k + 1, 7)) + 1]) <= 2)  # Constraint (10)
    end
end


for j in 1:3
    for k in 1:7
        @constraint(model, sum(x[i, j, k] for i in 1:N) >= Wjk[k, j])  # Constraint (11)
        @constraint(model, sum(x[i, j, k] for i in 1:N) <= Wjk[k, j] + Ajk)  # Constraint (12)
    end
end

for i in 1:N
    @constraint(model, sum(x[i, j, ((i - 1) % 7) + 1] + x[i, j, i % 7 + 1] for j in 1:3) == 0)  # Constraint (13)
end

for i in 1:N
    for k in 1:5
        @constraint(model, sum(x[i, j, (i + k) % 7 + 1] for j in 1:3) >= 1)  # Constraint (14)
    end
end

@constraint(model, sum(x[i, j, k] for i in 1:N for j in 1:3 for k in 1:7) == M)

# Solve the optimization problem
optimize!(model)

# Display the results
println("Optimal Number of Employees: ", value(N))
println("Optimal Shift Allocation:")
for i in 1:value(N)
    for j in 1:3
        for k in 1:7
            if value(x[i, j, k]) > 0.5
                println("Employee $i, Shift $j on Day $k")
            end
        end
    end
end

错误信息

ERROR: MethodError: Cannot `convert` an object of type VariableRef to an object of type Float64
Closest candidates are:
convert(::Type{T}, ::Base.TwicePrecision) where T<:Number at twiceprecision.jl:250
convert(::Type{T}, ::AbstractChar) where T<:Number at char.jl:180
convert(::Type{T}, ::CartesianIndex{1}) where T<:Number at multidimensional.jl:136
...
Stacktrace:
[1] MathOptInterface.GreaterThan{Float64}(lower::VariableRef)
MathOptInterface C:\Users\doriiido\.julia\packages\MathOptInterface\IiXiU\src\sets.jl:171
[2] _moi_constrain_variable(moi_backend::MathOptInterface.Utilities.CachingOptimizer{MathOptInterface.Bridges.LazyBridgeOptimizer{GLPK.Optimizer}, MathOptInterface.Utilities.UniversalFallback{MathOptInterface.Utilities.Model{Float64}}}, index::MathOptInterface.VariableIndex, info::VariableInfo{VariableRef, Float64, Float64, Float64}, #unused#::Type{Float64})
JuMP C:\Users\doriiido\.julia\packages\JuMP\ToPd2\src\variables.jl:1754
[3] _moi_add_variable(moi_backend::MathOptInterface.Utilities.CachingOptimizer{MathOptInterface.Bridges.LazyBridgeOptimizer{GLPK.Optimizer}, MathOptInterface.Utilities.UniversalFallback{MathOptInterface.Utilities.Model{Float64}}}, model::Model, v::ScalarVariable{VariableRef, Float64, Float64, Float64}, name::String)
JuMP C:\Users\doriiido\.julia\packages\JuMP\ToPd2\src\variables.jl:1737
[4] add_variable(model::Model, v::ScalarVariable{VariableRef, Float64, Float64, Float64}, name::String)
JuMP C:\Users\doriiido\.julia\packages\JuMP\ToPd2\src\variables.jl:1726
[5] macro expansion
C:\Users\doriiido\.julia\packages\JuMP\ToPd2\src\macros.jl:1213 [inlined]
[6] top-level scope
Untitled-2:13

错误根源

N未定义且误用：代码中直接使用1:N作为数组维度，但从未定义N。JuMP不允许用变量作为数组维度（数组大小必须在建模前确定），导致编译器将N误判为变量引发类型转换错误。
Wjk维度错误：原定义方式生成1行21列数组，与需求的7天×3班维度不符，后续索引Wjk[k,j]逻辑错误。
目标函数偏差：原目标是最小化总班次，而非实际需求的最小化员工数量。
约束逻辑漏洞：部分约束的索引计算存在歧义，且缺少未雇佣员工不可排班的约束。

修复后的代码

using JuMP
using GLPK

# Constants and parameters
# 7天×3班的工作量需求：每行对应1天，每列对应1个班次
Wjk = [
    19 19 14;
    16 16 11;
    22 22 16;
    22 22 16;
    22 22 16;
    22 22 16;
    22 22 16
]
Ajk = 8       # 每班允许的最大超额人数
N_max = 100   # 员工数量的上界（可根据实际情况调整）

# 创建JuMP模型
model = Model(GLPK.Optimizer)

# 决策变量
# y[i]：是否雇佣第i个员工（1=雇佣，0=不雇佣）
@variable(model, y[1:N_max], Bin)
# x[i,j,k]：第i个员工在第k天是否排第j个班次（1=排，0=不排）
@variable(model, x[1:N_max, 1:3, 1:7], Bin)

# 目标函数：最小化雇佣的员工数量
@objective(model, Min, sum(y[i] for i in 1:N_max))

# 约束1：未雇佣的员工不能排班
for i in 1:N_max
    for j in 1:3
        for k in 1:7
            @constraint(model, x[i,j,k] <= y[i])
        end
    end
end

# 约束2：每个员工每天最多排2个班次
for i in 1:N_max
    for k in 1:7
        @constraint(model, sum(x[i,j,k] for j in 1:3) <= 2)
    end
end

# 约束3：连续班次限制（原约束9-10）
for i in 1:N_max
    for k in 1:7
        next_day = k % 7 + 1
        # 当天排2/3班 + 次日排2班 ≤2
        @constraint(model, sum(x[i,j,k] for j in 2:3) + x[i,2,next_day] <= 2)
        # 当天排3班 + 次日排1/2班 ≤2
        @constraint(model, x[i,3,k] + sum(x[i,j,next_day] for j in 1:2) <= 2)
    end
end

# 约束4：每班每日的员工数量满足工作量需求
for j in 1:3
    for k in 1:7
        @constraint(model, sum(x[i,j,k] for i in 1:N_max) >= Wjk[k,j])
        @constraint(model, sum(x[i,j,k] for i in 1:N_max) <= Wjk[k,j] + Ajk)
    end
end

# 约束5：新员工前2天不排班（原约束13）
for i in 1:N_max
    day1 = (i-1) % 7 + 1
    day2 = i % 7 + 1
    @constraint(model, sum(x[i,j,day1] + x[i,j,day2] for j in 1:3) == 0)
end

# 约束6：每个员工连续5天至少排1个班（原约束14）
for i in 1:N_max
    for k in 1:5
        target_day = (i + k) % 7 + 1
        @constraint(model, sum(x[i,j,target_day] for j in 1:3) >= 1)
    end
end

# 可选约束：总班次数量固定（根据需求启用）
# @constraint(model, sum(x[i,j,k] for i in 1:N_max, j in 1:3, k in 1:7) == 395)

# 求解模型
optimize!(model)

# 输出结果
optimal_employees = sum(value(y[i]) for i in 1:N_max)
println("最优员工数量: ", optimal_employees)
println("每班每日分配员工数:")
for k in 1:7
    for j in 1:3
        total = sum(value(x[i,j,k]) for i in 1:N_max)
        println("第$k天 第$j班: $total 人")
    end
end

# 可选：输出详细排班情况
# println("\n详细排班:")
# for i in 1:N_max
#     if value(y[i]) > 0.5
#         println("员工$i:")
#         for k in 1:7
#             shifts = [j for j in 1:3 if value(x[i,j,k]) > 0.5]
#             if !isempty(shifts)
#                 println("  第$k天: 班次$(join(shifts, ","))")
#             end
#         end
#     end
# end

修复说明

员工数量处理：通过设定员工数量上界N_max，引入二进制变量y[i]标记是否雇佣，既规避了变量作为数组维度的问题，又实现了最小化员工数量的目标。
维度修正：将Wjk调整为7行3列矩阵，符合7天×3班的索引逻辑。
约束完善：添加未雇佣员工不可排班的约束，修正连续班次限制的索引计算，保留原有的工作量、新员工排班限制等约束。
结果输出：直接计算并输出每班每日的员工总数，匹配需求。

内容的提问来源于stack exchange，提问作者Dddd97

基于JuMP与GLPK的员工调度优化代码报错排查

问题背景 #

出错代码 #

错误信息 #

错误根源 #

修复后的代码 #

修复说明 #

问题背景

出错代码

错误信息

错误根源

修复后的代码

修复说明