Mindopt Modeling Solving Task Assignment Problems (II)

Multi-Task Assignment Problem

Example 1 (Dual-task assignment) only has two tasks for the assignment. Next, we will consider a more complex situation with more tasks.

1. Problem Description

Consider a multi-task assignment scenario where the employee needs to handle m tasks a day, the hourly compensation for each task i is r_i, and the hours spent on each task i should be at least ℓ_i but no more than u_i. Knowing that the employee works b hours each day, how should he allocate his time to receive the most compensation while fulfilling the factory requirements? This problem is a generalization of the dual-task assignment problem in the previous section.

2. Optimization Model

Introduce a variable x_i for the hours spent on each task i. From the problem description, the variables must satisfy the constraint and the bounds ℓ_i ≤ x_i ≤u_i, ∀i∈{1,2,⋯,m}. The objective is to maximize the total compensation . Putting them together, we obtain:

Compared with the dual-task assignment model, this model has the following differences:

The hourly compensation, the lower and upper bounds of the working hours for each task, and the daily total hours of employees are all parameterized as r_i, ℓ_i, u_i and b.
We have expanded 2 tasks to m tasks. Therefore, there are decision variables, m lower bounds, m upper bounds, and m hourly compensation rates. In order to express the operation relationship of multiple variables (parameters) in the model more concisely, we will use the set operator in MAPL coding.

3. MAPL Coding

MAPL modeling requires specific values of parameters (such as data). MAPL has two ways to read data:

The first way is to give the data directly in the modeling file .mapl, which is called the data is with the model.
The second way is that the model and data are declared in separate files (also known as model and data isolation).

The first one is easier to use when the mathematical planning problem model is smaller, while the second one is more suitable for larger problems.

This section focuses on the case where the data and model are in the same file.

Run the codes in the following cell:

clear model;  #Used when running multiple times to clear the model
option modelname model/multiTask; #Facilitate the generation of intermediate files in the same directory as Method 2.

#--------------------------
# multiTask.mapl
set Tasks := { 1 .. 10 };                    # Declare the set Tasks

param Reward[Tasks] := 1:2, 3: 3, 7:4 default 1;      # Declare the parameter Reward and assign a value
param LB[Tasks] := 1 : 1, 3: 1 default 0;            # Declare the lower bound LB and assign a value
param UB[Tasks] := 3: 4, 7: 2 default 8;            # Declare the upper bound UB and assign a value
param b := 8;                                         # Declare the total working hours of the employee

var x[i in Tasks] >= LB[i] <= UB[i];         # Declare the variable LB[i] <= x[i] <= UB[i]

maximize Total_Reward: sum{i in Tasks} Reward[i] * x[i];  # Declare the target function

subto Worker_time: sum{i in Tasks} x[i] == b;    # declaration constraint sum_i x[i] == 8
#--------------------------


print "-----------------Solving---------------";
option solver mindopt; # (Optional) Specify the solver for solving, the default is MindOpt
solve; 

print "-----------------Display---------------";
display; # Show Results

print "Total_Reward: ", sum {i in Tasks} Reward[i] * x[i];
print "Work time: ", sum {i in Tasks} x[i] ;

We can also save the modeling part codes in file multiTask.mapl. and call MAPL to load the file:

clear model;  #Used when running multiple times to clear the model

#--------------------------
model ./model/multiTask.mapl;
#--------------------------

print "-----------------Solving---------------";
#option solver mindopt; # (Optional) Specify the solver for solving, the default is MindOpt
option solver cbc; # Use Cbc solver

solve; 

print "-----------------Display---------------";
display; # Show Results

print "Total_Reward: ", sum {i in Tasks} Reward[i] * x[i];
print "Work time: ", sum {i in Tasks} x[i] ;

4. Results

The results of Method 1 and Method 2 are consistent:

-----------------Solving---------------
Running mindoptampl
wantsol=1
MindOpt Version 0.25.1 (Build date: 20230816)
Copyright (c) 2020-2023 Alibaba Cloud.

Start license validation (current time : 24-AUG-2023 16:07:57).
License validation terminated. Time : 0.004s

Model summary.
 - Num. variables     : 10
 - Num. constraints   : 1
 - Num. nonzeros      : 10
 - Bound range        : [1.0e+00,8.0e+00]
 - Objective range    : [1.0e+00,4.0e+00]
 - Matrix range       : [1.0e+00,1.0e+00]

Presolver started.
Presolver terminated. Time : 0.000s

Simplex method started.
Model fingerprint: ==gZ3F2dhdnZ

    Iteration       Objective       Dual Inf.     Primal Inf.     Time
            0     5.25006e+01      0.0000e+00      2.3400e+01     0.00s    
            1     2.40000e+01      0.0000e+00      0.0000e+00     0.00s    
Postsolver started.
Simplex method terminated. Time : 0.001s


OPTIMAL; objective 24.00
1 simplex iterations

Completed.
-----------------Display---------------
Primal Solution:
     x@1 = 2.00000000
     x@2 = 0.00000000
     x@3 = 4.00000000
     x@4 = 0.00000000
     x@5 = 0.00000000
     x@6 = 0.00000000
     x@7 = 2.00000000
     x@8 = 0.00000000
     x@9 = 0.00000000
    x@10 = 0.00000000
Total_Reward: 24
Work time: 8

----------------------------------------
-----------------Solving---------------
Running cbc
CBC 2.10.5Completed.
-----------------Display---------------
Primal Solution:
x@1      = 2.000000000000000E+00
x@2      = 0.000000000000000E+00
x@3      = 4.000000000000000E+00
x@4      = 0.000000000000000E+00
x@5      = 0.000000000000000E+00
x@6      = 0.000000000000000E+00
x@7      = 2.000000000000000E+00
x@8      = 0.000000000000000E+00
x@9      = 0.000000000000000E+00
x@10     = 0.000000000000000E+00

Dual Solution:
Worker_time_1 = 2.000000000000000E+00
Total_Reward: 24
Work time: 8

Print x@name is the value of the decision variable, followed by the dual solution.

This result indicates that when allocating 2 hours to task 1, 4 hours to task 3, and 2 hours to task 7, the maximum profit is achieved with an objective value of 24.00.

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Mindopt Modeling Solving Task Assignment Problems (II)

Multi-Task Assignment Problem

1. Problem Description

2. Optimization Model

3. MAPL Coding

4. Results

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