The Impact of the Learning and Forgetting Effect on the Cost of a Multi-Unit Construction Project with the Use of the Simulated Annealing Algorithm
Abstract
:1. Introduction
2. Materials and Methods
2.1. Project Scheduling Problems—The Overview
2.1.1. Multi-Mode Problems
2.1.2. Schedule Optimization for Multi-Unit Construction Projects
2.2. Learning and Forgetting Phenomena
2.2.1. LAF Models
- -
- —the time to produce the xth unit,
- -
- —the time to produce the first unit,
- -
- —the production count,
- -
- —the learning curve exponent.
2.2.2. LAF in Construction Industry
2.3. The Construction Project Model
2.3.1. The Model of the Learning and Forgetting Effect Used in the Research
- -
- —current performance,
- -
- —asymptotic limit of y value,
- -
- —amount of accumulated work performed,
- -
- —previous experience in the same unit as ,
- -
- —amount of work to be performed to achieve efficiency of ,
- -
- —forgetting factor,
- -
- —the ratio of the time elapsed from the first timestamp to the time that has elapsed since the last unit x was produced.
2.3.2. The Optimization Model
- The task of the contractor in the project is to construct a set of construction units:
- The set of working groups performing one type of work is as follows:
- Each unit Zi ∈ Z requires implementation of m activities which form the set:
- Each working group Bj can perform activity Oij ∈ Oi using a maximum of p different methods (technologies) of execution:Dj = {Dj1, …, Djk, …, Djp},
- -
- Dj—set of possible execution modes (methods) for performing activity j by a working group Bj,
- -
- Djk—k method of performing activity j by a working group Bj, a k = 1 ... p.
- It is assumed that the activity Oij ∈ Oi can be done by the working group Bj. The durations of activities from the set Oi are defined by a vector:
- Similarly, as the above, it is assumed that the activity Oij ∈ Oi can be implemented by the working group Bj. The cost of realization of the activity Oij by the working group Bj defines a variable cijk ≥ 0. The set of possible costs of activities ci from the set Oi is defined by the vector:
- The LAF effect is considered for the whole duration of activities from the set Oi within vector tu_i = [tu_i1, tu_i2, tu_i3, ..., tu_ik, ..., tu_im], where tu_ik is a real duration of activity Oij taking into account the LAF effect calculated in accordance with the formula:
- -
- —the efficiency (performance) of the jth working group in carrying out the ith job on the zth day in turn.
- The order of execution of the activities resulting from work technology is assumed such that:
- It is assumed that only one job can be performed at a time.
- It is assumed that the activity cannot be stopped during its performance (preemption is not allowed).
- the order π of execution of units (permutation):
- set of numbers of methods of activity execution (from k = 1 to k = p):
- -
- π(0) = 0,
- -
- F0,j = 0,
- -
- Fi,0 = 0.
- -
- direct costs , which are the sum of the costs of all activities in the project,
- -
- indirect costs (all costs that do not become a final part of the installation but are crucial for the operation of the contractor) depending on the duration of the project,
- -
- penalty costs (optional) charged in case of delays (failure to meet the directive deadlines for the completion of construction works),
- -
- cost bonuses (optional) awarded in case of finishing the works early (earlier than directive deadlines). Such bonuses reduce the overall cost of the project.
- -
- —daily indirect construction costs,
- -
- —deadline for completion of works in the project,
- -
- —deadline for completion of works on i unit,
- -
- —the finish time for construction works on i unit,
- -
- —daily cost of the penalty for exceeding the directive deadlines for finishing works on units,
- -
- —the contractor’s daily bonus for not exceeding the directive deadlines for completion of works in the units,
- -
- —a set of all possible permutations in a given project,
- -
- n—number of all units (objects),
- -
- m—number of works (activities) to be completed on each of the units.
2.4. The Optimization Method of the Multi-Unit Project Discussed in the Article
2.4.1. Simulated Annealing Algorithm
Algorithm 1: SA |
Step 0. Determine the initial solution π ∈ Π. Substitute πSA = π0, k = 0, T = T0.
Step 1. Perform steps 1.1–1.3 x times. Step 1.1. Substitute k: = k + 1. Choose at random π ∈ N(V, πk−1). Step 1.2. If c(π) > c(πSA) then substitute πSA = π. Step 1.3. If c(π) > c(πk-1) then substitute πk = π. Otherwise, accept solution π with a probability of p = exp((c(π) − c(πk-1))/T, where πk = π’. If solution π was accepted, and πk = πk-1 if solution π was not accepted. Step 2. Change the temperature T according to the defined pattern of cooling. Step 3. If T > TN return to the step 1; otherwise, STOP. |
2.4.2. The Algorithm for Solving the Optimization Problem in the Presented Model of a Multi-Unit Project
- Let π ∈ Π be any random permutation (π* the best solution found so far, π* = π to start with), let R be a randomly determined set of methods for performing activities (R* the best solution found so far, R* = R to start with), and let MaxIter be the accepted maximum number of iterations of the algorithm.
- Step 1. Find the solution of the optimization problem using the SA metaheuristic (permutation δ) minimizing the cost of the project with the assumption of R as a set of methods for the implementation of activities.
- Step 2. Find the solution of the optimization problem using the SA metaheuristic (set of ρ methods of carrying out activities) minimizing the cost of the project with the assumption of the order of implementation of the units expressed by the permutation ρ.
- Step 3. If U(δ, ρ) ≤ U(π*, R*), to π* = δ, R* = ρ. Assume that R = ρ.
- Step 4. If Termination_Condition, then STOP;otherwise, go to Step 1.
- the Nπ neighborhood contains permutations generated from π by the “insert” movement,
- the Boltzmann acceptance function was used,
- a geometric cooling scheme was adopted, i.e., Ti+1 = λTi and T0 = 60, λ = 0.99, and the number of considered solutions at a fixed temperature—7,
- the maximum number of iterations of the SA algorithm in step 2—500.
- sets of numbers of the methods of carrying out works generated from R by means of a movement consisting in changing a randomly selected method of carrying out activities by a value selected at random +1 or −1 are contained in the neighboring set NR,
- the Boltzmann acceptance function was used,
- a geometric cooling scheme was adopted, i.e., Ti+1 = λTi and T0 = 60, λ = 0.99, and the number of considered solutions at a fixed temperature—35,
- the maximum number of iterations of the SA algorithm in step 3—20,000.
3. Results
3.1. Testing the Accuracy of the Results Obtained Using the SA Algorithm
- —the value of the adopted objective function obtained by means of the SA + SA algorithm,
- —the value of the adopted objective function obtained by means of the RS algorithm,
- —the value of the adopted objective function obtained by means of the CR algorithm.
3.2. Case Study
- R1 = (2,2,2,2,1),
- R2 = (2,2,2,3,2),
- R3 = (2,1,2,2,2),
- R4 = (2,2,2,1,1),
- R5 = (3,3,2,2,2),
- R6 = (1,1,2,2,1),
- R7 = (1,1,3,2,1).
4. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
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Example Name: Number of Units × Number of Works | PRD(SA + SA)_RS (%) − Algorithm RS | PRD(SA + SA)_CR (%) − Algorithm CR |
---|---|---|
Examples n = 6 units | ||
6 × 3 | −6.19 | −3.73 |
6 × 5 | −7.36 | −4.62 |
6 × 7 | −9.92 | −6.69 |
mean PRD for size n = 6 | −7.82 | −5.01 |
Examples n = 10 units | ||
10 × 3 | −11.17 | −6.67 |
10 × 5 | −14.95 | −8.71 |
10 × 7 | −18.07 | −11.93 |
mean PRD for size n = 10 | −14.73 | −9.10 |
Examples n = 15 units | ||
15 × 3 | −16.74 | −10.14 |
15 × 5 | −10.68 | −3.68 |
15 × 7 | −11.14 | −2.93 |
mean PRD for size n = 15 | −12.85 | −5.58 |
Examples n = 20 units | ||
20 × 3 | −16.64 | −9.31 |
20 × 5 | −11.48 | −4.23 |
20 × 7 | −20.48 | −13.31 |
mean PRD for size n = 20 | −16.20 | −8.95 |
Examples n = 6, 10, 15, 20 units | ||
mean PRD (%) | −12.90 | −7.16 |
Modes (Number/Details) | Units i = | |||||||
---|---|---|---|---|---|---|---|---|
1 | 2 | 3 | 4 | 5 | 6 | 7 | ||
Earthworks (j = 1) | ||||||||
1 | Execution time [working days] | 11 | 9 | 7 | 17 | 8 | 9 | 8 |
Execution cost [in thous. EUR] | 5.5 | 4.5 | 3.5 | 8.5 | 4 | 4.5 | 4 | |
2 | Execution time [working days] | 10 | 7 | 8 | 18 | 9 | 11 | 9 |
Execution cost [in thous. EUR] | 6.05 | 5.79 | 3.06 | 8.03 | 3.56 | 3.68 | 3.56 | |
3 | Execution time [working days] | 12 | 10 | 6 | 13 | 10 | 10 | 9 |
Execution cost [in thous. EUR] | 5.04 | 4.05 | 4.08 | 11.12 | 3.20 | 4.05 | 3.56 | |
Foundations (j = 2) | ||||||||
1 | Execution time [working days] | 17 | 11 | 25 | 24 | 17 | 14 | 26 |
Execution cost [in thous. EUR] | 24.00 | 15.53 | 35.29 | 33.88 | 24.00 | 19.76 | 36.71 | |
2 | Execution time [working days] | 14 | 10 | 27 | 26 | 14 | 16 | 24 |
Execution cost [in thous. EUR] | 29.14 | 17.08 | 32.68 | 31.28 | 29.14 | 17.29 | 39.76 | |
3 | Execution time [working days] | 22 | 8 | 28 | 25 | 14 | 17 | 28 |
Execution cost [in thous. EUR] | 18.55 | 21.35 | 31.51 | 32.53 | 29.14 | 16.28 | 34.08 | |
Walls. slabs (j = 3) | ||||||||
1 | Execution time [working days] | 35 | 13 | 45 | 44 | 35 | 36 | 52 |
Execution cost [in thous. EUR] | 65.50 | 24.33 | 84.21 | 82.34 | 65.50 | 67.37 | 97.31 | |
2 | Execution time [working days] | 31 | 13 | 35 | 54 | 42 | 30 | 67 |
Execution cost [in thous. EUR] | 73.95 | 24.33 | 108.28 | 67.09 | 54.58 | 80.85 | 75.53 | |
3 | Execution time [working days] | 34 | 13 | 56 | 42 | 30 | 37 | 36 |
Execution cost [in thous. EUR] | 67.43 | 24.33 | 67.67 | 86.26 | 76.42 | 65.55 | 140.57 | |
Rafter framings, roofing (j = 4) | ||||||||
1 | Execution time [working days] | 12 | 10 | 17 | 12 | 11 | 19 | 10 |
Execution cost [in thous. EUR] | 35.50 | 29.58 | 50.29 | 35.50 | 32.54 | 56.21 | 29.58 | |
2 | Execution time [working days] | 13 | 8 | 21 | 12 | 11 | 18 | 11 |
Execution cost [in thous. EUR] | 32.77 | 36.98 | 40.71 | 35.50 | 32.54 | 59.33 | 26.89 | |
3 | Execution time [working days] | 11 | 7 | 14 | 14 | 12 | 14 | 10 |
Execution cost [in thous. EUR] | 38.73 | 42.26 | 61.07 | 30.43 | 29.83 | 76.28 | 29.58 | |
Windows and doors (j = 5) | ||||||||
1 | Execution time [working days] | 9 | 9 | 4 | 14 | 17 | 7 | 14 |
Execution cost [in thous. EUR] | 38.00 | 38.00 | 16.89 | 59.11 | 71.78 | 29.56 | 59.11 | |
2 | Execution time [working days] | 8 | 10 | 4 | 16 | 19 | 7 | 13 |
Execution cost [in thous. EUR] | 42.75 | 34.20 | 16.89 | 51.72 | 64.22 | 29.56 | 63.66 | |
3 | Execution time [working days] | 10 | 7 | 3 | 16 | 16 | 9 | 16 |
Execution cost [in thous. EUR] | 34.20 | 48.86 | 22.52 | 51.72 | 76.26 | 22.99 | 51.72 |
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Podolski, M.; Rosłon, J.; Sroka, B. The Impact of the Learning and Forgetting Effect on the Cost of a Multi-Unit Construction Project with the Use of the Simulated Annealing Algorithm. Appl. Sci. 2022, 12, 12667. https://doi.org/10.3390/app122412667
Podolski M, Rosłon J, Sroka B. The Impact of the Learning and Forgetting Effect on the Cost of a Multi-Unit Construction Project with the Use of the Simulated Annealing Algorithm. Applied Sciences. 2022; 12(24):12667. https://doi.org/10.3390/app122412667
Chicago/Turabian StylePodolski, Michał, Jerzy Rosłon, and Bartłomiej Sroka. 2022. "The Impact of the Learning and Forgetting Effect on the Cost of a Multi-Unit Construction Project with the Use of the Simulated Annealing Algorithm" Applied Sciences 12, no. 24: 12667. https://doi.org/10.3390/app122412667
APA StylePodolski, M., Rosłon, J., & Sroka, B. (2022). The Impact of the Learning and Forgetting Effect on the Cost of a Multi-Unit Construction Project with the Use of the Simulated Annealing Algorithm. Applied Sciences, 12(24), 12667. https://doi.org/10.3390/app122412667