Simulated annealing (
Aggregate production planning (APP) is considered as an important technique in operations management. Other elemental methods that are closely related to this included capacity requirements planning (
The APP problem has been researched widely since it was first conceptualised in the 1955 study by Holt et al. [
A mixed-integer linear programming model was proposed by Zhang et al. [
In recent times, with the influence of sophisticated modeling methods invented and increasing the number of assumptions, the APP problem has turned into a complicated and large-scale problem. In the research community, there is a tendency to resolve large and complex problems by the use of modern heuristic optimisation methods. This is mainly due to the time-consuming and unsuitability of classical techniques in many circumstances.
A fuzzy multiobjective APP model was introduced by Baykasoglu and Gocken [
Kumar and Haq [
Aungkulanon et al. [
A simulated annealing algorithm (SAA) has not been functional in solving multiobjective APP problems, even though most applications made use of metaheuristic algorithms on APP. In this paper, we first suggest a conventional multiobjective linear programming model for APP. Throughout the implementation of SA to the APP problems, it was observed that SA has an inadequacy with respect to big APP problems that have plenty of decision variables. To improve the SA efficiency for the production planning system, a modified SA is introduced that can resolve APP problems with multiobjective linear programming model, since a large number of companies intend to fulfill more than one objective while creating a response and flexibility. The objective of the model is to decrease the aggregate expenses of production and workforce, and simulation and real-life data were used to verify the effectiveness of this method.
The organisation of the paper is as follows. Section
We proposed mathematical model for APP problem and assumed that an industrial company manufacturing produces
Consider the following:
Geem et al. [
This creates a perfect state of harmony or a pleasurable harmony as governed by an aesthetic quality with the pitch of every musical device. In a similar manner, the optimisation technique searches for a globule solution as ascertained by an objective function through a series of values designated to every decision variable. In a musical orchestration, the set of pitches from every musical instrument accomplishes the aesthetic evaluation. The quality of harmony gets enriched with every practice. Each style of music is composed by musicians out of specific instruments. If all musical pitches create a perfect harmony, then that musical experience remains in every instrument player's memory, and the likelihood to create a good harmony the next time increases manyfold. If every player plays together with dissimilar notes, then a new musical harmony is composed. There are three rules of musical improvisation playing a completely random pitch from the workable sound range, playing a pitch from memory, or playing an adjoining pitch of a pitch from memory. These techniques rules are implemented in HSA and the same was explained to describe the process of HSA.
The HSA process can be summed up as follows.
Initialize the harmony memory (HM) which contains HMS vectors generated randomly, where
Improvise a new harmony. There are three rules for this: Harmony memory considering If Pitch adjusting rate ( If where BW is a bandwidth factor, which is used to control the local search around the selected decision variable in the new vector. Random initialization rule is as follows. If the condition where
Update harmony memory. If improvised harmony vector is better than the worst harmony, replace the worst harmony in the HM form:
Check the stopping criterion. If the stopping criterion is satisfied, the computation is terminated. Otherwise, repeat Steps
The steps of
Flowchart for
For complicated optimisation problems, simulated annealing (
Operations for SA.
Practitioners make extensive use of
Operation for modified SA.
Generate
Find the objective function for each
Sort the solution such as
Repeat this step Generate a new solution If Else if
Reduce the parameter
To assess the functioning of the
In this section, the General Company for Vegetable Oils is used as a case study to demonstrate the proposed model. This company produces ten types of products. Each product is represented by a letter (
Production and inventory costs in dollars.
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3285 | 385 | 451 | 1006 | 801 | 487 | 449 | 1007 | 496 | 739 |
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38 | 47 | 53.6 | 35.511 | 35.415 | 24.6 | 37.75 | 37.666 | 58.666 | 37 |
Hours required to produce one ton of product.
Product |
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Hours | 92 | 525 | 69 | 64 | 50 | 121 | 42 | 607 | 172 | 692 |
Forecast demand for all products.
Period |
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1 | 3049.1 | 539 | 340.6 | 100 | 606.4 | 23.1 | 1.7 | 1.2 | 3.1 | 0/74 |
2 | 1664.1 | 509 | 708.1 | 152 | 482.7 | 265 | 3.3 | 2 | 1.8 | 1.1 |
3 | 1236.4 | 35.4 | 700 | 138 | 496.8 | 14.8 | 7.4 | 1.7 | 2.3 | 0.47 |
4 | 782.5 | 40.8 | 650 | 77 | 429.9 | 25 | 8.7 | 2.5 | 2.9 | 0.76 |
5 | 914.4 | 275 | 439 | 56 | 324.7 | 15 | 215 | 2.4 | 2.1 | 2.3 |
6 | 652.9 | 379 | 619.1 | 50 | 652.9 | 12.4 | 29.1 | 1.3 | 2.7 | 0.71 |
The authors select HS because it provides better results than standard
After we used the
Results for each algorithm.
Algorithm |
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Time |
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HS | 7043102 | 5737053 | 157 s. |
SA | 7159657 | 5918976 | 17 s. |
MSA | 6993111 | 5814778 | 11 s. |
Due to the fact that
Production yield.
Product | P1 | P2 | P3 | P4 | P5 | P6 |
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2944.82 | 1664.1 | 1236.4 | 782.5 | 914.4 | 652.9 |
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53.9 | 50.9 | 35.4 | 40.8 | 27.5 | 37.9 |
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340.6 | 708.1 | 700 | 650.2 | 439 | 619.1 |
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100 | 152 | 138 | 77 | 56 | 50 |
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144.071 | 482.7 | 496.8 | 429.9 | 324.7 | 652.9 |
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23.101 | 26.501 | 14.801 | 25 | 14.5671 | 11.7534 |
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1.7 | 3.3 | 7.4 | 8.7 | 21.5 | 29.1 |
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1.19 | 2 | 1.7 | 2.5 | 3.69 | 0.7 |
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1.258 | 1.8 | 2.3 | 2.9 | 2.1 | 2.7 |
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0.741 | 1.1669 | 0.4614 | 0.722 | 3.0498 | 0.068 |
Inventory levels.
Product | P1 | P2 | P3 | P4 | P5 | P6 |
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36.50167 | 36.5017 | 36.5017 | 36.502 | 36.502 | 36.502 |
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1.29 | 2.59 | 3.63 | 4.15 | 4.15 | 5.27 |
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0 | 0 | 0 | 0 | 0 | 0 |
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0 | 0 | 0 | 0 | 0 | 0 |
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0.1612 | 0.1612 | 0.1612 | 0.1612 | 0.1612 | 0.1612 |
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0 | 0 | 0 | 0 | 0.2308 | 0 |
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0 | 0 | 0 | 0 | 0 | 0 |
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0.1434 | 0.1434 | 0.1434 | 1.1434 | 0.1946 | 0.1263 |
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0 | 0 | 0 | 0 | 0 | 0 |
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0.1 | 0.6 | 0.06 | 0.02 | 0.77 | 0 |
The rate of workforce level.
Workforce level | P1 | P2 | P3 | P4 | P5 | P6 |
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1915 | 1831 | 1602 | 1296 | 1190 | 1191 |
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0 | 0 | 0 | 0 | 0 | 1 |
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1400 | 84 | 229 | 306 | 106 | 0 |
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31079 | 0 | 0 | 0 | 0 | 0 |
Tables
Simulation is another criterion for analysing the effectiveness of the proposed modified
Optimal costs and time for each problem when
SA | HS | MSA | |||
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96231219.04 | 95850473.04 | 93276091.81 |
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44743742.22 | 44686669.41 | 44327446.38 | ||
Time | 4.8048 | 9.8436 | 0.6864 | ||
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250053722.9 | 247445920 | 241796947.2 |
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61096198.47 | 61096198.47 | 61096198.47 | ||
Time | 8.4552 | 17.4097 | 0.7956 | ||
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364563840.4 | 363660493.9 | 350843833.1 |
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48838028.24 | 48838028.24 | 48838028.24 | ||
Time | 13.1352 | 44.3042 | 0.4212 | ||
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178871907.1 | 176467812.5 | 173263161 |
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37988545.97 | 37988545.97 | 37988545.97 | ||
Time | 5.0856 | 14.821 | 0.3276 | ||
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272811050.9 | 271276691.3 | 260926577.2 |
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54821702.95 | 47805043.77 | 45726695.02 | ||
Time | 8.081 | 68.999 | 0.4056 | ||
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489020932.2 | 486493908 | 467749414.1 |
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55478086.78 | 50555865.02 | 50163753.84 | ||
Time | 16.9105 | 135.1904 | 0.4524 | ||
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209736151.9 | 208824558.5 | 202629622.2 |
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42890759.51 | 41396915.52 | 41396915.52 | ||
Time | 8.346 | 18.33 | 0.3276 | ||
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400060878.1 | 399731639.9 | 384906240.5 |
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62516501.51 | 54712694.41 | 46909627.02 | ||
Time | 11.4817 | 126.0956 | 0.421 | ||
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769395323.5 | 767172581.9 | 740635030.5 |
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87815572.15 | 85013218.68 | 85013218.68 | ||
Time | 16.3801 | 165.478 | 0.4683 |
From Figure
Optimal cost comparison
Comparison time running for each algorithm.
Simulated annealing provides a mechanism to escape local optima by allowing hill-climbing moves in hopes of finding a global optimum. However, it has a disadvantage which is the imperfections ability and unacceptable performance, especially a large constrained APP issue. Therefore, this algorithm works sequentially that the current solution will generate only one solution. To improve its performance and lessen its insufficiencies to problem-solving, a modified
Number of products,
Number of periods in the planning horizon,
Production cost per ton of product
Inventory carrying cost per ton of product
Hiring cost per worker in period
Firing cost per worker in period
Cost per man-hour of overtime labor per period
Cost of regular labor per period
Forecasted demand for product
Production of product
Inventory level of product
Man-hours of overtime labor per period
Workforce level per period
Hired workers per period
Fired workers per period
Maximum hiring in each period
Maximum firing in each period
Hours required to produce one ton of product
Working regular hours per period
Working overtime hours, which are allowed during per period
Hours required to produce one ton for product
The authors declare that there are no competing interests regarding the publication of this paper.