Mathematical description of a case study
Taking into account the above data, the production and transport capacity should be used in such a way as to meet the total needs of all four construction sites at the lowest total costs. The problem cannot be solved by means of a separate consideration of production costs and transport costs [12]. In order to find a solution, the following mathematical procedure needs to be used (Figure 2).
Figure 2. Flowchart of the proposed optimizing procedure.
By determining the production of the m-th precast plant by ym (m = 1, 2, 3, 4), the following equations can be formulated:
| (3.1) |
The production capacity of each precast production plant (both during regular and overtime) is limited, which is expressed by the following inequalities:
| (3.2) |
The demand for each construction site is expressed as follows:
| (3.3) |
Transportation costs can be expressed as follows:
| (3.4) |
where cmn is the cost of transporting one ton of product from the m-th precast production plant to the n-th construction site.
If production in the precast production plant No. 1 will remain in the range of:
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then the production costs will amount to:
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if the additional capacity of the precast production plant No. 1 is used, i.e. for:
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own production costs in the considered precast production plant are:
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The production costs of the products in the precast production plant No. 1 depend on the size of this production, which is generally expressed as follows:
 .
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The production costs of the precast production plant No. 2, 3 and 4 are expressed as follows:
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The final objective function takes the following form:
| (3.5) |
The minimum function value should be determined when conditions (3.1), (3.2) and (3.3) and xmn ≥ 0 are met.
Results achieved
Using non-linear optimization, the following results were obtained (Table 3).
Table 3. Final output data.
   Construction site
Precast production plant
| 1 | 2 | 3 | 4 | 5 (unused production capacity) | Production capacity [t/week] |
| 1 | 0 | 0 | 0 | 150 | 0 | 150 |
| 2 | 50 | 0 | 0 | 0 | 190 | 240 |
| 3 | 0 | 380 | 0 | 0 | 0 | 380 |
| 4 | 0 | 0 | 130 | 90 | 0 | 220 |
| 1’ | 30 | 0 | 0 | 0 | 0 | 30 |
| 2’ | 0 | 0 | 0 | 0 | 20 | 20 |
| 3’ | 120 | 0 | 0 | 0 | 0 | 120 |
| 4’ | 0 | 0 | 30 | 0 | 0 | 30 |
| Demand | 200 | 380 | 160 | 240 | 210 | 1190 |
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It is economically viable to produce during regular normal business hours:
- 150 tons of elements for building site No. 4 by the precast production plant No. 1,
- 50 tons of elements for building site No. 1 by the precast production plant No. 2,
- 380 tons of elements for building site No. 2 by the precast production plant No. 3,
- 130 tons of elements for the construction site No. 3 and 90 tons of elements for the construction site No. 4. – by the precast plant No. 4.
The solution is supplemented by the following overtime production:
- 30 tons of elements for building site No. 1 by the precast production plant No. 1,
- 120 tons of elements for building site No. 1 by the precast production plant No. 3,
- 30 tons of elements for building site No. 3 by the precast production plant No. 4.
Precast concrete production plant No. 2 does not carry out any overtime production.
Conclusion
The study established precast production and transportation planning model (PPTPM) enabling optimization of precast transport and production. A nonlinear programming was applied to search for an optimal precast production program meeting the minimum production and transport costs assumptions while limiting the production capacities of individual precast plants and the demand for construction sites. For experimental purposes, a case study of four precast plants carrying out transport to four construction sites was used. The diverse production capacity of each precast plant and the varied needs of each construction site were assumed. Finally, the optimized production and transportation programme was obtained with the lowest costs amounting to $ 147,410. The research demonstrates the effectiveness of the adopted assumptions to optimize transport and production of precast elements. Future research directions should focus on developing further models combining optimal scheduling for production, internal and external transport as well as planning and resource allocation strategies.
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REFERENCES
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