A mathematical model to optimize the volumetric capacity of trucks utilized in the transport of food products

Pedruzzi, Suzane; Nunes, Luiz Paulo Amorim; Rosa, Rodrigo de Alvarenga; Arpini, Bianca Passos

doi:10.1590/0104-530X1898-14

Suzane Pedruzzi

Departamento de Engenharia de Produção, Universidade Federal do Espírito Santo – UFES, CEP 29075-910, Vitória, ES, Brazil, e-mail: suzanepedruzzi@gmail.com; luizpauloan@gmail.com

Luiz Paulo Amorim Nunes

Departamento de Engenharia de Produção, Universidade Federal do Espírito Santo – UFES, CEP 29075-910, Vitória, ES, Brazil, e-mail: suzanepedruzzi@gmail.com; luizpauloan@gmail.com

Rodrigo de Alvarenga Rosa

Programa de Pós-graduação em Engenharia Civil, Universidade Federal do Espírito Santo – UFES, CEP 29075-910, Vitória, ES, Brazil, e-mail: rodrigo.a.rosa@ufes.br; bianca.p.arpini@gmail.com

Bianca Passos Arpini

Programa de Pós-graduação em Engenharia Civil, Universidade Federal do Espírito Santo – UFES, CEP 29075-910, Vitória, ES, Brazil, e-mail: rodrigo.a.rosa@ufes.br; bianca.p.arpini@gmail.com

Scenario	Destination			Trip			Vehicle					Load		Not occupied Space (m³)
Scenario	State	City	Number of Clients	Number	Date	Type	Type	m³	Length (m)	Width (m)	Height (m)	Nº Box /Pickings	m³	Not occupied Space (m³)
1	PE	Garanhuns Recife	2	1	03/sept	Transfer	Trailer	90	10	3	3	4	51	39
				2	13/sept	Transfer	Trailer	90	10	3	3	4	50	40
				3	19/sept	Transfer	Trailer	90	10	3	3	4	42	48
2	SE AL	Aracajú Maceió	2	1	03/sept	Transfer	Trailer	90	10	3	3	4	63	27
				2	13/sept	Transfer	Trailer	90	10	3	3	4	64	26
				3	19/sept	Transfer	Trailer	90	10	3	3	4	58	32
3	TO PA	Palmas Benevides	2	1	02/sept	Transfer	Trailer	90	10	3	3	4	35	55
				2	12/sept	Transfer	Trailer	90	10	3	3	4	36	54
				3	19/sept	Transfer	Trailer	90	10	3	3	4	58	32
4	PR SC	Pato Branco Chapecó	2	1	02/may	Transfer	Trailer	90	10	3	3	4	42	48
4	PR SC	Pato Branco Chapecó	2	2	17/may	Transfer	Trailer	90	10	3	3	6	44	46
5	BA	Feira de Santana Juazeiro	2	1	13/sept	Transfer	Trailer	90	10	3	3	6	51	39
5	BA	Feira de Santana Juazeiro	2	2	20/sept	Transfer	Trailer	90	10	3	3	4	50	40
6	ES	Guarapari Cachoeiro	2	1	02/aug	Sale	Truck	12	6	2	1	4	4	8
6	ES	Guarapari Cachoeiro	2	2	18/aug	Sale	Truck	12	6	2	1	4	6	6
7	BA	Itabuna Vitória da Conq. Lauro de Freitas	3	1	15/sept	Transfer/Sale	Trailer	90	10	3	3	6	57	33
7	BA	Itabuna Vitória da Conq. Lauro de Freitas	3	2	20/sept	Transfer/Sale	Trailer	90	10	3	3	6	50	40
8	PA	Benevides Castanhal Ananindeua Paragominas	4	1	05/feb	Transfer	Trailer	90	10	3	3	8	71	19
				2	06/feb	Transfer	Trailer	90	10	3	3	8	60	30
				3	28/feb	Transfer	Trailer	90	10	3	3	8	71	19

Scenario	Total Execution Time (s)	Time to Find Optimal Solution (s)	GAP (%)	OF	UB	LB	Nº Trips
1	1056.0	5.8	0.0	9222.1	-	-	2
2	99.0	9.4	0.0	19125.3	-	-	3
3	261.1	47.9	0.0	11475.4	-	-	2
4	93.0	4.1	0.0	21150.2	-	-	2
5	21.7	3.3	0.0	17775.2	-	-	2
6	52.6	1.8	0.0	450.3	-	-	1
7	953.0	8.2	0.0	16425.3	-	-	2
8	14400.0	5040.0	36.8	-	23,817.8	15,052.9	3

Scenario	Trip	Vehicle Volume (m³)	Company		Mathematical Model		Difference Model x Company
Scenario	Trip	Vehicle Volume (m³)	Not Occupied Volume (m³)	Not Occupied (%)	Not Occupied Volume (m³)	Not Occupied (%)	Not Occupied Volume (m³)	Not Occupied (%)
1	1	90.0	39.0	43.3	18.0	20.0	21.0	23.3
	2	90.0	40.0	44.4	20.0	22.2	20.0	22.2
	3	90.0	48.0	53.3	90.0	100.0	-42.0	100.0
2	1	90.0	27.0	30.0	27.0	30.0	0.0	0.0
	2	90.0	26.0	28.9	36.0	40.0	-10.0	-11.1
	3	90.0	32.0	35.6	22.0	24.4	10.0	11.1
3	1	90.0	55.0	61.1	90.0	100.0	-35.0	100.0
	2	90.0	54.0	60.0	24.0	26.7	30.0	33.3
	3	90.0	32.0	35.6	27.0	30.0	5.0	5.6
4	1	90.0	48.0	53.3	34.0	37.8	14.0	15.6
4	2	90.0	46.0	51.1	60.0	66.7	-14.0	-15.6
5	1	90.0	39.0	43.3	47.0	52.2	-8.0	-8.9
5	2	90.0	40.0	44.4	32.0	35.6	8.0	8.9
6	1	12.0	8.0	66.7	2.0	16.7	6.0	50.0
6	2	12.0	6.0	50.0	12.0	100.0	-6.0	100.0
7	1	90.0	33.0	36.7	29.0	32.2	4.0	4.4
7	2	90.0	40.0	44.4	44.0	48.9	-4.0	-4.4
8	1	90.0	19.0	21.1	22.0	24.4	-3.0	-3.3
	2	90.0	30.0	33.3	27.0	30.0	3.0	3.3
	3	90.0	19.0	21.1	19.0	21.1	0.0	0.0

Scenario	Trip		Not Occupied Space Analysis - Heterogeneous Fleet
		Company			Mathematical Model				Reduction Model x Company
		Vehicle Volume (m³)	Not Occupied Volume (m³)	Not occupied (%)	Vehicle Volume (m³)	Not Occupied Volume (m³)	Not occupied (%)	Freight Cost (R$)	Not Occupied Volume (m³)	Not occupied (%)
4A	1	90.0	48.0	53.3	54.0	10.0	18.5	5,590.2	38.0	34.8
4A	2	90.0	46.0	51.1	54.0	12.0	22.2	5,590.2	34.0	28.9
4B	1	90.0	39.0	43.3	54.0	10.0	18.5	3,897.2	29.0	24.8
4B	2	90.0	40.0	44.4	54.0	12.0	22.2	3,897.2	28.0	22.2
4C	1	90.0	48.0	53.3	54.0	10.0	18.5	3,987.2	38.0	34.8
4C	2	90.0	46.0	51.1	54.0	12.0	22.2	3,987.2	34.0	28.9

Scenario	Freight Cost per Vehicle (R$)	Number of trips		Total Cost Freight (R$)		Financial Gains Model x Company
Scenario	Freight Cost per Vehicle (R$)	Company	Model	Company	Model	R$	%
1	7,757.0	3	2	23,272.0	15,515.0	7,757.0	33
3	11,240.0	3	2	33,721.0	22,481.0	11,240.0	33
6	1,596.0	2	1	3,191.0	1,596.0	1,595.0	50
	Monthly reduction (R$)						20,593.0
	Estimated annual gains (R$)						247,116.0

M i n i m i z a r (\begin{array}{l} (\sum_{j \in D} φ_{j} (L_{j} W_{j} H_{j} n_{j} - \sum_{i \in C} p_{i} q_{i} r_{i} s_{i j})) + \\ ω (\sum_{i \in C} z_{i} + \sum_{i \in C} x_{i} + \sum_{i \in C} y_{i}) \end{array})

\begin{array}{l} x_{i} + p_{i} l x_{i} + q_{i} (l z_{i} - w y_{i} + h z_{i}) + r_{i} (1 - l x_{i} - l z_{i} + w y_{i} - h z_{i}) - \\ δ_{i k j} M \leq x_{k} + (1 - a_{i k}) M \forall i, k \in C : i \neq k \end{array}

\begin{array}{l} x_{k} + p_{k} l x_{k} + q_{k} (l z_{k} - w y_{k} + h z_{k}) + r_{k} (1 - l x_{k} - l z_{k} + w y_{k} - h z_{k}) - \\ δ_{i k j} M \leq x_{i} + (1 - b_{i k}) M \forall i, k \in C : i \neq k \end{array}

\begin{array}{l} y_{i} + q_{i} w y_{i} + p_{i} (1 - l x_{i} - l z_{i}) + r_{i} (l x_{i} + l z_{i} - w y_{i}) - \\ δ_{i k j} M \leq y_{k} + (1 - c_{i k}) M \forall i, k \in C : i \neq k \end{array}

\begin{array}{l} y_{k} + q_{k} w y_{k} + p_{k} (1 - l x_{k} - l z_{k}) + r_{k} (l x_{k} + l z_{k} - w y_{k}) - \\ δ_{i k j} M \leq y_{i} + (1 - d_{i k}) M \forall i, k \in C : i \neq k \end{array}

\begin{array}{l} x_{i} + p_{i} . l x_{i} + q_{i} . (l z_{i} - w y_{i} + h z_{i}) + \\ r_{i} . (1 - l x_{i} - l z_{i} + w y_{i} - h z_{i}) \leq L_{j} + (1 - s_{i j}) M \forall i \in C, j \in D \end{array}

\begin{array}{l} y_{i} + q_{i} w y_{i} + p_{i} (1 - l x_{i} - l z_{i}) + \\ r_{i} (l x_{i} + l z_{i} - w y_{i}) \leq W_{j} + (1 - s_{i j}) M \forall i \in C, j \in D \end{array}

\begin{array}{l} x_{i} + p_{i} l x_{i} + q_{i} (1 - l x_{i}) + (1 - c_{i k}) M \geq \\ x_{k} + p_{k} l x_{k} + q_{k} (1 - l x_{k}) \forall i, k \in C : i \neq k;; o_{i} < o_{k} \end{array}

\begin{array}{l} x_{i} + p_{i} l x_{i} + q_{i} (1 - l x_{i}) - x_{k} + (1 - e_{i k}) M \geq \\ (p_{k} l x_{k} + q_{k} (1 - l x_{k})) a s \forall i, k \in C : i \neq k \end{array}

\begin{array}{l} y_{i} + p_{i} l y_{i} + q_{i} (1 - l y_{i}) - y_{k} + (1 - e_{i k}) M \geq \\ (p_{k} l y_{k} + q_{k} (1 - l y_{k})) a s \forall i, k \in C : i \neq k \end{array}

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A mathematical model to optimize the volumetric capacity of trucks utilized in the transport of food products

Abstract