The scope of this research is to elaborate a strategy to minimize the logistic cost of the whey collection. The problem consists of the description of the whey collection basin and transport from CP (Cheese plant) to WPP (Whey processing plant). We started with an initial basic solution and proceeded with successive iterations to find the final optimal solution. Two numeric methods are proposed to solve iteratively the problem: the first one emulates the simplex method, the second one is an empirical solution to find the optimal route. Both are solved with an Excel and Google map software and do not require a dedicated LP program for calculus. The results demonstrate that both methods contribute to solve the transport problem and generate valuable information for the achievement of economic and environmental targets.
Milk and cheese production are important contributors of the Trentino A. A. (a mountain region in Nord-East Italy) economy. In 2015, 135.094 tons of milk were produced, mostly curled and strained for cheese production by a large number of small cheese plants scattered around the region, processing on average 14,000 liters of milk per day. The region is very sensitive to the circular economy paradigm and since a long time is trying to recycle the waste from agri-food activities. An important regional project is dedicated to recycle the whey, a polluting by-product of cheese production, representing the 85% - 90% of the milk transformed in cheese. Using appropriate technologies is possible to separate the whey components and sell in different market channels: animal feedstock, proteins, vitamins for human consumption, lactose for PHA and others. Simulation about milk quota removal in Italy [
Milk, cheese and whey productions are strictly correlated: 10 kg of fresh milk produce approximately 1 kg of cheese and 9 kg of whey; the total cheese production in Italy it is estimated a quantity of whey between 8 and 9 million tons; lactose is the most important component (40-45 gr. per Kg of whey), and is responsible of the high values of BOD (Biochemical Oxygen Demand) (BOD: 40,000 - 60,000 ppm) and COD (Chemical Oxygen Demand) (COD: 50,000 - 80,000 ppm) if released into water bodies [
Italy the largest whey quantity is used for animal feedstock (65% of the total consumption), another 20% is sold as infant formulas and the remaining 15% is used in chocolate, ice cream, bakery and confectionery industry [
In USA the whey powder used for animal feeding has a lower incidence (estimated 45%); most of the whey is sold to the dairy industry. A growth of the whey consumption is expected in the nutritional segment; used in nutritional formulations such as whey powder, demineralized whey, WPC and WPI, whose demand in the health, pharmaceutic and nutritional sectors is expected to grow in next years, an interesting development is expected also in the Biopolymer industry. The derived whey products are growing at a rate of 3% per year, mostly for whey powder and lactose. With the progress in whey processing technologies, new market opportunities are disclosed to operators and the logistic of transport, packaging, storage, conservation and the environmental impact are becoming growingly important for the competitiveness of the dairy chain.
The purpose is to afford the transport problem due to shipping growing volumes of liquid whey at minimum transport cost from CP (cheese production) to WPP (whey processing) by selecting the optimal route to reduce cost and environmental impact. Preliminary information is requested about the dimension of the whey collection basin, transport costs, distance from (CP) to processing plant, (WPP), type of road (high way, state, trigonal or provincial and communal roads), road conditions, traffic intensity during the day, number of city crossed, physical obstacles orographic nature^{1}. This information will be used to minimize both the transport costs, and the environmental impact caused by CO_{2} emissions [
This logistic problem requires to define:
1) The algebraic formulation of the objective function and constraints;
2) The balance condition that is the sum of the supplies of all the sources equal to the sum of the demands for all the destinations;
3) The selection of an iterative process to emulate the simplex algorithm, by starting with an initial basic feasible solution (IBFS),check for the demand- supply conditions and proceed iteratively to find the final optimal solution [
The remainder of this paper is structured as follows: Section 2 introduces the network theory, to find the optimum transport solution; Section 3 describes the whey supply in the basin with distances among CP and road network condition; Section 4 describes the case study based on the optimization approach; Section 5 reports the comments about results of simulation, policy implications and suggestions to improve the whey collection.
The contribution of the present paper to the transport problem is twofold. The first one is to introduce a novel formulation that extends a globally inclusive facility hierarchy problem [
^{2}The WPP will use different technologies as concentration, demineralization, ultrafiltration, crystallization, PHA to separate the whey ingredients.
Definition: k is the homogeneous commodity produced by a given CP and delivered through a route to the WPP that can be located in one of the CP of the network, assuming that any CP can be also a potential location of the WPP. The solution consists in finding the optimal route to minimize the transport costs and environmental impact caused by CO_{2} emissions [
The following data are required to define this problem:
k i quantity of whey produced by a given CP and delivered to the next CP to the end WPP, x_{i}_{,j}, variable that indicate if the link between nodes i and j is open; c_{ij} is the unit cost of shipping the whey along the a given branch of the network.
The logistic of transport is an allocation problem illustrated in
be located in one of the sources; the whey route from node i (departure) to node j (destination). The m sources can ship the product to any of the n destinations at per unit carrying cost c i j (unit transportation cost from source i to destination j). The transport problem requires to minimize the shipping cost of the whey commodity from a source CP defined the supply node i to a destination node j for j = 1..n defined the demand node. Along the transitory nodes it is completed the route from source i to destination node j. As the CP (cheese plant) are interchangeable with WPP (whey processing plant), the possible WPP locations are equal to CP plants [
The O.F is targeted to find the minimum transport cost of the product that passes through the branches from the initial node i to the final node j in a route constrained by the quantity of product shipped from origin to the destination. The
in opposite directions meaning that the flow direction is inverted .
With the route A, the branch connecting nodes 1 and 2 measures 6 and the flow is equal 5; the branch connecting nodes 2 and 5 measures 4 and the flow is 7. The final solution is given by a total flow equal to 12, the total distance is 10 and the total transport cost, assuming the unit transport cost c = 1 the total cost is 58 that is the value of the route A.
With route B the length of the branch 1 - 3 is 1 and allows a flow equal 5, the branch 3 - 4 measures 2 and allows a flow equal 4; the branch 4 - 5 measures 3 and allows a flow equal 3. The total flow is 12 equal to the solution 1 but the total distance is 22 and assuming a cost c = 1, the solution is a transport cost = 22 inferior to solution 1. Other routes can be hypothesized as the 1-3-4-2-5 or 1-3-4-1-2-5 but they are less efficient in normal route conditions to Solution 2.
The first step is to draw the graph of the road network for the collection basin reported in
environmental impact from CP to the WPP. Due to the orographic nature of this region, the road network is developed along vertical lines following the mountain compluvium lines: one is a highway and four state or province roads. The horizontal road development is represented by three state and province roads. This network configuration allows different route choices then the problem is to find the optimal route that will optimize the transport cost and environmental impact. The network indicates the concentration of ten CP in a restricted area of 45 × 31 square km that deliver the 55% of the total daily whey production; other six plants are distributed in an area of 93 × 61 square km., that is four times larger compared to the first one and offers only the 45% of the total daily whey supply. This non homogeneous distribution of the CP affects the transport costs and will be taken into account to select the optimal transport solution.
The map reported in
In
From the literature two empirical procedures are selected to solve the transport
Supply | daily delivery | cost |
---|---|---|
Castelfondo | 137.25 | 91.96 |
Cavalese | 172.30 | 115.44 |
Cavareno | 87.79 | 58.82 |
Coredo | 92.96 | 62.28 |
Fondo | 82.47 | 55.25 |
Latte Trento | 347.44 | 232.78 |
Lavarone | 33.60 | 22.51 |
Mezzana | 106.75 | 71.52 |
Predazzo | 106.48 | 71.34 |
Primiero | 114.36 | 76.62 |
Revò | 26.16 | 17.52 |
Romeno | 175.92 | 117.87 |
Rumo | 102.27 | 68.52 |
Terzolas | 146.64 | 98.25 |
Tuenno | 65.12 | 43.63 |
Val di Fassa | 58.71 | 39.34 |
Sabbionara | 0.00 | 0.00 |
TN Consorzio | 0.00 | 0.00 |
Total | 1856.20 | 1243.66 |
problem: The first one is a numerical solution that emulate the simplex method and proceeds with successive iterations starting with an initial minimum cost value, proceeding to the next minimum cost and finally it is obtained the final minimum cost value. At the beginning the minimum cost cell is selected the corresponding row (supply) and column (demand) is selected at the crossing of row (supply) and column (demand) and the residual is calculated by finding the positive difference between demand and supply (or vice versa) and proceeds by finding the next minimum cost. [
The second procedure is more empirically oriented, and uses the observation of the network to choose the preferred route.
The advantage is that if the network features and road conditions are known it is simpler to solve the problem and allow to perform easily many simulations. Alternatively one can use some algorithm (i.e. the Dijkstra algorithm) to find the shortest route.
The network is composed by i = 1, ・・・, m sources (CP suppliers) and j = 1 , ⋯ , n destinations (customers): each sources CP_{i} supplies x i j quantity of whey to the next j destinations at per unit shipping cost C i j that is the transport cost per unit of whey shipped from source i to destination j. Each source i for 1 ≤ i ≤ m delivers si quantity to n destinations (customers) and the destination demands d_{j} for 1 ≤ j ≤ n . The supply constraints i is the flow limit of whey from the origin supply node i to the n destination (consumption) nodes j; the demand constraints (see column) indicate the quantity of product from the all origin nodes i allocated to one destination node j.
The total supply is the sum of the product delivered by a given CP to destination, then:
s i = ∑ i = 1 m s i j ; the demand is the sum of d_{ij} units, offered by the all CP_{j}, then d j = ∑ i = 1 n d i j this is the demand at every location. If origins and destinations coincide the problem can be represented by a square table with source i equal to destination j so that m rows equal to n columns.
The transport problem consists in the formulation of the objective function Z that is the transport cost minimization bounded with the supply and demand constraints as indicated below:
Min Z = ∑ i = 1 m ∑ j = 1 n c i j ∗ x i j (for all branches)
subject to
∑ j = 1 n x i j ≤ s i for i = 1 , 2 , ⋯ , m (supply constraint; row m)
∑ i = 1 m x i j ≤ d j for j = 1 , 2 , ⋯ , n for j = 1 , 2 , ⋯ , n (demand constraint; column n)
∑ i = 1 m s i = ∑ j = 1 n d j (equality constraint for balance problem)
X i j ≥ 0 for all i, j (positive or non negatives)
The OF with supply and demand constraints are reported in extended notation below:
Min Z = c 11 x 11 + c 12 x 12 + ⋯ + c 1 n x 1 n + c 21 x 21 + c 22 x 2 , 2 + ⋯ + c 2 n x 2 n + ⋯ + c m 1 x m 1 + c m 2 x m 2 + ⋯ + c m n x m n _{ }
subject to:
1) Supply constraints (row 1…16)
x 11 + x 12 + ⋯ + x 1 n ≤ s 1 _{ }
x 21 + x 22 + ⋯ + x 2 n ≤ s 2 _{}
x m 1 + x m 2 + ⋯ + x m n ≤ s m
2) Demand constraints (column 1…16)
x 11 + x 21 + ⋯ + x m 1 ≤ d 1
x 12 + x 22 + ⋯ + x m 2 ≤ d 2 _{}
x 1 n + x 2 n + ⋯ + x m n ≤ d n
The
1) For each branch i-, j is reported: The distance d i j in km from node i to node j, adjusted with time varying with local network conditions affecting the transport difficulty. The adjusted distance in km X_{ij} is included in the OF.
2) c i j is the cost of shipping one unit of whey from i to j then c i j = 0.67 ∗ d i j expressed in €/100Kg.
The data of
F . O 10.05 x 12 + 26.13 x 13 + 22.78 x 14 + 30.82 x 15 + 32.83 x 16 + 33.5 x 17 + 30.15 x 18 + 26.80 x 19 + 22.78 x 1 , 10 + 43.55 x 1 , 11 + 64.32 x 1 , 12 + 75.04 x 1 , 13 + 88.44 x 1 , 14 + 93.8 x 11 , 5 + 76.38 x 11 , 6 + 10.05 x 2 , 1 + 18.76 x 2 , 3 + 18.09 x 2 , 4 + ⋯ + 68.34 x 16 , 15
castelf. c_{ij} | 30.82 | 26.13 | 20.77 | 10.72 | 0.00 | 6.03 | 8.71 | 10.72 | 21.44 | 20.77 | 38.19 | 55.61 | 64.99 | 73.03 | 88.44 | 71.69 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
fondo x_{ij} | 11.63 | 9.97 | 9.26 | 4.51 | 2.14 | 0.00 | 1.19 | 1.90 | 5.70 | 6.88 | 11.87 | 17.56 | 20.89 | 23.97 | 29.20 | 23.50 | 180.16 |
fondo c_{ij} | 32.83 | 28.14 | 26.13 | 12.73 | 6.03 | 0.00 | 3.35 | 5.36 | 16.08 | 19.43 | 33.50 | 49.58 | 58.96 | 67.67 | 82.41 | 66.33 | |
cavareno x_{ij} | 12.63 | 10.87 | 10.11 | 5.05 | 3.28 | 1.26 | 0.00 | 0.76 | 4.80 | 6.06 | 11.62 | 18.95 | 22.24 | 25.52 | 29.82 | 23.75 | 186.73 |
cavareno c_{ij} | 33.50 | 28.81 | 26.80 | 13.40 | 8.71 | 3.35 | 0.00 | 2.01 | 12.73 | 16.08 | 30.82 | 50.25 | 58.96 | 67.67 | 79.06 | 62.98 | |
rumeno x_{ij} | 22.78 | 19.75 | 19.24 | 8.10 | 8.10 | 4.05 | 1.52 | 0.00 | 8.10 | 10.63 | 21.77 | 38.99 | 46.08 | 52.66 | 58.23 | 42.53 | 362.53 |
rumeno c_{ij} | 30.15 | 26.13 | 25.46 | 10.72 | 10.72 | 5.36 | 2.01 | 0.00 | 10.72 | 14.07 | 28.81 | 51.59 | 60.97 | 69.68 | 77.05 | 56.28 | |
coredo x_{ij} | 10.70 | 8.83 | 9.90 | 5.08 | 8.56 | 6.42 | 5.08 | 4.28 | 0.00 | 4.28 | 9.90 | 18.46 | 22.21 | 27.56 | 29.43 | 23.01 | 193.71 |
coredo c_{ij} | 26.80 | 22.11 | 24.79 | 12.73 | 21.44 | 16.08 | 12.73 | 10.72 | 0.00 | 10.72 | 24.79 | 46.23 | 55.61 | 69.01 | 73.70 | 57.62 | |
tuenno x_{ij} | 6.37 | 5.06 | 5.81 | 3.19 | 5.81 | 5.44 | 4.50 | 3.94 | 3.00 | 0.00 | 7.50 | 13.49 | 16.12 | 19.87 | 21.18 | 16.68 | 137.95 |
tuenno c_{ij} | 22.78 | 18.09 | 20.77 | 11.39 | 20.77 | 19.43 | 16.08 | 14.07 | 10.72 | 0.00 | 26.80 | 48.24 | 57.62 | 71.02 | 75.71 | 59.63 | |
trento x_{ij} | 65.00 | 58.00 | 63.00 | 46.00 | 57.00 | 50.00 | 46.00 | 43.00 | 37.00 | 40.00 | 0.00 | 55.00 | 68.00 | 89.00 | 84.00 | 55.00 | 856.00 |
trento c_{ij} | 43.55 | 38.86 | 42.21 | 30.82 | 38.19 | 33.50 | 30.82 | 28.81 | 24.79 | 26.80 | 0.00 | 36.85 | 45.56 | 59.63 | 56.28 | 36.85 | |
cavalese x_{ij} | 47.61 | 44.63 | 47.11 | 38.68 | 41.16 | 36.70 | 37.19 | 38.19 | 34.22 | 35.71 | 27.28 | 0.00 | 18.35 | 18.35 | 38.68 | 50.58 | 554.43 |
cavalese c_{ij} | 64.32 | 60.30 | 63.65 | 52.26 | 55.61 | 49.58 | 50.25 | 51.59 | 46.23 | 48.24 | 36.85 | 0.00 | 24.79 | 24.79 | 52.26 | 68.34 | |
predazzo x_{ij} | 34.32 | 31.87 | 33.10 | 28.20 | 29.73 | 26.97 | 26.97 | 27.89 | 25.44 | 26.36 | 20.84 | 11.34 | 0.00 | 7.36 | 19.61 | 33.41 | 383.39 |
predazzo c_{ij} | 75.04 | 69.68 | 72.36 | 61.64 | 64.99 | 58.96 | 58.96 | 60.97 | 55.61 | 57.62 | 45.56 | 24.79 | 0.00 | 16.08 | 42.88 | 73.03 | |
val di fassa x_{ij} | 22.31 | 20.95 | 21.46 | 18.76 | 18.42 | 17.07 | 17.07 | 17.57 | 17.40 | 17.91 | 15.04 | 6.25 | 4.06 | 0.00 | 18.08 | 21.80 | 254.14 |
val di fassa c_{ij} | 88.44 | 83.08 | 85.09 | 74.37 | 73.03 | 67.67 | 67.67 | 69.68 | 69.01 | 71.02 | 59.63 | 24.79 | 16.08 | 0.00 | 71.69 | 86.43 | |
fiera prim x_{ij} | 46.08 | 34.89 | 45.09 | 39.17 | 43.45 | 40.49 | 38.84 | 37.85 | 36.21 | 37.19 | 27.65 | 25.67 | 21.07 | 35.22 | 0.00 | 33.57 | 542.44 |
fiera prim c_{ij} | 93.80 | 71.02 | 91.79 | 79.73 | 88.44 | 82.41 | 79.06 | 77.05 | 73.70 | 75.71 | 56.28 | 52.26 | 42.88 | 71.69 | 0.00 | 68.34 | |
lavarone x_{ij} | 11.02 | 10.15 | 10.73 | 9.09 | 10.35 | 9.57 | 9.09 | 8.12 | 8.32 | 8.61 | 5.32 | 9.86 | 10.54 | 12.48 | 9.86 | 0.00 | 143.13 |
lavarone c_{ij} | 76.38 | 70.35 | 74.37 | 62.98 | 71.69 | 66.33 | 62.98 | 56.28 | 57.62 | 59.63 | 36.85 | 68.34 | 73.03 | 86.43 | 68.34 | 0.00 | |
demand | 329.01 | 285.26 | 312.52 | 240.47 | 268.92 | 247.21 | 239.37 | 232.49 | 231.36 | 238.13 | 247.76 | 349.69 | 404.88 | 493.67 | 527.27 | 485.20 | 5133.21 |
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | ||
cost: c_{ij} (red digit are the minimim cost for each column) | |||||||||||||||||
source | destination (customers) | ||||||||||||||||
(supply) | mezzana | terzolas | rumo | revo | castelf. | fondo | cavareno | rumeno | coredo | tuenno | trento | cavalese | predazzo | val di fassa | fiera prim | lavarone | supply |
mezzana | 0.00 | 10.05 | 26.13 | 22.78 | 30.82 | 32.83 | 33.5 | 30.15 | 26.8 | 22.78 | 43.55 | 64.32 | 75.04 | 88.44 | 93.8 | 76.38 | |
terzolas | 10.05 | 0.00 | 18.76 | 18.09 | 26.13 | 28.14 | 28.81 | 26.13 | 22.11 | 18.09 | 38.86 | 60.3 | 69.68 | 83.08 | 71.02 | 70.35 | |
rumo | 26.13 | 18.76 | 0.00 | 14.74 | 20.77 | 26.13 | 26.8 | 25.46 | 24.79 | 20.77 | 42.21 | 63.65 | 72.36 | 85.09 | 91.79 | 74.37 | |
revo | 22.78 | 18.09 | 14.74 | 0.00 | 10.72 | 12.73 | 13.4 | 10.72 | 12.73 | 11.39 | 30.82 | 52.26 | 61.64 | 74.37 | 79.73 | 62.98 | |
castelf. | 30.82 | 26.13 | 20.77 | 10.72 | 0.00 | 6.03 | 8.71 | 10.72 | 21.44 | 20.77 | 38.19 | 55.61 | 64.99 | 73.03 | 88.44 | 71.69 | |
fondo | 32.83 | 28.14 | 26.13 | 12.73 | 6.03 | 0.00 | 3.35 | 5.36 | 16.08 | 19.43 | 33.5 | 49.58 | 58.96 | 67.67 | 82.41 | 66.33 | |
cavareno | 33.5 | 28.81 | 26.8 | 13.4 | 8.71 | 3.35 | 0.00 | 2.01 | 12.73 | 16.08 | 30.82 | 50.25 | 58.96 | 67.67 | 79.06 | 62.98 | |
rumeno | 30.15 | 26.13 | 25.46 | 10.72 | 10.72 | 5.36 | 2.01 | 0.00 | 10.72 | 14.07 | 28.81 | 51.59 | 60.97 | 69.68 | 77.05 | 56.28 | |
coredo | 26.8 | 22.11 | 24.79 | 12.73 | 21.44 | 16.08 | 12.73 | 10.72 | 0.00 | 10.72 | 24.79 | 46.23 | 55.61 | 69.01 | 73.7 | 57.62 | |
tuenno | 22.78 | 18.09 | 20.77 | 11.39 | 20.77 | 19.43 | 16.08 | 14.07 | 10.72 | 0.00 | 26.8 | 48.24 | 57.62 | 71.02 | 75.71 | 59.63 | |
trento | 43.55 | 38.86 | 42.21 | 30.82 | 38.19 | 33.5 | 30.82 | 28.81 | 24.79 | 26.8 | 0.00 | 36.85 | 45.56 | 59.63 | 56.28 | 36.85 | |
cavalese | 64.32 | 60.3 | 63.65 | 52.26 | 55.61 | 49.58 | 50.25 | 51.59 | 46.23 | 48.24 | 36.85 | 0.00 | 24.79 | 24.79 | 52.26 | 68.34 | |
predazzo | 75.04 | 69.68 | 72.36 | 61.64 | 64.99 | 58.96 | 58.96 | 60.97 | 55.61 | 57.62 | 45.56 | 24.79 | 0.00 | 16.08 | 42.88 | 73.03 | |
val di fassa | 88.44 | 83.08 | 85.09 | 74.37 | 73.03 | 67.67 | 67.67 | 69.68 | 69.01 | 71.02 | 59.63 | 24.79 | 16.08 | 0.00 | 71.69 | 86.43 | |
fiera prim | 93.8 | 71.02 | 91.79 | 79.73 | 88.44 | 82.41 | 79.06 | 77.05 | 73.7 | 75.71 | 56.28 | 52.26 | 42.88 | 71.69 | 0.00 | 68.34 | |
lavarone | 76.38 | 70.35 | 74.37 | 62.98 | 71.69 | 66.33 | 62.98 | 56.28 | 57.62 | 59.63 | 36.85 | 68.34 | 73.03 | 86.43 | 68.34 | 0.00 | |
demand | 0.00 |
s. to
supply constraint (nr = 16)
x 1 , 2 X 1 , 2 + X 1 , 3 + X 1 , 4 + X 1 , 5 + X 1 , 6 + X 1 , 7 + X 1 , 8 + X 1 , 9 + X 1 , 10 + X 1 , 11 + X 1 , 12 + X 1 , 13 + X 1 , 14 + X 1 , 15 + X 1 , 16 ≤ 310.63
x 16 , 1 + X 16 , 2 + X 16 , 3 + X 16 , 4 + X 16 , 5 + X 16 , 6 + ⋯ + X 16 , 55 ≤ 143.13
demand constraint (nr = 16)
x 1 , 1 + X 2 , 1 + X 3 , 1 + X 4 , 1 + X 5 , 1 + X 6 , 1 + X 7 , 1 + X 8 , 1 + X 9 , 1 + X 10 , 1 + X 11 , 1 + X 12 , 1 + ⋯ + X 13 , 1 + X 14 , 1 + ⋯ + X 16 , 1 ≤ 329.01
x 1 , 16 + X 2 , 16 + X 3 , 16 + X 4 , 16 + X 5 , 16 + ⋯ X 16 , 16 + ⋯ + ≤ 485.20
Procedure 1―To find the optimal solution is required the following steps:
Step 1―Define the OF: the transport cost minimization by selecting a route composed by nodes (CP) and branches, supply (row) and demand (column) constraints;
Step 2―Check for the balance condition: sum of row values equal to sum of column values;
Step 3―Find the minimum transport cost value c_{ij} in the transport
Step 4―First allocation: find the smaller value by comparing s_{i} and d_{j} referred to cell c_{ij}; (in our case min c_{ij} = 2.01 in cell 8, 7 and the smaller value between s i = 362.53 and d j = 239.37 is s i . Allocate X i j = supply value (in this case is 23,937 in the corresponding cell (i, j) and compute the difference: 362.53 - 239.37 = 123.16 that is the residual supply value.
Step 5―Second allocation: search for the next minimum cost C i j corresponding to value 3.35 in cell 7, 6 and s 7 = 185.73 and d 6 = 247.21 . The residual supply s 7 = 123.16 is is compared with d_{6} that is greater so Residual d j = 247.21 − 123.16 = 124.05 and supply go to 0; this value is the new reduced demand allocated in the new cell c i j . Proceeding with these operations, the supply and demand requirements are progressively allocated, allowing to compute the partial costs of allocation at each step.
The min C_{ij}_{ }from the 16 columns reported in
c 8 , 7 = 2.01 ; c 7 , 6 = 3.35 ; c 6 , 8 = 5.36 ; c 6 , 5 = 6.03 ; c 2 , 1 = 10 , 05 c 10 , 9 = 10.72 ; c 4 , 10 = 11.39 ; c 4 , 3 = 14.74 ; c 10 , 2 = 18.09 ; c 9 , 11 = 24.79 ; c 13 , 12 = 24.80 ; c 11 , 16 = 36.85 ; c 13 , 15 = 42.88
The cost minimizations obtained with successive iterations are reported in
Simulation 2: For this simulation, the all CP of the first route are excluded from route 2 and 3 (value = 0) a priori because inefficient in term of distance and time as suggested by the Google map and roads previously observed in
In
Minimum cost | Cell position | Demand (column) d_{j} | Supply (row) s_{i} | Minimum | Difference | Transport | |||||
---|---|---|---|---|---|---|---|---|---|---|---|
allocation | value | i = row; j = column | position | value | position | value | (d_{j}, s_{i}) | value-min | resid d_{j} | resid s_{i} | cost |
first allocation | 2.01 | 8 - 7 | d7 | 239.37 | s8 | 362.53 | d7 | 123.16 | 0.00 | 123.16 | 247.55 |
2.nd allocation | 3.35 | 7 - 6 | d6 | 247.21 | res s8 | 123.16 | s8 | 123.16 | 124.05 | 0 | 415.57 |
3.rd allocation | 5.36 | 6 - 8 | resid d | 124.05 | s6 | 180.16 | resid d | 124.05 | 0.00 | 56.11 | 300.74 |
4.th allocation | 6.03 | 6 - 5 | d5 | 268.92 | resid s | 56.11 | resid d | 212.81 | 212.81 | 0 | 1283.26 |
5.th allocation | 10.05 | 2 - 1 | s2 | 371.41 | resid d | 212.81 | resid s | 158.60 | 0.00 | 158.60 | 1593.92 |
6.th allocation | 10.72 | 10 - 9 | resid s | 158.60 | resid d | 231.36 | resid s | 158.60 | 72.76 | 0 | 780.01 |
7.th allocation | 11.39 | 4 - 10 | resid d | 72.76 | s4 | 54.96 | resid d | 17.80 | 17.80 | 0 | 202.72 |
8.th allocation | 12.73 | 6 - 4 | resid d | 17.80 | s6 | 180.16 | resid d | 17.80 | 0.00 | 162.36 | 2066.87 |
9.th allocation | 14.74 | 4 - 3 | resid s | 162.36 | d3 | 312.52 | resid d | 150.15 | 150.15 | 0 | 2213.28 |
10.th allocation | 16.08 | 13 - 14 | resid d | 150.15 | s13 | 383.39 | resid d | 150.15 | 0.00 | 233.24 | 3750.50 |
11.th allocation | 18.09 | 10 - 2 | resid s | 233.24 | d2 | 285.26 | resid s | 233.24 | 52.02 | 0 | 941.11 |
12.th allocation | 24.79 | 9 - 11 | resid d | 52.02 | s9 | 193.71 | resid d | 52.02 | 0.00 | 141.69 | 3512.43 |
13.th allocation | 24.79 | 13 - 12 | resid s | 141.69 | d12 | 349.69 | resid s | 141.69 | 208.01 | 0 | 5156.49 |
14.th allocation | 36.85 | 11 - 16 | resid d | 208.01 | s11 | 856.00 | resid d | 208.01 | 0.00 | 647.99 | 23,878.55 |
15.th allocation | 37.65 | 12 - 15 | resid s | 647.99 | d15 | 516.46 | resid d | 516.46 | 0.00 | 131.53 | 4952.10 |
simulation 2: x_{ij}: transport matrix with limited number of origin and destination locations imposed by selected transport road | |||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
source | destination (customers) | ||||||||||||||||
(supply) | mezzana | terzolas | rumo | revo | castelf. | fondo | cavareno | rumeno | coredo | tuenno | trento | cavalese | predazzo | val di fassa | fiera prim | lavarone | supply |
mezzana | 0.00 | 4.61 | 11.98 | 10.45 | 14.13 | 15.06 | 15.36 | 13.83 | 12.29 | 10.45 | 19.97 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 128.12 |
terzolas | 6.33 | 0.00 | 11.82 | 11.40 | 16.46 | 17.73 | 18.15 | 16.46 | 13.93 | 11.40 | 24.48 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 148.14 |
rumo | 11.48 | 8.24 | 0.00 | 6.48 | 9.12 | 11.48 | 11.77 | 11.19 | 10.89 | 9.12 | 18.54 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 108.32 |
revo | 2.56 | 2.03 | 1.66 | 0.00 | 1.20 | 1.43 | 1.51 | 1.20 | 1.43 | 1.28 | 3.46 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 17.77 |
castelf. | 18.17 | 15.41 | 12.25 | 6.32 | 0.00 | 3.56 | 5.14 | 6.32 | 12.64 | 12.25 | 22.52 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 114.56 |
fondo | 11.63 | 9.97 | 9.26 | 4.51 | 2.14 | 0.00 | 1.19 | 1.90 | 5.70 | 6.88 | 11.87 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 65.04 |
cavareno | 12.63 | 10.87 | 10.11 | 5.05 | 3.28 | 1.26 | 0.00 | 0.76 | 4.80 | 6.06 | 11.62 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 66.45 |
rumeno | 22.78 | 19.75 | 19.24 | 8.10 | 8.10 | 4.05 | 1.52 | 0.00 | 8.10 | 10.63 | 21.77 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 124.05 |
coredo | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 4.28 | 9.90 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 14.18 |
tuenno | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 7.50 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 7.50 |
trento | 0.00 | 0.00 | 0.00 | 46.00 | 57.00 | 0.00 | 0.00 | 0.00 | 37.00 | 40.00 | 0.00 | 55.00 | 68.00 | 89.00 | 84.00 | 55.00 | 531.00 |
cavalese | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 18.35 | 18.35 | 38.68 | 50.58 | 125.96 |
predazzo | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 20.84 | 11.34 | 0.00 | 7.36 | 19.61 | 33.41 | 92.55 |
val di fassa | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 17.40 | 17.91 | 15.04 | 6.25 | 4.06 | 0.00 | 18.08 | 21.80 | 100.54 |
fiera prim | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 36.21 | 37.19 | 27.65 | 25.67 | 21.07 | 35.22 | 0.00 | 33.57 | 216.58 |
lavarone | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 5.32 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 5.32 |
demand | 85.59 | 70.87 | 76.31 | 98.30 | 111.45 | 54.56 | 54.63 | 51.65 | 160.39 | 167.46 | 220.48 | 98.27 | 111.47 | 149.92 | 160.38 | 194.36 | 1866.09 |
Minimum cost | cell position | supply (row) s_{i} | demand (column) d_{j} | Minimum | difference | transport | |||||
---|---|---|---|---|---|---|---|---|---|---|---|
allocation | value | i = row; j = column | position | value | position | value | (d_{j}, s_{i}) | value-min | resid d_{j} | resid s_{i} | cost |
first allocation | 10.05 | 2-1 | s2 | 371.41 | d1 | 379.39 | 7.98 | 0.00 | 80.18 | ||
2.nd allocation | 18.09 | 10-2 | s10 | 137.95 | d2 | 335.17 | resid d_{j} | 7.98 | 0.00 | 129.97 | 2351.13 |
3.rd allocation | 14.74 | 4-3 | s4 | 54.96 | d3 | 346.56 | resid s_{i} | 129.97 | 216.59 | 0.00 | 3192.56 |
4.th allocation | 10.72 | 5-4 | s5 | 323.14 | d4 | 258.12 | resid d_{j} | 216.59 | 0.00 | 106.54 | 1142.16 |
6.th allocation | 20.77 | 3-5 | s3 | 278.46 | d5 | 294.30 | resid s_{i} | 106.54 | 187.76 | 0 | 3899.67 |
7.th allocation | 3.35 | 7-6 | s7 | 186.73 | d6 | 283.58 | resid d_{j} | 187.76 | 1.03 | 0.00 | 3.44 |
8.th allocation | 2.01 | 8-7 | s8 | 362.53 | d7 | 276.21 | resid d_{j} | 1.03 | 0.00 | 361.51 | 726.63 |
9.th allocation | 5.36 | 6-8 | s6 | 180.16 | d8 | 264.15 | resid s_{i} | 361.51 | 0.00 | 97.36 | 521.83 |
10.th allocation | 26.80 | 1-9 | s1 | 310.63 | d9 | 282.91 | resid s_{i} | 97.36 | 185.55 | 0.00 | 4972.83 |
11.th allocation | 10.72 | 9-10 | s9 | 193.71 | d10 | 280.90 | resid d_{j} | 185.55 | 0.00 | 8.16 | 87.45 |
12.th allocation | 24.79 | 13-12 | s13 | 383.39 | d12 | 673.52 | resid s_{i} | 8.16 | 665.36 | 0.00 | 16,494.32 |
13.th allocation | 45.56 | 13-11 | s13 | 383.39 | d11 | 0.00 | resid d_{j} | 665.36 | 281.97 | 0 | 12,846.45 |
The optimal transport values from the two roads are the following:
Simulation 1 = 51295;
Simulation 2 = 46319
This procedure can be adopted preferably in case some information are available ex ante as graph map, distance, road condition, or preference about the route that could facilitate the search for the optimal transport solution.
The problem is the same, minimization of the transport cost: c_{ij} is the unit cost of transport; the 16 nodes and branches are forming the network that shows possible alternative routes to ship the whey from nodes i (origin) to nodes j (destination), X_{ij} is the quantity of whey shipped from i to j. We start by solving the problem previously illustrated in
O F : 6 X 1 , 2 + X 1 , 3 + 4 X 2 , 5 − 3 X 3 , 2 + 2 X 3 , 4 + 4 X 4 , 1 + X 4 , 2 + 3 X 4 , 5 _{ }
s. t. (maximum flow through the node-archs; the values are selected with the branch capacity) node-arch 1: X 1 , 2 + X 1 , 3 − X 1 , 4 = (1 branch with negative versus, max. capacity = 5)
Node arch 2: X 2 , 5 − X 2 , 1 − X 2 , 3 − X 2 , 4 = − 4 (2 branches with negative versus, max. capacity = −4)
Node arch 3: − X 3 , 2 + X 3 , 4 − X 3 , 1 = 0 (1 branch with negative versus, max. capacity = −4
Node arch 4: X 4 , 1 + X 4 , 2 + X 4 , 5 − − X 4 , 3 = 1 (1 branch with negative versus, max. capacity = 1_{ }
Node arch 5: − X 5 , 2 − X 5 , 4 = − 3 (2 branches with negative versus, max. capacity = −3)
Route options: The
+1 if e exit from v (v is the tail of e = positive branch direction);
−1 if e entry in v ((v is the head of e = negative branch direction);
0, otherwise (not allowed flow).
The incidence matrix described in
Parameters
0 ⊆ V ; origin nodes
D ⊆ V ; destination nodes
T ⊆ V ; passage nodes
s_{i}, for iÎ0, supply product vertix i
d_{i}, for i ÎD, demand product vertix i
X_{ij} for i, jÎA flow of product on the arch i, j
u_{ij} for i, jÎA, capacity of the arch i, j (maximum admittable flow on the arch)
c i j ∗ x i j , (i, j)ÎA is the transport cost of the flow X_{ij} on the arch i, j
The second version of the transport problem is represented in
Node-arch A | X_{1,2} | X_{1,3} | X_{1,4} | X_{2,1} | X_{2,3} | X_{2,4} | X_{2,5} | X_{3,1} | X_{3,2} | X_{3,4} | X_{4,1} | X_{4,2} | X_{4,3} | X_{4,5} | X_{5,2} | X_{5,4} |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | 1 | 1 | −1 | |||||||||||||
2 | −1 | −1 | −1 | 1 | ||||||||||||
3 | −1 | 1 | 1 | |||||||||||||
4 | 1 | 1 | −1 | 1 | ||||||||||||
5 | −1 | −1 |
destination J = 1..n | ||||||
---|---|---|---|---|---|---|
1 | 2 | j | n | limit | ||
1 | c_{1,1}X_{11} | c_{1,2}X_{12} | ・・・ | c_{1,n}X_{1n} | s_{1} | |
2 | c_{21}X_{21} | c_{2,2}X_{22} | ・・・ | c_{2,n}X_{2n} | s_{2} | |
Source = s_{i}, row 1..m | i | |||||
(whey supply) | . | |||||
m | c_{m,1}X_{m1} | c_{m,2}X_{m2} | ・・・ | c_{m,n}X_{mn} | s_{m} | |
Demand | d_{1} | d_{2} | d_{n} | |||
d_{j} |
The values at the right and bottom sides of the transportation
A column indicates the whey supply from the sources s_{i} for i = 1, ⋯ , 16 to the m demand d_{j} for j = 1 , ⋯ , m that are the quantity supplied to each demand. In
C i j for 1 ≤ i ≤ m and 1 ≤ j ≤ n is the unit cost of shipping the whey from the i-th source to the j-th destination;
X i j is the quantity of whey shipped from i-th source to j-th destination;
C i j ∗ X i j is the total transport cost from source (node i) to destination (node j) for all i, j. pair combinations (from the first to the last allocation).The cost cells are distributed in continuous^{3}:
O F : C 1 , 1 X 1 , 1 = 464 X 11 ; C 1 , 2 X 1 , 2 = 513 X 1 , 2 ; C 3 , 4 = 685 X 3 , 4
For the equilibrium condition it is required:
∑ i = 1 m S i = ∑ i = 1 n d j then (see
750 + 1250 + 1000 = 800 + 650 + 700 + 850 = 3000 i.e. demand = supply then this problem is balanced.
This procedure is used to solve the problem of the minimum transport cost from the CP origins to WPP destination with cost simulations of some predefined routes to fulfill specific objectives of the operators. The
MATR. SOMMA. PRODUCT (B98:S115; B123:S140)
The first matrix B98:S115 reports the whey quantity multiplied by the adjusted distances (see
The route is specified with coefficient value = 1 for positive direction and −1 for the opposite direction of the branch while the non activated branch is indicated with nul value 0. The objective function minimizes the transport cost between a given CP (origin) and the WPP (destination); the total cost will be calculated by
Destination (demand) = row | Supply | |||||
---|---|---|---|---|---|---|
Source | 1 | 2 | 3 | 4 | limit | |
Source = row | 1 | c_{1,1} = 464 | c_{1,2} = 513 | 654 | 867 | 750 |
(supply) | 2 | 352 | 416 | 690 | 791 | 1250 |
3 | 995 | 682 | 388 | c_{3,4} = 685 | 1000 | |
demand | Allocation | 800 | 650 | 700 | 850 |
summing the costs of various branches composing the route: C T = ∑ ( C i j ∗ X i j ) (see
Compared the second with the first procedure, the transport cost with destination Trento are quite similar, the difference is 15.5% and is explained by the differences in route 2 and 3 followed to ship the product to Trento. Sustainability of the collection strategy is an important collateral effect of the transport: the available technologies to process the whey contribute to limit the dispersion of whey pollutants in the environment and demonstrate that economic and environmental targets of the whey processing can be simultaneously obtained. The production, processing and transportation of milk products, contributes with 2.7 percent to the global anthropogenic greenhouse gas emissions in Italy. According to the methodology of the IPCC [
The whey transport emission is computed by assuming that CO_{2} emission of a normal diesel truck with capacity of 20 ton and emission of 20 g CO_{2} eq per ton/ km. This coefficient has been used to compute the whey pollution due transport. The CO_{2} emissions are: route 1 = 15.20; route 2 = 12.05; route 3 = 245.29; total