In this work you will be able to find different logistics exercises, which allow, through the use of the "LINGO" program, to find the adequate routes for locating plants according to the different needs that arise in each exercise.
solution-problems-logistics-lingoThis mainly allows the user to greatly reduce costs, among the main ones are:
- Reduction of transportation costs Reduction of installation costs Reduction of operating costs
Currently LOGISTICS is in charge of the efficient distribution of the products of a certain company with a lower cost and excellent customer service, that is why in companies it has taken a very important role as well as the means and programs to be able to concisely define the variables that intervene in the distribution systems to generate the most viable routes and make our service more efficient and optimal.Each of the exercises that will be presented below contain different characteristics, be it plant capacity, costs, potential plants, etc, however the lingo program will allow us to select the best plants of all the possible ones for each exercise, as long as the model is correctly entered into the program.It is important to highlight from the beginning that the first step for any exercise is the determination of variables since, according to the number of variables that we model, it is the possibility that the program works correctly and provides the most optimal solutions to achieve the objectives of each respective year. Each of the exercises that will be seen in this work will be able to analyze step by step how they are solved, that is, in the following way:Each of the exercises that will be seen in this work will be able to analyze step by step how they are solved, that is, in the following way:Each of the exercises that will be seen in this work will be able to analyze step by step how they are solved, that is, in the following way:
- Presentation of the information of the exercise Determination of variables Solving the problem Description of how it is entered into the program Analysis of results Interpretation of results Outlining of results
We can also find a general conclusion about the importance of obtaining excellent results with the use of a good program such as Lingo.
GENERALITIES
Linear programming is a series of methods and procedures that allow the resolution of problems in an optimal way, either to maximize or minimize an objective, in such a way that the variables are subject to a series of restrictions expressed through a system of equations. This means that through its method of use calculations can be simplified and a result close to reality can be obtained.
The most widely used method for linear programming is LINGO, which is a tool that helps to formulate linear and non-linear problems for the resolution and channeling of the solution of the problems. It uses a mathematical modeling language, which allows it to be expressed naturally and easily understood.
The objective function is obtained from the sum of all the products of the unit cost by the number of goods sent from each origin to each destination, that is:
Subject to:
For this model it is assumed that there is a balance between supply and demand, that is, that equality is fulfilled:
The variables xij represent the number of units that are sent from the i-th source to the j-th destination. In this case, the number of origins is i = 1,2 and four destinations j = 1, 2, 3,4
According to Industrial Engineer Humberto Ángel Chávez Milla, he mentions that the restrictions are relationships between the decision variables and the available resources. The model constraints limit the value of the decision variables. They are generated when the available resources are limited, in addition to the restrictions, the No Negativity restriction of the Decision Variables, that is: Xi = 0.
The locations of plants, services and, in general, the design of the distribution and customer service system are decisions of enormous importance for the success or failure of a business project, if one takes into account that once the plant or warehouse is located, the decision to move to a more convenient location is not feasible, since this decision implies a considerable investment as well as radical changes in the operation of the business. (BALLOU, 2004)
ILLUSTRATION 4 PLANT LOCATION
In the following examples, cases of location of plants and distribution centers will be presented for the solution and optimization of services that different companies and / or plants require.
EXERCISE 1
The company Manufacturas Águila Real, dedicated to the production of fenders for automobiles, must decide on the construction of new plants to meet orders for export, as well as on the development of a more efficient distribution system. They currently have a single plant in San Luis Potosí, with a production capacity of 30,000 units. Due to an increase in demand, four possible locations for new plants are being considered: Durango, Mexico City, Toluca and Zacatecas. The following by units and the demands for the following year.
| ORIGIN | NUEVO LAREDO | BLACK STONES | CD. JUÁREZ | CAPABILITIES |
| DURANGO | 5 | two | 3 | 30,000 |
| TOLUCA | 8 | 6 | 6 | 20,000 |
| CDMX | 9 | 7 | 6 | 30,000 |
| ZACATECAS | 4 | 4 | 5 | 40,000 |
| SAINT LOUIS | 3 | two | 4 | 30,000 |
| DEMAND | 30,000 | 20,000 | 20,000 |
ILLUSTRATION 5 PLANT INFORMATION
The cost of locating the new plants is as follows (with annual operating and amortization expenses):
| DURANGO | TOLUCA | CDMX | ZACATECAS |
| 175,000 | 300,000 | 375,000 | 500,000 |
ILLUSTRATION 6 LOCATION COST
Formulate the problem of finding the plant locations that minimize the annual cost of transportation and location, satisfying the demands, as a linear programming problem with integer variables and solve it with the help of a solver of your choice.
SOLUTION
In the first place, it will begin by solving this problem manually and later it will be attached to the Lingo 14.0 software to evaluate the results.
This model aims to minimize the annual costs of transport and location, satisfying the demands. Schematically, the transportation problem can be represented as follows:
ILLUSTRATION 7 NETWORK MODEL
Declaration of decision variables
X = plants that supply localities
Y = total cost of each plant
- Now we will interpret the objective function for the minimization of transport costs
OBJECTIVE FUNCTION
Min = 5 * X11 + 2 * X12 + 3 * X13
+ 8 * X21 + 6 * X22 + 6 * X23
+ 9 * X31 + 7 * X32 + 6 * X33
+ 4 * X41 + 4 * X42 + 5 * X43
+ 3 * X51 + 2 * X52 + 4 * X53
+ 175000 * Y1 + 300000 * Y2 + 375000 * Y3 + 500000 * Y4;
Where:
Once the functions are located, we will proceed to place the capacity restrictions of the problem
! CAPACITY RESTRICTION;
X11 + X12 + X13 <= 30000 * Y1;
X21 + X22 + X23 <= 20000 * Y2;
X31 + X32 + X33 <= 30000 * Y3;
X41 + X42 + X43 <= 40000 * Y4;
! DEMAND RESTRICTION;
X11 + X21 + X31 + X41 <= 30000;
X12 + X22 + X32 + X42 <= 20000;
X13 + X23 + X33 + X43 <= 20000;
Y1 + Y2 + Y3 + Y4> = 1;
Incorporating it into Lingo as follows:
It is important to note that once the data is ready to be entered into the lingo program, the following signs must be carefully placed:
* = indicates multiplication
; = indicates closing of a certain amount
! = Highlight the actions to be taken
Model = openness to modeling
End = modeling closure
These signs are representative and mandatory for the program to function properly.
ILLUSTRATION 8 LINGO # 1
INTERPRETATION OF RESULTS
| CALCULATION OF VARIABLES | ||||
|
OF |
TO |
VARIABLE |
VALUE |
COST
REDUCED |
| Durango | Nuevo Laredo | X11 | 0.000000 | 5,000,000 |
| Durango | P. Black | X12 | 0.000000 | 2,000,000 |
| Durango | Cd juarez | X13 | 0.000000 | 3,000,000 |
| Toluca | Nuevo Laredo | X21 | 0.000000 | 8,000,000 |
| Toluca | P. Black | X22 | 0.000000 | 6,000,000 |
| Toluca | Cd juarez | X23 | 0.000000 | 6,000,000 |
| CDMX | Nuevo Laredo | X31 | 0.000000 | 9,000,000 |
| CDMX | P. Black | X32 | 0.000000 | 7,000,000 |
| CDMX | Cd juarez | X33 | 0.000000 | 6,000,000 |
| Zacatecas | Nuevo Laredo | X41 | 0.000000 | 4,000,000 |
| Zacatecas | P. Black | X42 | 0.000000 | 4,000,000 |
| Zacatecas | Cd juarez | X43 | 0.000000 | 5,000,000 |
| saint Louis | Nuevo Laredo | X51 | 0.000000 | 3,000,000 |
| saint Louis | P. Black | X52 | 0.000000 | 2,000,000 |
| saint Louis | Cd juarez | X53 | 0.000000 | 4,000,000 |
ILLUSTRATION 9 RESULTS OF LINGO # 1
The cost of locating the plants will be $ 175,000.00 pesos to satisfy the demand and lower the costs of transportation and operation of the car fenders. Meanwhile, the costs of the plants at the beginning, the sum of the 4 plants is $ 1,350,000.00 pesos with the calculation solved does not give a total of the 4 plants of $ 650,000 pesos giving a total difference of $ 700,000 pesos in the four plants of Durango, Toluca, CDMX and Zacatecas.
| COST CONCLUSION | |||||
| BEFORE | PLANT | VARIABLE | VALUE | REDUCED | DIFFERENCE |
| 175000 | Durango | Y1 | 1000000 | 0.000000 | 175000 |
| 300000 | Toluca | Y2 | 0.000000 | 125000 | 175000 |
| 375000 | CDMX | Y3 | 0.000000 | 200000 | 175000 |
| 500000 | Zacatecas | Y4 | 0.000000 | 325000 | 175000 |
| 1,350,000 | TOTALS | 650,000 | 700,000 |
Illustration 10 COST ANALYSIS # 1
EXERCISE 2
A manufacturer of computer equipment wants to determine the location of the plants that it will install to serve the national market. For this purpose I locate four possible locations. The following table shows monthly production capacities, setup costs (incurred one time only), and monthly operating costs.
| PLANT | MONTHLY CAPACITY (COMPUTERS) | COST OF OPERATIONS | INSTALLATION COST |
| P1 | 1700 | 700,000 | 6,400,000 |
| P2 | 2000 | 700,000 | 8,600,000 |
| Q3 | 1700 | 650,000 | 7,500,000 |
| Q4 | 2000 | 700,000 | 5,500,000 |
ILLUSTRATION 11 PLANT ANALYSIS # 2
The manufacturer expects the plants to serve four markets. The following table shows the unit shipping costs ($ / computers) from each of the plants to each of the markets, as well as the estimated monthly demand.
| PLANT | M1 | M2 | M3 | M4 |
| P1 | 5 | 3 | two | 6 |
| P2 | 4 | 7 | 8 | 10 |
| Q3 | 6 | 5 | 3 | 8 |
| Q4 | 9 | 8 | 6 | 5 |
| DEMAND | 1,700 | 1 000 | 1,500 | 1 200 |
ILLUSTRATION 12 UNIT SHIPPING COSTS
- For each of the localities, calculate the monthly amortization necessary to absorb the installation costs, if the interest rate is 2% per month and the investment is amortized in 10 years. Formulate the problem of finding the location of the plants that minimize the monthly costs of transportation (final product only), installation and operation combined common linear programming problem (integer) Solve the entire programming problem using the solver of your choice Prepare a brief report for the manufacturer's director with your recommendations, indicate to be sent from each plant to each of the markets.
SOLUTION
In the first place, it will begin by solving this problem manually and later it will be attached to the Lingo 14.0 software to evaluate the results.
- This model aims to minimize the monthly costs of transportation, installation and operation for the supply of new plants. Schematically, the transportation problem can be represented as follows: Section A mentions that installation costs have to be amortized, taking into account that these costs have a useful life. Two methods can be used to resolve amortization, one is manually, for this we will use the following formula:
Where:
I = amortization A = investment i = percentage
n = number of periods
And in the second way in Excel: = PAYMENT (rate, years, installation cost) these, the operating costs will be added and the total result will be the one we will use to carry out the exercise.
| INSTALLATION COST | AMORTIZATION | AMORTIZATION | COST OF OPERATION | TOTAL COST |
| 6400000 | - $ 712,489.78 | 712489.78 | 700000 | 1412489.78 |
| 8600000 | - $ 957,408.14 | 957408.14 | 700000 | 1657408.14 |
| 7500000 | - $ 834,948.96 | 834948.96 | 650000 | 1484948.96 |
| 5500000 | - $ 612,295.90 | 612295.9 | 700000 | 1312295.9 |
ILLUSTRATION 15 COST ANALYSIS # 2
- Declaration of decision variables
Xij = where the “i” (plants) will supply “j” (markets) Yi = total cost of each plant
The problem consists of minimizing costs and determining where the new plants will be located, since, with the variables, we will begin to add the unit shipping costs of the computers from each plant with each market, as well as the total costs resulting from each plant for this, the objective function will be proposed using the variables:
! OBJECTIVE FUNCTION
Min = 5 × 11 + 3 × 12 + 2 × 13 + 6 × 14 +
4 × 21 + 7 × 22 + 8 × 23 + 10 × 24 +
6 × 31 + 5 × 32 + 3 × 33 + 8 × 34 +
9 × 41 + 8 × 42 + 6 × 43 + 5 × 44 +
1412489.78 * y1 + 1657488.14 * y2 + 1484948.96 * y3 + 1312295.9 * y4;
Where:
Next, the restrictions will be made where the demand and the capacity of the markets and plants will be taken.
SUBJECT TO:
X11 + X21 + X31 + X41> = 1700; X12 + X22 + X32 + X42> = 1000; X13 + X23 + X33 + X43> = 1500; X14 + X24 + X34 + X44> = 1200, X11 + X12 + X13 + X14 <= 1700 * y1; X21 + X22 + X23 + X24 <= 2000 * y2; X31 + x32 + x33 + x34 <= 1700 * y3; X41 + X42 + X43 + X44 <= 2000 * y4; Y1 + Y2 + Y3 + Y4 = 4
- It is placed> = because the demand can be increased It is placed <= and it is multiplied by the total cost, since it seeks to reduce the cost according to the capacity of each plant. 4 is taken, so that 4 locations are being taken as reference.
| M4 |
Where:
X11 + X21 + X31 + X41
X12 + X22 + X32 + X42
X13 + X23 + X33 + X43
X14 + X24 + X34 + X44
Finally, the binary restriction of Y1, Y2, Y3 and Y4 is applied, this will help us to see if the 4 options are optimal
! BINARY RESTRICTIONS;
@BIN (Y1); @ BIN (Y2); @ BIN (Y3); @ BIN (Y4);
- Once all these points are well defined, we proceed to enter these data in the lingo program, remembering that after each number there is an asterisk that indicates multiplication of the different variables, the modalities that the program marks as the word "Model", put an exclamation point before highlighting any operation and with the ending "End" at the end of the exercise.
It is important to note that once the data is ready to be entered into the lingo program, the following signs must be carefully placed:
* = indicates multiplication
; = indicates closing of a certain amount
! = Highlight the actions to be taken
Model = openness to modeling
End = modeling closure
These signs are representative and mandatory for the program to function properly.
ILLUSTRATION 16 LINGO # 2
INTERPRETATION OF RESULTS
When we take into account the 10-year amortization with a 2% rate, the total cost was modified, this was due to the fact that the computers have an amortization for accounting purposes. This is how I do the table with the cost already including amortization:
|
PLANT |
MONTHLY CAPACITY (COMPUTERS) | COST OF OPERATIONS | INSTALLATION COST |
| P1 | 1700 | 700,000 | 6,400,000 |
| P2 | 2000 | 700,000 | 8,600,000 |
| Q3 | 1700 | 650,000 | 7,500,000 |
| Q4 | 2000 | 700,000 | 5,500,000 |
FIGURE 17 LINGO RESULTS # 2
Below is a table of the variables and their behavior to satisfy the market demand by the plants:
ILLUSTRATION 18 COST ANALYSIS # 2
It is shown that plant 1 supplied market 1 with 700, and market 3 with 1000 of estimated monthly demand. While silver 2 had no movements. But plant 3 supplied market 1 with 200 computers per month, and market 3 with 1500. And finally, plant 4 supplied market 1 with 800 units and market 4 with 1200 units.
- This is how then we can observe the operation and installation costs with the amortization already calculated. The cost of locating the plants will be $ 4,235,135.00 pesos to satisfy the demand and lower the installation and operation costs for the computer units.
Meanwhile, the costs of the plants at the beginning, the sum of the 4 plants is $ 5,867,222.78 pesos with the calculation solved, it does not give a total of the 4 plants of $ 5,845,243.00 pesos, giving a total difference of $ 21,979.78 pesos in the four plants.
FIGURE 19 COST ANALYSIS
EXERCISE 3
A television manufacturer wants to determine the location for assembly plants that will serve the domestic market. At present, almost all of the parts are imported, so the manufacturer has three warehouses located in Mexicali, Toluca and Matamoros respectively.
Below are the transportation costs (for each set of pieces to assemble a television) from the warehouses to the potential plants (Guadalajara, Tijuana, Monterrey and Mexico City are explored), as well as the capacities from monthly production and the Costs (monthly) generated by the operation and installation of said plants.
| PLANT | MONTHLY CAPACITY (TELEVISIONS) | MONTHLY COST (INSTALLATION AND OPERATION COSTS) |
| Guadalajara | 1700 | 140,000 |
| Tijuana | 2000 | 140,000 |
| Monterrey | 1700 | 130000 |
| Mexico City | 2000 | 140,000 |
ILLUSTRATION 20 PLANT ANALYSIS # 3
| TABLE 2 DESTINATION COST OF TRANSPORTATION | |||
| PLANT | MEXICALI | TOLUCA | KILL US |
| Guadalajara | 7 | 4 | 10 |
| Tijuana | 4 | 8 | 6 |
| Monterrey | 7 | 8 | 5 |
| Mexico City | 10 | two | 7 |
| CAPACITY | 700 | 900 | 450 |
ILLUSTRATION 21 PLANT LOCATION ANALYSIS
The manufacturer expects the plants in Mexico to serve four potential markets. The following table shows the unit shipping costs ($ / televisions) from each of the plants to each of the markets, as well as the estimated monthly demand. Formulate the problem.
| TABLE 3 DESTINATION COST OF TRANSPORTATION | ||||
| PLANT | CDMEX | PUEBLA | MERIDA | MONTERREY |
| Guadalajara | 5 | 3 | two | 6 |
| Tijuana | 4 | 7 | 8 | 10 |
| Monterrey | 6 | 5 | 3 | 8 |
| Mexico City | 9 | 8 | 6 | 5 |
| CAPACITY | 1700 | 1000 | 1500 | 1200 |
ILLUSTRATION 22 CAPACITY ANALYSIS # 3
SOLUTION
In the first place, it will begin by solving this problem manually and later it will be attached to the Lingo 14.0 software to evaluate the results.
This model aims to minimize transportation, installation and operating costs to meet demand. Schematically, the transportation problem can be represented as follows:
Declaration of decision variables.
Xij: It is the number of televisions that will be sent from warehouse i (Mexicali, Toluca, Matamoros) to plant j (Guadalajara, Tijuana, Monterrey, DF)
Zij: It is the number of televisions that will be sent from plant i (Guadalajara, Tijuana, Monterrey, DF) to demand center j (Mexico, Puebla, Mérida, Monterrey).
Yi: It is the number of plants that will be built to facilitate the shipment of televisions to the final customer (1, 2, 3,4).
- The next step is to determine the objective function of the exercise (maximize or minimize) according to the respective needs of the case study, for this exercise it is to minimize because the objective of the manufacturers is to reduce costs by choosing the appropriate routes of travel for their respective transport of televisions.
! OBJECTIVE FUNCTION
Min = 7 × 11 + 4 × 12 + 7 × 13 + 10 × 14
+ 4 × 21 + 8 × 22 + 8 × 23 + 2 × 24
+ 10 × 31 + 6 × 32 + 5 × 33 + 7 × 34
+ 5z11 + 3z12 + 2z13 + 6z14
+ 4z21 + 7z22 + 8z23 + 10z24
+ 6z31 + 5z32 + 3z33 + 8z34
+ 9z41 + 8z42 + 6z43 + 5z44
+ 140000y1 + 140000y2 + 130000y3 + 140000y4;
Where:
| MATAMOROS |
X11 + X12 + X13 + X14
X21 + X22 + X23 + X24
X31 + X32 + X33 + X34
| Merida |
Next the part of the z towards the potential markets mentioned in the table.
z11 + z12 + z13 + z14
z21 + z22 + z23 + z24
z31 + z32 + z33 + z34
z41 + z42 + z43 + z44
With the definition of the Y we will place the indicator of the costs of the plants, which are the installation and operating expenses:
The next step is to establish the restrictions set by the information provided (Capacity, Plants and Demand). The importance of establishing these restrictions lies in the fact that the Lingo program marks it as a parameter to produce results no greater or less than what In order to meet the cost reduction objective, the company needs to deliver the right amount of televisions to the customer in a timely manner.
The sum of the different variables must not exceed or fall behind what the company requires, for this reason each of these sums must be subject to a restriction either less than or greater than a certain amount.
SUBJECT TO:
CAPACITY RESTRICTION: X11 + x12 + x13 + x14 = 700; x21 + x22 + x23 + x24 = 900; x31 + x32 + x33 + x34 = 450; z11 + z12 + z13 + z14 = 1700 * y1; z21 + z22 + z23 + z24 = 2000 * y2; z31 + z32 + z33 + z34 = 1700 * y3; z41 + z42 + z43 + z44 = 2000 * y4;
- In the Table, mention the warehouse capacities for sending the televisions in Guadalajara, Tijuana, Monterrey and Mexico City The same happens with the plant capacity mentioned in the table and that shows us its capacity from Guadalajara, Tijuana, Monterrey to Mexico City.
RESTRICTION OF THE DEMAND; z11 + z21 + z31 + z41> = 1700; z12 + z22 + z32 + z42> = 1000; z13 + z23 + z33 + z43> = 1500; z14 + z24 + z34 + z44> = 1200;
- The table shows the demand of the potential markets that will offer Guadalajara, Tijuana, Monterrey to Mexico City, this may be greater than those destined
PLANT RESTRICTION:
y1 + y2 + y3 + y4 <= 4;
- The location of the plants from Guadalajara, Tijuana, Monterrey to Mexico City is determined, which can be less than 4 as shown in the following equation
Once all these points are well defined, we proceed to enter these data in the lingo program, remembering that after each number there is an asterisk that indicates the multiplication of the different variables, the modalities that the program marks as the word "Model", put an exclamation point before highlighting any operation and with the ending "end" at the end of the exercise.
ILLUSTRATION 24 LINGO # 3
INTERPRETATION OF RESULTS
The following table shows the variables and the behavior they suffered to meet the demand:
FIGURE 25 COST ANALYSIS # 3
The cost of locating the plants will be $ 439,950.00 pesos to meet demand and lower transportation and operating costs for television parts
Meanwhile the costs of the plants at the beginning, the sum of the 4 plants is $ 550,000.00 pesos with the calculation solved does not give a total of the 4 plants of $ 529,000.00 pesos giving a total difference of $ 20,200.00 pesos in the four plants of Guadalajara, Tijuana, Monterrey and Mexico City
| COST ANALYSIS | |||||
| BEFORE | PLANTS | VARIABLE |
REDUCED |
DIFFERENCE |
|
|
$ 140,000 |
GUADALAJARA |
Y1 |
one |
$ 133,200 |
$ 6,800 |
|
$ 140,000 |
TIJUANA |
Y2 |
one |
$ 140,000 |
- |
|
$ 130,000 |
MONTERREY |
Y3 |
one |
$ 126,600 |
$ 3,400 |
|
$ 140,000 |
CDMX |
Y4 |
0 |
$ 130,000 |
$ 10,000 |
|
$ 550,000 |
TOTALS |
$ 529,800 |
$ 20,200 |
ILLUSTRATION 26 COST REDUCTION # 3
EXERCISE # 4
The Indument Group wishes to locate one or two plants for the production of Refractories from magnesite, having determined San Luis Potosí and Zacatecas as possible locations for the plants. The raw material comes from two deposits located in Oaxaca and Zacatecas. The final product (refractory bricks) is demanded in plants located in Torreón and Monterrey, in addition, it can be exported through the ports of the raw material (in thousands of $ per shipment of 50 tons), as well as the maximum annual offers (in tons), from the reservoirs to the possible locations, for the plants are summarized in the following table.
| ORIGIN | DESTINATION
SAN LUIS P. |
ZACATECAS | OFFER |
| OAXACA | 6 | 7 | 8000 |
| ZACATECAS | two | 0 | 13000 |
ILLUSTRATION 27 PLANT ANALYSIS # 4
The following table shows the transportation costs (in thousands of $ per shipment of 50 tons) of the final product to the demand centers, as well as the estimated annual demand.
|
ORIGIN |
DESTINATION KEEP |
MONTERREY |
SALINAS CRUZ |
TAMPICO |
| SAN LUIS P. | two | 3 | 9 | 4 |
| ZACATECAS | two | 5 | 8 | 5 |
| DEMAND (TON) | 1500 | 2000 | 1500 | 2000 |
ILLUSTRATION 28 PLANT ANALYSIS # 4
The company wants to explore the possibilities of setting up a large company, or two medium-sized ones in the selected locations, so it calculates the installation and operation costs under the scenarios shown below.
| SCENARIOS | CAPACITY
(TON / YEAR) |
COST OF
INSTALLATION (THOUSANDS $) |
COST OF
OPERATION (THOUSANDS $ / YEAR) |
| SAN LUIS P. (1) | 8000 | 10,000 | 800 |
| SAN LUIS P. (2) | 4000 | 6,000 | 500 |
| ZACATECAS (3) | 8000 | 12,000 | 900 |
| ZACATECAS (4) | 4000 | 6,500 | 550 |
ILLUSTRATION 29 CAPACITIES AND COSTS # 4
- Calculate the Annual Amortization of the installation costs for each of the proposed scenarios, consider an interest rate of 20% per year and 15 years of investment useful life. Formulate the linear programming problem (with integer variables) to explore if it is convenient one or two plants, and a distribution policy that minimizes installation, operation and transportation costs (consider inputs and finished product), knowing that one ton of magnesite produces 0.95 ton of finished product (on average). The formulation should include: definition of decision variables, objective function, restrictions and ranges of existence Solve the linear programming problem using the Solver of your choice
SOLUTION
In the first place, it will begin by solving this problem manually and later it will be attached to the Lingo 14.0 software to evaluate the results.
- This model aims to minimize the monthly costs of transportation, installation and operation. Schematically, the transportation problem can be represented as follows:
ILLUSTRATION 30 NETWORK MODEL # 4
- Declaration of decision variables that the company wishes to take in reference to the number of doors and windows of each type to be produced in the period considered.
X1 = quantity of inputs from the Oaxaca supplier.
X2 = quantity of inputs from the Zacatecas supplier
- The following determines the availability of resources for the amount of inputs used in total production, it cannot exceed the maximum amount that the company can acquire:
(1) San Luis Potosí maximum quantity is 8000/50 (ton) = 160
(2) San Luis Potosí maximum quantity is 4000/50 (ton) = 80
(3) Zacatecas maximum quantity is 8000/50 (ton) = 160
(4) Zacatecas maximum quantity is 8000/50 (ton) = 80
- Now the objective function will be determined
! OBJECTIVE FUNCTION
Min = 6 × 11 + 6 × 12 + 7 × 13 + 7 × 14 + 2 × 21 + 2 × 22
+ 2z11 + 2z12 + 2z13 + 2z14
+ 3z21 + 3z22 + 5z23 + 5z24
+ 9z31 + 9z32 + 8z33 + 8z34
+ 4z41 + 4z42 + 5z43 + 5z44
+ 975.63y1 + 605.38y2 + 1110.76y3 + 664.16y4
Where:
TOTAL COST AMORTIZED
Y1 + y2 + y3 + y4
Once the objective function is obtained, let us analyze the restrictions of each section, as well as supply, capacity, demand, and demand for input, obtaining it in the following way:
SUBJECT TO:
! OFFER RESTRICTION
Oaxaca
X11 + X12 + X13 + X14 <= 160 Zacatecas
X21 + X22 + X23 + X24 <= 260
! CAPACITY RESTRICTION
Z11 + Z12 + Z13 + Z14> = 30 Torreón
Z21 + Z22 + Z23 + Z24> = 40 Monterrey Z31 + Z32 + Z33 + Z34> = 30 Salinas
z41 + Z42 + Z43 + Z44> = 40 Tampico
! DEMAND RESTRICTION
Z11 + Z21 + Z31 + Z41 <= 160 * Y1 San Luis
Z12 + Z22 + Z32 + Z42 <= 80 * Y2 San Luis
Z13 + Z23 + Z33 + Z43 <= 160 * Y3 Zacatecas
Z14 + Z24 + Z34 + Z44 <= 80 * Y4 Zacatecas
! SUPPLY DEMAND RESTRICTION
0.95 (Z11 + Z12 + Z13 + Z14) = X11 + X21 0.95 (Z21 + Z22 + Z23 + Z24) = X12 + X22 0.95 (Z31 + Z32 + Z33 + Z34) = X13 + X23
0.95 (Z41 + Z42 + Z43 + Z44) = X14 + X24
! PLANT RESTRICTION
Y1 + Y2 <= 2;
saint Louis
Zacatecas
Y3 + Y4 <= 2;
- Bringing the data into lingo
ILLUSTRATION 34 31 LINGO # 4
INTERPRETATION OF RESULTS
Below is the table of variables and their behavior to satisfy market demand by plants as well as suppliers:
Suppliers will satisfy the plants 133 tons with the supplier from Zacatecas, which in turn will give 28.5 tons to the San Luis Potosí 1 plant, 38 tons to the San Luis Potosí 2 plant, 28.5 tons to the Zacatecas 1 plant and finally 38 tons to the Zacatecas plant 2.
Meanwhile, the plants will satisfy the demand from the CEDIS: from Torreón with 30 from the San Luis Potosí 1 plant, 40 from the San Luis Potosí 2 plant, 30 from the Zacatecas 1 plant and 40 from the Zacatecas 2 plant.
This is how then we can observe the operation and installation costs with the amortization already calculated
ILLUSTRATION 33 COST DIFFERENCE # 4
The cost of locating the plants will be $ 1,718.63 pesos to meet demand and lower installation and operation costs for the tons shipped.
Meanwhile, the costs of the plants at the beginning, the sum of the 4 plants is $ 3,355.93 pesos, with the calculation resolved, it does not give a total of the 4 plants of $ 3,115.93 pesos, giving a total difference of $ 240.00 pesos in the four plants.
EXERCISE # 5
Aceros Industriales successfully ventured into the production of special steels for industry. At this moment, it has a single plant in Mexico City with a production capacity of 5000 tons per year.Due to an increase in demand and transportation costs, it is studying the possibility of opening new plants and designing a new distribution system, in particular, the towns of Puebla, Monterrey and Zacatecas are being considered as candidates to locate new plants. The following table shows the capacities (in tons), the transportation costs (in thousands of $ for each shipment of 50 tons), and the demands (in tons) for the following year
DESTINATION (COST OF TRANSPORTATION)
|
ORIGIN |
CDMX |
GUADALAJARA |
SAINT
LUIS |
PUEBLA |
SONORA |
CAPABILITIES |
| CDMX | 0 | 8 | 4 | one | 10 | 5000 |
| PUEBLA | one | 9 | 3 | 0 | eleven | 12000 |
| ZACATECAS | 6 | 4 | one | 7 | 3 | 12000 |
| MONTERREY | 7 | 12 | 5 | 8 | 4 | 12000 |
| DEMAND | 2000 | 5000 | 4000 | 6000 | 4000 |
ILLUSTRATION 34 TRANSPORTATION COSTS # 5
In addition to the costs of transporting the final product, the company must consider the costs of installation, operation and transport of inputs (mainly scrap that must be transported from the ports of Manzanillo and Tampico). The following table shows the transportation costs (in thousands of $ per 50 ton shipment), the installation cost (one-time outlay in thousands of $) and the operating costs (in thousands of $ per year)
DESTINATION (COST OF TRANSPORTATION)
| ORIGIN | CDMX | PUEBLA | ZACATECAS | MONTERREY | MAXIMUM OFFER (TON PER YEAR) |
| MANZANILLO | 6 | 7 | 5 | 8 | 15000 |
| TAMPICO | 4 | 5 | 4 | one | 20000 |
| INST COST | 24000 | 21000 | 28000 | ||
| OPERATION COST | 12000 | 9000 | 11000 |
ILLUSTRATION 35 TRANSPORTATION COSTS OF PLANTS # 5
- Calculate the annual amortization of the installation cost for each of the candidate locations to locate a plant, consider an interest rate of 15% per year and 10 years of useful life of the investment Formulate the programming problem to find the most convenient locations of the plants and distribution policy that simultaneously minimize installation, operation and transportation costs (inputs and terminal product) if a ton of scrap produces 0.95 ton of terminal product on average Solve the problem with the solver of your choice
SOLUTION
In the first place, it will begin by solving this problem manually and later it will be attached to the Lingo 14.0 software to evaluate the results.
This model aims to minimize the monthly costs of transportation, installation and operation. Schematically, the transportation problem can be represented as follows:
Declaration of decision variables
X (X11, X12 + X13, X14, X21, X22, X23, X24 ) 2 plants in Manzanillo and Tampico that will supply 4 cities CDMX, Puebla, Zacatecas and Monterrey (the new opening of these is desired)
Z that correspond to the 5 cities where the finished product is supplied (CDMX, Guadalajara, San Luis, Puebla, Sonora), it should be noted that 3 variables Y1, Y2, Y3 are also applied in which the annual amortizations of the 3 plants to be opened (Puebla, Zacatecas, Monterrey) of which are obtained as follows
- Since it asks for amortization, we will have the support of the Excel program where the following formula will be applied:
= PAYMENT (%); (PERIOD); (OPERATION COST) + COST OF
INSTALLATION
- Now we will proceed to obtain the objective function of the variables of X that are the Manzanillo and Tampico plants, but this time their order is according to the transportation cost of the 4 supply plants CDMX, Puebla, Zacatecas and Monterrey
OBJECTIVE FUNCTION;
Min = 6 * X11 + 7 * X12 + 5 * X13 + 8 * X14
+ 4 * X21 + 5 * X22 + 4 * X23 + 1 * X24
+ 8 * Z12 + 4 * Z13 + 1 * Z14 + 10 * Z15
+ 1 * Z21 + 9 * Z22 + 3 * Z23 + 11 * Z25
+ 6 * Z31 + 4 * Z32 + 1 * Z33 + 7 * Z34 + 3 * Z35
+ 7 * Z41 + 12 * Z42 + 5 * Z43 + 8 * Z44 + 4 * Z45
+ 16782 * Y1 + 13184 * Y2 + 16574 * Y3;
Where:
| Monterrey |
| TOTAL COST |
y1 + y2 + y3
The total cost of amortization is already direct
After the application of the objective function, the restrictions will be carried out, for this, a new data must be added that is 0.95 (it is what produces a ton of finished product scrap, this will be applied to the variables of Z (CDMX, Guadalajara, San Luis, Puebla and Sonora)
The application is as follows in which it seeks to obtain the demand restriction
! DEMAND RESTRICTION;
X11 + X21 = 0.95 * Z11 + 0.95 * Z12 + 0.95 * Z13 + 0.95 * Z14 + 0.95 * Z15;
X12 + X22 = 0.95 * Z21 + 0.95 * Z22 + 0.95 * Z23 + 0.95 * Z24 + 0.95 * Z25;
X13 + X23 = 0.95 * Z31 + 0.95 * Z32 + 0.95 * Z33 + 0.95 * Z34 + 0.95 * Z35;
X14 + X24 = 0.95 * Z41 + 0.95 * Z42 + 0.95 * Z43 + 0.95 * Z44 + 0.95 * Z45;
As the following process, the demand of the 5 destinations is applied, this is obtained as follows
DEMAND / 50 = (TRANSPORTATION COST IN $ PER SHIPPING OF 50 TON)
Example
CDMX
2000/50 = 40
For the application of product demand it is as follows;
! PRODUCT DEMAND RESTRICTION;
Z11 + Z21 + Z31 + Z41> = 40; > = It is applied to see which of the 5 Z12 + Z22 + Z32 + Z42> = 100; destinations will have greater demand if Z13 + Z23 + Z33 + Z43> = 80; the plants of
Z14 + Z24 + Z34 + Z44> = 120; Puebla Monterrey and Zacatecas
Z15 + Z25 + Z35 + Z45> = 80;
In this process, the X variables of the 2 plants in Manzanillo and Tampico are added, but this time consecutively
Now a process is applied where the maximum supply of tons of the 2 plants Manzanillo and Tampico is sought
MAX OFFER / 50 = (TRANSPORTATION COST IN $ PER SHIPPING OF 50 TON)
Example =
15000/50 = 300
The application is as follows
! SCRAP OFFERING RESTRICTION;
X11 + X12 + X13 + X14 <= 300; destinations will have a greater offer if they reach
X21 + X22 + X23 + X24 <= 400; supply the Puebla plants
Monterrey and Zacatecas
- In this process, the Z variables of the CDMX, Puebla, Zacatecas and Monterrey plants that supply 5 CDMX cities, Guadalajara, San Luis Puebla and Sonora are added.
Z11 + Z12 + Z13 + Z14 + Z15
Subsequently, the capacity is obtained as follows
Example =
CAPACITY / 50 = (TRANSPORTATION COST IN $ PER SHIPPING OF 50 TON)
- It should be noted that the variables Y1, Y2, Y3 are applied to the 3 cities that want to open the new plants
Z21 + Z22 + Z23 + Z24 + Z25 = 240 * y1;
Z31 + Z32 + Z33 + Z34 + Z35 = 240 * y2;
Z41 + Z42 + Z43 + Z44 + Z45 = 240 * y3;
The application is as follows in which the capacity is searched
! CAPACITY RESTRICTION;
Z11 + Z12 + Z13 + Z14 + Z15 = 100;
Z21 + Z22 + Z23 + Z24 + Z25 = 240 * y1;
Z31 + Z32 + Z33 + Z34 + Z35 = 240 * y2;
Z41 + Z42 + Z43 + Z44 + Z45 = 240 * y3;
Y1 + Y2 + Y3 <= 3;
Finally, the binary restriction of Y1, Y2, Y3 is applied, this will help us to see if the 3 options are optimal
! BINARY RESTRICTIONS;
@BIN (Y1); @ BIN (Y2); @ BIN (Y3);
VIII. Interpreting it in LINGO as follows:
ILLUSTRATION 37 LINGO # 5
INTERPRETATION OF RESULTS
Below is the table of variables and their behavior to satisfy market demand by plants as well as suppliers:
| CALCULATION OF VARIABLES | ||||
| OF | TO | VARIABLE | VALUE | REDUCED COST |
| Manzanillo | CDMX | X11 | 0.000000 | 1,000,000 |
| Manzanillo | Puebla | X12 | 0.000000 | 2,789,474 |
| Manzanillo | Zacatecas | X13 | 1,510,000 | 0.000000 |
| Manzanillo | Monterrey | X14 | 0.000000 | 6,000,000 |
| Tampico | CDMX | X21 | 9,500,000 | 0.000000 |
| Tampico | Puebla | X22 | 0.000000 | 1,789,474 |
| Tampico | Zacatecas | X23 | 7,700,000 | 0.000000 |
| Tampico | Monterrey | X24 | 2,280,000 | 0.000000 |
| CDMX | Guadalajara | Z12 | 0.000000 | 8,000,000 |
| CDMX | saint Louis | Z13 | 0.000000 | 7,000,000 |
| CDMX | Puebla | Z14 | 1,000,000 | 0.000000 |
| CDMX | Sonora | Z15 | 0.000000 | 1,300,000 |
| Puebla | CDMX | Z21 | 0.000000 | 2,000,000 |
| Puebla | Guadalajara | Z22 | 0.000000 | 1,000,000 |
| Puebla | saint Louis | Z23 | 0.000000 | 7,000,000 |
| Puebla | Sonora | Z25 | 0.000000 | 1,500,000 |
| Zacatecas | CDMX | Z31 | 0.000000 | 2,000,000 |
| Zacatecas | Guadalajara | Z32 | 1,000,000 | 0.000000 |
| Zacatecas | saint Louis | Z33 | 1,400,000 | 0.000000 |
| Zacatecas | Puebla | Z34 | 0.000000 | 2,000,000 |
| Zacatecas | Sonora | Z35 | 0.000000 | 2,000,000 |
| Monterrey | CDMX | Z41 | 4,000,000 | 0.000000 |
| Monterrey | Guadalajara | Z42 | 0.000000 | 5,000,000 |
| Monterrey | saint Louis | Z43 | 0.000000 | 1,000,000 |
| Monterrey | Puebla | Z44 | 2,000,000 | 0.000000 |
| Monterrey | Sonora | Z45 | 1,800,000 | 0.000000 |
ILLUSTRATION 38 ANALYSIS OF RESULTS # 5
The cost of locating the plants will be $ 33,229.00 pesos to satisfy the demand and lower the costs of transportation and operation of the steels
According to the results obtained, the Zacatecas and Monterrey plants would be the plants that would benefit the most if the transportation costs of the Manzanillo and Tampico plants were opened with the amortization made (16782 Y1 13184 Y2 16574 Y3) and with the reduction in costs, the following results
| COST CONCLUSION | |||
| PLANT | VARIABLE | VALUE | REDUCED |
| CDMX | |||
| PUEBLA | Y1 | 0.000000 | 16782.00 |
| ZACATECAS | Y2 | 1,000,000 | 14564.00 |
| MONTERREY | Y3 | 1,000,000 | 17990.00 |
ILLUSTRATION 39 COST REDUCTION # 5
CONCLUSION:
Currently LOGISTICS is in charge of the efficient distribution of the products of a certain company with a lower cost and excellent customer service, that is why in companies it has taken a very important role as well as the means and programs to be able to define concisely the variables that intervene in the distribution systems to generate the most viable routes and make our service more efficient and optimal.
Logistics revolves around creating value: value for the company's customers and suppliers, and value for the company's shareholders. Value in logistics is primarily expressed in terms of time and place. Products and services have no value unless they are in the possession of customers when (time) and where (place) they want to consume them. A good logistics management views each activity in the supply chain as a contribution to the process of adding value. If only little value can be added to it, then it may be questioned whether such activity should exist. However, value is added when customers prefer to pay more for a product or service than it costs to put it in their hands. (BALLOU, 2004, p. 13)
Markets have become more demanding, companies have to compete with others at a regional, national or global level and must serve each and every one of their clients in the best way, in addition, the appearance of new information technologies has brought as As a consequence, shorter transaction times and costs have forced companies to take logistics management more seriously if they want to remain competitive.
Companies have also continuously dealt with movement and storage (transport-inventory) activities. The novelty of this field lies in the concept of coordinated management of related activities, instead of the historical practice of managing them separately, in addition to the concept that logistics adds value to essential products or services for customer satisfaction and for sales (BALLOU, 2004, p. 3)
Selecting an appropriate supply chain and logistics strategy requires some of the same creative process necessary to develop an appropriate corporate strategy. Innovative approaches to logistics and supply chain strategy can represent a competitive advantage (BALLOU, 2004, p. 35)
That is the use case of the LINGO programming solver, it is a complete tool designed to make linear, non-linear (convex and non-convex / Global), quadratic, restricted quadratic programming models (to name a few), and in turn it can give us a better solution to problems in terms of maximizing or minimizing resources with a powerful language for expressing optimization models, a full-featured environment for build and edit problems, and a set of built-in fast solvers.
On the other hand, once LINGO has created a document with data about a model, we can modify it to our liking, deleting any information that does not interest us, adding any kind of explanations, titles, comments, etc., On the contrary, it that we write in a document must be correct in the LINGO language, otherwise when trying to solve the problem we will get error messages instead of the desired solution.
The exercises previously developed were elaborated through the use of this programming solver, the use of variables and the importance of each of them were correctly understood, achieving the correct model in conjunction with the restrictions that mark the wording of each one for that once entering data the program the solver could run and throw the optimal results with the corresponding costs.
Knowing the interpretation of the results will help us optimize the distribution system and the coherence between its different nodes, specifically in the transport activity, where a procedure is proposed that favors the design of network systems that contributes to improving the effectiveness of physical distribution, making it possible to offer a high level of balanced service with the lowest possible cost, all of which contributes to the fulfillment of business objectives and the achievement of customer satisfaction, essential elements, fundamentally, for the maintenance and growth of the companies in general.
BIBLIOGRAPHY
BALLOU, RH (2004). LOGISTICS, SUPPLY CHAIN ADMINISTRATION, FIFTH EDITION. MEXICO: PEARSON, EDUCATION.
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