Docstoc

Distribution

Document Sample
Distribution Powered By Docstoc
					Distribution/Logistics Examples
On each example worksheet, read the comments at the bottom of the sheet, then
click Tools Solver... to examine the decision variables, constraints, and objective.
To find the optimal solution, click the Solve button.
An important group of Solver applications is based on distribution or network models.
The amount of money that companies save each year by applying linear programming
towards their distribution problems is enormous.

In this series we will look at a simple transportation problem in worksheet Transport1, then
extend it to a 2-level multi-product model in worksheets Transport2 and Transport3.

We'll also examine a frequently encountered class of problems called 'knapsack'
problems, in worksheet Knapsack. As an example we will look at a truck that has to
transport different kinds of gas.

In the Facility worksheet we will look at a facility location problem, where a company has
to decide if it's profitable to close down one or more of their plants and save overall costs.

Finally, in the Prodtran worksheet, we will examine a combination production and
transportation model where the number of products made in plants depends on choices
made in distribution. This kind of combination is often possible, but many users prefer to
split these models up into smaller ones to simplify the problem.
                                           6c23297b-696d-4da5-b26b-dda7bd16bdc1.xls


Transportation Problem 1
Minimize the costs of shipping goods from factories to customers, while not exceeding
the supply available from each factory and meeting the demand of each customer.

Cost of shipping ($ per product)
                                          Destinations
                  Customer 1      Customer 2       Customer 3      Customer 4     Customer 5
Factory 1           $1.75           $2.25           $1.50            $2.00          $1.50
Factory 2           $2.00           $2.50           $2.50            $1.50          $1.00
Number of products shipped
                  Customer 1      Customer 2       Customer 3      Customer 4     Customer 5         Total     Capacity
Factory 1             0               0               0               0               0               0        60,000
Factory 2             0               0               0               0               0               0        60,000
Total                 0               0               0               0               0
Demand              30,000          23,000          15,000          32,000          16,000
Total cost of shipping                      $0



Problem
A company wants to minimize the cost of shipping a product from 2 different factories to 5 different customers.
Each factory has a limited supply and each customer a certain demand. How should the company distribute the
product?


Solution
1) The variables are the number of products to ship from each factory to the customers. These are given the
name Products_shipped in worksheet Transport1.
2) The logical constraint is
                Products_shipped >= 0 via the Assume Non-Negative option
The other two constraints are
                Total_received >= Demand
                Total_shipped <= Capacity
3) The objective is to minimize cost. This is given the name Total_cost.


Remarks
This is a transportation problem in its simplest form. Still, this type of model is widely used to save many
housands of dollars each year.
In worksheet Transport2 we will consider a 2-level transportation, and in worksheet Transport3 we expand this to
a multi-product, 2-level transportation problem.




                                                                    Page 2
                                          6c23297b-696d-4da5-b26b-dda7bd16bdc1.xls


Transportation Problem 2 (2-stage-transport)
Minimize the costs of shipping goods from factories to warehouses and customers, and
warehouses to customers, while not exceeding the supply available from each factory or
the capacity of each warehouse, and meeting the demand from each customer.

Cost of shipping ($ per product)
                                          Destinations
                Warehouse 1     Warehouse 2      Warehouse 3    Warehouse 4
Factory 1          $0.50           $0.50            $1.00          $0.20
Factory 2          $1.50           $0.30            $0.50          $0.20

                 Customer 1      Customer 2       Customer 3     Customer 4     Customer 5
Factory 1          $1.75           $2.50            $1.50          $2.00          $1.50
Factory 2          $2.00           $2.50            $2.50          $1.50          $1.00

                 Customer 1      Customer 2       Customer 3     Customer 4     Customer 5
Warehouse 1        $1.50           $1.50            $0.50          $1.50          $3.00
Warehouse 2        $1.00           $0.50            $0.50          $1.00          $0.50
Warehouse 3        $1.00           $1.50            $2.00          $2.00          $0.50
Warehouse 4        $2.50           $1.50            $0.20          $1.50          $0.50
Number of products shipped
                Warehouse 1     Warehouse 2      Warehouse 3    Warehouse 4         Total
Factory 1           0              20,000            0             15,000         35,000
Factory 2         45,000             0             11,000            0            56,000
Total             45,000           20,000          11,000          15,000
Capacity          45,000           20,000          30,000          15,000

                 Customer 1      Customer 2       Customer 3     Customer 4     Customer 5       Total
Factory 1         10,000              0                  0         15,000             0        25,000     Factory
Factory 2           0                 0                  0           0                0          0        Capacity
                                                Total products shipped out of factory 1        60,000     60,000
                                                Total products shipped out of factory 2        56,000     60,000

               Customer 1      Customer 2       Customer 3     Customer 4      Customer 5    Total
Warehouse 1         0              23,000            0             17,000          5,000       45,000
Warehouse 2       20,000             0               0               0               0         20,000
Warehouse 3         0                0               0               0            11,000       11,000
Warehouse 4         0                0             15,000            0               0         15,000
Total             30,000           23,000          15,000          32,000         16,000
Demands           30,000           23,000          15,000          32,000         16,000
Total cost of shipping           $237,000



Problem
A company has 2 factories, 4 warehouses and 5 customers. It wants to minimize the cost of shipping its
product from the factories to the warehouses, the factories to the customers, and the warehouses to the
customers. The number of products received by a warehouse from the factory should be the same as the
number of products leaving the warehouse to the customers. How should the company distribute the products?


Solution




                                                                  Page 3
                                          6c23297b-696d-4da5-b26b-dda7bd16bdc1.xls


1) The variables are the number of products to ship from the factories to the warehouses, the factories to the
customers, and the warehouses to the customers. These are defined in worksheet Transport2 as
Factory_to_warehouse, Factory_to_customer, Warehouse_customer.
2) The logical constraints are all defined via the Assume Non-Negative option:
                Factory_to_warehouse >= 0
                Factory_to_customer >= 0
                Warehouse_customer >= 0
The other constraints are
                Total_from_factory <= Factory_capacity
                Total_to_customer >= Demand
                Total_to_warehouse <= Warehouse_capacity
                Total_to_warehouse = Total_from_warehouse
3) The objective is to minimize cost, given by Total_cost.


Remarks
Please note that the last constraint must be an '=' , because otherwise products would start piling up at the
warehouse. It would be possible to make this a multi-period model where storage at the warehouses would be
possible and even desired, if transportation prices would fluctuate during the different time periods. In
worksheet Transport3 we will look at a multi-product situation.




                                                                   Page 4
                                        6c23297b-696d-4da5-b26b-dda7bd16bdc1.xls


Transportation Problem 3 (2-stage-transport, multi-commodity)
Minimize the costs of shipping 3 different goods from factories to warehouses and customers, and
warehouses to customers, while not exceeding the supply available from each factory or
the capacity of each warehouse, and meeting the demand from each customer.

Cost of shipping ($ per product)
                                                     Destinations
                              Warehouse 1    Warehouse 2     Warehouse 3    Warehouse 4
Factory 1      Product 1        $0.50           $0.50          $1.00           $0.20
               Product 2        $1.00           $0.75          $1.25           $1.25
               Product 3        $0.75           $1.25          $1.00           $0.80
Factory 2      Product 1        $1.50           $0.30          $0.50           $0.20
               Product 2        $1.25           $0.80          $1.00           $0.75
               Product 3        $1.40           $0.90          $0.95           $1.10

                               Customer 1     Customer 2     Customer 3      Customer 4     Customer 5
Factory 1      Product 1        $2.75           $3.50          $2.50           $3.00         $2.50
               Product 2        $2.50           $3.00          $2.00           $2.75         $2.60
               Product 3        $2.90           $3.00          $2.25           $2.80         $2.35
Factory 2      Product 1        $3.00           $3.50          $3.50           $2.50         $2.00
               Product 2        $2.25           $2.95          $2.20           $2.50         $2.10
               Product 3        $2.45           $2.75          $2.35           $2.85         $2.45

                               Customer 1     Customer 2     Customer 3      Customer 4     Customer 5
Warehouse 1    Product 1        $1.50           $0.80          $0.50           $1.50         $3.00
               Product 2        $1.00           $0.90          $1.20           $1.30         $2.10
               Product 3        $1.25           $0.70          $1.10           $0.80         $1.60
Warehouse 2    Product 1        $1.00           $0.50          $0.50           $1.00         $0.50
               Product 2        $1.25           $1.00          $1.00           $0.90         $1.50
               Product 3        $1.10           $1.10          $0.90           $1.40         $1.75
Warehouse 3    Product 1        $1.00           $1.50          $2.00           $2.00         $0.50
               Product 2        $0.90           $1.35          $1.45           $1.80         $1.00
               Product 3        $1.25           $1.20          $1.75           $1.70         $0.85
Warehouse 4    Product 1        $2.50           $1.50          $0.60           $1.50         $0.50
               Product 2        $1.75           $1.30          $0.70           $1.25         $1.10
               Product 3        $1.50           $1.10          $1.50           $1.10         $0.90
Number of products shipped
                              Warehouse 1    Warehouse 2     Warehouse 3    Warehouse 4        Total
Factory 1      Product 1          0               0              0              0                  0
               Product 2          0               0              0              0                  0
               Product 3          0               0              0              0                  0
Factory 2      Product 1          0               0              0              0                  0
               Product 2          0               0              0              0                  0
               Product 3          0               0              0              0                  0
Total          Product 1          0               0              0              0
               Product 2          0               0              0              0
               Product 3          0               0              0              0
Capacity       Product 1        35,000          20,000         30,000         15,000
               Product 2        30,000          25,000         15,000         24,000
               Product 3        20,000          20,000         25,000         20,000




                                                                Page 5
                                           6c23297b-696d-4da5-b26b-dda7bd16bdc1.xls


                                   Customer 1   Customer 2      Customer 3     Customer 4      Customer 5         Total
Factory 1       Product 1              0             0               0                0             0               0
                Product 2              0             0               0                0             0               0
                Product 3              0             0               0                0             0               0
Factory 2       Product 1              0             0               0                0             0               0
                Product 2              0             0               0                0             0               0
                Product 3              0             0               0                0             0               0

                                                                                                                 Capacity
                               Total products shipped out of factory 1        Product 1             0             90,000
                                                                              Product 2             0            100,000
                                                                              Product 3             0             80,000
                               Total products shipped out of factory 2        Product 1             0             75,000
                                                                              Product 2             0             65,000
                                                                              Product 3             0             90,000


                                   Customer 1   Customer 2      Customer 3     Customer 4      Customer 5         Total
Warehouse 1     Product 1             0             0               0              0              0                 0       0
                Product 2             0             0               0              0              0                 0       0
                Product 3             0             0               0              0              0                 0       0
Warehouse 2     Product 1             0             0               0              0              0                 0       0
                Product 2             0             0               0              0              0                 0       0
                Product 3             0             0               0              0              0                 0       0
Warehouse 3     Product 1             0             0               0              0              0                 0       0
                Product 2             0             0               0              0              0                 0       0
                Product 3             0             0               0              0              0                 0       0
Warehouse 4     Product 1             0             0               0              0              0                 0       0
                Product 2             0             0               0              0              0                 0       0
                Product 3             0             0               0              0              0                 0       0
Total           Product 1             0             0               0              0              0
                Product 2             0             0               0              0              0
                Product 3             0             0               0              0              0
Demands         Product 1           30,000        23,000          15,000         32,000         16,000
                Product 2           20,000        15,000          22,000         12,000         18,000
                Product 3           25,000        22,000          16,000         20,000         25,000

Total cost of shipping                     $0



Problem
This model builds on model Transport2. Again, a company wants to minimize cost of shipping, but this time
there are 3 products to distribute. How should the company distribute the products?


Solution
The solution to the problem is identical to the one in Transport2. Notice that we have used the 'Insert Name
Define' command to extend the model to a multiproduct problem. This way the variables and constraints are
still the same as in Transport2.


Remarks
Notice that this model delivers the same result as three separate models for the three products. There will be




                                                                   Page 6
                                          6c23297b-696d-4da5-b26b-dda7bd16bdc1.xls


times however, that there are constraints that apply to more than one product. In that case it would not be
desirable to have three different models and maybe even impossible. For an extension of this model, where the
number of products made in the factories depends on the demand and distribution rather than being constant,
see the worksheet Prodtran in this workbook.




                                                                  Page 7
                                             6c23297b-696d-4da5-b26b-dda7bd16bdc1.xls


Partial Loading (Knapsack Problem)
A fuel truck with 4 compartments needs to supply 3 different types of gas to a customer.
When demand is not filled, the company loses $0.25 per gallon that is not delivered.
How should the truck be loaded to minimize loss?

Truck Specifications
                      Comp. 1        Comp. 2       Comp. 3        Comp. 4
Size (gallons)          1200             800         1300           700
Loading of Compartments (1=yes, 0=no)
                      Comp. 1        Comp. 2       Comp. 3        Comp. 4
Gas 1                     0              0             0             0
Gas 2                     0              0             0             0
Gas 3                     0              0             0             0
Total                     0              0             0             0
Amount (gallons)
                      Comp. 1        Comp. 2       Comp. 3        Comp. 4         Total        Demand           Loss
Gas 1                     0              0             0             0              0           1800         $450.00
Gas 2                     0              0             0             0              0           1500         $375.00
Gas 3                     0              0             0             0              0           1000         $250.00
                                                                                             Total Loss     ########
Maximum Amount (gallons)
                      Comp. 1        Comp. 2       Comp. 3        Comp. 4
Gas 1                     0              0             0             0
Gas 2                     0              0             0             0
Gas 3                     0              0             0             0



Problem
A fuel truck needs to supply 3 different kinds of gas to a customer. When demand is not filled the company
loses $0.25 per gallon that is not delivered. The truck has 4 separate compartments of different size. How
should the truck be loaded to minimize loss?


Solution
1) The variables are the decisions to fill the compartments for each type of gas, and the amounts to be put in
if the compartment is filled. In worksheet Knapsack, these are given the name Gallons_loaded and
Loading_decisions.
2) The logical constraints are
                   Gallons_loaded >= 0 via the Assume Non-Negative option
                   Loading_decisions = binary
Since there can only be one kind of gas in any compartment we have
                   Total_decisions <= 1
The size limitations of the truck give
                   Gallons_loaded <= Maximum_gallons
We don't want to load more than needed. This gives
                   Total_gallons <= Demand
3) The objective is to minimize the loss. This is given the name Total_loss.


Remarks
It is often possible to have different objectives in these types of problems. We might, for instance, want to
minimize the wasted space in the truck in this example. Knapsack problems are characterized by a series of
0-1 integer variables with a single capacity constraint. If someone goes camping and his backpack can hold




                                                                    Page 8
                                         6c23297b-696d-4da5-b26b-dda7bd16bdc1.xls


only a certain amount of weight, what items should the camper bring? He should try to optimize the value
of the items while not exceeding the weight allowed by the backpack. There is a wide set of problems that
fall into this category.




                                                                 Page 9
                                            6c23297b-696d-4da5-b26b-dda7bd16bdc1.xls


Facility Location
A company currently ships its product from 5 plants to 4 warehouses. It is considering closing
one or more plants to reduce cost. What plant(s) should the company close, in order to
minimize transportation and fixed costs?

Transportation Costs (per 1000 products)
                     Plant 1        Plant 2        Plant 3        Plant 4         Plant 5
Warehouse 1         $4,000         $2,000         $3,000         $2,500         $4,500
Warehouse 2         $2,500         $2,600         $3,400         $3,000         $4,000
Warehouse 3         $1,200         $1,800         $2,600         $4,100         $3,000
Warehouse 4         $2,200         $2,600         $3,100         $3,700         $3,200
Open/close decision variables
                     Plant 1        Plant 2        Plant 3        Plant 4         Plant 5
Decision                 0              0              0              0              0
Number of products to ship (per 1000)
                     Plant 1        Plant 2        Plant 3        Plant 4         Plant 5         Total        Demand
Warehouse 1             0              0              0              0              0               0             15
Warehouse 2             0              0              0              0              0               0             18
Warehouse 3             0              0              0              0              0               0             14
Warehouse 4             0              0              0              0              0               0             20
Total                   0              0              0              0              0
Capacity                0              0              0              0              0
Distr. Cost            $0             $0             $0             $0             $0
Fixed Cost             $0             $0             $0             $0             $0
Total Cost             $0             $0             $0             $0             $0             $0



Problem
A company currently ships products from 5 plants to 4 warehouses. The company is considering the option of
closing down one or more plants. This would increase distribution cost but perhaps lower overall cost. What
plants, if any, should the company close?


Solution
1) The variables are the decisions to open or close the plants, and the number of products that should be
shipped from the plants that are open to the warehouses. In worksheet Facility these are given the names
Open_or_close and Products_shipped.
2) The logical constraints are
                  Products_shipped >= 0 via the Assume Non-Negative option
                  Open_or_close = binary
The products made can not exceed the capacity of the plants and the number shipped should meet the
demand. This gives
                  Products_made <= Capacity
                  Total_shipped >= Demand
3) The objective is to minimize cost. This is given the name Total_cost on the worksheet.


Remarks
It is often possible to increase the capacity of a plant. This could be worked into the model with additional 0-1
or binary integer variables. The Solver would find out if it would be profitable to extend the capacity of a plant.
It could also be interesting to see if it would be profitable to open another warehouse. An example of this can
be found, in somewhat modified form, in the capacity planning model in the Finance Examples workbook.




                                                                    Page 10
                                        6c23297b-696d-4da5-b26b-dda7bd16bdc1.xls


Production Transportation Problem (2-stage-transport, multi-commodity)
Minimize the costs of producing 3 different goods, and shipping them from factories to warehouses and
customers, and warehouses to customers, while not exceeding the supply available from each factory or
the capacity of each warehouse, and meeting the demand from each customer.

Cost to make products
                    Product 1         Product 2           Product 3
Factory 1              $4                 $5                 $3
Factory 2              $2                 $8                 $6


                    Product 1         Product 2           Product 3         Cost
Factory 1               0                  0                  0              $0
Factory 2               0                  0                  0              $0
                                                       Total Cost            $0

Cost of shipping ($ per product)
                                                            Destinations
                                     Warehouse 1        Warehouse 2    Warehouse 3     Warehouse 4
Factory 1         Product 1             $0.50              $0.50            $1.00         $0.20
                  Product 2             $1.00              $0.75            $1.25         $1.25
                  Product 3             $0.75              $1.25            $1.00         $0.80
Factory 2         Product 1             $1.50              $0.30            $0.50         $0.20
                  Product 2             $1.25              $0.80            $1.00         $0.75
                  Product 3             $1.40              $0.90            $0.95         $1.10
                                     Customer 1          Customer 2     Customer 3      Customer 4      Customer 5
Factory 1         Product 1             $2.75              $3.50            $2.50         $3.00           $2.50
                  Product 2             $2.50              $3.00            $2.00         $2.75           $2.60
                  Product 3             $2.90              $3.00            $2.25         $2.80           $2.35
Factory 2         Product 1             $3.00              $3.50            $3.50         $2.50           $2.00
                  Product 2             $2.25              $2.95            $2.20         $2.50           $2.10
                  Product 3             $2.45              $2.75            $2.35         $2.85           $2.45
                                     Customer 1          Customer 2     Customer 3      Customer 4      Customer 5
Warehouse 1       Product 1             $1.50              $0.80            $0.50         $1.50           $3.00
                  Product 2             $1.00              $0.90            $1.20         $1.30           $2.10
                  Product 3             $1.25              $0.70            $1.10         $0.80           $1.60
Warehouse 2       Product 1             $1.00              $0.50            $0.50         $1.00           $0.50
                  Product 2             $1.25              $1.00            $1.00         $0.90           $1.50
                  Product 3             $1.10              $1.10            $0.90         $1.40           $1.75
Warehouse 3       Product 1             $1.00              $1.50            $2.00         $2.00           $0.50
                  Product 2             $0.90              $1.35            $1.45         $1.80           $1.00
                  Product 3             $1.25              $1.20            $1.75         $1.70           $0.85
Warehouse 4       Product 1             $2.50              $1.50            $0.60         $1.50           $0.50
                  Product 2             $1.75              $1.30            $0.70         $1.25           $1.10
                  Product 3             $1.50              $1.10            $1.50         $1.10           $0.90

Number of products shipped
                                     Warehouse 1        Warehouse 2    Warehouse 3     Warehouse 4        Total
Factory 1         Product 1                0                  0               0              0              0
                  Product 2                0                  0               0              0              0
                  Product 3                0                  0               0              0              0
Factory 2         Product 1                0                  0               0              0              0
                  Product 2                0                  0               0              0              0




                                                                  Page 11
                                         6c23297b-696d-4da5-b26b-dda7bd16bdc1.xls


                  Product 3                0                   0               0              0              0
Total             Product 1                0                   0               0              0
                  Product 2                0                   0               0              0
                  Product 3                0                   0               0              0
Capacity          Product 1              35,000             20,000           30,000         15,000
                  Product 2              30,000             25,000           15,000         24,000
                  Product 3              20,000             20,000           25,000         20,000

                                      Customer 1          Customer 2       Customer 3    Customer 4     Customer 5       Total
Factory 1         Product 1                0                   0               0              0              0             0
                  Product 2                0                   0               0              0              0             0
                  Product 3                0                   0               0              0              0             0
Factory 2         Product 1                0                   0               0              0              0             0
                  Product 2                0                   0               0              0              0             0
                  Product 3                0                   0               0              0              0             0

                                                                                                                        Capacity
                                 Total products shipped out of factory 1                Product 1            0             0
                                                                                        Product 2            0             0
                                                                                        Product 3            0             0
                                 Total products shipped out of factory 2                Product 1            0             0
                                                                                        Product 2            0             0
                                                                                        Product 3            0             0

                                      Customer 1          Customer 2       Customer 3    Customer 4     Customer 5       Total
Warehouse 1       Product 1                0                   0               0              0              0             0
                  Product 2                0                   0               0              0              0             0
                  Product 3                0                   0               0              0              0             0
Warehouse 2       Product 1                0                   0               0              0              0             0
                  Product 2                0                   0               0              0              0             0
                  Product 3                0                   0               0              0              0             0
Warehouse 3       Product 1                0                   0               0              0              0             0
                  Product 2                0                   0               0              0              0             0
                  Product 3                0                   0               0              0              0             0
Warehouse 4       Product 1                0                   0               0              0              0             0
                  Product 2                0                   0               0              0              0             0
                  Product 3                0                   0               0              0              0             0
Total             Product 1                0                   0               0              0              0
                  Product 2                0                   0               0              0              0
                  Product 3                0                   0               0              0              0
Demands           Product 1              30,000             23,000           15,000         32,000         16,000
                  Product 2              20,000             15,000           22,000         12,000         18,000
                  Product 3              25,000             22,000           16,000         20,000         25,000

Total cost of shipping                              $0
Total cost of production                            $0
                    Total Cost                      $0




Problem
A company wants to minimize the cost of shipping three different products from factories to warehouses and customers
and from warehouses to customers. The production of each product at each plant depends on the distribution. How many
products should each factory produce and how should the products be distributed in order to minimize total cost while
meeting demand?



                                                                   Page 12
                                          6c23297b-696d-4da5-b26b-dda7bd16bdc1.xls



Solution
Notice that this is an extension of the transportation model as seen in the Transport3 worksheet. This time the factories do
not produce a fixed amount. The amounts produced are now variables.
1) The variables are the number of products to make in the factories, the number of products to ship from factories to
warehouses, factories to customers, and warehouses to customers. In worksheet Prodtran these are given the names
Products_made, Factory_to_warehouse, Factory_to_customer, and Warehouse_to_customer.
2) The logical constraints are all defined via the Assume Non-Negative option:
                   Products_made >= 0
                   Factory_to_warehouse >= 0
                   Factory_to_customer >= 0
                   Warehouse_to_customer >= 0
The other constraints are
                   Total_from_factory <= Factory_capacity
                   Total_to_customer >= Demand
                   Total_to_warehouse <= Warehouse_capacity
                   Total_to_warehouse = Total_from_warehouse
3) The objective is to minimize cost. This is defined in the worksheet as Total_cost.


Remarks
This is one of the more complex models in this series of examples. If the number of products, factories and warehouses
becomes large, the number of variables in a model like this one becomes very large. Also bear in mind the degree of
coordination between business units that may be needed in order to implement the optimal solution. For these reasons,
some users prefer to split problems like this one into a set of smaller, simpler models.




                                                                  Page 13

				
DOCUMENT INFO
Shared By:
Stats:
views:165
posted:2/23/2008
language:English
pages:13