Computational Methods for Management and Economics Carla Gomes Module 8a The transportation model

The transportation and assignment problems Special types of linear programming problems. The structure of these problems leads to algorithms – streamlined versions of the simplex method - more efficient than the standard simplex method.

The transportation problem

Prototype example : P&T Company Main product of P&T Company – canned peas –3 canneries –4 distribution centers Shipping costs – major expense in management –GOAL – minimize shipping costs

P&T Company Distribution Problem

Shipping Data CanneryOutputWarehouseAllocation Bellingham75 truckloadsSacramento80 truckloads Eugene125 truckloads Salt Lake City 65 truckloads Albert Lea100 truckloadsRapid City70 truckloads Total300 truckloadsAlbuquerque85 truckloads Total 300 truckloads

Current Shipping Plan Warehouse From \ To SacramentoSalt Lake CityRapid CityAlbuquerque Cannery Bellingham75000 Eugene Albert Lea001585

Shipping Cost per Truckload Warehouse From \ To Sacramento Salt Lake City Rapid CityAlbuquerque Cannery Bellingham $464$513$654$867 Eugene Albert Lea Total shipping cost= 75($464) + 5($352) + 65($416) + 55($690) + 15($388) + 85($685) = $165,595

Terminology for a Transportation Problem P&T Company Problem Truckloads of canned peas Canneries Warehouses Output from a cannery Allocation to a warehouse Shipping cost per truckload from a cannery to a warehouse General Model Units of a commodity Sources Destinations Supply from a source Demand at a destination Cost per unit distributed from a source to a destination

Characteristics of Transportation Problems The Requirements Assumption –Each source has a fixed supply of units, where this entire supply must be distributed to the destinations. –Each destination has a fixed demand for units, where this entire demand must be received from the sources. The Feasible Solutions Property –A transportation problem will have feasible solutions if and only if the sum of its supplies equals the sum of its demands. The Cost Assumption –The cost of distributing units from any particular source to any particular destination is directly proportional to the number of units distributed. –This cost is just the unit cost of distribution times the number of units distributed.

The Transportation Model Any problem (whether involving transportation or not) fits the model for a transportation problem if: 1.It can be described completely in terms of a table that identifies all the sources, destinations, supplies, demands, and unit costs, and 2.satisfies both the requirements assumption and the cost assumption. The objective is to minimize the total cost of distributing the units.

The P&T Co. Transportation Problem Transportation Tableau Unit Cost Destination (Warehouse): SacramentoSalt Lake CityRapid CityAlbuquerqueSupply Source (Cannery) Bellingham$464$513$654$86775 Eugene Albert Lea Demand

Network Representation This graph is “bipartite.” That is, the nodes are partitioned into two parts and arcs have one endpoint in each part.

The Transportation Problem is an LP 1.Decision Variable: Since we have to determine how much electricity is sent from each plant to each city; x ij = Amount of commodity produced at source i and sent to destination j x 13 = truckloads produced at cannery 1 and sent to warehouse 3

Network Representation This graph is “bipartite.” That is, the nodes are partitioned into two parts and arcs have one endpoint in each part. x 11 x 12 x 13 x 14 x 21 x 22 x 23 x 24 x 31 x 32 x 33 x 34 Shipping cost – source i destination j

Objective function Since we want to minimize the total cost of shipping from canneries to wharehouses; Minimize Z = $464x 11 + $513x 12 + $654x 13 + $867x 14 + $352x 21 + $416x 22 + $690x 23 + $791x 24 + $995x 31 + $682x 32 + $388x 33 + $685x 34

Supply Constraints Since each supply point has a limited production capacity; Cannery 1:x 11 + x 12 + x 13 + x 14 = 75 Cannery 2: x 21 + x 22 + x 23 + x 24 = 125 Cannery 3: x 31 + x 32 + x 33 + x 34 = 100

Demand Constraints Since each supply point has a limited production capacity; Warehouse 1:x 11 + x 21 + x 31 = 80 Warehouse 2: x 12 + x 22 + x 32 = 65 Warehouse 3: x 13 + x 23 + x 33 = 70 Warehouse 4: x 14 + x 24 + x 34 = 85

Sign Constraints Since a negative amount of truckloads can not be shipped all Xij’s must be non negative; Xij >= 0 (i= 1,2,3; j= 1,2,3,4)

The Transportation Problem is an LP Let x ij = the number of truckloads to ship from cannery i to warehouse j (i = 1, 2, 3; j = 1, 2, 3, 4) Minimize Cost = $464x 11 + $513x 12 + $654x 13 + $867x 14 + $352x 21 + $416x 22 + $690x 23 + $791x 24 + $995x 31 + $682x 32 + $388x 33 + $685x 34 subject to Cannery 1:x 11 + x 12 + x 13 + x 14 = 75 Cannery 2: x 21 + x 22 + x 23 + x 24 = 125 Cannery 3: x 31 + x 32 + x 33 + x 34 = 100 Warehouse 1:x 11 + x 21 + x 31 = 80 Warehouse 2: x 12 + x 22 + x 32 = 65 Warehouse 3: x 13 + x 23 + x 33 = 70 Warehouse 4: x 14 + x 24 + x 34 = 85 and x ij ≥ 0 (i = 1, 2, 3; j = 1, 2, 3, 4)

The Transportation Problem is an LP with special matrix A structure Let x ij = the number of truckloads to ship from cannery i to warehouse j (i = 1, 2, 3; j = 1, 2, 3, 4) Minimize Cost = c 11 x 11 + c 12 x 12 + c 13 x 13 + c 14 x 14 + c 21 x 21 + c 22 x 22 + c 23 x 23 + c 24 x 24 + c 31 x 31 + c 32 x 32 + c 33 x 33 + c 34 x 34 subject to Source1:x 11 + x 12 + x 13 + x 14 = supply1 Source 2: x 21 + x 22 + x 23 + x 24 = supply2 Source 3: x 31 + x 32 + x 33 + x 34 = supply3 Destination 1:x 11 + x 21 + x 31 = demand1 Destination 2: x 12 + x 22 + x 32 = demand2 Destination 3: x 13 + x 23 + x 33 = demand3 Destination 4: x 14 + x 24 + x 34 = demand4 and x ij ≥ 0 (i = 1, 2, 3; j = 1, 2, 3, 4)

The Node-Arc Incidence Matrix (for directed graphs) a b c d e A Directed Graph Have a row for each node a b c d e Have a column for each arc Put a 1 in row i- column j if arc j starts at node i. Put a -1 in row i- column j if arc j ends at node i. What would happen if arc (4,2) became arc (2,4)?

On Incidence Matrices If the constraint matrix for a linear program is a node-arc incidence matrix (at most one 1 and at most one –1 per column), then the linear program solves in integer optima. Node arc incidence matrix shows up in Linear Programs. The constraint matrix of a transportation problem is a node-arc incidence matrix in disguise.

The Transportation Problem is an LP Let x ij = the number of truckloads to ship from cannery i to warehouse j (i = 1, 2, 3; j = 1, 2, 3, 4) Minimize Cost = $464x 11 + $513x 12 + $654x 13 + $867x 14 + $352x 21 + $416x 22 + $690x 23 + $791x 24 + $995x 31 + $682x 32 + $388x 33 + $685x 34 subject to Cannery 1:x 11 + x 12 + x 13 + x 14 = 75 Cannery 2: x 21 + x 22 + x 23 + x 24 = 125 Cannery 3: x 31 + x 32 + x 33 + x 34 = 100 Warehouse 1:-x 11 - x 21 - x 31 = -80 Warehouse 2: - x 12 - x 22 -x 32 = -65 Warehouse 3: -x 13 - x 23 - x 33 = -70 Warehouse 4: -x 14 - x 24 - x 34 = -85 and x ij ≥ 0 (i = 1, 2, 3; j = 1, 2, 3, 4)

The Node-Arc Incidence Matrix: Transportation Problem is an LP

Spreadsheet Formulation Note: excel does not have a specialized simplex algorithm to solve transportation problems.

Integer Solutions Property As long as all its supplies and demands have integer values, any transportation problem with feasible solutions is guaranteed to have an optimal solution with integer values for all its decision variables. Therefore, it is not necessary to add constraints to the model that restrict these variables to only have integer values.

On the integrality Property The fact that solutions to the transportation problem are integral is an amazing property. In general, solutions to IP are fractional. But solutions to the transportation problem are integral. Structure of the matrix of technological coefficients (A matrix) and the fact that the RHS are integral – in general, if there is at most one 1 and at most one –1 in any column of the constraint matrix, then every basic feasible solution is integer (so long as RHS is integral.) For many applications, we want to restrict variables to be integer valued.

General Description of a Transportation Problem 1.A set of m supply points from which a good is shipped. Supply point i can supply at most s i units. 2.A set of n demand points to which the good is shipped. Demand point j must receive at least d i units of the shipped good. 3.Each unit produced at supply point i and shipped to demand point j incurs a variable cost of c ij.

Features of this transportation problem The constraint matrix is (or can be made to be) the node arc incidence matrix of the network If supplies/demands are integral, then the flows are also integral. If the total supply is equal to the total demand, then all supply and demand constraints hold with equality Very efficient special purpose solution techniques exist Applications to shipment of goods

Features of this transportation problem The constraint matrix is (or can be made to be) the node arc incidence matrix of the network If supplies/demands are integral, then the flows are also integral. If the total supply is equal to the total demand, then all supply and demand constraints hold with equality Very efficient special purpose solution techniques exist Applications to shipment of goods

x ij = number of units shipped from supply point i to demand point j

Balanced Transportation Problem If Total supply equals to total demand, the problem is said to be a balanced transportation problem:

Balancing a TP if total supply exceeds total demand If total supply exceeds total demand, we can balance the problem by adding dummy demand point. Since shipments to the dummy demand point are not real, they are assigned a cost of zero.

Balancing a transportation problem if total supply is less than total demand If a transportation problem has a total supply that is strictly less than total demand the problem has no feasible solution. There is no doubt that in such a case one or more of the demand will be left unmet. Generally in such situations a penalty cost is often associated with unmet demand and as one can guess this time the total penalty cost is desired to be minimum

Distribution System at Proctor and Gamble Proctor and Gamble needed to consolidate and re-design their North American distribution system in the early 1990’s. –50 product categories –60 plants –15 distribution centers –1000 customer zones Solved many transportation problems (one for each product category). Goal: find best distribution plan, which plants to keep open, etc. Closed many plants and distribution centers, and optimized their product sourcing and distribution location. Implemented in Saved $200 million per year. For more details, see 1997 Jan-Feb Interfaces article, “Blending OR/MS, Judgement, and GIS: Restructuring P&G’s Supply Chain”, downloadable from course web site.

Dummy Destination Balancing a TP if total supply exceeds total demand

Northern Airplane (Production Scheduling) Northern Airplane Company produces commercial airplanes. The last stage in production is to produce the jet engines and install them. –The company must meet the delivery deadline indicated in column 2. An option is to produce some engines one month or more before they are scheduled for installation and store them. –Production and storage costs vary from month to month. Maximum Production Unit Cost of Production ($million) Unit Cost of Storage(*) ($thousand) Month Scheduled Installations Question: How many engines should be produced in each of the four months so that the total of the production and storage costs will be minimized? (*) storage cost is incurred at the end of the month for just those engines that are being held over to the next month;

Source i - production of jet engines in month i (i = 1,2,3 4); Destination j – installation of jet engines in month j (j = 1,2,3 4) x ij = number of engines produce in month i to be installed in month j c ij = cost associated with each unit of x ij Cost per unit for production + storage i < j i >j ???

Northern Airplane (Production Scheduling) Cost per unit distributed Destinations (installation in month j) Source (production in month i)1234SUPLLY ? 2? ? 3?? ? 4???1.130? Demand Question: How many engines should be produced in each of the four months so that the total of the production and storage costs will be minimized? (*) storage cost is incurred at the end of the month for just those engines that are being held over to the next month;

Transportation Tableau Cost per unit distributed Destinations (installation in month j) Source (production in month i)1234 5(D) (*)SUPLLY M MM MMM Demand (*) the dummy destination can be seen as a slack variable that represents the unused production capacity. Cost is zero because it is the cost of distributing to a fictional destination. Note that it would be inappropriate to assign M since we do not want to force the corresponding values to be zero. In fact these values need to sum 30.

Dummy Source Balancing a transportation problem if total supply is less than total demand

Metro Water (Distributing Natural Resources) Metro Water District is an agency that administers water distribution in a large goegraphic region. The region is arid, so water must be brought in from outside the region. –Sources of imported water: Colombo, Sacron, and Calorie rivers. –Main customers: Cities of Berdoo, Los Devils, San Go, and Hollyglass. Cost per Acre Foot BerdooLos DevilsSan GoHollyglass Available (million acre feet) Colombo River $160$130$220$17050 Sacron River Calorie River —50 Min. Needed Requested  Question: How much water should Metro take from each river, and how much should they send from each river to each city?

What’s the problem with the previous table to look like a transportation tableau? It is not clear what the demands are at the destinations The amount to be received at each destination is a decision variable, with an upper bound and a lower bound – in the transportation model it should be a constant. Also, here we have excess demand  dummy source

What’s the problem with the previous table to look like a transportation tableau? Hollyglass  upper-bound = Total supply – min requested = ( ) – ( ) = 60 How do we solve the problem of having constant demands?

Metro Water Transportation Tableau Berdoo (min) (1) Berdoo (extra) (2) Los Devils (3) San Go (4) Hollyglass (5) Available (million acre feet) Colombo River (1)$160 $130$220$17050 Sacron River (2) Calorie River (3) M50 Dummy (4) M070M050 Demand