Operations Management Transportation Models Quantitative Methods for Managerial Decision-Making ACN 309-5

Quantitative Methods for Managerial Decision-Making Outline TRANSPORTATION MODELING DEVELOPING AN INITIAL SOLUTION The Northwest-Corner Rule The Intuitive Lowest-Cost Method THE STEPPING-STONE METHOD SPECIAL ISSUES IN MODELING Demand Not Equal to Supply Degeneracy Quantitative Methods for Managerial Decision-Making ACN 309-5

Quantitative Methods for Managerial Decision-Making Learning Objectives After you read these notes, you should be able to Identify or Define: Transportation modeling Facility location analysis Explain or be able to use: Northwest-corner rule Stepping-stone method Quantitative Methods for Managerial Decision-Making ACN 309-5

Transportation Problem DesMoines (100 unit capacity) Fort Lauderdale (300 units capacity) Cleveland (200 units required) Evansville Albuquerque (300 units required) Boston Quantitative Methods for Managerial Decision-Making ACN 309-5

Transportation Problem How much should be shipped from several sources to several destinations Sources: Factories, warehouses, etc. Destinations: Warehouses, stores, etc. Transportation models Find lowest cost shipping arrangement Used primarily for existing distribution systems Quantitative Methods for Managerial Decision-Making ACN 309-5

A Transportation Model Requires The origin points, and the capacity or supply per period at each The destination points and the demand per period at each The cost of shipping one unit from each origin to each destination Quantitative Methods for Managerial Decision-Making ACN 309-5

Transportation Problem Graphical Solution 2 n Supply Quantity Source Quantity Shipped Destination ai i x mn j bj a1 1 b1 x11 a2 x22 b2 : 2n am xmn bn x1n x12 x21 Demand Quantity m xm2 xm1 Quantitative Methods for Managerial Decision-Making ACN 309-5

Transportation Problem Solution Steps Define problem Set up transportation table (matrix) Summarizes all data Keeps track of computations Develop initial solution Northwest corner rule Find optimal solution Stepping stone method Quantitative Methods for Managerial Decision-Making ACN 309-5

Quantitative Methods for Managerial Decision-Making Transportation Costs From To (Destination) (Sources) Albuquerque Boston Cleveland Des Moines $5 $4 $3 Evansville $8 Fort Lauderdale $9 $7 Quantitative Methods for Managerial Decision-Making ACN 309-5

Quantitative Methods for Managerial Decision-Making Transportation Table Destination Source Supply Demand 1 2 : m a . . n b Quantity demanded or required Quantitative Methods for Managerial Decision-Making ACN 309-5

Quantitative Methods for Managerial Decision-Making Transportation Table Destination Source 1 2 . . n Supply x 11 c 12 1n a 21 22 2n : m m1 m2 mn Demand b Cost of supplying 1 unit from sources to destinations Quantitative Methods for Managerial Decision-Making ACN 309-5

Quantitative Methods for Managerial Decision-Making Transportation Table Destination Source Supply Demand 1 2 : m a . . n b x 11 12 1n 21 22 2n m1 m2 mn Quantity supplied from sources to destinations Quantitative Methods for Managerial Decision-Making ACN 309-5

Quantitative Methods for Managerial Decision-Making Transportation Table To Albuquerque Boston Cleveland Factory From (A) (B) (C) Capacity Des Moines 5 4 3 100 (D) Evansville 8 4 3 300 (E) Fort Lauderdale 9 7 5 300 (F) Warehouse 300 200 200 700 Requirements Quantitative Methods for Managerial Decision-Making ACN 309-5

Initial Solution Using the Northwest Corner Rule To From Albuquerque (A) Boston (B) Cleveland (C) Factory Capacity Des Moines (D) 100 Evansville (E) 200 300 Fort Lauderdale (F) Warehouse Requirements 700 5 8 9 7 4 3 Quantitative Methods for Managerial Decision-Making ACN 309-5

The Stepping Stone Method Select any unused square to evaluate Begin at this square. Trace a closed path back to the original square via squares that are currently being used (only horizontal or vertical moves allowed) Place + in unused square; alternate - and + on each corner square of the closed path Calculate improvement index: add together the unit cost figures found in each square containing a +; subtract the unit cost figure in each square containing a -. Repeat steps 1-4 for each unused square Quantitative Methods for Managerial Decision-Making ACN 309-5

Stepping-Stone Method: Tracing a Closed Path - Des Moines to Cleveland From Albuquerque (A) Boston (B) Cleveland (C) Factory Capacity Des Moines (D) 100 Evansville (E) 200 300 Fort Lauderdale (F) Warehouse Requirements 700 5 8 9 7 4 3 Start + - Quantitative Methods for Managerial Decision-Making ACN 309-5

The Intuitive Lowest Cost Method Identify the cell with the lowest cost. Arbitrarily break any ties for the lowest cost. Allocate as many units as possible to that cell without exceeding the supply or demand. Then cross out that row or column (or both) that is exhausted by this assignment. Find the cell with the lowest cost from the remaining cells. Repeat steps 2 & 3 until all units have been allocated. Quantitative Methods for Managerial Decision-Making ACN 309-5

Initial Solution Using the Intuitive Lowest-Cost Method Second, cross out column C To Albuquerque Boston Cleveland Factory From (A) (B) (C) Capacity First, cross out top row Des Moines 5 4 3 100 100 (D) Evansville 8 4 3 Third, cross out row E 100 300 (E) 200 Fort Lauderdale 9 7 5 300 (F) 300 Warehouse 300 200 200 700 Requirements Quantitative Methods for Managerial Decision-Making ACN 309-5

Quantitative Methods for Managerial Decision-Making Specialized Methods Linear programming model is difficult to formulate & solve Special purpose methods Are easier to formulate Are faster to compute Give integer solutions Methods Stepping-stone MODI See your CD Tutorial © 1995 Corel Corp. Quantitative Methods for Managerial Decision-Making ACN 309-5

Special Issues in the Transportation Model Demand not equal to supply Called ‘unbalanced’ problem Add dummy source if demand > supply Add dummy destination if supply > demand Degeneracy in Stepping Stone Method Too few shipping routes (cells) used Number of occupied cells should be: m + n - 1 Create artificially occupied cell (0 value) Represents fake shipment Quantitative Methods for Managerial Decision-Making ACN 309-5

Transportation Table Demand Not Equal Supply To From Albuquerque (A) Boston (B) Cleveland (C) Factory Capacity Des Moines (D) 250 Evansville (E) 300 Fort Lauderdale (F) Warehouse Requirements 200 700 5 8 9 7 4 3 Dummy 150 New Des Moines capacity Quantitative Methods for Managerial Decision-Making ACN 309-5

Quantitative Methods for Managerial Decision-Making Degeneracy To From Albuquerque (A) Boston (B) Cleveland (C) Factory Capacity Des Moines (D) 100 Evansville (E) 200 300 Fort Lauderdale (F) Warehouse Requirements 700 5 8 9 7 4 3 Quantitative Methods for Managerial Decision-Making ACN 309-5

Degeneracy - Continued To Albuquerque Boston Cleveland Factory From (A) (B) (C) Capacity Des Moines 5 4 3 100 100 (D) Evansville 8 4 3 200 100 300 (E) Fort Lauderdale 9 7 5 200 200 (F) Warehouse 300 100 200 700 Requirements Quantitative Methods for Managerial Decision-Making ACN 309-5