Introduction to Management Science 8th Edition by Bernard W.Taylor III Chapter 6 Transportation, Transshipment, and Assignment Problems Chapter 6 - Transportation, Transshipment, and Assignment Problems 1 Chapter Topics The Transportation Model Computer Solution of a Transportation Problem The Assignment Model Computer Solution of the Assignment Model Chapter 6 - Transportation, Transshipment, and Assignment Problems 2 Overview Part of a larger class of linear programming problems known as network flow models. Possess special mathematical features that enabled development of very efficient, unique solution methods. Methods are variations of traditional simplex procedure. Detailed description of methods is contained in CD-ROM Module B, Transportation and Assignment Solution Methods. Text focuses on model formulation and solution with Excel and QM for windows. Chapter 6 - Transportation, Transshipment, and Assignment Problems 3 Chapter 6 .The Transportation Model Characteristics A product is transported from a number of sources to a number of destinations at the minimum possible cost. Transshipment.Transportation. All constraints are equalities in a balanced transportation model where supply equals demand. Each source is able to supply a fixed number of units of the product. The linear programming model has constraints for supply at each source and demand at each destination. and each destination has a fixed demand for the product. Constraints contain inequalities in unbalanced models where supply does not equal demand. and Assignment Problems 4 . Omaha 3. St. Kansas City 2.Transportation. Cincinnati $ 6 $ 8 $10 7 11 11 12 4 5 5 Chapter 6 . Louis C. Kansas City 2. Des Moines Transport Cost from rain Elevator to Mill ($/ton) A. Chicago B.Transportation Model Example Problem Definition and Data Problem: How many tons of wheat to transport from each grain elevator to each mill on a monthly basis in order to minimize the total cost of transportation? Data: Grain Elevator 1. Omaha 3. Chicago B. Transshipment. Des Moines Total Supply 150 175 275 600 tons Mill A. Cincinnati Total Demand 200 100 300 600 tons rain Elevator 1. St. and Assignment Problems . Louis C. to each mill j. and Assignment Problems 6 . i. 3.Transportation Model Example Model Formulation (1 of 2) Minimize Z = $6x1A + 8x1B + 10x1C + 7x2A + 11x2B + 11x2C + 4x3A + 5x3B + 12x3C subject to: x1A + x1B + x1C = 150 x2A + x2B + x2C = 175 x3A + x3B + x3C = 275 x1A + x2A + x3A = 200 x1B + x2B + x3B = 100 x1C + x2C + x3C = 300 xij u 0 xij = tons of wheat from each grain elevator. Transshipment.B.C Chapter 6 .Transportation. 2. i = 1. j = A. 1 Network of Transportation Routes for Wheat Shipments Chapter 6 . and Assignment Problems 7 .Transportation Model Example Model Formulation (2 of 2) Figure 6.Transportation. Transshipment. Transportation.Transportation Model Example Computer Solution with Excel (1 of 3) Exhibit 6.1 Chapter 6 . Transshipment. and Assignment Problems 8 . Transshipment.Transportation Model Example Computer Solution with Excel (2 of 3) Exhibit 6.2 Chapter 6 . and Assignment Problems 9 .Transportation. Transshipment.Transportation Model Example Computer Solution with Excel (3 of 3) Exhibit 6.3 Chapter 6 . and Assignment Problems 10 .Transportation. and Assignment Problems 11 .4 Chapter 6 . Transshipment.Transportation Model Example Computer Solution with Excel QM (1 of 3) Exhibit 6.Transportation. Transportation Model Example Computer Solution with Excel QM (2 of 3) Exhibit 6.Transportation. Transshipment. and Assignment Problems 12 .5 Chapter 6 . Transshipment. and Assignment Problems 13 .6 Chapter 6 .Transportation Model Example Computer Solution with Excel QM (3 of 3) Exhibit 6.Transportation. Transshipment.7 Chapter 6 .Transportation. and Assignment Problems 14 .Transportation Model Example Computer Solution with QM for Windows (1 of 3) Exhibit 6. 8 Chapter 6 .Transportation.Transportation Model Example Computer Solution with QM for Windows (2 of 3) Exhibit 6. and Assignment Problems 15 . Transshipment. Transportation Model Example Computer Solution with QM for Windows (3 of 3) Exhibit 6. Transshipment.9 Chapter 6 .Transportation. and Assignment Problems 16 . Transshipment. Intermediate transshipment points are added between the sources and destinations. Items may be transported from: Sources through transshipment points to destinations One source to another One transshipment point to another One destination to another Directly from sources to to destinations Some combination of these Chapter 6 .Transportation. and Assignment Problems 17 .The Transshipment Model Characteristics Extension of the transportation model. Transshipment.T P bl ip fi i i l x pl ( f ) Extension of the transportation model in which intermediate transshipment points are added between sources and destinations. .Transportation. l $ i G i l v . and Assignment Problems 18 .O .N . i Chapter 6 . Data: . Transportation. Transshipment. and Assignment Problems 19 .Transshipment Model Example Problem Definition and Data (2 of 2) Figure 6.2 Network of Transshipment Routes Chapter 6 . x47 .Transportation.x38 = 0 x14 + x24 . Transshipment.Transshipment Model Example Model Formulation Minimize Z = $16x13 + 10x14 + 12x15 + 15x23 + 14x24 + 17x25 + 6x36 + 8x37 + 10x38 + 7x46 + 11x47 + 11x48 + 4x56 + 5x57 + x58 subject to: x13 + x14 + x15 = 300 x23+ x24 + x25 = 300 x36 + x37 + x38 = 200 x46+ x47 + x48 = 100 x56 + x57 + x58 = 300 x13 + x23 .x57 . and Assignment Problems 20 .x36 .x37 .x56 .x48 = 0 x15 + x25 .x46 .x58 = 0 xij u 0 Chapter 6 . and Assignment Problems 21 . Transshipment.10 Chapter 6 .Transshipment Model Example Computer Solution with Excel (1 of 2) Exhibit 6.Transportation. Transshipment.11 Chapter 6 .Transportation. and Assignment Problems 22 .Transshipment Model Example Computer Solution with Excel (2 of 2) Exhibit 6. The Assignment Model Characteristics Special form of linear programming model similar to the transportation model. Supply at each source and demand at each destination limited to one unit. Transshipment. In a balanced model supply equals demand. In an unbalanced model supply does not equal demand.Transportation. and Assignment Problems 23 . Chapter 6 . Transshipment. Supply is always one team of officials.1 Chapter 6 .Transportation. and Assignment Problems 24 . Data: Table 6. demand is for only one team of officials at each game.Assignment Model Example Problem Definition and Data Problem: Assign four teams of officials to four games in a way that will minimize total distance traveled by the officials. Transshipment. and Assignment Problems xij u 0 25 .Transportation.Assignment Model Example Model Formulation Minimize Z = 210xAR + 90xAA + 180xAD + 160xAC + 100xBR + 70xBA + 130xBD + 200xBC + 175xCR + 105xCA + 140xCD + 170xCC + 80xDR + 65xDA + 105xDD + 120xDC subject to: xAR + xAA + xAD + xAC = 1 xBR + xBA + xBD + xBC = 1 xCR + xCA + xCD + xCC = 1 xDR + xDA + xDD + xDC = 1 xAR + xBR + xCR + xDR = 1 xAA + xBA + xCA + xDA = 1 xAD + xBD + xCD + xDD = 1 xAC + xBC + xCC + xDC = 1 Chapter 6 . Assignment Model Example Computer Solution with Excel (1 of 3) Exhibit 6. and Assignment Problems 26 .Transportation.12 Chapter 6 . Transshipment. Assignment Model Example Computer Solution with Excel (2 of 3) Exhibit 6.13 Chapter 6 .Transportation. and Assignment Problems 27 . Transshipment. Assignment Model Example Computer Solution with Excel (3 of 3) Exhibit 6.14 Chapter 6 .Transportation. Transshipment. and Assignment Problems 28 . Assignment Model Example Computer Solution with Excel QM Exhibit 6.Transportation. Transshipment. and Assignment Problems 29 .15 Chapter 6 . Transshipment.Assignment Model Example Computer Solution with QM for Windows (1 of 2) Exhibit 6. and Assignment Problems 30 .Transportation.16 Chapter 6 . Transshipment.17 Chapter 6 . and Assignment Problems 31 .Transportation.Assignment Model Example Computer Solution with QM for Windows (2 of 2) Exhibit 6. Transportation. Transshipment. and Assignment Problems 32 .Example Problem Solution Transportation Problem Statement Determine linear programming model formulation and solve using Excel: Plant 1 2 3 Demand (tons) A $ 8 15 3 150 Construction site B C $ 5 $ 6 10 12 9 10 70 100 Supply (tons) 120 80 80 Chapter 6 . and Assignment Problems 33 . Transshipment.Transportation.Example Problem Solution Model Formulation Minimize Z = $8x1A + 5x1B + 6x1C + 15x2A + 10x2B + 12x2C + 3x3A + 9x3B + 10x3C subject to: x1A + x1B + x1C = 120 x2A + x2B + x2C = 80 x3A + x3B + x3C = 80 x1A + x2A + x3A e 150 x1B + x2B + x3B e 70 x1C + x2C + x3C e 100 xij u 0 Chapter 6 . Transportation. and Assignment Problems 34 .Example Problem Solution Computer Solution with Excel Chapter 6 . Transshipment.