Chapter 7 Transportation, Assignment & Transshipment Problems Part 1 ISE204/IE252 Prof. Dr. Arslan M. ÖRNEK
Remember from ISE 203: A Transportation Example Warehouse supply of Retail store demand Television Sets: for television sets: 1 - Cincinnati 300 A - New York 150 2 - Atlanta 200 B - Dallas 250 3 - Pittsburgh 200 C - Detroit 200 Total 700 Total 600 Unit Shipping Costs: To Store From Warehouse A B C 1 $16 $18 $11 2 14 12 13 3 13 15 17 2
Transportation Problem: Characteristics ■ A transportation problem aims to find the best way to fulfill the demand of n demand points using the capacities of m supply points . ■ A product is transported from a number of sources to a number of destinations at the minimum possible cost . ■ Each source is able to supply a fixed number of units of the product, and each destination has a fixed demand for the product. ■ The linear programming model has constraints for supply at each source and demand at each destination. ■ I n a balanced transportation model supply equals demand. 3
7.1 Formulating Transportation Problems How many tons of wheat to transport from each grain elevator to each mill in order to minimize the total cost of transportation? Grain Elevator Supply Mill Demand 1. Kansas City 150 A. Chicago 200 2. Omaha 175 B. St. Louis 100 3. Des Moines 275 C. Cincinnati 300 Total 600 tons Total 600 tons Transport Cost from Grain Elevator to Mill ($/ton) Grain Elevator A. Chicago B. St. Louis C. Cincinnati 1. Kansas City $ 6 $ 8 $ 10 2. Omaha 7 11 11 3. Des Moines 4 5 12 4
Transportation Model Example Transportation Network Routes Figure 6.1 Network of Transportation Routes for Wheat Shipments 5
Transportation Model Example Linear Programming Model Formulation x ij = tons of wheat from each grain elevator, i, i = 1, 2, 3, to each mill j = A,B,C Minimize Z = $6x 1A + 8x 1B + 10x 1C + 7x 2A + 11x 2B + 11x 2C + 4x 3A + 5x 3B + 12x 3C subject to: x 1A + x 1B + x 1C = 150 x 2A + x 2B + x 2C = 175 x 3A + x 3B + x 3C = 275 x 1A + x 2A + x 3A = 200 x 1B + x 2B + x 3B = 100 x 1C + x 2C + x 3C = 300 x ij 0 6
Transportation Model Example Optimal Solution Figure 6.2 Transportation Network Solution 7
7.1 Formulating Transportation Problems Example 1: Powerco has three electric power plants that supply the electric needs of four cities. • The associated supply of each plant and demand of each city are known. • The cost of sending 1 million kwh of electricity from a plant to a city depends on the distance the electricity must travel. • Formulate an LP to minimize cost.
Solution: Decision Variable: x 14 = Amount of electricity produced at plant 1 and sent to city 4 Since we want to minimize the total cost of shipping from plants to cities; Objective Function: Minimize Z = 8X 11 +6X 12 +10X 13 +9X 14 +9X 21 +12X 22 +13X 23 +7X 24 +14X 31 +9X 32 +16X 33 +5X 34
Solution (cont): Since each supply point has a limited production capacity; X 11 +X 12 +X 13 +X 14 <= 35 X 21 +X 22 +X 23 +X 24 <= 50 Supply Constraints X 31 +X 32 +X 33 +X 34 <= 40 Since each demand point has a demand to satisfy; X 11 +X 21 +X 31 >= 45 X 12 +X 22 +X 32 >= 20 Demand Constraints X 13 +X 23 +X 33 >= 30 X 14 +X 24 +X 34 >= 30 Sign Constraints X ij >= 0 (i= 1,2,3; j= 1,2,3,4)
Solution (cont): LP formulation:
Network representation of Optimal Solution:
General Description of a Transportation Problem 1. A set of m supply points with a supply of at most s i units. 2. A set of n demand points with a demand of at least d j units. 3. Each unit produced at supply point i and shipped to demand point j incurs a variable cost of c ij .
General Formulation of a Transportation Problem x ij = number of units shipped from supply point i to demand point j
General Formulation of a Transportation Problem • If a problem has these constraints and is a maximization problem, then it is still a transportation problem. • If total supply equals to total demand, the problem is said to be a balanced transportation problem :
General Formulation of a Balanced Transportation Problem It is desirable to formulate a problem as a balanced transportation problem (due to the ease of solution procedures).
Balancing a Transportation Problem if total supply exceeds total demand If total supply exceeds total demand, we can balance the problem by adding a dummy demand point . Since shipments to the dummy demand point are not real, they are assigned a cost of zero .
Balancing a TP if total supply exceeds total demand Example 1: Suppose that in the Powerco problem, the demand for city 1 were reduced to 40 million kwh. To balance the Powerco problem, we would add a dummy demand point (City 5) with a demand of Total Supply – Total Demand = 125 - 120 = 5 million kwh. From each plant, the cost of shipping 1 million kwh to the dummy is 0.
Optimal Solution of the Balanced Powerco Problem
Transportation tableau A transportation problem is specified by the supply, the demand, and the shipping costs. So the relevant data can be summarized in a transportation tableau.
Transportation tableau Optimal transportation tableau for Powerco:
Balancing a transportation problem if total supply is less than total demand If total supply < total demand The problem has no feasible solution. When total supply is less than total demand, it is sometimes desirable to allow the possibility of leaving some demand unsatisfied . In such a situation, a penalty is often associated with unmet demand.
• Example 2 (Handling Shortages): 23
• Solution: 24
25
Sailco Corporation must determine how many sailboats should be produced during each of the next four quarters (one quarter is three months). Demand : first quarter, 40 sailboats; second quarter, 60 sailboats; third quarter, 75 sailboats; fourth quarter, 25 sailboats. Sailco must meet demand on time. At the beginning of the first quarter, Sailco has an inventory of 10 sailboats. At the beginning of each quarter, Sailco must decide how many sailboats should be produced during the current quarter. For simplicity, we assume that sailboats manufactured during a quarter can be used to meet the demand for that quarter.
During each quarter, Sailco can produce up to 40 sailboats at a cost of $400 per sailboat. By having employees work overtime during a quarter, Sailco can produce additional sailboats at a cost of $450 per sailboat. At the end of each quarter (after production has occurred and the current quarter’s demand has been satisfied), a carrying or holding cost of $20 per sailboat is incurred. Formulate a balanced transportation problem to minimize the sum of production and inventory costs during the next four quarters.
Capacity of each OT supply point = 150 = 200 (Total demand) – 10 (initial inventory) – 40 (regular time production capacity)
7.2 Finding Basic Feasible Solution for TP A balanced TP with m supply points and n demand points is easier to solve than a regular LP, although it has m + n equality constraints. If a set of values for the x ij ’s satisfies all but one of the constraints of a balanced transportation problem, then the values for the x ij ’s will automatically satisfy the other constraint. This means that only m+n-1 constraints are linearly independent. m+n-1 basic variables
Methods to find the bfs for a balanced TP There are three basic methods: 1. Northwest Corner Method 2. Minimum Cost Method 3. Vogel’s Method
1. Northwest Corner Method (NWC) To find the bfs by the NWC method: Begin in the upper left (northwest) corner of the transportation tableau and set x 11 as large as possible (the limitations for setting x 11 will be the demand of demand point 1 and the supply of supply point 1. Your x 11 value can not be greater than the minimum of this two values).
Example: Set x 11 =3 (meaning demand of demand point 1 is satisfied by supply point 1). 5 6 2 3 5 2 3 3 2 6 2 X 5 2 3 35
After we check the east and south cells, we see that we can go east (meaning supply point 1 still has capacity to fulfill some demand). 3 2 X 6 2 X 3 2 3 3 2 X 3 3 2 X X 2 3 36
After applying the same procedure, we see that we can go south this time (meaning demand point 2 needs more supply by supply point 2). 3 2 X 3 2 1 2 X X X 3 3 2 X 3 2 1 X 2 X X X 2 37
Finally, we will have the following bfs: x 11 =3, x 12 =2, x 22 =3, x 23 =2, x 24 =1, x 34 =2 3 2 X 3 2 1 X 2 X X X X X 38
Example: 39
Example: Supply and demand equal, cross only one, not both! 40
Example: Degenerate solution m+n-1 = 3+4-1=6 basic 1 variables 41
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