Spring 2015 Mathematics in Management Science Transportation Problems Tableaus Shipping Solutions NorthWest Corner Rule Stepping Stone Algorithm
Transportation Problems Problems where group of suppliers must meet the needs/demands of a group of users of these supplies. (Think corporate sized users, not individuals.) Have cost associated with shipping from a particular supplier to particular “consumer”. Objective is to minimize total shipping cost.
Example The military buys fuel from many refineries, and needs to it delivered to multiple bases.
Example During World War II, supplies needed to be shipped from multiple US ports to many different European ports. (This is the origin of these types of problems.)
Example A supermarket chain buys bread from a supplier that makes its bread in multiple bakeries. The bread must be shipped from individual bakeries to individual stores.
Bakery Example Chain gets bread deliveries from a bakery chain that does its baking in different places. Each store needs certain number of loaves/day. Supplier bakes enough to exactly meet demands. How many loaves ship (b to s) to stay within the demands and to minimize the cost?
Transportation Problems Have suppliers, users, shipping costs. Each supplier has a fixed amount that they can supply. Each user has a fixed amount they need/demand. The total capacity of the suppliers exactly matches the needs of the users.
Transportation Problems Inputs: individual capacities of the suppliers, individual needs of the users, and shipping costs from any one supplier to any one user. This information is collected in a table called a tableau.
Transportation Problems Have suppliers, users, shipping costs. Tableau: displays info with rim conditions. Each cell has two entries: costs and amount to ship. Shipping solution: obeys all constraints. NorthWest Corner Rule: an algorthim for obtaining a shipping solution. Indicator Values. Stepping Stone Algorithm.
Mines & Plants Two mines (extract/supply ore) and three plants (process/demand ore). Mine A supplies 7 m tons, B 3 m tons. Plant 1 needs 2 tons, 2&3 each 4 tons. Shipping costs are from A to P1,P2,P , 1, 3 from B to P1,P2,P , 5, 12
Tableaus The tableau is a table with one row per supplier (mines—labeled with A,B) and one column per consumer/demander (plants—labeled with 1,2,3).
Plant 1 Plant 2 Plant 3 Mine A Mine B Suppliers on left; demanders on top. Next add rim conditions: Supply amounts in last column. Demand amounts in last row. Check that amounts agree!
Plant 1 Plant 2 Plant 3 Mine A 7 Mine B
Plant 1 Plant 2 Plant 3 Mine A 7 Mine B Next add shipping costs. Cell (m,p) is cost to ship from mine m to plant p.
Plant 1 Plant 2 Plant 3 Mine A Mine B
Plant 1 Plant 2 Plant 3 Mine A Mine B Need shipping solution to start with.
NorthWest Corner Rule Locate top far-left-hand cell. Ship via this cell with smaller of the two rim cells (call the value s) ; circle entry in tableau. Cross out row or column with rim value s, & reduce other rim value for this cell by s. When a single cell remains, shud be tie for rim conditions of row and column involved; put this amount into cell and circle.
Plant 1 Plant 2 Plant 3 Mine A Mine B Here!
Plant 1 Plant 2 Plant 3 Mine A Mine B
Plant 1 Plant 2 Plant 3 Mine A Mine B
Plant 1 Plant 2 Plant 3 Mine A Mine B
Plant 1 Plant 2 Plant 3 Mine A Mine B Cost= =57
Plant 1 Plant 2 Plant 3 Mine A Mine B Check Indicator Values
Example A supermarket chain has three stores. Their bread supplier makes their bread in three bakeries. The stores require 3 dozen, 7 dozen, and 1 dozen loaves of bread. The bakeries are able to produce 8 dozen, 1 dozen, and 2 dozen loaves. Note: (Needs) = 11 = (Supplies) The tableau will be a table with one row per supplier (bakeries) labeled with Roman numerals and one column per consumer (stores) labeled with Arabic numerals.
Tableau The tableau for this problem is the following: Stores 2 13 Ba k e ri e s I II III Each square is called a cell and the top-right corner of each cell shows the shipping cost for the given supplier → user.
Stores 2 13 Ba k e ri e s I II III Cells are labeled by row and column: e.g., I-2 or III-1. Shipping costs are per unit (here 1 dozen loaves). The numbers along the bottom are the needs of the corresponding users. (Store 2 needs 7 doz.) The numbers along the right are the capacities of the corresponding suppliers. (Bakery III can supply 2 doz.) Collectively the numbers along the bottom and right are called the rim conditions.
Plant 1 Plant 2 Plant 3 Mine A Mine B Check Indicator Values
Plant 1 Plant 2 Plant 3 Mine A Mine B
Plant 1 Plant 2 Plant 3 Mine A Mine B
Plant 1 Plant 2 Plant 3 Mine A Mine B
Plant 1 Plant 2 Plant 3 Mine A Mine B
Plant 1 Plant 2 Plant 3 Mine A Mine B
Plant 1 Plant 2 Plant 3 Mine A Mine B
Plant 1 Plant 2 Plant 3 Mine A Mine B
Plant 1 Plant 2 Plant 3 Mine A Mine B
Plant 1 Plant 2 Plant 3 Mine A Mine B
Plant 1 Plant 2 Plant 3 Mine A Mine B
Plant 1 Plant 2 Plant 3 Mine A Mine B
Plant 1 Plant 2 Plant 3 Mine A Mine B Mine C
Plant 1 Plant 2 Plant 3 Mine A Mine B Mine C