Operations Research I Transportation problems Ing Lenka Skanderov

Operations Research I Transportation problems Ing. Lenka Skanderová, Ph. D.

• • Typical example Introduction Model creation Initial BF solution Optimality test Factory Ship Warehouse Goods Operations research I 2/60

P & T Company • • Introduction Model creation Initial BF solution Optimality test • Product: canned peas • Canneries: • Bellingham, Washington • Eugene, Oregon • Albert Lea, Minesota • Warehouses • Sacramento, California • Salt Lake City, Utah • Rapid City, South Dakota, • Albuquerque, New Mexico • Shipping costs are the major expense Operations research I COST MINIMIZATION 3/60

• • P & T Company Table Introduction Model creation Initial BF solution Optimality test Shipping Cost ($) per Truckload Warehouse Cannery 1 2 3 4 Output 1 464 513 654 867 75 2 352 416 690 791 125 3 995 682 388 685 100 Allocation 80 65 70 85 Constraints of canneries Constraints of warehouses Operations research I 4/60

P & T Company Model creation • • Introduction Model creation Initial BF solution Optimality test • Minimize • Subject to: Typical structure of constraint coefficients in the transportation problem Operations research I 5/60

P & T Company Model creation • • Introduction Model creation Initial BF solution Optimality test • Minimize • Subject to: Do not forget the nonnegative coefficients! Operations research I 6/60

• • P & T Company Graph representation 464 513 [75] [125] C 1 654 867 352 Introduction Model creation Initial BF solution Optimality test W 1 [-80] W 2 [-65] W 3 [-70] W 4 [-85] 416 C 2 690 995 [100] C 3 682 388 685 Operations research I 791 7/60

Transportation problem Assumptions • • Introduction Model creation Initial BF solution Optimality test • The requirements assumption: Each source has a fixed supply of units, where this entire supply must be distributed to the destinations. Similarly, each destination has a fixed demand for units, where this entire demand must be received from the sources. • 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. Therefore, this cost is just the unit cost of distribution times the number of units distributed. Operations research I 8/60

Transportation problem Feasible solution • • Introduction Model creation Initial BF solution Optimality test The feasible solution property: A transportation problem will have feasible solutions if and only if Operations research I 9/60

Transportation problem General structure • • Introduction Model creation Initial BF solution Optimality test Any linear programming problem that fits the special formulation is of the transportation problem type, regardless of its physical context Operations research I 10/60

• • Transportation problem General structure Introduction Model creation Initial BF solution Optimality test Cost per Unit Distributed Destination Supply Source Demand Operations research I 11/60

• • Transportation problem Example Introduction Model creation Initial BF solution Optimality test • Northern airplane company builds the airplanes for various airline companies. • The company has some contracts to the airplanes and the production of the jet engines must be scheduled. Month Scheduled Maximum Unit Cost of Installations Production Unit Cost of Storage 1 10 25 1. 08 0. 015 2 15 35 1. 11 0. 015 3 25 30 1. 10 0. 015 4 20 10 1. 13 Cost is expressed in milions of dollars Operations research I 12/60

• • Transportation problem Example Introduction Model creation Initial BF solution Optimality test • The company must supply engines for installation as mentioned in the 2 nd column of the table • The company can produce more engines than scheduled and these ones must be storaged. Month Scheduled Maximum Unit Cost of Installations Production Unit Cost of Storage 1 10 25 1. 08 0. 015 2 15 35 1. 11 0. 015 3 25 30 1. 10 0. 015 4 20 10 1. 13 Cost is expressed in milions of dollars Operations research I 13/60

• • Transportation problem Supply and demand Introduction Model creation Initial BF solution Optimality test • The goal: to minimize the costs for production and storage of the jet engines Month Scheduled Maximum Unit Cost of Installations Production Unit Cost of Storage 1 10 25 1. 08 0. 015 2 15 35 1. 11 0. 015 3 25 30 1. 10 0. 015 4 20 10 1. 13 • Task: • Identify supply and demand Operations research I 14/60

• • Transportation problem Supply and demand Introduction Model creation Initial BF solution Optimality test • The goal: to minimize the costs for production and storage of the jet engines Month Scheduled Maximum Unit Cost of Installations Production Unit Cost of Storage 1 10 25 1. 08 0. 015 2 15 35 1. 11 0. 015 3 25 30 1. 10 0. 015 4 20 10 1. 13 demand supply • Task: • Create table using Big M method Operations research I 15/60

Transportation problem Supply and demand • • Introduction Model creation Initial BF solution Optimality test • The goal: to minimize the costs for production and storage of the jet engines We need „=“ • What we will use? Operations research I 16/60

Transportation problem Slack variable • • Introduction Model creation Initial BF solution Optimality test • The goal: to minimize the costs for production and storage of the jet engines We need „=“ • What we will use? SLACK VARIABLE • Slack variable express the number of built engines which have not been used • How many jet engines have not been used? Operations research I 17/60

• • Transportation problem Slack variable Introduction Model creation Initial BF solution Optimality test • The number of built engines which have not been used: Month Scheduled Maximum Unit Cost of Installations Production Unit Cost of Storage 1 10 25 1. 08 0. 015 2 15 35 1. 11 0. 015 3 25 30 1. 10 0. 015 4 20 10 1. 13 Operations research I 18/60

• • Transportation problem Table Introduction Model creation Initial BF solution Optimality test dummy destination Cost per Unit Distributed Destination Source Supply 1 2 3 4 5(D) 1 1. 080 1. 095 1. 110 1. 125 0 25 2 M 1. 110 1. 125 1. 140 0 35 3 M M 1. 100 1. 115 0 30 4 M M M 1. 130 0 10 10 15 25 20 30 Demand Operations research I 19/60

Trensportation and assignment problems Streamlined simplex method Operations research I

• • Example Water for cities Introduction Model creation Initial BF solution Optimality test • How to supply cities from three rivers to satisfy the needs of cities and minimize the total cost to the district Cost (Ten of $) per Acre Foot Berdoo Los Devils San Go Hollyglass Supply Colombo River 16 13 22 17 50 Sacron River 14 13 19 15 60 Calorie River 19 20 23 - 50 Minimum needed 30 70 0 10 Requested 50 70 30 infinity Operations research I 21/60

• • Example Supply and demand Introduction Model creation Initial BF solution Optimality test • Identify supply and demand • Where is the problem? Cost (Ten of $) per Acre Foot Berdoo Los Devils San Go Hollyglass Supply Colombo River 16 13 22 17 50 Sacron River 14 13 19 15 60 Calorie River 19 20 23 - 50 Minimum needed 30 70 0 10 Requested 50 70 30 infinity Operations research I 22/60

• • Example Supply and demand Introduction Model creation Initial BF solution Optimality test Cost (Ten of $) per Acre Foot Berdoo Los Devils San Go Hollyglass Supply Colombo River 16 13 22 17 50 Sacron River 14 13 19 15 60 Calorie River 19 20 23 - 50 Minimum needed 30 70 0 10 Requested 50 70 30 infinity ? demand • Problem: Which row is the demand? Minimum need or Requested? Operations research I 23/60

• • Example Upper and lower bounds for demand Introduction Model creation Initial BF solution Optimality test • Identify the lower and upper bounds for each city Cost (Ten of $) per Acre Foot Berdoo Los Devils San Go Hollyglass Supply Colombo River 16 13 22 17 50 Sacron River 14 13 19 15 60 Calorie River 19 20 23 - 50 Minimum needed 30 70 0 10 lower bound Requested 50 70 30 infinity upper bound Upper bounds are identified based on supplies Operations research I 24/60

• • Example Upper bound for Hollyglass Introduction Model creation Initial BF solution Optimality test Cost (Ten of $) per Acre Foot Berdoo Los Devils San Go Hollyglass Supply Colombo River 16 13 22 17 50 Sacron River 14 13 19 15 60 Calorie River 19 20 23 - 50 Minimum needed 30 70 0 10 lower bound Requested 50 70 30 infinity upper bound Operations research I 25/60

• • Example Upper bound for Hollyglass Introduction Model creation Initial BF solution Optimality test Cost (Ten of $) per Acre Foot Berdoo Los Devils San Go Hollyglass Supply Colombo River 16 13 22 17 50 Sacron River 14 13 19 15 60 Calorie River 19 20 23 - 50 Minimum needed 30 70 0 10 lower bound Requested 50 70 30 60 upper bound Operations research I 26/60

• • Example Dummy source Introduction Model creation Initial BF solution Optimality test Cost (Ten of $) per Acre Foot Berdoo Los Devils San Go Hollyglass Supply Colombo River 16 13 22 17 50 Sacron River 14 13 19 15 60 Calorie River 19 20 23 - 50 Minimum needed 30 70 0 10 Requested 50 70 30 60 • Problem: Demands must be constants. Not bounded decision variables Operations research I 27/60

• • Example Dummy source Introduction Model creation Initial BF solution Optimality test Cost (Ten of $) per Acre Foot Berdoo Los Devils San Go Hollyglass Supply Colombo River 16 13 22 17 50 Sacron River 14 13 19 15 60 Calorie River 19 20 23 - 50 Minimum needed 30 70 0 10 Requested 50 70 30 60 The difference indicates the use of dummy supply Operations research I 28/60

Introduction Model creation Initial BF solution Optimality test • • Example Table with dummy source Cost (Ten of $) per Acre Foot Berdoo Los Devils San Go Hollyglass Colombo River 16 13 22 17 50 Sacron River 14 13 19 15 60 Calorie River 19 20 23 M 50 Dummy source 0 0 50 Demand 50 70 30 60 Operations research I Supply • • No provision has been made to supply Hollyglass with Calorie River water. We will use Big M to penalize the solution, where the Hollyglass is supplied by Calorie River 29/60

Introduction Model creation Initial BF solution Optimality test • • Example Don‘t forget minimum request Cost (Ten of $) per Acre Foot Berdoo Los Devils San Go Hollyglass Supply Colombo River 16 13 22 17 50 Sacron River 14 13 19 15 60 Calorie River 19 20 23 - 50 Dummy source 0 0 (!) 0 0 50 30 (!) 70 0 10 50 70 30 60 Minimum request Demand Operations research I 30/60

Example – where is the penalization? • • Introduction Model creation Initial BF solution Optimality test Cost (Ten of $) per Acre Foot Berdoo (min) Berdoo (extra) Los Devils San Go Hollyglass Supply Colombo River 16 16 13 22 17 50 Sacron River 14 14 13 19 15 60 Calorie River 19 19 20 23 M 50 Dummy source M 0 0 50 Minimum request 30 0 70 0 10 Demand 30 20 70 30 60 Operations research I 31/60

• • Vogel‘s approximation method 1 st step Introduction Model creation Initial BF solution Optimality test • For each row and column: • Calculate difference between two smallest values Destination 1 2 3 4 5 Supply Diff. 1 16 16 13 22 17 50 3 2 14 14 13 19 15 60 1 3 19 19 20 23 M 50 0 4 M 0 0 50 0 Demand 30 20 70 30 60 Operations research I 32/60

• • Vogel‘s approximation method 1 st iteration Introduction Model creation Initial BF solution Optimality test • For each row and column: • Calculate difference between two smallest values Destination 1 2 3 4 5 Supply Diff. 1 16 16 13 22 17 50 3 2 14 14 13 19 15 60 1 3 19 19 20 23 M 50 0 4 M 0 0 50 0 Demand 30 20 70 30 60 Diff. 2 14 0 19 15 Operations research I 33/60

• • Vogel‘s approximation method 1 st iteration Introduction Model creation Initial BF solution Optimality test • For each row and column: • Largest difference is circled, smallest unit cost in the same row or column is enclosed in the box Destination 1 2 3 4 5 Supply Diff. 1 16 16 13 22 17 50 3 2 14 14 13 19 15 60 1 3 19 19 20 23 M 50 0 4 M 0 0 50 0 Demand 30 20 70 30 60 Diff. 2 14 0 19 15 Operations research I 30 < 50 Eliminate column 4 34/60

• • Vogel‘s approximation method 2 nd iteration Introduction Model creation Initial BF solution Optimality test • For each row and column calculate the difference between two smallest values • Select the largest difference and in the same row or column select the smallest unit cost Destination 1 2 3 5 Supply Diff. 1 16 16 13 17 50 3 2 14 14 13 15 60 1 3 19 19 20 M 50 0 4 M 0 50 -30 0 Demand 30 20 70 60 Diff. 2 14 0 15 Operations research I Eliminate row 4 35/60

• • Vogel‘s approximation method 3 rd iteration Introduction Model creation Initial BF solution Optimality test • For each row and column calculate the difference between two smallest values • Select the largest difference and in the same row or column select the smallest unit cost Destination 1 2 3 5 Supply Diff. 1 16 16 13 17 50 3 2 14 14 13 15 60 1 3 19 19 20 M 50 0 Demand 30 20 70 40 Diff. 2 2 0 2 Operations research I Eliminate row 1 36/60

• • Vogel‘s approximation method 4 th iteration Introduction Model creation Initial BF solution Optimality test • For each row and column calculate the difference between two smallest values • Select the largest difference and in the same row or column select the smallest unit cost Destination 1 2 3 5 Supply Diff. 2 14 14 13 15 60 1 3 19 19 20 M 50 0 Demand 30 20 20 40 Diff. 5 5 7 M-15 Eliminate column 5 Operations research I 37/60

Vogel‘s approximation method 5 th iteration • • Introduction Model creation Initial BF solution Optimality test • For each row and column calculate the difference between two smallest values • Select the largest difference and in the same row or column select the smallest unit cost Destination 1 2 3 Supply Diff. 2 14 14 13 20 1 3 19 19 20 50 0 Demand 30 20 20 Diff. 5 5 7 Operations research I Eliminate row 2 38/60

Vogel‘s approximation method solution Destination • • Introduction Model creation Initial BF solution Optimality test Variable values: 1 2 3 Supply Diff. 3 19 19 20 50 0 Demand 30 20 0 Diff. 5 5 7 The value of the remaining variables is 0 Operations research I 39/60

Russell‘s approximation method Principle • • Introduction Model creation Initial BF solution Optimality test Destination 1 2 3 4 5 S 1 16 16 13 22 17 50 2 14 14 13 19 15 60 3 19 19 20 23 M 50 4 M 0 0 50 D 30 20 70 30 60 Operations research I 22 19 M M M 19 M 23 M 40/60

Russell‘s approximation method 1 st iteration • • Introduction Model creation Initial BF solution Optimality test Destination 1 2 3 4 5 S 1 16 16 13 22 17 50 22 2 14 14 13 19 15 60 19 3 19 19 20 23 M 50 M 4 M 0 0 50 M D 30 20 70 30 60 M 19 M 23 M Operations research I 41/60

Russell‘s approximation method 1 st iteration • • Introduction Model creation Initial BF solution Optimality test Destination 1 2 3 4 5 S 1 16 16 13 22 17 50 22 2 14 14 13 19 15 60 19 3 19 19 20 23 M 50 M 4 M 0 0 50 M D 30 20 70 30 60 M 19 M 23 M Operations research I 42/60

Russell‘s approximation method 2 nd iteration • • Introduction Model creation Initial BF solution Optimality test Destination 1 2 3 4 5 S 1 16 16 13 22 17 50 22 2 14 14 13 19 15 60 19 3 19 19 20 23 M 50 M D 30 20 70 30 10 19 19 20 23 M Operations research I 43/60

Russell‘s approximation method 3 rd iteration • • Introduction Model creation Initial BF solution Optimality test Destination 1 2 3 4 S 1 16 16 13 22 40 22 2 14 14 13 19 60 19 3 19 19 20 23 50 23 D 30 20 70 30 19 19 20 23 Operations research I 44/60

Russell‘s approximation method 4 th iteration • • Introduction Model creation Initial BF solution Optimality test Destination 1 2 3 4 S 2 14 14 13 19 60 19 3 19 19 20 23 50 23 D 30 20 30 30 19 19 20 23 Operations research I 45/60

Russell‘s approximation method Result • Initial BF solution from Russell‘s approximation: Operations research I Introduction Model creation Initial BF solution Optimality test • • Destination 1 2 3 4 5 S 1 16 16 13 22 17 50 2 14 14 13 19 15 60 3 19 19 20 23 M 50 4 M 0 0 50 D 30 20 70 30 60 46/60

• • Russell‘s approximation method Initial transportation simplex tableau Iteration Destination 0 1 2 16 2 14 14 3 19 19 4(D) M 30 Demand Operations research I 3 16 1 Source Introduction Model creation Initial BF solution Optimality test 0 13 13 4 40 30 5 22 17 10 19 60 M 50 23 0 M 0 0 20 70 30 60 30 50 15 20 20 Supply 50 50 47/60

Russell‘s vs. Vogel‘s approximation • Initial BF solution from Russell‘s approximation: • • Introduction Model creation Initial BF solution Optimality test • Initial BF solution from Vogel‘s approximation: Better initial BF solution • The Russell‘s approximation often provides better results than Vogel‘s. For large problems, both methods are usually used. Operations research I 48/60

Optimality test • • Introduction Model creation Initial BF solution Optimality test • Initial BF solution from Russell‘s approximation: 1 allocation 2 allocations 3 allocations Operations research I 49/60

• • Optimality test Introduction Model creation Initial BF solution Optimality test Destination 1 2 3 4 5 S 1 16 16 13 22 17 50 2 14 14 13 19 15 60 3 19 19 20 23 M 50 4 M 0 0 50 D 30 20 70 30 60 Operations research I 50/60

An iteration Operations research I • • Introduction Model creation Initial BF solution Optimality test 51/60

• • Completed initial transportation simplex tableau Iter. Destination 0 Source Demand Introduction Model creation Initial BF solution Optimality test 1 2 3 13 16 1 16 2 14 30 14 3 19 0 19 4(D) M +2 M+3 +2 0 0 20 +3 13 4 40 22 30 19 20 23 M+4 17 +4 +2 M 5 0 +1 30 -1 15 M 0 30 20 70 30 60 19 19 18 23 22 Operations research I Supply 10 -2 M-22 50 50 -5 60 -5 50 0 50 -22 52/60

Init. transportation simplex tableau Chain reaction Iter. Destination 0 1 Source 2 Demand Operations research I Introduction Model creation Initial BF solution Optimality test • • 3 13 40 13 70 30 4 + - 5 22 17 10 +4 - + +1 50 60 15 19 30 Supply -2 60 53/60

Init. transportation simplex tableau Chain reaction Iter. Destination 0 1 Source 2 Demand Introduction Model creation Initial BF solution Optimality test • • 3 13 40 13 30 + - 4 22 17 10 +4 15 19 +1 70 30 + 5 Supply 50 - 60 -2 60 • Cells (2, 5) and (1, 3) = recipient cells (they recieve additional allocation from donor celss) • Cells (1, 5) and (2, 3) = donor cells Operations research I 54/60

Init. transportation simplex tableau Chain reaction Iter. Destination 0 1 Source 2 Demand Introduction Model creation Initial BF solution Optimality test • • 3 13 40 13 30 + - 4 22 17 10 +4 15 19 +1 70 30 + 5 Supply 50 - 60 -2 60 • The recipient and donor cells correspond to the basic variables in the current BF solution Operations research I 55/60

Init. transportation simplex tableau Chain reaction Iter. Destination 0 1 Source 2 Demand Introduction Model creation Initial BF solution Optimality test • • 3 13 40 13 30 + - 4 22 17 10 +4 15 19 +1 70 30 + 5 Supply 50 - 60 -2 60 • When we find the chain reaction, the donor cell having the smallest allocation automaticaly provides the leaving basic variable. Operations research I 56/60

Init. transportation simplex tableau New BF solution Iter. Destination 0 Source 3 1 13 2 13 Demand Introduction Model creation Initial BF solution Optimality test • • 50 70 20 4 + - 5 22 17 - 50 +4 15 19 +1 30 Supply 10 -2 60 60 • Add the value of the leaving basic variable to the allocation for each recipient cell • Subtract the same amount from the allocation for each donor cell Operations research I 57/60

Init. transportation simplex tableau New BF solution Operations research I • • Introduction Model creation Initial BF solution Optimality test 58/60

• • Transportation simplex tableau Init. transportation simplex tableau Optimalsolution • Optimality test Iter. Destination 0 Source Demand Introduction Model creation Initial BF solution Initial BFtest solution Optimality 1 1 16 2 14 3 19 4(D) M 2 3 +4 +2 30 M+3 50 22 13 20 19 20 0 23 13 16 +4 14 19 0 +2 20 +3 4 M M+2 5 17 +7 0 +4 +1 30 +2 15 M 0 30 20 70 30 60 19 19 20 22 22 Operations research I Supply 40 M-22 20 50 -7 60 -7 50 0 50 -22 59/60

• • Transportation simplex tableau Init. transportation simplex tableau Optimalsolution • Optimality test Iter. Destination 0 Source Demand Introduction Model creation Initial BF solution Initial BFtest solution Optimality 1 1 16 2 14 3 19 4(D) M 2 3 +4 +2 30 M+3 50 22 13 20 19 20 0 23 13 16 +4 14 19 0 +2 20 +3 4 M M+2 5 17 +7 0 +4 +1 30 +2 15 M 0 30 20 70 30 60 19 19 20 22 22 Operations research I Supply 40 M-22 20 50 -7 60 -7 50 0 50 -22 60/60

Init. transportation simplex tableau Literature Optimal solution • • Introduction Model creation Initial BF solution Initial BFtest solution Optimality • Optimality test [1] Literature: Hillier and Lieberman: Introduction to Operations Research, 8 th edition, 2005 Operations research I 60/60