Transportation, Transshipment, and Assignment Problems Prof. Yongwon Seo (seoyw@cau.ac.kr) College of Business Administration, CAU
Transportation, Transshipment, and Assignment Problems TRANSPORTATION PROBLEM
Transportation Model: Eample Problem Definition and Data 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? 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 2. Omaha 3. Des Moines $ 6 7 4 $ 8 11 5 $ 10 11 12 3
Transportation Model: Schematic Diagram 150 1 6 8 A 200 10 7 175 2 11 11 B 100 4 5 205 3 12 C 300 4
Transportation Model: Formulation Minimize Z = $6 1A 8 1B 10 1C 7 2A 11 2B 11 2C 4 3A 5 3B 12 3C subject to: 1A 1B 1C = 150 2A 2B 2C = 175 3A 3B 3C = 275 1A 2A 3A = 200 1B 2B 3B = 100 1C 2C 3C = 300 ij 0 ij = tons of wheat from each grain elevator, i, i = 1, 2, 3, to each mill j, j = A,B,C 5
Transportation Table To From A. Chicago B. St.Louis C. Cincinnati Supply 1. Kansas 6 8 10 150 2. Omaha 7 11 11 175 3. Des Moines 4 5 12 275 Demand 200 100 300 600 6
Ecel Solver Objective function =D5D6D7 Decision variables in cells C5:E7 =C7D7E7 Cost array in cells K5:M7 7
Ecel Solver Supply constraints Demand constraints 8
9 Solution
Schematic Diagram of a Transportation Problem E) Harley s Sand and Gravel Pit supplies topsoil for three residential housing developments from three different farms. Unit Transportation Cost Farms Project 10
11 Transportation Table for Harley s Sand and Gravel
LP formulation 11 21 31 12 22 32 13 23 33 Minimize Z 4 subject to Farm A : Farm B : Farm C : 11 21 31 11 2 12 22 32 12 8 13 23 33 13 100 200 200 5 21 22 9 23 7 31 6 32 3 33 Project 1: Project 2 : Project 3: 11 12 13 21 22 23 31 32 33 50 150 300 12 All variables 0
LP formulation : General Form 13 j i all for m j D n i S to subject c Minimize ij n i j ij m j i ij n i m j ij ij, 0,, 1,,, 1,, 1 1 1 1
14 Using Ecel to solve transportation prob.
15 When the Shipping Route between Farm B and Project 1 Is Prohibited
Transportation, Transshipment, and Assignment Problems TRANSSHIPMENT PROBLEM
Transshipment Problems A transportation problem in which some locations are used as intermediate shipping points, thereby serving both as origins and as destinations. Involve the distribution of goods from intermediate nodes in addition to multiple sources and multiple destinations. 17
18 Transshipment: Eample
Transshipment: Formulation Minimize Z = $16 13 10 14 12 15 15 23 14 24 17 25 6 36 8 37 10 38 7 46 11 47 11 48 4 56 5 57 12 58 subject to: 13 14 15 = 300 23 24 25 = 300 36 46 56 = 200 37 47 57 = 100 38 48 58 = 300 13 23 36 37 38 = 0 14 24 46 47 48 = 0 15 25 56 57 58 = 0 ij 0 19
Transshipment: Ecel Solver =SUM(B6:B7) Objective function =SUM(B6:D6) Cost arrays =SUM(C13:C15) =SUM(C13:E13) Constraints for transshipment flows; i.e., shipments in = shipments out 20
Transshipment: Ecel Solver Transshipment constraints in cells C20:C22 21
22 Transshipment: Solution
E. The manager of Harley s Sand and Gravel Pit has decided to utilize two intermediate nodes as transshipment points for temporary storage of topsoil. Farm A Farm B Farm C 23
Formulation Minimize Z 3 subject to Node1: Node 2 : Node 3: Node 4 : Node 5: Node 6 : Node 7 : Node8: ij 14 24 34 14 15 46 47 48 15 25 35 24 25 56 57 58 14 2 100 200 200 34 35 50 150 300 0, for all i, j 15 4 46 56 47 57 48 5 48 58 58 24
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Transportation, Transshipment, and Assignment Problems ASSIGNMENT PROBLEM
Assignment Problems Involve the matching or pairing of two sets of items such as jobs and machines, secretaries and reports, lawyers and cases, and so forth. Have different cost or time requirements for different pairings. Special form of linear programming model similar to the transportation model. Supply at each source and demand at each destination limited to one unit. 27
Assignment Model Problem: Assign four teams of officials to four games in a way that will minimize total distance traveled by the officials. Supply is always one team of officials, demand is for only one team of officials at each game. 28
Formulation Minimize Z = 210 AR 90 AA 180 AD 160 AC 100 BR 70 BA 130 BD 200 BC 175 CR 105 CA 140 CD 170 CC 80 DR 65 DA 105 DD 120 DC subject to: AR AA AD AC = 1 ij 0 BR BA BD BC = 1 CR CA CD CC = 1 DR DA DD DC = 1 AR BR CR DR = 1 AA BA CA DA = 1 AD BD CD DD = 1 AC BC CC DC = 1 29
Ecel Solver Objective function Decision variables, C5:F8 =C5D5E5F5 =D5D6D7D8 Mileage array 30
Ecel Solver Simple LP 31
32 Solution
Assignment Problem: Eample A manager has prepared a table that shows the cost of performing each of five jobs by each of five employees (see Table 68). According to this table, job I will cost $15 if done by Al. $20 if it is done by Bill, and so on. The manager has stated that his goal is to develop a set of job assignments that will minimize the total cost of getting all four jobs done. It is further required that the jobs be performed simultaneously, thus requiring one job being assigned to each employee. In the past, to find the minimumcost set of assignments, the manager has resorted to listing all of the different possible assignments (i.e., complete enumeration) for small problems such as this one. But for larger problems, the manager simply guesses because there are too many possibilities to try to list them. For eample, with a 5X5 table, there are 5! = 120 different possibilities; but with, say, a 7X7 table, there are 7! = 5,040 possibilities. 33
Problem 34 j i all for to Subject Z Minimize ij, 0, 1 1 1 1 1 1 16 18 20 15 55 45 35 25 15 52 42 32 22 12 51 41 31 21 11 55 54 53 52 51 25 24 23 22 21 15 14 13 12 11 55 54 12 11 Integer?
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MultiCriteria DecisionMaking Models Prof. Yong Won Seo (seoyw@cau.ac.kr) College of Business Administration, CAU
MultiCriteria DecisionMaking Study of problems with several criteria, i.e., multiple criteria, instead of a single objective when making a decision. Three techniques discussed: goal programming, the analytical hierarchy process and scoring models. Goal programming is a variation of linear programming considering more than one objective (goals) in the objective function. The analytical hierarchy process develops a score for each decision alternative based on comparisons of each under different criteria reflecting the decision makers preferences. Scoring models are based on a relatively simple weighted scoring technique. 37
Applications of MCDA Some of the MCDM methods are: Analytic hierarchy process (AHP) Goal programming (GP) Data Envelopment Analysis (DEA) Inner Product of Vectors (IPV) Multiattribute value theory (MAVT) Multiattribute utility theory (MAUT) Multiattribute global inference of quality (MAGIQ) ELECTRE (Outranking) PROMETHÉE (Outranking) The evidential reasoning approach Dominancebased Rough Set Approach (DRSA) Aggregated indices randomization method (AIRM) Nonstructural fuzzy decision support system (NSFDSS) Grey relational analysis (GRA) Superiority and inferiority ranking method (SIR method) (source:: Wikipedia) 38
Goal Programming Goal Programming (GP) A variation of linear programming that allows multiple objectives (goals) soft (goal) constraints or a combination of soft and hard (nongoal) constraints There are priorities in the goals. (prioritized goal programming model) In order to obtain an acceptable solution when there are conflicts, it becomes necessary to make tradeoffs: satisfying hard constraints and achieving higher levels of certain goals with sacrificing other goals.
Standard LP Eample Beaver Creek Pottery Company Eample Maimize Z = 40 1 50 2 subject to: 1 1 2 2 40 hours of labor (per day) 4 1 3 2 120 pounds of clay (per day) 1, 2 0 Where: 1 = number of bowls produced 2 = number of mugs produced 40
Modified Problem Labor : overtime allowed (but not desirable) Storage space can be added (but not desirable) The company has the following objectives, listed in order of importance: 1. To avoid layoffs, the company does not want to use fewer than 40 hours of labor per day 2. The company would like to achieve a satisfactory profit level of $1600 per day 3. Because the clay must be stored in a special place so that it does not dry out, the company prefers not to prepare more than 120 pounds on hand each day 4. Because of high overtime cost, the company would like to minimize the amount of overtime. 41
Goal Constraints Labor goal: 1 2 2 d 1 d 1 = 40 (hours/day) Profit goal: 40 1 50 2 d 2 d 2 = 1,600 ($/day) Material goal: 4 1 3 2 d 3 d 3 = 120 (lbs of clay/day) 42
Objective Function Minimize P 1 d 1, P 2 d 2, P 3 d 3, P 4 d 1 Add one by one based on priorities 1. Min labor constraint (priority 1 less than 40 hours labor) Minimize P 1 d 1 2. Add profit goal constraint (priority 2 achieve profit of $1,600): Minimize P 1 d 1, P 2 d 2 3. Add material goal constraint (priority 3 avoid keeping more than 120 pounds of clay on hand) Minimize P 1 d 1, P 2 d 2, P 3 d 3 4. Add overtime constraint (priority 4 minimum overtime): Minimize P 1 d 1, P 2 d 2, P 3 d 3, P 4 d 1 43
Graphical Interpretation Minimize P 1 d 1, P 2 d 2, P 3 d 3, P 4 d 1 subject to: 1 2 2 d 1 d 1 = 40 40 1 50 2 d 2 d 2 = 1,600 4 1 3 2 d 3 d 3 = 120 1, 2, d 1, d 1, d 2, d 2, d 3, d 3 0 44
Graphical Interpretation Minimize P 1 d 1, P 2 d 2, P 3 d 3, P 4 d 1 subject to: 1 2 2 d 1 d 1 = 40 40 1 50 2 d 2 d 2 = 1,600 4 1 3 2 d 3 d 3 = 120 1, 2, d 1, d 1, d 2, d 2, d 3, d 3 0 45
Graphical Interpretation Minimize P 1 d 1, P 2 d 2, P 3 d 3, P 4 d 1 subject to: 1 2 2 d 1 d 1 = 40 40 1 50 2 d 2 d 2 = 1,600 4 1 3 2 d 3 d 3 = 120 1, 2, d 1, d 1, d 2, d 2, d 3, d 3 0 46
Graphical Interpretation Minimize P 1 d 1, P 2 d 2, P 3 d 3, P 4 d 1 subject to: 1 2 2 d 1 d 1 = 40 40 1 50 2 d 2 d 2 = 1,600 4 1 3 2 d 3 d 3 = 120 1, 2, d 1, d 1, d 2, d 2, d 3, d 3 0 47
Graphical Interpretation Minimize P 1 d 1, P 2 d 2, P 3 d 3, P 4 d 1 subject to: 1 2 2 d 1 d 1 = 40 40 1 50 2 d 2 d 2 = 1,600 4 1 3 2 d 3 d 3 = 120 1, 2, d 1, d 1, d 2, d 2, d 3, d 3 0 48