AEC 851 LINEAR PROGRAMMING APPLICATIONS LOGISTIC AND TRANSPORTATION MODELS TRANSPORTATION PROBLEMS

Transcription

1 AEC 85 LINEAR PROGRAMMING APPLICATIONS LOGISTIC AND TRANSPORTATION MODELS LOGISTIC AND TRANSPORTATION PROBLEMS P These issues are important because of the impact it has on economic decisions and the flow of items P Examples: < How to meet logistic needs < What is the optimum flow of goods from areas of surplus to areas of deficit < What impact will transportation have on trade flows < How to minimize the overall cost of moving items from one region to another < Others (food safety, customer satisfaction, etc.) P Will be examining several transportation models < Simple transportation model < Trans-shipment model < Lockset Method (optional)

2 SIMPLE TRANSPORTATION MODEL A 5(+) B 3(+) C 2(+) 4(-) Y 5(-) Z (-) (Simple Transportation Model Continued) MODEL ASSUMPTIONS:. Need to determine the net supply (or plant output) and demand for each region 2. Supply demand 3. There is a central shipping and receiving point in each region 4. The transportation cost between shipping points is known 5. The objective is to minimize total transportation costs 6. There are no economies of scale in transportaion costs Cost per unit distance (e.g., mile) can decrease as the distance increase Cost per unit can not decrease as volumn increases

3 (Simple Transportation Model Continued) 2 A 5(+) 3 4 B 3(+) 4 3 C 2(+) 2 3 4(-) Y 5(-) Z (-) Use a MODI Chart to help organize this information for analysis (Simple Transportation Model Continued) MODI CHART Supply A B C Y Z

4 (Simple Transportation Model Continued) From a MODI Chart to a Linear Programming Tableau Constraint C j MA A MA B MA C MIN MIN Y MIN Z S I G N B i (Simple Transportation Model Continued) Linear Programming Results Optimum Solution Objective Function = -2, Level of Activities Activity Level (A! ) 2 2 (B! ) 2 4 (A! Y) 3 6 (C! Y) 2 8 (B! Z) Slack Levels Row Level 2(B). 4 5 Shadow Prices Row Price (A) 2(B) 3(C) 3 4() -2 5(Y) -4 6(Z) -3 Supply Demand Non-Optimum Activities Activity Cost 3 (C! ) 4 7 (A! Z) 2 9 (C! Z) 2

5 (Simple Transportation Model Continued) 2 2 A 5(+) 3 B 3(+) 2 C 2(+) 4(-) Y 5(-) Z (-) TRANS-SHIPMENT MODEL P An extension of the simple transportation model < The same assumptions apply to the trans-shipment model P Used to minimize total system cost that examines both transportation and processing costs < Still has points of origin (excess supply) and destination (unmet demand) < Considers both the transportation costs to the processing plants and the transportation costs on to the end user < Considers the processing costs < Address the capacity of the processing plants < With the use of heuristics, it is possible to address: The optimum number of plants needed The optimum size of each plant The optimum location for each of the plants

6 (Trans-Shipment Model Continued) 8 A 5(+) B 3(+) 6 Y Z 32(-) H I C 2(+) 4(-) 8(-) 3 (Trans-Shipment Model Continued) MODI CHART - Supply A B C H I Max Cap = 6 Min Cap = 455 Max Cap = 5 Min Cap = 375 Variable Processing Costs for Plant H ( 7 )=5 Variable Processing Costs for Plant I ( 8 )=8 In processing, the yield is 8% for both plants

7 (Trans-Shipment Model Continued) MODI CHART - 2 Plants H I Y Z Note: This is built as sub-modules (Trans-Shipment Model Continued) TABLEAU PROBLEM Trans-Shipment Model TABLEAU Iter. CONSTRAINTS C j SupplyA SIGN B i A º B º C º A º B º C º Proc. Proc. H º H º H º I º I º I º H H H I I I H I Y Z Y Z SupplyB 3. 3SupplyC 2 4Amt.IntoH Amt.IntoI MaxCapH 6 7MaxCapI 5 8MinCapH 455 9MinCapI 375 Amt Out Of H -.8 Amt Out of I Demand 32 3 Demand Y 4 4 Demand Z 8

8 (Trans-Shipment Model Continued) OPTIMUM RESULTS Optimum Solution Objective Function = -3,379.5 Optimum Solution Activities Activity in Solution Level (A to Plant H) 5 5 (B to Plant I) 3. 6 (C to Plant I) (Proc at Plant H) 5 8 (Proc at Plant I) 5 (Plant H to Y) 32 (Plant H to Z) 8 2 (Plant I to ) 32 3 (Plant I to Y) 8 (Trans-Shipment Model Continued) OPTIMUM FLOWS A 5(+) B 3.(+) 32 Y Z 4(-) 5 H 5(-) I C 2(+) (-)

9 (Trans-Shipment Model Continued) OPTIMUM RESULTS CONT. Constraints shadow prices and slacks Row Shadow Price Excess Supply A 2 2 Supply B 5 3 Supply C. 4 Amount into Plant H 9 5 Amount into Plant I 6 Max capacity Plant H 7 Max capacity Plant I.6 8 Min capacity Plant H 45 9 Min capacity Plant I 25 Amount out of Plant H 3 Amount out of Plant I 37 2 Demand Demand Y Demand Z -39 (Trans-Shipment Model Continued) P Some observations about the results < Activities in solutions The flows seem reasonable The low cost plant (H) is not at full capacity The high cost plant (I) is at full capacity < Shadow prices It possible, expand production in region B Region C is the less desirable to expand production Explore expanding plant I Region is the most costly demand to fulfill P What decisions would you explore?

10 ETENSIONS OF THE TRANSSHIPMENT MODEL P Use to determine plant locations and size < Make use of heuristics Start with a large number of options and use shadow prices to narrow alternatives P Spatial equilibrium model < Used to examine shifts in supply, and/or demand, and/or transportation costs < Two approaches:. Transshipment model in a looping process Factors and Trends of Regional Shift of Production: Analysis of the Pork Sector, B.B. Adhikari, Quadratic programming The Impact of Liberalized Trade on the World Rice Market, Arkansas Research Report 53, Gail Grammar, et. al. LOCKSET MODEL P Used to address routing problems where the capacity of the transportation systems is constrained < Examples: Milk truck routes Where to locate warehouses School bus routes Others P The method may not always find the global optimum but it is relative simple and it finds very good solutions P Since this model addresses a repetitive problem, the savings generated can be substantial

11 (LOCKSET Model Continued) Lockset Model Example 7 D D 4 D D 5 5 D 2 Load Size Warehouse Truck Capacity = 3 (LOCKSET Model Continued) LOCKSET Final Routes 7 D D 4 D D 5 5 D 2 Load Size Warehouse Truck Capacity = 3