ARE 621 Quantitative Methods for Resource Economics

Transcription

1 ARE 621 Quantitative Methods for Resource Economics Lecture 7 Tatiana (Tanya) Borisova, 2030 Ag Sci Bld., tatiana.borisova@mail.wvu.edu 1

2 Plan Mid-Term Exam General Algebraic Modeling System (GAMS) GAMS solution to MIP warehouse location problem 2

3 One hour Mid-Term Exam Four sections: 1. Multiple choice 10 out of 13 questions 20 points total 2. True / False 5 questions 10 points total 3

4 Reminder: term-paper outline due on Oct. 10 4

5 Mid-Term Exam (cont.) Four sections (cont.) 3. Short Answer 2 out of 4 concepts 10 points total 4. Problem 6 out of 9 problems 60 points total Do not forget to sign your name 5

6 General Algebraic Modeling System (GAMS) 6

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11 Warehouse Location Problem Acme Block Company considers construction of warehouses to ship concrete blocks from 2 plants to 2 suburban locations. Annual demand at each sub-urban locations is: Westwood th. tons, and Eastwood th. tons. Acme has two plants. Plant 1 can produce 50 th. tons per year, and plant 2 can produce 75 th tons per year. Three warehouse locations that Acme is considering are referred to as A, B, and C. They are described on the next slide. Only one warehouse can be built. Based on McCarl and Spreen 11

12 Warehouse Location Characteristics Warehouse Location Annual Capacity (th. ton) Construction Cost ($) Life Span (years) A Unlimited B C Based on McCarl and Spreen 12

13 Warehouse Location Problem Delivery Cost Per unit Warehouse Demand Shipping Points Supply Warehouse 1 2 A B C A B C Westwood Eastwood Based on McCarl and Spreen 13

14 Warehouse Location Problem: MIP Model Based on McCarl and Spreen 14

15 Obj. F. Construction cost Warehouse Location Problem: MIP Model Transp. cost from plants to warehouses Transp. cost from warehouses to markets Transp. cost from plants to markets Supply constr. Demand constr. Wareh. capacity const. Intermediate node const. Config. const. Non-negativity / integer const. Based on McCarl and Spreen 15

16 Warehouse Location: MIP Formulation Indices i plant index j - demand point index k - warehouse location index Based on McCarl and Spreen 16

17 Data Supply Demand Per unit transportation costs Plant to warehouse Warehouse to markets Plant to markets Warehouse location characteristics Capacity Construction cost Life span 17

18 Variables Vk 0-1 variable indicating whether k-th warehouse is constructed Xik - continuous variable indicating the quantity shipped from supply point i to warehouse k; Ykj - a continuous variable indicating the quantity shipped from warehouse k to demand point j; Zij - a continuous variable indicating the quantity shipped from supply point i directly to demand point j. 18

19 MIP Model: Equations Objective function - total cost Min Construction cost + Transp. cost from plants to warehouses + Transp. cost from warehouses to markets + Transp. cost from plants to markets Constraints Supply Demand Warehouse Supply and Demand balance Warehouse capacity Configuration only 1 warehouse Non-negativity / integer / binary 19

20 Warehouse Location: GAMS Sets Indices i plant index SETS SUPPLYL plants /S1,S2/ k - warehouse location index WAREHOUSE warehouse locations /A,B,C/ j - demand point index MARKET demand locations /D1,D2/ Based on McCarl and Spreen 20

21 GAMS Sets 21

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23 Data Definitions Scalars Parameters Tables 23

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25 GAMS Parameters Annual demand at each sub-urban locations is: Westwood th. tons, and Eastwood th. tons. Plant 1 can produce 50 th. tons per year, and plant 2 can produce 75 th tons per year. PARAMETERS SUPPLY(SUPPLYL) quantity available at each supply point /S1 50, S2 75/ DEMAND(MARKET) / D1 75, D2 50/; quantity demanded 25

26 GAMS: Parameter Definition GAMS reserved word Name of Data Item Index Description Verbal description Name of each data set index Parameter Value semicolon 26

27 Data Definition: Tables Delivery Cost Per unit Warehouse Demand Shipping Points Supply Warehouse 1 2 A B C A B C Westwood Eastwood Based on McCarl and Spreen 27

28 GAMS: Tables 28

29 GAMS Tables 29

30 GAMS Tables 30

31 Remaining Information: Warehouse location characteristics Warehouse Annual Capacity (th. ton) Investment Cost (th. $) Life Span (years) A Unlimited B C Based on McCarl and Spreen 31

32 Add Remaining Data 32

33 GAMS Variables 33

34 GAMS Variables 34

35 GAMS Variables 35

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37 GAMS Equations: Declaration 37

38 GAMS Equations: Definition Limits of Supply Available Xik - continuous variable indicating the quantity shipped from supply point i to warehouse k; Zij - a continuous variable indicating the quantity shipped from supply point i directly to demand point j. 38

39 GAMS Equations: Definition Min Requirements at Demand Market Ykj - a continuous variable indicating the quantity shipped from warehouse k to demand point j; Zij - a continuous variable indicating the quantity shipped from supply point i directly to demand point j. Based on McCarl and Spreen 39

40 GAMS Equations: Definition Shipment balance constraints Xik - continuous variable indicating the quantity shipped from supply point i to warehouse k; Ykj - a continuous variable indicating the quantity shipped from warehouse k to demand point j; Based on McCarl and Spreen 40

41 GAMS Equations: Definition Warehouse capacity constraint Vk 0-1 variable indicating whether k-th warehouse is constructed Ykj - a continuous variable indicating the quantity shipped from warehouse k to demand point j; Based on McCarl and Spreen 41

42 Recall: Warehouse Data 42

43 GAMS Equations: Definition Configuration constraint Vk 0-1 variable indicating whether k-th warehouse is constructed Amk = 1 bm = 1 Based on McCarl and Spreen 43

44 GAMS Equations: Definition Vk 0-1 variable indicating whether k-th warehouse is constructed Xik - continuous variable indicating the quantity shipped from supply point i to warehouse k; Ykj - a continuous variable indicating the quantity shipped from warehouse k to demand point j; Zij - a continuous variable indicating the quantity shipped from supply point i directly to demand point j. Objective function: 44

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47 GAMS Model Two ways to define a model: Or 47

48 SOLVE Statement 48

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68 Suggested Reading GAMS Tutorial by R. Rosenthal McCarl and Spreen, GAMS book, Ch. 5 Thompson and Thore - handout 68

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70 Lab Work: Correct GAMS program for Warehouse Location Problem Few words about homework 70