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ARE 621 Quantitative Methods for Resource Economics
Lecture 7
Tatiana (Tanya) Borisova, 2030 Ag Sci Bld.,
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Plan
• Mid-Term Exam
• General Algebraic Modeling System (GAMS)
• GAMS solution to MIP warehouse location problem
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Mid-Term Exam
• One hour
• Four sections:
1. Multiple choice • 10 out of 13 questions• 20 points total
2. True / False• 5 questions• 10 points total
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Reminder: term-paper outline –due on Oct. 10
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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
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General Algebraic Modeling System (GAMS)
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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 -- 75 th. tons, and Eastwood -- 50 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.
Warehouse Location Problem
Based on McCarl and Spreen
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Warehouse Location Characteristics
WarehouseLocation
Annual Capacity (th. ton)
Construction Cost ($)
Life Span (years)
A Unlimited 500 10
B 60 720 12
C 70 680 10
Based on McCarl and Spreen
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Warehouse Location Problem
Delivery Cost Per unit
Shipping Points Supply Warehouse
1 2 A B C A 1 6 - - - B 2 3 - - -
Warehouse
C 8 1 - - - Westwood 4 7 4 3 5 Demand Eastwood 8 6 6 4 3
Based on McCarl and Spreen
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Warehouse Location Problem: MIP Model
Based on McCarl and Spreen
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Construction cost
Transp. cost from plants to warehouses
Transp. cost from warehouses to markets
Transp. cost from plants to markets
Supply constr.
Demand constr.
Intermediate node const.Wareh. capacityconst.
Config.const.
Obj. F.
Non-negativity / integer const.
Warehouse Location Problem: MIP Model
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Warehouse Location: MIP Formulation
Indices
i – plant index
j - demand point index
k - warehouse location index
Based on McCarl and Spreen
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Data• Supply
• Demand
• Per unit transportation costsPlant to warehouseWarehouse to marketsPlant to markets
• Warehouse location characteristicsCapacityConstruction costLife span
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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.
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MIP Model: Equations
• Objective function - total costMin 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
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Warehouse Location: GAMS Sets
Indices
i – plant index
k - warehouse location index
j - demand point index
SETS
SUPPLYL plants /S1,S2/
WAREHOUSE warehouse locations /A,B,C/
MARKET demand locations /D1,D2/
Based on McCarl and Spreen
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GAMS Sets
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Data Definitions
• Scalars• Parameters• Tables
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GAMS Parameters• Annual demand at each sub-urban locations is: Westwood -- 75 th.
tons, and Eastwood -- 50 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) quantity demanded / D1 75, D2 50/;
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GAMS: Parameter Definition
GAMS reserved word
Verbal description
Name of Data Item Index
Description
Name of each data set index
Parameter Value
semicolon
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Data Definition: Tables
Delivery Cost Per unit
Shipping Points Supply Warehouse
1 2 A B C A 1 6 - - - B 2 3 - - -
Warehouse
C 8 1 - - - Westwood 4 7 4 3 5 Demand Eastwood 8 6 6 4 3
Based on McCarl and Spreen
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GAMS: Tables
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GAMS Tables
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GAMS Tables
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Remaining Information:
• Warehouse location characteristics
Warehouse Annual Capacity (th. ton)
Investment Cost (th. $)
Life Span (years)
A Unlimited 500 10
B 60 720 12
C 70 680 10
Based on McCarl and Spreen
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Add Remaining Data
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GAMS Variables
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GAMS Variables
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GAMS Variables
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GAMS Equations: Declaration
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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.
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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
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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
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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
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Recall: Warehouse Data
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GAMS Equations: Definition
• Configuration constraint
Vk – 0-1 variable indicating whether k-th warehouse is constructedAmk = 1bm = 1
Based on McCarl and Spreen
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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:
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GAMS Model
• Two ways to define a model:
Or
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SOLVE Statement
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Suggested Reading
• GAMS Tutorial by R. Rosenthalhttp://www.gams.com/docs/gams/Tutorial.pdf
• McCarl and Spreen, GAMS book, Ch. 5http://agecon2.tamu.edu/people/faculty/mccarl-bruce/books.htm
• Thompson and Thore - handout
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Lab Work:
• Correct GAMS program for Warehouse Location Problem
• Few words about homework