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An Optimization Model in Support of Biomass CoFiring in Coal Fired Power Plants Sandra D. Eksioglu, Hadi Karimi Industrial Engineering Department Clemson University Sun Grant Conference Feb. 2015 Auburn, AL
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An Optimization Model in Support of Biomass Co-‐Firing in Coal Fired Power Plants
Sandra D. Eksioglu, Hadi Karimi Industrial Engineering Department Clemson University
Sun Grant Conference Feb. 2015 Auburn, AL
Outline
I. Motivations II. Research objectives III. Problem description
IV. Optimization model formulation V. A case study
VI. Conclusions and future research
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Motivation Co-‐Ciring biomass is an attractive renewable energy option:
Ø Increases renewable energy without major capital investments and investments in the infrastructure
Ø Low risk option
Ø Reduces emissions of CO2, SO2 and NO2 emissions Ø 5% (15%) co-‐Ciring would reduce CO2 emissions by 5.4% (18.2%)
Ø Minimizes waste -‐ such as, wood waste, agricultural waste -‐ and the environmental problem associated with waste disposal
Ø It is a near term market solution for biomass
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Motivation Renewable electricity generation is expected to increase due to the following reasons (EIA):
Ø Increasing demand for electricity Ø Programs encouraging renewable energy use (e.g. PTC, RPS etc.)
Ø PTC: An income tax credit of 2.3 cents/kilowa;-‐hour Ø The implementation of new environmental rules dampens future
coal use (e.g. Cross-‐State Air Pollution Rule of EPA)
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20.00 40.00 60.00 80.00
100.00 120.00 140.00
2008 2012 2016 2020 2024 2028 2032 2036 2040
Gen
erat
ed E
lect
. (B
ill. K
wh)
Year
Renewable electricity generation by biomass (Billion kilowatt-‐hours), EIA, 2012
Renewable electircity generated by biomass (Billion kilowatthours)
Literature review Ø Currently, 40 of 560 coal-‐Cired power plants in USA co-‐Cire biomass.
Ø Two major barriers for adopting co-‐Ciring are: o Additional plant investment costs o High cost of biomass transportation and inventory holding
Ø The literature is mainly focused on analyzing its technological and economical feasibility.
Ø There are no studies which integrate logistics and investment decisions in coal-‐Cired power plants.
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Research Objectives Ø Develop an optimization model in support of biomass co-‐Ciring decisions in coal-‐Cired power plants.
The model captures the: o Additional costs and savings o Loss of process efCiciencies due to co-‐Ciring
Ø Evaluate the impact of Production Tax Credit (PTC) on renewable electricity production.
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Decision variables o 𝑿↓𝒊𝒋 the amount of biomass (in tons/year) delivered to coal plant j
from supplier i Amount of biomass needed at plant j is: ∑𝑖∈𝑆↑▒𝑋↓𝑖𝑗
o 𝑩↓𝒋 the percentage of coal displaced in plant j
o 𝒀↓𝒋 binary variable which takes the value 1 if 𝐵↓𝑗 ≤4% and takes
the value 0 otherwise
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A Non-‐Linear Optimization Model
Model Formulation Modeling of plant efCiciency loss due to co-‐Ciring [1].
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Coal Pant j: 𝑻𝑪↓𝒋 capacity 𝒇↓𝒋 plant utilization 𝝆↓𝒋↑ efKiciency
Heat input Electricity output
𝑸↓𝒋↑ = 𝑻𝑪↓𝒋 ∗𝒇↓𝒋 /𝝆↓𝒋↑
𝑴↓𝒋↑𝒄𝒐𝒂𝒍 = 𝑸↓𝒋↑ ∗𝑶𝑯↓𝒋 ∗𝑪↑𝒘𝒃 /𝑳𝑯𝑽↓𝒋↑𝒄𝒐𝒂𝒍
Model Formulation cont. Biomass has a lower heating value than coal. Let ∆𝑴↓𝒋↑𝒄𝒐𝒂𝒍 be the amount of coal displaced.
Energy balance equation: 𝑳𝑯𝑽↓𝒋↑𝒃𝒎 ∗∑𝐢∈𝑺↑▒𝑿↓𝒊𝒋 = 𝑳𝑯𝑽↓𝒋↑𝒄𝒐𝒂𝒍 ∗∆𝑴↓𝒋↑𝒄𝒐𝒂𝒍 The amount of biomass required is:
∑𝑖∈𝑆↑▒𝑋↓𝑖𝑗 = ∆𝑴↓𝒋↑𝒄𝒐𝒂𝒍 𝑳𝑯𝑽↓𝒋↑𝒄𝒐𝒂𝒍 /𝑳𝑯𝑽↓𝒋↑𝒃𝒎
The efCiciency loss of boilers when biomass is used (Tillman 2000):
𝑬𝑳↓𝒋 = 0.0044*𝐵↓𝑗↑2 + 0.0055 ∑𝑖∈𝑆↑▒𝑋↓𝑖𝑗 =∆𝑴↓𝒋↑𝒄𝒐𝒂𝒍 ∗𝑳𝑯𝑽↓𝒋↑𝒄𝒐𝒂𝒍 /𝑳𝑯𝑽↓𝒋↑𝒃𝒎 ∗𝝆↓𝒋↑𝒃 /𝝆↓𝒋↑𝒃 − 𝑬𝑳↓𝒋
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Problem Description Plant investment costs Ø If the percentage of coal displaced is 𝐵↓𝑗 ≤4%, capital investment $50/KWbm (Caputo et al. 2005):
𝐼↓𝑗↑𝐶𝐴𝑃 =50,000∗𝑀↓𝑗↑𝑏𝑚 /𝑀↓𝑗↑𝑐𝑜𝑎𝑙 −∆𝑀↓𝑗↑𝑐𝑜𝑎𝑙 𝑇𝐶↓𝑗 ∗𝑓↓𝑗 ∗𝐿𝐻𝑉↓𝑗↑𝑏𝑚 /𝐿𝐻𝑉↓𝑗↑𝑐𝑜𝑎𝑙 = 50,000∗ 𝑇𝐶↓𝑗 ∗𝑓↓𝑗 ∗𝐿𝐻𝑉↓𝑗↑𝑏𝑚 /𝐿𝐻𝑉↓𝑗↑𝑐𝑜𝑎𝑙 ∗𝐵↓𝑗 /1− 𝐵↓𝑗 = 𝑰↓𝒋↑𝒄𝒂𝒑 (𝑩↓𝒋 /𝟏− 𝑩↓𝒋
)
Ø If 𝐵↓𝑗 >4%, new investments required: Biomass storage: 𝐼↓𝑗↑𝑆 =136578∗(𝑇𝐶↓𝑗 ∗𝑓↓𝑗 ∗𝐿𝐻𝑉↓𝑗↑𝑏𝑚 /𝐿𝐻𝑉↓𝑗↑𝑐𝑜𝑎𝑙 ∗𝐵↓𝑗 /1− 𝐵↓𝑗 )↑0.5575 = 𝑰↓𝒋↑𝒔 ∗(𝑩↓𝒋 /𝟏− 𝑩↓𝒋 )↑𝟎.𝟓𝟓𝟕𝟓 Biomass handling: 𝐼↓𝑗↑𝐻 =55780∗(𝑇𝐶↓𝑗 ∗𝑓↓𝑗 ∗𝐿𝐻𝑉↓𝑗↑𝑏𝑚 /𝐿𝐻𝑉↓𝑗↑𝑐𝑜𝑎𝑙 ∗𝐵↓𝑗 /1− 𝐵↓𝑗 )↑0.9554 = 𝑰↓𝒋↑𝒉 ∗(𝑩↓𝒋 /𝟏− 𝑩↓𝒋 )↑𝟎.𝟗𝟓𝟓𝟒 Compressors and driers: 𝐼↓𝑗↑𝐶𝐷 =13646∗(𝑇𝐶↓𝑗 ∗𝑓↓𝑗 ∗𝐿𝐻𝑉↓𝑗↑𝑏𝑚 /𝐿𝐻𝑉↓𝑗↑𝑐𝑜𝑎𝑙 ∗𝐵↓𝑗 /1− 𝐵↓𝑗 )↑0.5575 =𝑰↓𝒋↑𝑪𝑫 ∗(𝑩↓𝒋 /𝟏− 𝑩↓𝒋 )↑𝟎.𝟓𝟓𝟕𝟓
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Problem Description Ø Operating costs
o Biomass purchase cost: ∑𝑖∈𝑆↑▒𝑐↓𝑖↑𝑏𝑚 𝑋↓𝑖𝑗↑
o Transportation cost: ∑𝑖∈𝑆↑▒𝑐↓𝑖𝑗↑ 𝑋↓𝑖𝑗↑
Ø Savings o Coal displacement
𝑆↓𝑗↑𝑝 = 𝑐↓𝑗↑𝑐𝑜𝑎𝑙 ∗∆𝑀↓𝑗↑𝑐𝑜𝑎𝑙 = 𝜎↓𝑗↑𝑝 * ∑𝑖∈𝑆↑▒𝑋↓𝑖𝑗↑
o Production Tax Credit (PTC): 2.3 cents/KWhbm 𝑆↓𝑗↑𝑡𝑎𝑥 =23∗𝐿𝐻𝑉↓𝑗↑𝑏𝑚 /𝐶↑𝑤𝑏 ∗𝑀↓𝑗↑𝑏𝑚 = 𝜎↓𝑗↑𝑡 ∗ ∑𝑖∈𝑆↑▒𝑋↓𝑖𝑗↑
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Biomass market price $/ton
Unit transporting cost from supplier i to plant j
Coal market price $/ton
Objective function • 𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒:𝑍= ∑𝑗∈𝐶↑▒█■ (( 𝜎↓𝑗↑𝑝 + 𝜎↓𝑗↑𝑡 ) ∑𝑖∈𝑆↑▒𝑋↓𝑖𝑗 ) − ∑𝑖∈𝑆,𝑗∈𝐶↑▒(𝑐↓𝑖𝑗↑𝑡 + 𝑐↓𝑖↑𝑏𝑚 ) 𝑋↓𝑖𝑗 • −∑𝑗∈𝐶↑▒(𝐼↓𝑗↑𝑠 + 𝐼↓𝑗↑𝑐𝑑 )(1− 𝑌↓𝑗 )(𝐵↓𝑗 /1− 𝐵↓𝑗 )↑0.5575
• − ∑𝑗∈𝐶↑▒𝐼↓𝑗↑ℎ (1− 𝑌↓𝑗 )(𝐵↓𝑗 /1− 𝐵↓𝑗 )↑0.9554 • − ∑𝑗∈𝐶↑▒█■𝐼↓𝑗↑𝑐𝑎𝑝 (𝐵↓𝑗 /1− 𝐵↓𝑗 )𝑌↓𝑗
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Savings due to coal displacement and PTC
Biomass procurement and transportation costs
Storage and compressor- drier costs for Bj >4
Handling costs for Bj >4
Capital investments costs for Bj ≤4
A Non-‐Linear Optimization Model
A Non-‐Linear Optimization Model Constraints • Subject to:
• ∑𝑗∈𝐶↑▒𝑋↓𝑖𝑗 ≤ 𝑠↓𝑖 for all 𝑖∈𝑆 (1)
• ∑𝑖∈𝑆↑▒𝑋↓𝑖𝑗 ≤ (𝑄↓𝑗↑0 ∗𝑂𝐻↓𝑗 ∗𝐶↑𝑤𝑏 ∗𝜌↓𝑗↑𝑏 )/ 𝐿𝐻𝑉↓𝑗↑𝑐𝑜𝑎𝑙 /( 1∕𝐵↓𝑗 − 𝛼↓𝑗 )∗(𝜌↓𝑗↑𝑏 −0.0044𝐵↓𝑗↑2 − 0.0055) for all 𝑗∈𝐶 (2)
• 𝐵↓𝑗 −0.04 ≤𝑀(1− 𝑌↓𝑗 ) for all 𝑗∈𝐶 (3)
0.04− 𝐵↓𝑗 <𝑀𝑌↓𝑗 for all 𝑗∈𝐶 (4)
• 𝑋↓𝑖𝑗 ∈ 𝑅↑+ for all 𝑖∈𝑆, 𝑗∈𝐶 (5)
• 𝐵↓𝑗 ∈[0, 1] for all 𝑗∈𝐶 (6) • 𝑌↓𝑗 ∈{0, 1} for all 𝑗∈𝐶 (7)
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A Linear Optimization Model
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Consider plant j could displace coal (by mass) either at a rate of Bj = 1%, or 2%, or 3%, etc.
Let l = 1, . . . , |L| index the set of all potential values that Bj can take. Ll denote the l−the element of this set
Decision variables: 𝒀↓𝑙𝒋 binary variable which takes the value 1 if facility j displaces 𝐿↓𝑙 = 𝐵↓𝑗 % coal, and takes the value 0 otherwise
𝑿↓𝒊𝒋 the amount of biomass (in tons/year) delivered to coal plant j from supplier i
A Linear Optimization Model
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For a Cixed value of Bj : 𝑀↓𝑙𝑗↑𝑏𝑚 = (𝑄↓𝑗↑0 ∗𝑂𝐻↓𝑗 ∗𝐶↑𝑤𝑏 ∗𝜌↓𝑗↑𝑏 )/ 𝐿𝐻𝑉↓𝑗↑𝑐𝑜𝑎𝑙 /( 1∕𝐵↓𝑗 − 𝛼↓𝑗 )∗(𝜌↓𝑗↑𝑏 −0.0044𝐵↓𝑗↑2 − 0.0055) is a constant.
∑𝑖∈𝑆↑▒𝑋↓𝑖𝑗 ≤ (𝑄↓𝑗↑0 ∗𝑂𝐻↓𝑗 ∗𝐶↑𝑤𝑏 ∗𝜌↓𝑗↑𝑏 )/ 𝐿𝐻𝑉↓𝑗↑𝑐𝑜𝑎𝑙 /( 1∕𝐵↓𝑗 − 𝛼↓𝑗 )∗(𝜌↓𝑗↑𝑏 −0.0044𝐵↓𝑗↑2 − 0.0055) (2)
∑𝑖∈𝑆↑▒𝑋↓𝑖𝑗 ≤∑𝑙∈𝐿↑▒𝑀↓𝑙𝑗↑𝑏𝑚 𝑌↓𝑙𝑗 for all 𝑗∈𝐶 (2*)
𝐼↓𝑙𝑗↑ = 𝑰↓𝒋↑𝒄𝒂𝒑 (𝑩↓𝒋 /𝟏− 𝑩↓𝒋 ) is a constant.
∑𝑗∈𝐶↑▒∑𝑙∈𝐿↑▒𝐼↓𝑙𝑗↑ 𝑌↓𝑙𝑗↑
• 𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒:𝑍= ∑𝑗∈𝐶↑▒█■ (( 𝜎↓𝑗↑𝑝 + 𝜎↓𝑗↑𝑡 ) ∑𝑖∈𝑆↑▒𝑋↓𝑖𝑗 ) − ∑𝑖∈𝑆,𝑗∈𝐶↑▒(𝑐↓𝑖𝑗↑𝑡 + 𝑐↓𝑖↑𝑏𝑚 ) 𝑋↓𝑖𝑗 • −∑𝑗∈𝐶↑▒∑𝑙∈𝐿↑▒𝐼↓𝑙𝑗↑ 𝑌↓𝑙𝑗↑
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Savings due to coal displacement and PTC
Biomass procurement and transportation costs
Investment costs
A Linear Optimization Model
Subject to: ∑𝑗∈𝐶↑▒𝑋↓𝑖𝑗 ≤ 𝑠↓𝑖 for all 𝑖∈𝑆 (1) ∑𝑖∈𝑆↑▒𝑋↓𝑖𝑗 ≤∑𝑙∈𝐿↑▒𝑀↓𝑙𝑗↑𝑏𝑚 𝑌↓𝑙𝑗 for all 𝑗∈𝐶
(2*) ∑𝑙∈𝐿↑▒𝑌↓𝑙𝑗 ≤1 for all 𝑗∈𝐶 (8) 𝑋↓𝑖𝑗 ∈ 𝑅↑+ for all 𝑖∈𝑆, 𝑗∈𝐶 (5) 𝑌↓𝑙𝑗 ∈{0, 1} for all 𝑙∈𝐿,𝑗∈𝐶 (9)
A Case Study
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Biomass supply in the state of Mississippi [4]: • Knowledge Discovery Framework (KDF) • Woody biomass; available for different price targets
Truck transportation [5]: • Distance Cixed cost (DFC): $3.01/(tons) • Distance variable cost (DVC): $0.112/(tons mile) • 𝑐↓𝑖𝑗↑𝑡 =𝐷𝐹𝐶+𝐷𝑉∗𝑑↓𝑖𝑗
A Case Study
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Coal plants data [6]: • National Energy Technology Laboratory • Power plants of capacity greater than 1MW
Plant Name Primary Fuel Fuel Type Nameplate
Capacity (MW) Capacity Factor
Red Hills COAL Lignite coal 514 0.7213
Henderson COAL Bituminous coal 59 0.1078
R D Morrow COAL Bituminous coal 400 0.7281
Victor J Daniel Jr COAL Bituminous coal 2,229 0.4986
Jack Watson COAL Bituminous coal 1,216 0.3544
Product LHV (BTU/Ton)
Woody biomass 16,811,000
Bituminous coal 22,460,600
Lignite coal 19,536,300
Solved using GAMS and BONMIN, CPLEX on a personal computer. Ø It took a few seconds to solve all the problem instances.
Ø Increasing the size of |L| impacts the quality of the solutions found from LP formulation.
Numerical Results
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Relationship between the size of set |L| and the quality of solutions of LP formulation.
0.00
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2.00
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3.00
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Gap
|L|
Relative gap between NLP and LP solutions
$80/ton
$100/ton
Ø Increasing the target price of biomass for a Cixed PTC of 2.3cents /kwh : o Market price ≤ $120/ton:
o ProKits increase since savings from PTC are greater than the additional costs from investment and transportation
o Renewable electricity production increases. o Market price > $120/ton:
o Renewable electricity production remains constant.
Numerical Results
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Relationship between biomass price, investment costs, logistics costs, proCits, biomass use.
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5
0 20 40 60 80 100 120 140 160 180
40 70 100 130 160 190
Biom
ass used (1,000,000 tons)
Prfots/Costs ($1,000,000)
Price of Biomass ($/ton)
Total ProCits
Investment Costs
Logsitic Costs
Biomass Used
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Biom
ass Usage (1000 Tons)
Total ProKit (M
illion $)
Tax credit (cents/kwh)
Obj. Fun. Val Tax Saving Logsitics Costs Used Biomass
The impact of tax credits on proCits and biomass usage for biomass target price of $100/ton
Ø In Mississippi: CoCiring not proCitable for PTC ≤𝟎.𝟕 cents/kwh Ø For PTC ≥ 2.2
o Renewable electricity production does not increase o ProCits continue increasing
Numerical Results
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The impact of tax credits on proCits and biomass usage for biomass target price of $100/ton
Ø South Carolina: CoCiring not proCitable for PTC ≤𝟐.𝟏 cents/kwh Ø For PTC ≥ 2.6
o Renewable electricity production does not increase o ProCits continue increasing
Numerical Results
0
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-‐600
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Bio
mas
s use
d (1
000
Ton)
Tota
l pro
fit ($
1000
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Tax credit (¢/kwh)
Total Profit
Tax Saving
LogsiQcs Costs
Biomass Used
Summary of Cindings Ø Tax credits necessary to increase the production of the renewable energy.
Ø Tax credit should not be “one size Cits all”. PTC could be a function of the amount of renewable electricity produced.
Ø Biomass availability in the USA differs by region. To optimize renewable energy production, the tax rate
should be customized by region.
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Develop solution algorithms to solve the LP approximation model: • 𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒:𝑍= ∑𝑗∈𝐶↑▒█■ (( 𝜎↓𝑗↑𝑝 + 𝜎↓𝑗↑𝑡 ) ∑𝑖∈𝑆↑▒𝑋↓𝑖𝑗 ) − ∑𝑖∈𝑆,𝑗∈𝐶↑▒(𝑐↓𝑖𝑗↑𝑡 + 𝑐↓𝑖↑𝑏𝑚 ) 𝑋↓𝑖𝑗
• −∑𝑗∈𝐶↑▒∑𝑙∈𝐿↑▒𝐼↓𝑙𝑗↑ 𝑌↓𝑙𝑗↑
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Current Research
Subject to: ∑𝑗∈𝐶↑▒𝑋↓𝑖𝑗 ≤ 𝑠↓𝑖 for all 𝑖∈𝑆 (1) ∑𝑖∈𝑆↑▒𝑋↓𝑖𝑗 ≤∑𝑙∈𝐿↑▒𝑀↓𝑙𝑗↑𝑏𝑚 𝑌↓𝑙𝑗 for all 𝑗∈𝐶
(2*) ∑𝑙∈𝐿↑▒𝑌↓𝑙𝑗 ≤1 for all 𝑗∈𝐶 (8) 𝑋↓𝑖𝑗 ∈ 𝑅↑+ for all 𝑖∈𝑆, 𝑗∈𝐶 (5) 𝑌↓𝑙𝑗 ∈{0, 1} for all 𝑙∈𝐿,𝑗∈𝐶 (9)
Current Research Ø Investigate the impact of tax schemes on
renewable electricity production: o Tax rate as a function of biomass usage
o Tax rate as a function of plant capacity
o Tax rate as a function of biomass available in the region
Ø Initial results:
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Tax rate
X
X
Tax Savings ($)
2.3 (cents/kwh) α
γ 2γ 3γ
2α
nα ….
15 20 25 30 35 40
0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2
Biom
ass used x
100,000
Value of α
Biomass Used (γ = 250,000)
The unit PTC is smaller than 2.3/kwh
Future Research Ø Consider supply chains that span larger regions. In this case the in-‐bound supply chain will have a hub-‐and-‐spoke structure.
Ø Develop models which optimize costs and the social impacts of co-‐Ciring: Ø Use a Stackelberg game model:
Ø Leader: Decision makers at the federal level that identify a PTC structure
Ø Followers: Coal plants that decide on the amount of coal to displace.
Ø The goal is to a PTC which optimizes the social beneCits without sacriCicing power plants proCits. 26
Questions?
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References [1] De, S., M. Assadi. 2009. Impact of cofring biomass with coal in power plants: A techno-‐economic assessment. Biomass and Bioenergy, Vol. 33, 283-‐293.
[2] Sondreal, E.A., S.A. Benson, J.P. Hurley, M.D. Mann, J.H. Pavlish, M.L. Swanson. 2001. Review of advances in combustion technology and biomass cofring. Fuel Process Technology, 71 7-‐38.
[3] Caputo, A.C., M. Palumbo, P.M. Pelagagge, F. Scacchia. 2005. Economics of biomass energy utilization in combustion and gasiCication plants: effects of logistic variables. Biomass and Bioenergy, Vol. 28, 35-‐51.
[4] Knowledge discovery framework. US Department of Energy. https://bioenergykdf.net.
[5] Searcy, E., P. Flynn, E. Ghafoori, A. Kumar. 2007. The relative cost of biomass energy transport. Applied Biochemistry and Biotechnology, Vol.137, 639-‐652.
[6] National Energy Technology Laboratory. http://www.netl.doe.gov/energyanalyses/hold/technology.html.
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Ø Tax credits are necessary for production of the renewable energy o For Mississippi, cents 0.7/kwh is a lower bound on PTC
Ø Increasing PTC impacts positively the amount of renewable energy produced o At $100/ton: When PTC is less than cents 2.1/kwh, only some of the available biomass is used
Numerical Results
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The impact of tax credit and targeted price on biomass usage.
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ass usage (1000 Tons)
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Develop solution algorithms to solve the LP approximation model: • 𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒:𝑍= ∑𝑗∈𝐶↑▒█■ (( 𝜎↓𝑗↑𝑝 + 𝜎↓𝑗↑𝑡 ) ∑𝑖∈𝑆↑▒𝑋↓𝑖𝑗 ) − ∑𝑖∈𝑆,𝑗∈𝐶↑▒(𝑐↓𝑖𝑗↑𝑡 + 𝑐↓𝑖↑𝑏𝑚 ) 𝑋↓𝑖𝑗
• −∑𝑗∈𝐶↑▒∑𝑙∈𝐿↑▒𝐼↓𝑙𝑗↑ 𝑌↓𝑙𝑗↑
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Current Research
Subject to: ∑𝑗∈𝐶↑▒𝑋↓𝑖𝑗 ≤ 𝑠↓𝑖 for all 𝑖∈𝑆 (1) ∑𝑖∈𝑆↑▒𝑋↓𝑖𝑗 ≤∑𝑙∈𝐿↑▒𝑀↓𝑙𝑗↑𝑏𝑚 𝑌↓𝑙𝑗 for all 𝑗∈𝐶
(2*) ∑𝑙∈𝐿↑▒𝑌↓𝑙𝑗 ≤1 for all 𝑗∈𝐶 (8) 𝑋↓𝑖𝑗 ∈ 𝑅↑+ for all 𝑖∈𝑆, 𝑗∈𝐶 (5) 𝑌↓𝑙𝑗 ∈{0, 1} for all 𝑙∈𝐿,𝑗∈𝐶 (9)
∑𝑖∈𝑆,𝑗∈𝐶↑▒█■ 𝐶↓𝑖𝑗 𝑋↓𝑖𝑗
Complicating constraints:
Lagrangean relaxation model:
𝑀𝑎𝑥:𝑍↑𝜆 =∑𝑖∈𝑆,𝑗∈𝐶↑▒█■ 𝐶↓𝑖𝑗 𝑋↓𝑖𝑗 −∑𝑗∈𝐶↑▒∑𝑙∈𝐿↑▒𝐼↓𝑙𝑗↑ 𝑌↓𝑙𝑗↑ + ∑𝑖∈𝑆↑▒( 𝑠↓𝑖 −█■ 𝐶↓𝑖𝑗 𝑋↓𝑖𝑗 ) λ↓𝑖
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Current Research
Subject to: ∑𝑖∈𝑆↑▒𝑋↓𝑖𝑗 ≤∑𝑙∈𝐿↑▒𝑀↓𝑙𝑗↑𝑏𝑚 𝑌↓𝑙𝑗 for all 𝑗∈𝐶
(2*) ∑𝑙∈𝐿↑▒𝑌↓𝑙𝑗 ≤1 for all 𝑗∈𝐶 (8) 𝑋↓𝑖𝑗 ∈ 𝑅↑+ for all 𝑖∈𝑆, 𝑗∈𝐶 (5) 𝑌↓𝑙𝑗 ∈{0, 1} for all 𝑙∈𝐿,𝑗∈𝐶 (9)
For a give facility j and a given set of multipliers λ↓𝑖 :
𝑀𝑎𝑥:𝑍↑𝑗𝜆 =∑𝑖∈𝑆↑▒(█■ 𝐶↓𝑖𝑗 − λ↓𝑖 )𝑋↓𝑖𝑗 −∑𝑗∈𝐶↑▒∑𝑙∈𝐿↑▒𝐼↓𝑙𝑗↑ 𝑌↓𝑙𝑗↑ Theorem: The LP-‐relaxa7on of this problem provides the op7mal solu7on.
Lemma: In an op7mal solu7on to this problem plants receive shipments from a single supplier.
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Current Research
Subject to: ∑𝑖∈𝑆↑▒𝑋↓𝑖𝑗 ≤∑𝑙∈𝐿↑▒𝑀↓𝑙𝑗↑𝑏𝑚 𝑌↓𝑙𝑗 (2) ∑𝑙∈𝐿↑▒𝑌↓𝑙𝑗 ≤1 (8) 𝑋↓𝑖𝑗 ∈ 𝑅↑+ for 𝑖∈𝑆 (5) 𝑌↓𝑙𝑗 ∈{0, 1} for 𝑙∈𝐿 (9)
The dual of Lagrangean relaxa7on model can easily be solved using a simple inspec7on procedure: 𝐼𝑓:𝑚𝑎𝑥↓𝑖 {0, 𝐶↓𝑖𝑗 − λ↓𝑖 } = 𝐶↓𝑖∗𝑗 − λ↓𝑖∗ >0, 𝑡ℎ𝑒𝑛: █■𝐼𝑓:𝑚𝑎𝑥 {0, 𝑚𝑎𝑥↓𝑙 {𝑀↓𝑙𝑗 (𝐶↓𝑖∗𝑗 − λ↓𝑖∗ )− 𝐼↓𝑙𝑗 }} = 𝑀↓𝑙∗𝑗 (𝐶↓𝑖∗𝑗 − λ↓𝑖∗ )− 𝐼↓𝑙∗𝑗 >0, 𝑡ℎ𝑒𝑛: ↓
𝑌↓𝑙∗𝑗↑ =1; 𝑌↓𝑙𝑗↑ =0 𝑓𝑜𝑟 𝑙 ∈ L/{l*} 𝑋↓𝑖∗𝑗↑ = 𝑀↓𝑙𝑗 ; 𝑋↓𝑖𝑗↑ =0 𝑓𝑜𝑟 𝑖 ∈ I/{i*}
We are working on fine tuning the algorithm. 33
Current Research
