Eat Fast, Just Better.
An Organic Food Supply Network
Maj Tony DeMarco
Maj Art Terry
Fast Food Nation
• Americans eat ~ 25% of all meals at fast food
restaurants.
• Among the various reasons that Americans enjoy fast
food, the top three according to the USDA in 2006 are:
• TASTE
• NUTRITION
• CONVENIENCE
Nutrition
Health
Choices?
Possible New Alternatives
• Provide fresh, organic food in a fast food
setting.
• Examine a network of organic farms and
potential restaurant locations within a given
state.
• Operate a network to minimize the delivery
time of the food…(freshness, cost…) to find
the top 3 build sites among 10 locations.
Maryland
The Farms
Farm nodes were abstracted to zipcode locations
The Potential Restaurants
Potential restaurant nodes were centered around the
Baltimore and Capital Beltways
The Potential Restaurants
9
8
7
6
4
5
3
2
1
0
No
de
Location (Google Maps)
Name
Zip
0
38.833289,-76.876831
Forestville
20747
1
38.948729,-76.860352
Mitchellville
20716
2
38.993572,-76.889191
Greenbelt Park
20768
3
39.024518,-76.952362
Hillandale
20903
4
39.030919,-77.140503
North Bethesda
20852
5
38.974357,-77.180328
Potomac
20854
6
39.208847,-76.598053
Brooklyn Park
21225
7
39.254588,-76.68457
Arbutus
21227
8
39.305613,-76.743622
Woodlawn
21207
9
39.374649,-76.746368
Pikesville
21208
The Network
The arcs, from all farms to potential restaurants, are the
travel times along state roads.
Complete Bipartite Graph
Farm 1
Restaurant
1
Farm 2
Restaurant
2
…
Farm
…
Some More Information
• Only 3 restaurant nodes are within the budget
for final construction.
• Each restaurant must be supplied with Meat,
Dairy and Produce.
• Farms may produce 1, some or all of the
necessary supplies.
• The operator seeks to minimize time.
Attacks?
A very large corporate
fast food provider is
aware of the potential
competition and has
successfully lobbied
the MD state and
federal government
into instituting organic
food inspections. Such
inspections will delay
food shipment times
from inspected farms.
The Model
• Inner Problem
– Minimize the total transit time of the supply
network.
– Subject to:
• The total number of restaurants that can be built within
budget.
• Each restaurant must receive a supply of each type
(produce, meat/poultry and dairy).
The Model
• Outer Problem
– Maximize the solution to the inner problem.
– Subject to:
• There is a maximum of one inspection station for each
farm.
• The total number of inspection stations must not
exceed a certain number.
Mathematical Formulation
• SETS
– r  R potential restaurant locations
– f  F farm locations
– g  G food type, G =
{‘produce’,’meat/poultry’,’dairy’}
Mathematical Formulation cont…
• GIVEN DATA
– tfr
– pfg
– delay
– restaurants
– stations
transit time from farm f to
restaurant r [seconds]
1 if farm f can produce food
type food g, 0 otherwise
inspection station delay
[seconds]
number of restaurants to
build
number of inspection stations
Mathematical Formulation cont…
• DECISION VARIABLES
– br
– sfrg
– if
1 if build on restaurant location r,
0 otherwise
1 if farm f provides restaurant r
with food type g, 0 otherwise
1 if there is an inspection i
station for farm f, 0 otherwise
Mathematical Formulation cont…
• INNER PROBLEM
min
b,s
s.t.
 (t
fr
 delay·i f ) s frg
f ,r , g
b
r
 restaurants
r
s
frg
 br g , r
f
s frg  p fg r , f , g
Mathematical Formulation cont…
• OUTER PROBLEM
max [solution to inner problem]
i
s.t i f  1 f
i
f
f
 stations
Mathematical Formulation cont…
• Notice the inner problem is not an LP but a
MIP
• The dual-trick not used to solve the max-min
problem.
• Bender's decomposition.
Bender’s Algorithm
• 1) Set UB= “+infinity,” LB= “-infinity,” X=0.
• 2) Solve inner problem to get s variables and optimal objective, obj,
corresponding to attack X.
• 3) If obj>LB then set LB=obj, record X*=X as current incumbent attack.
• 4) Add new cut to outer problem based on optimal flows s from (2).
• 5) Solve master problem to get attack X and objective master_obj
corresponding to all flows so far.
• 6) If master_obj<UB then set UB=master_obj.
• 7) If LB>UB then goto (2).
• 8) Set X=X*, Solve inner problem for flows s* and objective (=LB)
corresponding to attack X*.
• 9) Report X* as optimal attack, and corresponding coefficients s*, with
value obj.
GAMS Bender’s Implementation
set iter the iteration in Bender's algorithm /iter1*iter50/;
set cutset(iter) a dynamic set which will define the new constraint for the iter;
parameters
coeff(iter,f,r,g)
x(f)
;
this corresponds to the s vector for the inner solution
current attack plan
equations
constraint(iter)
total_stations
;
constraint(cutset)..
master_obj =l=
sum((f,r,g),(arcdata(f,r,'time')+delay*i(f))*coeff(cutset,f,r,g))
;
total_stations..
sum(f,i(f)) =l= stations
;
GAMS Bender’s Implementation
ub = INF;
lb = -INF;
converged = 0;
loop(f,ibar(f)=0);
loop(f,x(f)=0);
cutset(iter)=no;
loop(iter$(not converged),
solve InnerProblem minimizing obj using MIP;
if(obj.l>lb,lb=obj.l; loop(f,x(f)=ibar(f)));
cutset(iter)=yes;
loop((f,r,g),coeff(iter,f,r,g)=s.l(f,r,g));
solve OuterProblem maximizing master_obj using MIP;
if(master_obj.l<ub,ub=master_obj.l);
if(lb>=ub,converged=1);
loop(f,ibar(f)=i.l(f));
);
loop(f,ibar(f)=x(f));
solve InnerProblem minimizing obj using MIP;
Analysis
• What locations are best with no attacks?
• With attacks? Attacks increased in intensity
from 1 to 6 hour inspections, ranging from 1
to 12 sites being inspected.
• What are the best build (most resilient) sites?
No Attacks…
•
Solution to Inner Problem with no inspection stations...
•
•
•
•
•
•
•
•
•
•
•
•
•
Build at
Mitchellville
Shipping PRODUCE
from
Woodmore_Farms
Shipping MEAT_POULTRY
from
Good_Fortune_Farm
Shipping DAIRY
from Maryland_Sunrise_Farm_LLC
Build at
Hillandale
Shipping PRODUCE
from
Organic_Acres
Shipping MEAT_POULTRY
from
Nicks_Organic_Farm_LLC
Shipping DAIRY
from Maryland_Sunrise_Farm_LLC
Build at
Potomac
Shipping PRODUCE
from
Nicks_Organic_Farm_LLC
Shipping MEAT_POULTRY
from
Nicks_Organic_Farm_LLC
Shipping DAIRY
from
Castle_Henry_Farm
Total shipping time: 12258.00 ~ 3.4 hours
1 Attack…
•
•
•
Solution to Problem with
1.00 inspection stations
Place inspection stations at:
Nicks_Organic_Farm_LLC
•
•
•
•
•
•
•
•
•
•
•
•
•
Build at
Mitchellville
Shipping PRODUCE
from
Shipping MEAT_POULTRY
from
Shipping DAIRY
from
Build at
Hillandale
Shipping PRODUCE
from
Shipping MEAT_POULTRY
from
Shipping DAIRY
from
Build at
Pikesville
Shipping PRODUCE
from
Shipping MEAT_POULTRY
from
Shipping DAIRY
from
Total shipping time: 15312.00 ~ 4.25 hours
Woodmore_Farms
Good_Fortune_Farm
Maryland_Sunrise_Farm_LLC
Organic_Acres
Maryland_Sunrise_Farm_LLC
Maryland_Sunrise_Farm_LLC
Rocky_Knoll
Maryland_Sunrise_Farm_LLC
Bellevale_Farms_Inc
2 Attacks…
•
•
•
•
Solution to Problem with
2.00 inspection stations
Place inspection stations at:
Maryland_Sunrise_Farm_LLC
Nicks_Organic_Farm_LLC
•
•
•
•
•
•
•
•
•
•
•
•
•
Build at
Hillandale
Shipping PRODUCE
Shipping MEAT_POULTRY
Shipping DAIRY
Build at
Woodlawn
Shipping PRODUCE
Shipping MEAT_POULTRY
Shipping DAIRY
Build at
Pikesville
Shipping PRODUCE
Shipping MEAT_POULTRY
Shipping DAIRY
Total shipping time: 19155.00 ~ 5.32
from
from
from
Organic_Acres
Good_Fortune_Farm
Castle_Henry_Farm
from
from
from
Rocky_Knoll
Country_Pleasures_Farm
Bellevale_Farms_Inc
from
from
from
Rocky_Knoll
Country_Pleasures_Farm
Bellevale_Farms_Inc
Operator Resilience Curve
Inspection Site Effectiveness
90000
80000
70000
60000
Delay
1 Hour
50000
2 Hour
40000
3 Hour
4 Hour
30000
5 Hour
20000
6 Hour
10000
0
0
1
2
3
4
5
6
7
Number Of Inspection Sites
8
9
10
11
12
Frequency of Potential Build Sites with Varying Delays
14
12
10
8
6
1 Hour Delay
2 to 6 hour delay
4
2
0
Total Frequency of Potential Build Sites
60
50
40
30
Frequency
20
10
0
Top Four Sites
9
8
7
6
4
5
3
2
1
0
No
de
Location (Google Maps)
Name
Zip
0
38.833289,-76.876831
Forestville
20747
1
38.948729,-76.860352
Mitchellville
20716
2
38.993572,-76.889191
Greenbelt Park
20768
3
39.024518,-76.952362
Hillandale
20903
4
39.030919,-77.140503
North Bethesda
20852
5
38.974357,-77.180328
Potomac
20854
6
39.208847,-76.598053
Brooklyn Park
21225
7
39.254588,-76.68457
Arbutus
21227
8
39.305613,-76.743622
Woodlawn
21207
9
39.374649,-76.746368
Pikesville
21208
Graphic Visualization of Attacks
Some Attack Characteristics
• The network seeks to make use of close,
multi-item producers.
• Attacks on farms that deliver multiple items
are the most effective at interfering with the
network.
• Protecting them is important. How?
Network Solution Under 7 Attacks
8 New Attack Placements
Updated Solution
Recommended Build Sites
9
8
7
6
4
5
3
2
1
0
No
de
Location (Google Maps)
Name
Zip
0
38.833289,-76.876831
Forestville
20747
1
38.948729,-76.860352
Mitchellville
20716
2
38.993572,-76.889191
Greenbelt Park
20768
3
39.024518,-76.952362
Hillandale
20903
4
39.030919,-77.140503
North Bethesda
20852
5
38.974357,-77.180328
Potomac
20854
6
39.208847,-76.598053
Brooklyn Park
21225
7
39.254588,-76.68457
Arbutus
21227
8
39.305613,-76.743622
Woodlawn
21207
9
39.374649,-76.746368
Pikesville
21208
Other Considerations
• Effects of different minimum product delivery
times?
– We were unable to implement this correctly. It would
make for an interesting follow-on analysis.
• Effects of customer volume in the model:
– Suppose it is known that customer flow in locations is
projected with an accepted accuracy.
– Costumer flow will directly impact profitability.
– Arcs to those more profitable restaurants are now
offset by a number corresponding to the projected
profitability of the restaurant.
Effects
• Making Forestville, Artibus and Woodlawn our
top three respective forecasted profit makers:
Without Profit Adjustment
8
7
6
5
4
3
2
1
0
With
10
8
6
4
2
0
*Limits the usefulness of total shipping time as a cost factor
Conclusion
• The Network provides an adequate abstraction to
study respective delays between certain
locations.
• The inner problem being a MIP makes for a much
more computationally expensive model.
• Min Cost flow should be further explored in the
context of the problem, where build site
construction is attacked, rather than the farms.
• How does the adjacency of restaurants affect the
model?
Questions/Discussion