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

No de Location (Google Maps)NameZip , Forestville , Mitchellville , Greenbelt Park , Hillandale , North Bethesda , Potomac , Brooklyn Park , Arbutus , Woodlawn , Pikesville The Potential Restaurants

The Network The arcs, from all farms to potential restaurants, are the travel times along state roads.

Complete Bipartite Graph Restaurant 1 Restaurant 2 Farm 1 Farm 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 – t fr transit time from farm f to restaurant r [seconds] – p fg 1 if farm f can produce food type food g, 0 otherwise – delay inspection station delay [seconds] – restaurantsnumber of restaurants to build – stations number of inspection stations

Mathematical Formulation cont… DECISION VARIABLES – b r 1 if build on restaurant location r, 0 otherwise – s frg 1 if farm f provides restaurant r with food type g, 0 otherwise – i f 1 if there is an inspection i station for farm f, 0 otherwise

Mathematical Formulation cont… INNER PROBLEM

Mathematical Formulation cont… OUTER PROBLEM

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.

Benders 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 Benders 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) this corresponds to the s vector for the inner solution x(f) 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 Benders 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: ~ 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 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 Maryland_Sunrise_Farm_LLC Shipping DAIRY from Maryland_Sunrise_Farm_LLC Build at Pikesville Shipping PRODUCE from Rocky_Knoll Shipping MEAT_POULTRY from Maryland_Sunrise_Farm_LLC Shipping DAIRY from Bellevale_Farms_Inc Total shipping time: ~ 4.25 hours

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 from Organic_Acres Shipping MEAT_POULTRY from Good_Fortune_Farm Shipping DAIRY from Castle_Henry_Farm Build at Woodlawn Shipping PRODUCE from Rocky_Knoll Shipping MEAT_POULTRY from Country_Pleasures_Farm Shipping DAIRY from Bellevale_Farms_Inc Build at Pikesville Shipping PRODUCE from Rocky_Knoll Shipping MEAT_POULTRY from Country_Pleasures_Farm Shipping DAIRY from Bellevale_Farms_Inc Total shipping time: ~ 5.32

Operator Resilience Curve

Frequency of Potential Build Sites with Varying Delays

Total Frequency of Potential Build Sites

Top Four Sites No de Location (Google Maps)NameZip , Forestville , Mitchellville , Greenbelt Park , Hillandale , North Bethesda , Potomac , Brooklyn Park , Arbutus , Woodlawn , 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 No de Location (Google Maps)NameZip , Forestville , Mitchellville , Greenbelt Park , Hillandale , North Bethesda , Potomac , Brooklyn Park , Arbutus , Woodlawn , Pikesville

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: *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