A Distribution Network for McDonalds in Massachusetts€¦ · · 2017-05-15A Distribution Network...

A Distribution Network for McDonalds in Massachusetts – A Project Report

A Distribution Network for McDonalds in

Massachusetts

Project Report

Supply Chain Management SOM 758

05/09/2000

Ann Bergquist and Andrew Hall Supply Chain Management SOM 758 1 May 2000 Ann Bergquist and Andrew Hall


Table of Contents Table of Contents................................................................................................ 2 Management Summary....................................................................................... 3 McDonalds’ Current Process .............................................................................. 3 Proposed New Approach .................................................................................... 4 The Model........................................................................................................... 4

Burger Demand............................................................................................... 4 Warehouse capacity........................................................................................ 5 Transportation ................................................................................................. 6 US Highways – A Node and Link Representation. .......................................... 6 Model Implementation – Meta Models............................................................. 7 The 1st model – The Minimum Cost Model...................................................... 7

Common Subscripts .................................................................................... 7 Variables...................................................................................................... 7 Parameters .................................................................................................. 7 The Objective Function................................................................................ 8 The Constraints ........................................................................................... 8 Problem Size ............................................................................................... 8

The 2nd Model.................................................................................................. 9 Common Subscripts .................................................................................... 9 Variables...................................................................................................... 9 Parameters .................................................................................................. 9 The Objective Function.............................................................................. 10 Constraints................................................................................................. 10

Available Solvers........................................................................................... 11 Rescoping the Project ................................................................................... 11 Lindo solving method .................................................................................... 15 Results from The LTM Model ........................................................................ 16 Changes to the model which might help........................................................ 17 Using the existing results .............................................................................. 18 Conclusions................................................................................................... 19

Supply Chain Management SOM 758 2 May 2000 Ann Bergquist and Andrew Hall


Management Summary We set out to model a Distribution Network for McDonalds patty distribution in Massachusetts. Our objectives were to achieve a lowest cost Distribution Network with 100% service level and to determine whether to have In House distribution or to continue to partner with the patty manufacturer. In particular we expected to construct a model or models to help us to decide on the number, location and capacity of warehouses and the flow of materials i.e. number and size of trucks and feasible routes. We wrote two models, the first to decide where to establish warehouses and what size of warehouse to build. The second model was to help decide how many trucks and which routes to send them on. Our models proved to be too big for the available computing resource to process. We reduced the scope of our problem geographically and by removing redundant elements of the network. We then implemented the first model in Lindo but discovered that an definitive Integer Solution would not be forthcoming in any reasonable time frame. The main results we obtained came from the LP relaxation of the first model from which we were able to conclude that just one large warehouse was optimal and should be located at a particular node. We concluded that a combination of better algorithms, greater computing power (possibly through the application of parallel processing) and the reduction of the number of binary variables through more detailed examination of the network topology would help in the solution of our problems.

McDonalds’ Current Process Patties are delivered direct from the manufacturer to each of the 239 McDonalds restaurants in Massachusetts. Two deliveries are made each week of a fixed order size. The cost is currently $1.87 per pound of patties in either quarter pound or one eighth of a pound sizes. Each restaurant typically holds six days of inventory to cover the lead time of 3 or 4 days and to allow for uncertainty in the demand for burgers at the restaurant. The current method has a number of drawbacks. McDonalds Corp has little control over the delivery process and in practice has become reliant on one manufacturer of beef patties. The amount of storage space available at each restaurant is limited and with the introduction of new menu items and the consequent need to store a wider variety of foodstuff storage is now at a premium.



Although the Managers are very good at running systems the large amount of stock held at each restaurant and the weight of the boxes of patties has resulted in poor date control in restaurants with wastage and potential food safety problems.

Proposed New Approach A new supply chain model is being considered in which McDonalds will maintain its own warehouses and control its own distribution of beef patties in Massachusetts. This will allow daily deliveries of amounts determined from the sales records of each restaurant and as a consequence of the shorter lead time and variable amounts of the orders it is expected that the restaurants will only need to hold 2 days inventory which will result in improved freshness and less wastage. Demand variability will be spread across the 239 restaurants but with a lower overall level of safety stock. It will also allow McDonalds to source patties from more than one manufacturer and to source Patties at a lower cost, currently $1.73 per pound delivered to a warehouse or warehouses in Massachusetts.

The Model Burger Demand We have not approached McDonalds for demands at each restaurant and so our demands will be estimated on the following basis: The USDA estimates that Beef consumption per capita in the USA in 1988 was 63.8 lbs. Beef.com and industry web site estimates that 70% of sales are retail leaving 19.14 pounds to be consumed in restaurants or as take out. They further estimate that 77% of food service beef is in the form of Burgers and that McDonalds has 43% of the Burger market. This means that the US Average McDonalds Burger consumption per capita in 1998 was 6.4 pounds. The Massachusetts Department of Revenue, Division of Local Services, Municipal Data Bank gives us the population of each of 351 towns in Massachusetts and the median income in each town. From this we have established that the population in Massachusetts is 6,147,132 (1998) and therefore that the overall demand for McDonalds’ burgers in Massachusetts per year is 38,972,817 lbs. We will be treating Beefburgers as “Inferior” goods in the economic sense and will allocated demand in an inverse ratio to the median income. Then we will aggregate the demand from 351 towns over 239 restaurants.



Warehouse capacity In order to allow for disruption in deliveries from the manufacturer McDonalds plan to hold 3 days of stock at warehouse. This corresponds to 315,000 lbs capacity or 183 cubic meters of beefburgers. Daily turnover will be 61 cubic meters. Three scales of warehouse are to be considered and modeled, Large, Medium and Small in the following dimensions which assume that beef will be stacked to a height of 2 metres on pallets (each 2 metres square), in the cold store .

Area Storage Staff Large 390 m2 192 m3 4 Medium 246 m2 96 m3 3 Small 210 m2 48 m3 2

The Model was simplified by assuming that salaries and utilities are fixed costs. 5 potential sites were identified in Massachusetts and estimates were worked up for the cost of land, building costs, refrigerated storage, handling equipment , computer equipment utilities, taxes, wages and salaries. A Large Warehouse costs approximately $1,000 per day; a Medium Sized Warehouse approximately $ 800 per day and a Small Warehouse $ 600 per day The following diagram gives an idea of the proposed layout of each facility:

Truck Parking Area

Loading and Unloading Dock

Administrative Offices

Cold Storage 3 Rows 2 Aisles



Metres

Truck Parking Area

Loading and Unloading Dock

Cold Storage

Administrative Offices

Cost Per Day

Large 18 x 12 2½ x 12 8 x 12 4 x 12 $ 1,000Medium 12 x 12 2½ x 12 8 x 6 4 x 6 $ 800Small 12 x 12 2½ x 12 4 x 6 2 x 6 $ 600 Transportation The model assumes that road transportation is to be used with either tractor trailer deliveries into the large warehouses or smaller truck deliveries into the medium and small warehouses. Outgoing will be one of three size trucks, a 6 meter truck, a 4 meter truck or a 2½ meter truck. Each truck will be driven by a single driver who will handle all deliveries at restaurants. The truck driver’s workday will be constrained to a maximum of 10 hours and a minimum of 6 hours, given that the routing plan is to be a daily routine. The State of Massachusetts web site (http://www.detma.org/lmi/wages/oesstw98.html) gave us an hourly rate for drivers of $ 11.85 and we will assume a diesel cost of $ 1.60 per gallon The following table summarizes the capacities and costs for each truck size.

Volume Cost Insurance Gas Capacity Lbs

6 Meter 22 m3 $ 50,000 $ 3,000 4.5 mpg 41,475 4 Meter 14 m3 $ 40,000 $ 2,800 6.0 mpg 26,250

2½ Meter 8 m3 $ 30,000 $ 2,600 9.0 mpg 15,000 US Highways – A Node and Link Representation. The US Department of Transportation (US DOT) provides a GIS database of the major highways in the USA. This has been reduced to the 1953 nodes and 1578 links in Massachusetts. McDonalds has a Web site which provided the locations of each restaurant in Massachusetts in GIS coordinates.



Model Implementation – Meta Models. At the time of the Project Proposal, we had 1,953 nodes and 1,578 links. We had demand of 105,000 lbs per day. In order to ship this on just large trucks we would need 3 large trucks; to ship on just medium sized trucks we would need 4 medium sized trucks and to ship on just small trucks we would need 7 small trucks. The 1st model – The Minimum Cost Model Common Subscripts

( 3,2,1∈g ) implies a large truck; ( )7,6,5,4∈g implies a medium sized truck and implies a small truck, i.e. 1( 14,13,12,11, )10,9,8∈g 14≤≤ g

d denotes a warehouse location such that 1 15≤≤ d , where d determines the size (3 sizes) and location (5 locations) of the warehouse e and f are start and end nodes such that: 578,11 ≤≤ e , where e is startnode

578,11 ≤≤ f , where f is endnode Variables

=dw warehouse d i.e. }1,0{∈dw =dw 1 when warehouse d is open

=efgl link efg i.e. }1,0{∈efgl =efgl 1 when the link between node e and f is used by one of the trucks g AND the reverse links

=fegl link feg }1,0{∈fegl

=efgx the amount lbs sent on link between node e and f, on truck number g

=fegx the amount lbs sent on link between node f and e, on truck number g

Parameters

=dwc the cost of opening warehouse d

=dwa the capacity of warehouse d

=efglc the cost of using a link = lc feg

(i.e. the gas cost for the truck corresponding to the size of truck numbered g) Supply Chain Management SOM 758 7 May 2000 Ann Bergquist and Andrew Hall


=gla capacity for the truck corresponding to size g

=lmd the demand per day at McDonalds l such that : 25416 ≤≤ l

The Objective Function

Min ∑ ∑ ∑∑ ∑ ∑∑= = == = ==

++578,1

1

578,1

1

14

1

578,1

1

578,1

1

14

1

15

1* * *

e f gfegfeg

e f gefgefg

ddd llcllcwwc

The Constraints

Warehouse capacity for d = 1 to 15 0*578,1

1

14

1

578,1

1

14

1≤−− ∑∑∑∑

====dd

ffdg

gfdfg

gwawxx

McDonalds’ demand ∑ for l = 16 to 254

lf

fgf

lfgg

mdxx =− ∑∑∑====

578,1

1lg

14

1

578,1

1

14

1

Interchanges for i = 255 to 1,578 014

1

14

1

578,1

1

578,1

1=−∑ ∑ ∑∑

= = ==g g ffig

fifg xx

Truck capacity 0* ≤− efggefg llax for all g and e and f and 0* ≤− feggfeg llax for all g and e and f Problem Size The Warehouse capacity and McDonalds’ demand and Interchange constraints number 1,578 and the number of binary variables is 15. The Truck capacity constraints number (1,578 * 14) *2 = 44,184 with 44,184 binary variables. The overall problem size is 45,762 constraints and 44,199 binary variables out of a total of 88,382 variables.



The 2nd Model Common Subscripts Depending on the decision from the Min Cost Model on size of truck to use:

=G implies a large truck; ( 3,2,1∈g ) ( )7,6,5,4∈g =G implies a medium sized truck and =G implies a small truck, i.e. 1( 14,13,12,11,10,9, )8∈g 3≤≤ g OR OR

74 ≤≤ g

148 ≤≤ g Depending on the decision from the Min Cost Model on which warehouses to use: d denotes a warehouse location such that 1 15≤≤ d , where d represents one of 3 sizes at one or more of 5 locations. So the set of warehouses W is some combination of up to 5 locations and 3 sizes. e and f are start and end nodes such that: 578,11 ≤≤ e , where e is startnode

1 578,1≤≤ f , where f is endnode Variables gt = truck g i.e. 1}1,0{∈gt =gt when truck g is used

gd = driver hours for truck g

=efgl link efg i.e. }1,0{∈efgl =efgl 1 when the link between node e and f

is used by truck g; AND the reverse links =fegl link feg }1,0{∈fegl

=efgx the amount lbs sent on link between node e and f, on truck number g

=fegx the amount lbs sent on link between node f and e, on truck number g

Parameters

=dwa the capacity of warehouse d 151 ≤≤ d , where d determines the size (3 sizes) and location (5 locations) of the warehouse

=efglc the monetary cost of using a link with truck g = lc feg

=efglt the time to go from node e to node f, with truck g = lt in hours. feg



gtc = fixed cost for truck g

gta = capacity for truck g Sal = salary per hour

=lmd the demand per day at McDonalds l 25416 ≤≤ l , where l determines the location

The Objective Function

Min ∑ ∑ ∑∑ ∑ ∑∑∑= = ∈= = ∈∈∈

+++578,1

1

578,1

1

578,1

1

578,1

1* * **

e f Ggfegfeg

e f Ggefgefg

Ggg

Gggg llcllcSaldtct

Constraints

Warehouse capacity for all d0*578,1

1

578,1

1≤−− ∑∑∑∑

=∈=∈dd

ffdg

Ggfdfg

Ggwawxx W∈

McDonalds’ demand for l = 16 to 254

lf

fGgf

lfgGg

mdxx =− ∑∑∑∑=∈=∈

578,1

1lg

578,1

1

Interchanges for i = 255 to 1,578 0578,1

1

578,1

1=−∑ ∑ ∑∑

∈ ∈ ==Gg Gg ffig

fifg xx

Truck capacity 0* ≤− efggefg llax for all g G∈ and e and f and 0* ≤− feggfeg llax for all g G∈ and e and f

Start-and-End in same node for all g∑ ∑= =

=578,1

1

578,1

1e efegefg ll G∈

Drivers hours 0***578,1

1

578,1

1=−++ ∑∑

==g

efegfeg

eefgefg dtimeUnloadltlltl

6

10

≥

≤

g

g

dd

for all g

Truck constraint for all g 0156,3*578,1

1

578,1

1≤−+ ∑∑

==g

efeg

eefg tll



Whilst we could tally up the number of constraints and variables for this model it suffices to say that each value exceeds that in the Min Cost Model. Available Solvers UMASS OIT do not have Lindo in any form. ECS have Lindo on a VAX machine; Kappa; which is shortly to be phased out! The version on Lindo on Kappa handles 6,500 constraints, 14,900 Integer (binary and other) variables and 120,000 general variables. Lindo in its largest incarnation (incomputerization???) handles problems with 32,000 constraints at a cost of $995. So there is no way to solve our original Minimum Cost model with Lindo. Rescoping the Project It was clear that we would have to reconsider the scope of the project given the available ECS Lindo with its 6,500 constraint capacity. We decided to restrict the scope geographically and to segment the Minimum Cost Model into a Large Truck Model (LTM) and a Medium Truck Model (MTM). We restricted the model to Massachusetts east of east of Worcester and we removed nodes and links that were performing no singular purpose, so for example:

Node Link

Became:



Node Link

As long as the central node was not a McDonalds’ location. Before the model was created Excel was used to check for clusters in the transportation network. When the entire network was displayed it was evident that the network consisted of several clusters that were not connected to each other. (See illustrative example below)

This would make the model infeasible and had to be taken care of. If separate clusters existed in the model it is a possibility that the solution would contain 2 or more warehouses located in each separate cluster, when in fact only one might be needed if the entire network was connected. Therefore links had to be created between the clusters in order to have a network where all the nodes can be reached from any point in the network. Our data had nine clusters so links were created to join them together at the shortest geographical distance between them. This left us with 496 nodes which we numbered our nodes in base 24 from AA to VA and 654 links which we numbered 1 to 654 in decimal. We did not have a Geographical representation tool and we were not prepared to risk the time sink that learning how to do it in C might involve, so we modeled the resulting network in Excel to ensure that we had no clusters and to facilitate interpretation of the model. Supply Chain Management SOM 758 12 May 2000 Ann Bergquist and Andrew Hall


SI 527CT SJ 523 523 523SI CT 73SM SJ SI SJ SF 518SI AG SJ 521SF BO AG SJ SF 516DR 71 AG SA 520 DK SJ 541 SA DK 522SJ 526 CG 541 541 AH SA DK SL 528CG 542 542 542AH SA DK 519DQ CG 540 AH SB 156SA DQ 510DP 540 SA DQ DP 509RY 540AI SA DQ RY AI SA DQ 511 511 511RY 513DR AI SA AI The section given above show a small part of the representation with a node SB linked by link 156 to node SA which in turn is linked to node DK through link number 520 and so on. This re-scoping left us with 109 McDonalds’ restaurants and a total Daily demand of 55,000 lbs. After the model was described in Word and the logical representation had been rationalized; the entire model with all the variables, parameters, constraints and the objective function had to be created as a text file, which could be used as an input file to Lindo. Since the number of variables, parameters and constraints was very large it was impossible to write them by hand. Our approach to create the constraints and the objective function was to use Excel. Creating the constraints and the objective functions was done in Excel. Each link was given a unique ID consisting of one letter and the link number. The letter was either A, B, M or O. A and M for the first truck to and from links and B and N for the second truck to and from links. So for example could be represented by A109 or B109 and could be represented by M109 or N109.

efgx

fegx The binary link variables l and were created by prefixing a “B” to the related link variable name, so if the link ID starts with a B (followed by the other letter) that means the variable is binary i.e. if the link is used it takes a value of 1 otherwise it is set to 0. Each link was then associated with a cost depending on the distance of the link and which truck (large or medium) was using the link.

efg fegl



Whether or not a warehouse is used was also modeled by binary variables (L1-3, M1-3 and S1-3) and each one associated with a cost if the warehouse was opened. Adding the links associated with its costs and the warehouses associated with its costs created the objective function. MIN 1.422 BN653 +1.178 BA654 + 1.178 BB654 + … + 1.178BM654 + 1.178 BN654 + 10 L1 +10 L2 + 10 L3 + 8M1 +8M2 +8M3 + 6S1 + 6S2 + 6S3 Since the links with their numbers and their costs were held in Excel it was a simple task to create formulas which gave us the objective function. The results of the formulas in Excel were copied into a Word document; creating a table which was then converted to text. Because Lindo only can read 80 characters in each line a line break had to be entered at every 80th character. After that was done the objective function was ready to be copied into WordPad and saved as a “.txt” file. This text file was later diverted and read into Lindo in order to solve our model. In addition to the objective function the file also consisted of all the constraints. The constraints were created in Excel in about the same way as the objective function. The first set of constraints make sure that the amount on the links coming in to the warehouse node subtracted with the amount going out from the warehouse does not exceed the capacity of the warehouse. An example of the constraint where S1 is 1 if the warehouse is used and A112, B112 etc are the amount of patties on each link: 18900 S1 + A112 + B112 - M112 - N112 + A214 + B214 - M214 - N214 + M194 + N194 - A194 - B194 + M197 + N197 – A197 - B197>=0 The second set of constraints are for the interchanges, i.e. the nodes that not are a warehouse nor a McDonalds restaurant. These constraints have a “demand” of zero. A189 + B189 – M189 - N189 + M186 + N186 - A186 -B186 + M621 + N621 - A621 -B621 =0 The third set of the constraints make sure that each McDonalds gets the amount of patties to match the demand. An example of this constraint is: A173 + B173 - M173 - N173 + M648 + N648 - A648 - B648 =410 The trucks also have a capacity of how large amount of patties that they can deliver with one truck on each link. This creates the following constraint for each Supply Chain Management SOM 758 14 May 2000 Ann Bergquist and Andrew Hall


link. If the link BA1 is used (BA1 = 1) then the amount transported on that link, A1 has to be less than 41,475 lbs. -41475 BA1 + A1<=0 Once the constraints had been created in Excel as formulas, they were pasted into a Word document in the same way as the objective function and finally converted into text in the text file in WordPad. The advantage of having the entire model in a text file that is diverted into Lindo when the model is to be solved, is that you only have to write the model once. Another advantage is that small changes in the model easily can be made in the file and then you will be able divert the new file to Lindo and run the model again. A Lindo Batch file was then written to “TAKE” the model in from the text file and to save the model as a Lindo format model. Lindo solving method We wrote two models for Lindo – LTM and MTM – one for two large trucks and the other for three medium sized trucks. When Lindo solves an Integer Program it uses a Branch and Bound procedure. First Lindo solves a Linear Program, which means that it ignores the integer and binary variables. This gives a relaxation of the problem and an objective function value lower than the optimal solution eventually will be. The objective function value is therefore set as the lower bound value of the solution. The next step consists of Lindo allocating a value of 1 to all of the binary variables with a value > 0 in the Linear Program. With the fixed values on the binary variables the problem is recalculated as a Linear Program and the objective function value is set as an upper bound solution to the problem. This procedure is repeated for each possible combination of values for the binary variables. If the solution is feasible Lindo has to check whether the new objective function value is improved from the upper bound. In the case of an improved value of the upper bound the revised problem is used as the best current optimum and the new objective function value is set as the upper bound. Eventually Lindo will have checked each branch and have a final optimal solution to the entire Integer Program. The issue with our model was that the number of branches was too big considering the solution algorithm that Lindo uses. Theoretically the largest number of branches that Lindo had to go through in order to find the optimal solution is given by:



• 654 links, 2 links between each node, at least 2 large trucks were needed • 496 nodes • (654 * 2 * 2) choose 496 => 2616 choose 496 ( r choose k )

! NUMBER LARGE VERYA !2120!496

!2616k)!(rk!

r!==

−

Since the number of combinations is a very large number of thousands and thousands of millions and in the worst case Lindo has to go through each one of them the processing time was expected to be very long. This was the reason for our problems when running the model on the Vax machines via Telnet. After only a couple of hours the Telnet connections were dropped and with that the connections to our Lindo processes. Within the short time frame that the model was running it was not possible to get a solution. Therefore we had to acquire another version of Lindo that was capable of handling our model on a PC. This version of Lindo was capable of handling 32,000 variables and 16,000 constraints with 3,200 Integer variables. When running our model there were no interruptions or connections dropped. However, it was still not possible to get the final optimal solution because the number of branches that Lindo had to search through. The process was interrupted by us after 2.7 million branches had been solved because we realized that the process could go on for weeks without any major improvements in the results. Results from The LTM Model We were not getting results in a hurry even on a Pentium III 600 Mhz PC! The left hand panel represents the state of the solver after 90 hours and 21 minutes and shows the best IP of 137.182 (1,127.182 given a cost of 1,000 for a large warehouse). The right hand panel shows the state 31 hours later when we stopped the process. We had not found a better solution than the one found after four minutes of 137.182.



Changes to the model which might help. As the LTM model stands there is no clear choice between the A links and the B links; or between the use of Truck 1 or Truck 2 on any given link and so the solution can oscillate between Truck 1 and Truck 2. In order to remove this condition we could increase the cost of B links fractionally to say 0.001 greater than the cost of A links. Links could be removed by careful examination of the topology of our Supply Chain. So for example at the tip of one of the “peninsulas” in our model (TX links to the rest of the chain) we could remove the links B90, N90, B558, N558, B557, N557, B 556 and N556 without compromising the result TN 556TO TP 558 TN TO TP TO TX 90TP 557 557 557TO TX TX Both of the models ran to the point of giving an Integer solution, although MTM has more integer variables than our version of Lindo can handle. Never the less, in both cases a single large warehouse at L1 was indicated. So it would be reasonable to remove the other warehouse options and run the LTM model again for the best link usages.


A look at the simple math of the case shows that a single large warehouse is the best solution. Activating all of the links would cost $ 416 per day, a large


warehouse costs $ 1,000 per day and a medium sized warehouse $800 dollars per day. If a medium sized warehouse is used 2 are needed so, only if $416 + $1,000 were less than 2 * $ 800 would medium sized warehouses be options. Using the existing results The full set of results is given in the appendix. The following excerpt shows the objective function value of $1,127.182; that the warehouse L1 was selected at a cost of 1000. BA10 and BA11 taking on the value 1, show that links A10 and A11 were selected. The values next to these variables in the 5th column show the amount in lbs moving across each link. The reduced cost of the non-binary variables is zero in each case as they do not appear in the Objective. The reduced costs in column 3 are the costs of activating each link. LTM Solution LTM Solution OBJECTIVE FUNCTION VALUE 1) 1127.182 VARIABLE VALUE REDUCED

COST VARIABLE VALUE REDUCED

COST L1 1 1000 BA10 1 0.342 A10 2503 0BA11 1 0.969 A11 2303 0 289 of the 2616 available links were activated at an average cost of 44 cents each. On the assumption that the very best model would improve the average cost on the link by 10% the Objective Function value of the very best model would be $1,114. So on this basis we might reduce our costs by $13 per day or $4,745 per annum. This model has given us a significant result even without running to conclusion; that is that we should use warehouse L1, which was the result of the LP relaxation as well as the first IP solution. The costs of gasoline are insignificant relative to the cost of the warehouse and the potential savings from running the model to conclusion are small. On the downside, the model ignores the return journey to the warehouse for each truck and it allows for disjointed routes (nothing about the formulation enforces a continuous route for a truck!) We would need a solution from the second stage model to draw any further conclusions and this model, even rationalized will take many hours to run on our version of Lindo. Supply Chain Management SOM 758 18 May 2000 Ann Bergquist and Andrew Hall



Conclusions Solving Integer Programs of this size is difficult. It is difficult because the problems are np-hard which is to say that branch and bound algorithms have to solve problems which have more than n2 iterations. As a result model formulation must seek to reduce the number of binary variables. Reducing the number of binary variables involves knowledge of the topology which is difficult to acquire without good geographical representation tools. Computing power is needed. Parallel processing could provide this power, cheaply, and fortunately the branch and bound algorithm is ideally suited to solution in parallel processes. Better algorithms exist for the solution of Integer Programs than the one Lindo is using and we believe that solutions to our formulations are available in reasonable time frames with the appropriate technology and algorithms. Our main conclusion that warehouse L1 should be used is based upon the results of the LP relaxation of the problem.

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