11

ISyE 6203Location

John H. Vande Vate

Fall 2011

22

HDT Case

• Read Case• Answer three questions:

– Compare the direct costs of the alternatives– Considering cash flow impacts, make a recommendation

about which alternative to pursue– Since the buyer pays for the trucks as they are delivered,

analyze whether or not it would be advantageous for HDT to pay overtime to speed up production assuming they ship the trucks via Baltimore as soon as they are ready. Assume that HDT’s cost of borrowing is 12% per year. Make reasonable assumptions about the impacts of overtime on production.

• Due Thursday September 15th

33

Next 2 Weeks• No class Tuesday, September 13th.

– Please use this time to prepare for your conference calls next week.

• Class on Thursday, September 15th: – Please have your answers for the HDT case prepared to

present. Moinul Islam will respond to your ideas. • Class on Tuesday, September 20th:

– Greg Andrews, EMIL-SCS Managing Director, will cover Air Freight 101. Mr. Andrews has 30 years experience in the industry and is a wealth of information we don’t often get in academic programs

• Class on Thursday, September 22nd: – My PhD student Moinul Islam, will start talking about

models for locating consolidation facilities

44

Mid Term

• It looks like the mid-term should fall in the first week of October.

• I will post old exams and solutions

55

Questions about Projects?

66

Agenda• Context• Academic Development with Euclidean Distance

– Single location– Several locations

• Discussion of Distances– Euclidean– Surface of the Earth– Over-the-road

• Making it Practical– Discussion of Frequency & Pool Points– Line Haul & Delivery

• Long Digression on Traveling Salesman Problem

• Apply to Our Company• More complex model

Today

Moin on

9/22

When I get back

77

Context• We will not discuss Site Selection

– Zoning– Road access– Utilities– Tax incentives– Labor– ….

• We will focus on– Guidelines for the site selection process– Number and general location of facilities– Which county or region– Part of Supply Chain design exercise

88

Academic Model

• Problem: Locate a point (x,y) on the plane to minimize the total (weighted) distance to a fixed set of customer locations (xi,yi)

• di(x,y) =

• Min or Min

2 2( ) ( )i ix x y y

( , )ii

d x y ( , )i ii

w d x y

99

Convexity The Hessian of d(x,y), describing the local

curvature of the function, is

Its first principal minors are

Its second principal minor is 0 As these are all non-negative, d is Convex

2

2

2 2 3/2

( ) ( )( )

( )( ) ( )

(( ) ( ) )

b y a x b y

a x b y a x

a x b y

2 2

2 2 3/2 2 2 3/2

( ) ( )

(( ) ( ) ) (( ) ( ) ) and b y a x

a x b y a x b y

1010

Solution

• The partials of

2 2

2 2

2( ) ( )( , ) 12 ( , )( ) ( )

( )( , )( , )

( , ) ( , ) ( ) ( )

i i

ii i

i

i

i i ii i

x x x xf x yx d x yx x y y

i i

y yf x yy d x y

i

f x y d x y x x y y

are

and

1111

Solution• So a local minimum is a global minimum and

we just need to find (x, y) that satisfies

• Solving for x and y yields

( , )

( , )

( , ) 0

( , ) 0

i

i

i

i

x xfx d x y

i

y yfy d x y

i

x y

x y

/ ( , )

1/ ( , )

/ ( , )

1/ ( , )

i ii

ii

i ii

ii

x d x y

d x y

y d x y

d x y

x

y

1212

Solution• Note that if all the distances di(x,y) are equal

reduces to

But this is rarely the answer

/ ( , )

1/ ( , )

/ ( , )

1/ ( , )

i ii

ii

i ii

ii

x d x y

d x y

y d x y

d x y

x

y

and i i

i i

x y

n nx x y y

1313

Simple Example

x (X1, Y1) = (1, 0)

x (X2, Y2) = (0, 1)

• Solution: y = 1 – x so long as 0 ≤ x ≤ 1

• So (½, ½) is a solution in this case

2222

2222

)1(

1

)1(

)1()1(

1

2222

),(

),(

)1()1(),(

yx

y

yx

yy

yx

x

yx

xx

yxf

yxf

yxyxyxf

1414

Finding x and y• Start at

• Compute the distances di = di(x,y)

• Compute the new values of x and y

• repeat

and i i

i i

x y

n nx x y y

/

1/

/

1/

i ii

ii

i ii

ii

x d

d

y d

d

x

y

1515

Example• Using Euclidean Distance from Our Company

stores97 83708 -88.20 41.45 8 0.12 7.51 0.13 7.02 0.14 98 38101 -97.26 25.54 11 0.09 11.19 0.09 11.43 0.09 99 33909 -80.09 26.04 18 0.06 16.68 0.06 15.96 0.06

100 35813 -86.35 34.42 9 0.12 7.26 0.14 6.42 0.16 Average -94.63 36.39 Total Distance Total 1/Distance Total Distance Total 1/Distance Total Distance Total 1/Distance

1,439 10.74 1,425 10.39 1,419 10.56 Longitude Latitude Longitude Latitude Longitude Latitude

-93.42 36.06 -92.59 35.97 -92.06 35.95 Iteration 1 Iteration 2 Initial Iteration

1717

Academic Model: Several Facilities

• Problem: Locate m points (xj,yj) on the plane to minimize the total (weighted) distance to the set of customer locations (ac,bc) that each point serves. Each customer must be served by some point

• dc(xj,yj) =

• zcj = 1 if customer c is assigned to point j,

0 otherwise• Min • s.t. Each customer assigned to one of the points

2 2( ) ( )c c j c jw a x b y

,

( , )cj c j jc j

z d x y

1818

Single Sourcing

• Only 1 Point can serve a Customer• Each Customer assigned to 1 point

for each Customer c

• Zcj {0, 1} Binary

1cjj

z

1919

Split Sourcing Model• Dc = Demand at Customer c

• dc(xj,yj) =

• wc typically Dc

• zcj = Fraction of demand at Customer c supplied by point j

for each Customer c• zcj non-negative, but not necessarily binary

• zcj Dc is the volume supplied to customer c from point j• Only makes sense if there is some other constraint, e.g., capacity of the

facilities that forces this

2 2( ) ( )c c c j c jD w a x b y

1cjj

z

2020

Split Sourcing Model• Dc = Demand at Customer c

• dc(xj,yj) =

• wc typically Dc

• zcj = Fraction of demand at Customer c supplied by point j

for each Customer c• Facility capacity constraints

2 2( ) ( )c c c j c jD w a x b y

1cjj

z

CapacityFacility cc

cj Dz

2121

Simple Versions

• If we know where the points should be located, how can we solve the problem?

• If we know which customers should be assigned to which points, how can we solve the problem?

2222

The ALT Heuristic

• If we know where the points should be located, how can we solve the problem?

• Fix the locations of the points and determine the best customer assignments

• If we know which customers should be assigned to which points, how can we solve the problem?

• Fix the customer assignments, find the best locations for the points

• Repeat

2323

The ALT Heuristic

• Fix the locations of the points and determine the best customer assignments

• Fix the customer assignments, find the best locations for the points

• Repeat

2424

The ALT Heuristic

• It is a Heuristic – no guarantee it finds a best answer

• It is simple and effective

• The idea has broader applicability than this academic problem

• We will come back to this broader idea

• First, let’s make it more practical

2525

Making It Practical

• Distances– Euclidean

– Surface of the Earth

– Over-the-Road

• A small Challenge• Frequency & Pool Points• Line Haul & Delivery

– Digression on TSP

• Challenge

2626

Euclidean Distance • In the US actual distance is something like

60*Euclidean Distance • Not appropriate for distances more than about

20 miles• Better estimate for larger distances 69 * ArcCos(Sin(latA * pi / 180) * Sin(latB * pi / 180) + Cos(latA * pi / 180) * Cos(latB * pi / 180) * Cos((lonA - lonB) * pi / 180))• I’ve built this into OurCompany.xls as Distance(LonA, LatA, LonB, LatB)• We also have rlDistance(OZip, DZip)

2727

Working with Distance• Complicated to find/simplify the roots of the

partials. Just use Solver or any Convex Optimization tool like Minos or various Matlab packages

• So, the ALT heuristic becomes– Fix the locations of the facilities and find the best

assignments of the customers to them– Fix the assignments and use Solver or Minos to find

the best locations for the facilities • Note once the assignments are made, you can locate each

facility separately to best serve the customers assigned to it

2828

Over-The-Road Distances• Available from many sources

– CzarLite– MapPoint– PC Miler (rlPCMiler(OZip, DZip) in Radical Tools)– ….

• Can be slow, tedious to work with– Distance computations are more demanding

• Makes sense to use approximations – At least until the last few iterations– Generally the differences don’t matter, but we do want to avoid

inaccessible locations and account for geographic barriers (Great Lakes, Grand Canyon, Mississippi River, …or Alps, Baltic, North, Aegean, Adriatic, …)

• May want to use actual transport cost and time for toll roads or ferries, etc.

2929

Questions?

3030

Small Challenge

• Relocate the Indianapolis Assembly operations to minimize the daily average CWT-miles (in terms of Distance) of transportation for Our Company

• Where should those operations go?

• The Sheet Location2 of OurCompanyLocation.xls will help you get started.

3131

Small Challenge part 2

• Relocate the Indianapolis Assembly operations to minimize the daily average CWT-miles (in terms of PCMiler distance) of transportation for Our Company

• Compare the solutions

• See Location2OTR of OurCompanyLocation.xls

3232

Big Challenge• Consider the challenge of locating the cross dock to

minimize:– Transportation

• TL (EOQ) from plants to Cross Dock• LTL (EOQ) from Cross Dock to Stores

– Inventory• Pipeline • Cycle stocks

• As the location changes, the transportation rates change and so do the EOQ (extended) values.

• Work out an approach with a small number (5) stores. What issues do you encounter? What resolutions would you propose?

• Open ended

3333

Frequency Revisited • Most companies have 100’s or 1000’s or

even 10’s of thousands of SKUs• Especially in retail, there is often a battle

between – Breadth – Having many SKUs available for

customers– Depth – Having more of each SKU available

• The two compete for– Limited shelf space– Limited working capital

• Frequency of replenishment is often fixed by limits on Depth

3434

Cross Docks

• Cross docks or “Pool Points” help– DC ships to a pool point (line haul)– Pool point distributes to stores (delivery)– No inventory at the pool point

• Frequency ensures low cycle stock

• Consolidation reduces transportation impacts– Aggregating stores for line haul– Multi-stop or mode shift for delivery

3535

Location

• How many pools should we have and which stores should each one serve?

• Frequency is fixed, say once per week

• Sets cycle stocks, e.g., half a week at the stores

• Our goal is to manage transportation and operating expenses at the pools

3636

Transportation

• Line Haul– Weekly volume to a pool dictated by

weekly demand at the stores it serves– Distance dictated by location of the pool

• Delivery– Delivery distance

• TSP Tour• LTL/parcel rates

– Distance & assignments dictated by pool locations

4646

Agenda

• Deciding the number and locations of cross docks/pool points– A simple approximation for delivery costs– A simple ALT type heuristic– A more complex model that considers both

line haul and delivery in making assignments (MIP for assigning stores)

– Industrial MIP modeling tools (AMPL)

4747

Approximating Delivery

• Something simple– Sum of distances from Pool to Stores

– Greater than the Minimum Spanning Tree

– Probably larger than the TSP

– Easy to work with

• Different cost per mile for Delivery than for Line Haul– Usually several smaller vans

– Usually more surface streets

– More time spent stopping

4848

Minimize Weekly Transport Costs

• Delivery– Delivery $/mile * Distance from Store to

Pool

• Line Haul– Line Haul $/mile * Distance from Pool to

Cross Dock*# of Truckloads

• Shipments to the Cross Dock– Line Haul $/mile * Distance from each

Plant to Cross Dock*# of Truckloads

4949

Strategy

• Separate Costs & Assignments to facilitate ATL heuristic

• Calculate Costs from each Pool to each Store • “Best” Assignments use cheapest Pool• Keep 2 Assignments

– “Current” Assignments – what was best before– “Best” Assignments – using cheapest pool

• Use Costs to improve Pool and Cross Dock locations given “Current” Assignments

• Use “Best” Assignments to update Assignments

5050

Challenge 7

• Use the ALT Heuristic Idea to find good locations for the Cross Docks and the Pools

• Repeat the exercise for different #’s of Pools and show how Transportation Costs change with the # of Pools.

• Identify shortcomings of this approach• Identify alternative SC structures to

consider

5151

Questions?

5252

Challenge 3

• Team Presentations

5353

Challenge 3

• Minimum of each rate range– 147 – 404 at 170.19/CWT

• Discounted cost/shipment at 147 lbs = $75.05

• Total Transport Cost = $6,382 – 85+ = 12,500/147 shipments

• Cycle inventory cost = $817– 147 lbs/5 lbs/unit/2 * $300/unit * 18.5%

– Why divide by 2?

• Pipeline cost = $1900– 12,500 lbs/5 lbs/unit * $300/unit/365 days/year *

5 days * 18.5%

– Total Cost: $9,100

5454

Challenge 3

• Minimum of each rate range– 501 – 851 at $137.85/CWT

• Total Cost: $9,854

– 1,001 – 1,681 at $117.38/CWT• Total Cost: $11,866

– 2,001 – 4,346 at $98.70/CWT• Total Cost: $16,723

– …

5555

Challenge 3• EOQ calculations

– At min charge: $75.02 for up to 146 lbs

– EOQ Units: 82

– EOQ Weight: 410 lbs – exceeds limit

– Shipment size: 145 lbs or 29 units

– Transport Cost: $6,468 = 2,500/29*75.02

– Cycle Inventory Cost: $806 = 29/2*$300*18.5%

– Pipeline Inventory: $1,900

– Total Cost: $9,174

300%*5.18500,2*02.75$*2

5656

Challenge 3• EOQ calculations

– At 500 lbs shoulder

– EOQ units = 136

– Applicable units = 100

– Total Cost = $9,850

• Recommended shipment size: 150 lbs or 30 units at $170.19/CWT

300%*5.18500,2*02.75$*2

5757

Challenge 4• Detailed Analysis by Store for each Plant• I used

– Min Charge– 500 lb– 1,000 lb– 2,000 lb– 5,000 lb– 10,000 lb– FTL

• Distinct value densities drive distinct solutions– Green Bay: Every store used Min Charge– Denver: With 2 exceptions, every store used 10,000 or FTL– Indianapolis: Every store used 500 or 1,000 lb shipments

5858

Our Company Our CompanyIncome Statement (all figures in 000's) Balance Sheet (all figures in 000's)

% of RevenueRevenue 450,000$ 100% AssetsCOGS 303,990$ 67.6% Current Assets:

Raw Materials 300,000$ 66.7% Cash and cash equivalents 5,000$ Transportation 2,580$ 0.6% Trade accounts receivables 18,493$ Labor 1,410$ 0.3% Inventory 6,881$

Gross Margin 146,010$ 32.4% At Green Bay 5$

SG&A 26,200$ 5.8% At Indianapolis 20$ Other Operating Expenses 500$ 0.1% At Denver 27$ Operating Income 119,310$ 26.5% At Stores 5,164$ Other Income -$ 0.0% Pipeline 1,666$ Financial Expenses 6,878$ 1.5% Other current assets 2,500$ Income before Taxes 112,432$ 25.0%Provision for Taxes 27,996$ 6.2% Total Current Assets 32,874$ Consolidated Net Income 84,436$ 18.8% Property & Equipment, net 260,000$

Other Assets 1,000$

Total Assets 293,874$ NOPAT 91,315$ Invested Capital 263,889$ Liabilities & Stockholders' EquitySPEED 1.71 Current Liabilities:ROIC 35% Accounts Payable 24,985$ DSO 15 days Short-term debt 6,881$ DII 8 daysDPO 30 days Total Current Liabilities 31,866$ Cash-to-Cash Cycle (7) days Long-term debt 130,000$

Other Liabilities 250Cost of Capital 6.4%Inventory Carrying Cost 18.5% Total Liabilities 162,116$ Cost of Inventory 1,272$

Total Stockholder's Equity 131,758$

Total Liabilities and Stockholders' Equity 293,874$

5959

Summary • Direct Full Truckloads

– $158 million in inventory– $435 thousand in transportation– Gross Margin % = 33%– SPEED = 1.08– ROIC = 23%

• Single EOQ to All Stores– $15.8 million in inventory– $2.54 million in transportation– Gross Margin % = 32.5%– SPEED = 1.65– ROIC = 34%

• Different EOQ shipments– $15.2 million in inventory– $2.47 million in transportation– Gross Margin % = 32.5%– SPEED = 1.65– ROIC = 34%

• LTL EOQ – $6.88 million in inventory– $2.58 million in transportation– Gross Margin % = 32.4%– SPEED = 1.71– ROIC = 35%

Working harder & harder to achieve smaller and

smaller impacts

6060

Changing the Rules• From FTL to LTL• Changing the network: Consolidation

– Challenge 6: October 4th

– Challenge 7: October ?• Combining LTL and network revisions

– Small & Big challenges• Comment: ALT heuristic is a heuristic, but

data are estimates. Does it make sense to work hard to get exact answers to approximate questions? Do the easiest thing first, work harder if it’s worth it.

6161

Review

• Consolidation for balance

• Consolidation for transport: Pool points

• Simple model – ignore inbound costs

• Complex model – consider inbound costs

6262

A More Complex Model

• The current model assigns each store to the Pool closest to it

• It ignores the costs of getting product to the Pool

• Example: Pool 8 only serves 2 Stores so the Line Haul is very expensive

• Objective: Develop an Assignment model that considers this cost as well.

6363

A More Complex Model

• Integer Programming Model– Linear Programming Model– Variables must be 0 or 1

• Who is familiar with Linear Programming?– Read Chapters 1 & 2 of Chvatal

• Who is familiar with Integer Programming?– Read Sections 2.4 – 2.9 of Der-San Chen et al.

6464

A More Complex Model

• Three tasks in building an LP or IP or MIP model:– The Decision Variables: What we can change– The Objective Function: What we are trying to

accomplish– The Constraints: The limits to our choices

• The Objective Function and Constraints must by Linear– Sum of Constant*Decision Variable

6565

A More Complex Model

• The Objective– Minimize Transportation Costs from the Cross

Dock to the Stores• Cross Dock to the Pool

– Line Haul Cost * Distance * # of Trucks/week

• Pool to the Store– Delivery Cost*Distance

• The Decision Variables– Assignments of Stores to Pools

• zij = 1 if Store i assigned to Pool j, 0 otherwise

6666

A More Complex Model

• The Constraints– Each Store is assigned to 1 Pool

– For each Store i

– zij is Binary (0 or 1)

1ijj

z

6767

The Objective• Minimize Transportation Costs from the Cross Dock to

the Stores– Cross Dock to the Pool

• Line Haul Cost * Distance * # of Trucks/week– Line Haul Cost is a constant, e.g., $1/mile– Once we fix locations of the Pool and the Cross Dock,

Distance from the Cross Dock to each Pool is a constant (This is the advantage of the ALT heuristic)

– # of Trucks/week • Must be enough to carry one week sales at the stores served by

the pool.• Must be at least one• Doesn’t need to be an integer since we can bring extra one week

and fewer the next. Though there is some inventory impact

6868

New Variables & Constraints

• Add new variables to capture this– Tj = the average number of trucks we run from the

Cross Dock to Pool j each week. This should be restricted to integer values.

• Need new Constraints to enforce these definitions– There are enough trucks to carry the load to each

Pool each week

6969

Challenge

• Formulate linear constraints to model these conditions

7272

Questions?

7474

ODBC

• Open Data Base Connectivity:– Read model data from Access, Excel, SQL Server,

…

table PoolTable IN "ODBC"

"DSN=OptimizationData"

"SQL=SELECT Pool, Distance AS LineHaulDist from PoolData":

POOLS <-[Pool], LineHaulDist;

read table PoolTable;

Ample name for the table

For reading

Access via ODBC

ODBC Data Source Name

Reading instructions

Mapping to AMPL model ParametersCommand to read the

table

7575

Model

• Pools– set POOLS;– param LineHaulDist{POOLS};

• Stores– set STORES;

• Delivery (Pool to Store)– set DELIVERYLEGS dimen 2;– param DeliveryDist{DELIVERYLEGS};

Model requires a set of Pools

Each pool has data called

LineHaulDist

Model requires a set of Stores

Model requires a set of pairs (Pool, Store)

Each pair has data called

DeliveryDist

7676

Model

• Other Data – param LineHaulCost := 1.00;– param DeliveryCost := 1.75;– param LbsPerTruck := 30000;– param WeightPerWeekPerStore := 5*10*55;

• Better if this is read from a table, but…• Model Variables

– var Open{POOLS} binary;– var Trucks{POOLS} integer;– var Assign{DELIVERYLEGS} binary;

A binary variable for each pool

An integer variable for each pool

A binary variable for each (Pool, Store) pair

7777

Model

• minimize TotalTransportCost: sum{pool in POOLS}

LineHaulCost*LineHaulDist[pool]*Trucks[pool]+

sum{(pool,store) in DELIVERYLEGS} DeliveryCost*DeliveryDist[pool,store]*

Assign[pool,store];

• s.t. AssignEachStore{store in STORES}:• sum{(pool,store) in DELIVERYLEGS}

Assign[pool,store] = 1;

The objective is to minimize total

transportation cost

Total Line Haul cost

Total delivery cost

A constraint for each store

Assign the store to exactly 1 pool

7878

Model

• s.t. EnoughTrucksToCarryLoad{pool in POOLS}:• Trucks[pool] >= WeightPerWeekPerStore*sum{(pool,

store) in DELIVERYLEGS}Assign[pool,store]/LbsPerTruck;

• s.t. AtLeastOneTruckToOpenPool{pool in POOLS}:• Trucks[pool] >= Open[pool];

• s.t. DontAssignStoresToClosedPools{(pool,store) in DELIVERYLEGS}:

• Open[pool] >= Assign[pool, store];

A constraint for each pool

Enough trucks to carry all the demand it serves

And at least one truck if the pool is open

If the pool is closed, it can’t serve any stores

7979

Solve & Report

• option solver cplex• solve;

• table DeliveryOut OUT "ODBC"• "DSN=OptimizationData"• "DeliveryOut":• {pool in POOLS, store in STORES:

Assign[pool, store] > 0.01}->[pool, store];

• write table DeliveryOut;

Select a solver

Solve the problem

Write out the solution

If the pool is closed, it can’t serve any stores

Just write out the “active” assignments

8080

Challenge

• For you to think about (no presentation):– What if we had more than one Cross Dock?– How would the model change?

8181

Zone Skipping

• Original applications from bulk mail

• Where do we give the mail to the USPS?– Close to origin?

• Our cost to get mail to the USPS is low

• Mail crosses several zones so rates are high

– Close to destination?• Our cost to get mail to the USPS is high

• Mail delivered “locally” so rates are low

8282

Zone Skipping

• LTL application• We send (or manage) large volume of

small LTL shipments from a limited number of origins

• Where should we give the freight to the LTL carrier?– Close to the origin? – Close to the destination?

• We pay TL rates to get it to the LTL terminal

8383

A Model• Fixed set of candidate LTL terminals to

deliver mail to• Includes our origin (LTL door-to-door)• Appropriately aggregated shipments

– Origin– Destination– Class– Weight– Ship date– Special handling requirements

Aggregate as much as is appropriate e.g., destination region (3-digit zip) and weight range (e.g., LTL ranges)

Rarely a dynamic model so we want averages (per day or perhaps longer depending on service commitment)

Some concern about variability/

8484

A Model

• How do we get shipments (of this category) from this terminal to this destination (e.g. 3-digit zip)?– LTL door-to-door?– TL to LTL terminal A and LTL from there– TL to LTL terminal B and LTL from there– …

• Very similar to our pool point model– LTL from terminal to destination is analogous to delivery – TL to LTL terminal is analogous to Line Haul– New option is just go LTL all the way

• Zij = 1 if we assign destination j to LTL terminal i• Zi = 1 if we deliver LTL direct to destination I

1 ij

ij zz

8585

A Model• LTL Cost from LTL terminal to customer is

analogous to “Delivery cost”. Now we can use rating engine to estimate it

• TL Cost from our origin to the LTL terminal is analogous to Line Haul Cost, based on distance and # of trucks per period

• LTL Direct cost from our origin to the destination is new. Use the rating engin to produce it.

• Model decides which ones to pay