IHE 890 Engineering Supply Chain Systems

Instructor: Pratik J. Parikh, Ph.D.

April 12, 2011

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Facility Location Models  Number of Facilities (Single or Multiple)  Continuous or Discrete Location Space

 Capacitated or Uncapacitated  Distance Measures

–  Rectilinear, Euclidean, Actual distance

 Objective: fixed cost plus transportation  Objective Functions

–  Minisum –  Minimax

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A Distribution Problem (Transportation Cost Minimization)

  The Hottest Mexican Restaurant has restaurants in 5 Midwestern cities. All orders of tortillas are from the Laredo Tortilla Factory, which has warehouses in 6 cities.

  The 5 consuming cities and the shipping costs (in dollars per dozen tortillas) from the warehouse to these cities are given below:

Unit Shipping Cost ($/dozen)

Whse/City Minneapolis Salina Kansas Lincoln Wichita Tulsa $3.00 $9.00 $6.00 $5.00 $7.00 Okla $6.00 $4.00 $5.00 $4.00 $5.00

Denver $7.00 $3.00 $7.00 $1.00 $2.00 St. Louis $3.00 $7.00 $9.00 $2.00 $2.00 Lawrence $4.00 $3.00 $3.00 $8.00 $4.00 Omaha $5.00 $1.00 $2.00 $5.00 $9.00

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 Assume DC locations are known for this example.   The demand for each restaurant and the tortillas available at

each DC are:

Distribution Problem (Example )

  Let’s model it as a linear programming problem. –  the objective function is the total transportation cost (to be minimized), –  subject to the demand-supply constraints.

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A Mathematical Model

Indices and Sets i, index for warehouses. i∈I = {Tulsa, Okla, Denver, St. Louis,

Lawrence, Omaha}

j, index for customers. j∈J = {Minneapolis, Salina, Kansas, Lincoln, Wichita }

Parameters cij: cost of transporting one unit from warehouse i to customer j ai: supply capacity at warehouse i bi: demand at customer j Decision Variables xij: number of units transported from warehouse i to customer j

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A Mathematical Model (contd.)  Objective:

∑∑= =

=m

i

n

jijij xcZ

1 1

Costtion Transporta Total Minimize

i) seat warehoun restrictio(supply m1,2,...,i ,

Subject to

1=≤∑

=

n

jiij ax

j)market at t requiremen (demandn 1,2,...,j ,1

=≥∑=

m

ijij bx

ns)restrictio negativity-(nonn 1,2,...,ji, ,0 =≥ijx

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Setting Up Excel Sheet   Step 1: Set up

the EXCEL spreadsheet

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  Notice that there are two sections. The first section shows the unit shipping

costs. The cells have been formatted as currency with 2 decimal places (Select by highlighting the cells, then click on ‘Format’- ‘Cell’- ‘Currency’ ).

  The second section shows the allocation and shipping costs. The optimal allocations have been assigned to cells B20:F25 (at this time, these cells are all empty). These are the decision variables.

  The demand and supply have been entered in cells B27:F27 and cells H20:H25, respectively. Also, row 28 has been formatted as “currency” with 2 decimal places, and all other cells formatted as number with 2 decimal places.

Setting Up Excel Sheet (Notes)

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Set Up Formula: Sums of Cells   Step 2: Enter the formulae for the sum of demand (cells B26:F26) and the

sum of supply (cells G20:G25), respectively. –  For example, B26=SUM(B20:B25); copy and paste the formula from C26:F26 . –  G20=SUM(B20:F20); copy and paste the formula from G21:G25 .

  To find out if supply is sufficient, enter the formulae of the total system demand and the total system supply.

–  Total system supply H26=SUM(H20:H25) –  Total system demand G27=SUM(B27:F27) –  The sum of supply is H26=423. Similarly, compute the sum of demand. The sum

is G27=370. In this case, there will be excess supply.

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Shipments from ... Shipments to …   Step 3: Enter the formula for cell G20=SUM(B20:F20), the total shipment from

Tulsa, as shown. Note that cells B20:F20 = the allocations from Tulsa to Minneapolis, Salina, Kansas, Lincoln, and Wichita, respectively. Copy this formula and paste it onto cells G21:G25.

  Step 4: Likewise, enter the formula for cell B26=SUM(B20:B25), the shipments to Minneapolis; copy and paste the formula onto cells C26:F26.

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Shipping Costs   Step 5: Enter the formula for cell B28=SUMPRODUCT(B3:B8,B20:B25),

the total shipping cost to Minneapolis. Copy and paste the formula onto cells C28:F28.

  Step 6: Enter the formula for cell G28=SUM(B28:F28), the total system cost.

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Using Solver To Optimize   Step 7 Click the “Data” tab, and

choose “Solver” on the right-most side. You should see this:

  Step 8: Enter the following (the “Set Target Cell” is $G$28, the grand total cost):

  By changing cells B20:F25 (the cells highlighted in light green is our allocation table).

  Select the Min button to minimize the grand total cost.

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If you do not see “Solver” in the “Data” tab, then click on the “Office Button” on top left of the window. Click on “Excel Options -> Add-ins on left -> Solver Add-in the window -> ok at the bottom.

Adding Constraints   Step 9: Add the following constraints (one at a time):   Since total capacity exceeds demand, the shipment from each source should

be less than or equal to its capacity: G20:G25 ≤ H20:H25, i.e.

  Since total demand is less than total capacity, the total shipment to each destination should be equal to its demand, B26:F26 = B27:F27

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Options: Linear, Non-negative   Step 10:After entering all constraints, set the option as shown:

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  Step 11: Click the ‘Solve’ button! Solution

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Modeled and solved a distribution problem with transportation cost minimization as objective using Excel Solver!

  Suppose, the warehouse in Omaha becomes unavailable.

 Originally, the sum of supply was 423. –  With Omaha gone, the total supply is now 351 units. –  Since total demand is 370 units, 19 (=370-351) units of demand will not

be satisfied.

 Replace Omaha by “Lost Sales,” with capacity equal to the demand not satisfied, i.e., 19 units.

  Suppose the unit cost of unsatisfied demand is $30 for the restaurants in Salina and Kansas, and $20 for the other locations.

Distribution Problem with Lost Sales

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  Lincoln & Wichita will have shortages (4 & 15 units, respectively).

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Location of a City’s Fire Stations   A city is made up of 11 of neighborhoods. A fire station can be placed in any

neighborhood and is able to handle the fires for both its neighborhood and any adjacent neighborhood (any neighborhood with a non-zero border with its home neighborhood).

  The problem is to find the minimum number of fire stations used to cover all neighborhoods; facility location and number!

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The Mathematical Model

Indices and Sets i,j neighborhood indices, i = 1…11 Parameters aij 1, if neighborhood j can be covered by a fire station located at i 0, otherwise Decision Variables xi 1, if fire-station i is opened, 0, otherwise

Integer Program for fire station location problem

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iallforx

jallforxats

x

i

iiij

ii

}1,0{

1..

max

∈

≥∑

∑

Location of a City’s Fire Stations   Let xi be the decision variable representing whether or not a fire station will

be built in neighborhood i. If xi = 1, build; 0, otherwise

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Cover neighborhood 1 by placing one or more fire stations at one of these 4 neighborhoods

Excel Implementation

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Maximal Covering Problem (coverage-based measure)

  Now, suppose we can only set up p=2 fire stations, what is the maximal number of neighborhoods that can be covered using p facilities?

  This is a p-covering problem. Model xi = 1 if fire-station i is opened, 0 otherwise yj = 1 if neighborhood j is covered, 0 otherwise aij = 1 if neighborhood j can be covered by a fire-station at i; 0 otherwise   Solve the above problem using Excel

Integer Programming Model for the

p-covering problem

jiallforyx

px

jallforyxats

y

ji

ii

ijiij

jj

,}1,0{,

..

max

∈

=

≥

∑

∑

∑

p-Median/Center Problems (distance-based measure)

  P-Center Problems –  To locate n new facilities, called centers, on a network with respect to m

customers, so as to minimize the maximal distance –  Example: locate n emergency vehicles within a neighborhood so that the

maximum distance is minimized

  P-Median Problem –  To locate n new facilities, called medians, on a network with respect to m

existing facilities with vertex locations, so as to minimize a sum of weighted distances between each existing facility

–  Example: locating 2 distribution centers that serve 100 stores

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p-Median Model Indices and Sets:

J: set of feasible plant locations, indexed by j I: set of markets, indexed by i

Data Di demand of market i No capacity limitations for plants At most p plants are to be opened dij distance between market i and plant j

Decision Variables

yj = 1 if plant is located at site j; 0 otherwise xij = 1 if market i is supplied from plant site j; 0 otherwise

Model Explanation

Min

. .

,

1

, {0,1} ,

ij iji j

jj G

ij j

ijj

ij j

d x

s ty p

x y i I j J

x i I

x y i I j J

∈

=

≤ ∀ ∈ ∈

= ∀ ∈

∈ ∀ ∈ ∈

∑∑

∑

∑

Minimize the total demand-weighted distance

There are a total of p plants

If the plant is located at j, only then it can serve

Every market has to be served

Yes or no decision, integrality

Constraints

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Di

p-Center Model In p-center problem, we are minimizing maximum distance between a market and a plant, or between fire stations and all the houses served by the fire stations. Replace the objective function in p-Median problem with

Min w

where w = max {dijxij : i is a market assigned to plant j}

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p-Center Model Minimize the Maximum Distance w

There are a total of p centers

Every market has to be served

Yes or no decision, integrity

Min . .

,

1

, {0,1} ,

jj J

ij j

ijj

ij ijj

ij j

ws ty p

x y i I j J

x i I

w d x

x y i I j J

∈

=

≤ ∀ ∈ ∈

= ∀ ∈

≥

∈ ∀ ∈ ∈

∑

∑

∑ Definition of w: it is max value across all distances for i-j pair

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JjIi ∈∀∈∀ ,

If the plant is located at j, only then it can serve

Example Problem

1

2 3

6

5

4

3

4

6 6

3

7

4 2 7

Find the 2-median and 2-center solutions to the following 6 city problem a)  Write the mathematical model b)  Implement using excel solver

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Demand = 1 unit

2-Median Problem Solution

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1 and 4 are not connected, so large distance!

2-Center Problem Solution

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Optimal Solutions

 Optimal solution to the 2-median problem –  Open 3 and 6 –  Total Distance: 13

 Solution to the 2-center problem –  Open 3 and 6 –  Maximal distance is 4

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Assumptions Ø  The number of warehouses to locate is fixed beforehand.

Ø  If a warehouse is located at site j: §  There is no fixed cost §  There is no capacity limit

Now, we relax this assumptions.

The p-median & p-Center Problems

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Uncapacitated Facility Location Problem   Suppose the cost of setting up the facility is different in different cities.   In the above 6 city problem, suppose the setup costs are (10, 8,15,11,8,12).   The transportation cost is 1 per unit distance.   How many warehouses should be setup, where should we setup them, and which

customer do they serve?   Our goal is to minimize the total cost.

3 1

2 3

6

5

4

3

4

6 6

7

4 2 6

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{ }{ }

minimize

subject to 1

,

0,1

0,1 ,

j j ij ij ijj J i I j J

ijj J

ij j

j

ij

f y c d x

x i I

x y i I j J

y j J

x i I j J

∈ ∈ ∈

∈

+

= ∀ ∈

≤ ∀ ∈ ∀ ∈

∈ ∀ ∈

∈ ∀ ∈ ∀ ∈

∑ ∑∑

∑

Uncapacited Facility Location Model

Fixed cost + Transportation Cost

Served at least once

If open, then serve

Yes/no decision

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≥

Solution to the Uncapacitated FLP

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Summary of Learning Objectives

 Solving FLPs in Excel  p-median and p-center problems  Uncapacitated FLP