Review of Part I
Preparation for Exam
John H. Vande Vate
Fall 2009
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Review
• Transportation costs are generally concave
– Economies of scale
– Consolidation reduces transport costs
• But there are other financial/operational issues to
balance
– Operating Expense
• We focus on transport, but handling, labor, …
– Capital
• We focus on working capital in inventory
– Time
• We focus on OTD
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2
Review
• Illustrated (some of) these trade-offs in our
“Case Study”
– Estimating inventory costs
• “cycle” inventory driven by mode
• Pipeline inventory driven by time and total demand
– The impact of crossdocking/consolidation on
inventory
– Trading off inventory vs transportation
• EOQ for 1-to-1
• EPQ for 1-to-many
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3
Review
• After that we have to work harder …
• Review of network models and useful extensions
– Modeling!
• Pool points – Consolidating for speed
– Load driven systems
• Zone skipping – Consolidating for cost (and
speed)
– Service or schedule driven systems
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Review
• Multi-Stop routes & Milk runs
– Digression into column generation
– Application to multi-stop routes
• Location
– Where to put consolidation/distribution
facilities
• Landed cost models
– Incremental vs systems views
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Exam Overview
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Typically 4-5 questions
There will be modeling!
Open books, open notes
Calculator ok
No computers
No internet
No collaboration
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There will be modeling
• /* The objective: Minimize Transportation Cost in
$/year */
• Minimize TransportCost:
• Sum{(orig, dest) in LANES}TruckCosts[orig,
dest]*Trucks[orig, dest];
• Minimize
 TruckCost[i, j ] * Trucks[i, j ]
( i , j ) inLANES
• You may provide answers using either notation.
• Be sure your answers are clear and unambiguous.
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Define your variables – what are they?
Set out the units (e.g., $/mile, lbs/Truckload, …)
Be specific about indices of summation
Comment on what a constraint is designed to do
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Last Year’s Exam
• Question 1 (25 points) Basic
understanding. Can you perform the
kind of simple analysis we did for our
case study company
• Question 2 (25 points) Elaboration of
basic concepts. Apply the EPQ idea
when costs include weight breaks and
freight includes a mix of products
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Last Year’s Exam
• Question 3: Extending a basic model.
Expand our basic consolidation model to
address different products with different
weights and dimensions
• Question 4: Theory through example.
Did you understand the basic tenets of
location?
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Last Year’s Exam
• Consider the operations of a company, similar to the one we
discussed in the lecture of August 25th, that sells computers and
TVs through 100 stores across the country.
• Assume the average distance to a store (from Indianapolis) is
1,000 miles and that a truck can travel 500 miles per day.
• As consumers adopted the new flat panel televisions, the business
of the company has changed so that its stores sell 20 TVs and 10
computers (consisting of a CPU and a monitor) each day. (Assume
250 days in the year).
• The company closed down the operations in Denver and now
produces
– CPUs weighing 5 lbs and costing $300 each in Green Bay and
– Flat Panel TVs and Monitors weighing 10 lbs and costing $400 each
in Indianapolis.
• The distance from Green Bay to Indianapolis is 500 miles.
• The company uses all full truck load shipments (a truck holds
35,000 lbs.) to ship everything to Indianapolis where it is
consolidated and shipped in full truckload shipments to the stores.
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Units/Truck = 35000 lbs per truck
5 lbs per unit
= 7000 units
Value of TL = $300*7000
= $2.1million
• Assuming an inventory holding cost
of 15% and a
Inventory in Green Bay $1.05million
Question 1
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transportation cost of $1.50/mile compute:
The capital required to run theUnits/day
system =including
100 storesthe
* 10capital:
cpus/day
= 1000 cpus
at Green Bay: __$1,050,000 _____________
Worth $300,000
at the cross dock in
Time units spend in transit is 1 days
Indianapolis:__$1,800,000_____________
So pipeline inventory $300,000
at each store:__$750,000______________
in-transit between
– Green Bay and Indianapolis: __$300,000 (or $240,000 is more
accurate)___________
– Indianapolis and each store: __$30,000____________
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What’s in truck to store?
20 TVs weigh
10 Computers weigh
So this "basket" weighs
Since a truck holds
200lbs and cost
150lbs and cost
350lbs and costs
35,000 lbs, it can hold
$
$
$
8,000
7,000
15,000
100
• Value on a truck to a store
– 100*$15,000 = $1.5 million
• Capital at a store: $750,000
• Capital at Indianapolis cross dock
– $1.05 million from Green Bay
– $0.75 million staged for store
– $1.8 million total
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Question 1
• Assuming an inventory holding cost of 15% and a
1 Store sells $15,000/day
transportation cost of $1.50/mile
compute:
Time units spend in transit is 2 days
• The capital required toSorun
the system
the
capital:
pipeline
inventoryincluding
$30,000 per
store
• at Green Bay: __$1,050,000 _____________
• at the cross dock in
Indianapolis:__$1,800,000_____________
• at each store:__$750,000______________
• in-transit between
– Green Bay and Indianapolis: __$300,000_________
– Indianapolis and each store: __$30,000____________
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Capital
At Green Bay: $1.05 million
At Indianapolis: $1.8 million
At each Store: $0.75 million
Between GB and Indianapolis: $0.3 million
Between Indianapolis and Stores: $3 million
Total: $81.15 million
Question 1 Cont’d
•
The total cost of operating the system
including:
Holding cost: $12.2 million
–
Annual transportation costs: __$402,000
(401,786)_ and
From Green Bay to Indianapolis:
– Units
Annual
costs:
per yearinventory
= 100 stores holding
* 10 units per
day *250 days = 250,000
units/year
_$12,172,500 (or $12,163,500)_
Trucks/year = 250,000/7000 = 35.7
$/year = 35.7*$1.50/mile*500miles
= $26,800
From Indianapolis to Store
A truck holds 100 days of sales
We visit each store 2.5 times per year
100 stores * $1500/visit *2.5 visits/year
= $375,000
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Question 2
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Consider the company described in Question 1. The company is exploring
the option of replacing truck load shipments from Indianapolis to its stores
with LTL shipments.
As a first approximation to the magnitude of the opportunity, the company
has assembled an estimate of average LTL rates for shipments to customers
1,000 miles away. For simplicity, they have averaged out geographic
aspects (e.g., shipping to customers in Florida is more expensive) and come
out with average costs based only on the weight of the shipment:
Weight
Cost per CWT ($/100lbs)
500 – 1000 lbs
$14
1000 – 5000 lbs
$12
5000 – 20000 lbs
$ 9
> 20000 lbs
$ 7.25
The minimum charge for any shipment is $100. So for example, a 500 lb
shipment nominally costs $70 = $14/CWT*5 CWT, but the carrier won’t
accept less than $100 for any shipment so the actual cost is $100.
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Thinking
• What are the options?
• Claim: Ship either
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$100 shipping cost
500 lbs
1,000 lbs
5,000 lbs or
20,000 lbs
• Why?
If you ship 800 lbs,
transport costs are
the same as if you
ship 500 lbs but
inventory costs are
higher!
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Calculations
• What are we shipping? Basket of
products
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weighs 3.5 CWT
Worth $15,000
Holding cost/year: $2,250
Annual demand at a store: 250
• Minimum Charge:
• $100/14 = 7.14 CWT
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The Numbers
With shipments of
Total Transport to the
stores is
Unit Costs are
Shipment Size in
Baskets is
Store Inventory is
Total Cost
500 lbs
14 $
1,225,000
1.43
1,071,429
1,385,714
1000 lbs
12 $
1,050,000
2.86
2,142,857
1,371,429
5000 lbs
9$
787,500
14.29
10,714,286
2,394,643
7.25 $
634,375
57.14
42,857,143
7,062,946
20000 lbs
Should we use
LTL? Yes!
What constitutes a shipment?
2.86 baskets
2.86*20 = 57 TVs
2.86*10 = 29 computers18
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Minimum Charge
• We exceeded it. No additional
calculation needed.
• If we hadn’t?
EPQ with fixed
transport cost of $100
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Question 3
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In class, we outlined a model to help identify which candidate consolidation points to use and to assign
customers to those consolidation points to minimize the costs of transportation while meeting a “service
constraint” imposed in terms of a minimum number or frequency of trucks to each opened consolidation point.
This question asks you to flesh out that formulation for a setting in which we sell several different products. Let
PRODS denote the set of products we sell.
Each customer has a projected demand for each product given in the parameter Demand. Let CUSTS denote the
set of customers and, for each customer c and product p, let Demand[c, p] be the customers demand for that
product in units per year.
Each product has a cubic ft. per unit given in the parameter Cubes, i.e., Cubes[p] is the cubic feet occupied by
one unit of product p.
Each product has a unit weight given in the parameter Weight, i.e., Weight[p] is the weight in pounds of one unit
of product p.
Let CONSOLS denote the set of candidate consolidation points and suppose the
LTL (less-than-truckload) costs for shipping each customer’s annual demand for
each product from each candidate consolidation point are given in the parameter
LTL, i.e., LTL[c, p, k] is the LTL cost for shipping all of customer c’s annual
demand for product p to the customer from consolidation point k.
The LTL costs for direct shipments for the annual demand of each product from our plant to each customer are
given in the parameter Direct, i.e., Direct[c, p] is the LTL cost for shipping all of customer c’s annual demand
for product p directly to the customer from the plant.
The cost to send a truck from our plant to each candidate consolidation point are given in a parameter
TruckCost, i.e., TruckCost[k] is the cost to send one truck from the plant to consolidation point k.
A truck can hold (with the load factor calculated in) up to 30,000 lbs and up to 3,000 cubic feet (again this
number incorporates the load factor).
Our service requirement stipulates that we send at least 112 trucks a year to each open consolidation point.
Formulate a linear, mixed integer program to model the problem of minimizing the total cost of
transportation while meeting the service requirement. Do NOT consider multi-stop routes in your answer.
Be sure to use the parameters described above. Be sure to clearly define your variables and their units
(e.g., lbs, $, hours).
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AMPL Model
Set PRODS; /* The set of Products */
Set CUSTS; /* The set of customers */
Set CONSOLS; /* The set of candidate
consolidation points */
Param Demand{CUSTS, PRODS}; /*
customer’s demand for each product in
units per year. */
Param Cubes{PRODS}; /* the cubic feet
occupied by one unit of each product */
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AMPL Model
Param Weight{PRODS}; /* the weight in
pounds of one unit of each product */
Param LTL{CUSTS, PRODS,
CONSOLS}; /* the LTL cost for
shipping all of a customer annual
demand for product p to the customer
from consolidation point k. */
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AMPL Model
Param Direct{CUSTS, PRODS}; /* the
LTL cost for shipping all of a
customer’s annual demand for each
product directly to the customer from
the plant. */
Param TruckCost{CONSOLS}; /* the
cost to send one truck from the plant to
consolidation point */.
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AMPL Model
Var Open{CONSOLS} binary; /*
Whether each consol is open or not */
Var Trucks{CONSOLS} integer >= 0; /*
How many trucks we send to each
consol */
Var Assign{CUSTS, PRODS,
CONSOLS} >= 0; /* Fraction of
demand for each product to each
customer we ship via each consol –
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note we allow fractions */
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AMPL Model
Var DirectShip{CUSTS, PRODS} >= 0;
/* Fraction of demand for each product
to each customer we ship direct from
the plant */
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AMPL Model
Minimize Transport Cost:
Sum{c in CUSTS, p in PRODS,
k in CONSOLS}
LTL[c, p, k]*Assign[c, p, k] +
Sum{k in CONSOLS}
TruckCost[c]*Trucks[c] +
Sum{c in CUSTS, p in PRODS}
Direct[c, p]*DirectShip[c, p];
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AMPL Model
s.t. MeetAllDemandForEachProductAtEachCustomer
{c in CUSTS, p in PRODS}:
sum{k in CONSOLS}Assign[c, p, k] + DirectShip[c, p] =
1;
s.t. DontAssignCustomersToClosedConsols
{c in CUSTS, p in PRODS, k in CONSOLS}:
Assign[c, p, k] <= Open[k];
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AMPL Model
s.t. MeetAllDemandForEachProductAtEachCustomer
{c in CUSTS, p in PRODS}:
sum{k in CONSOLS}Assign[c, p, k] + DirectShip[c, p] =
1;
s.t. DontAssignCustomersToClosedConsols
{c in CUSTS, p in PRODS, k in CONSOLS}:
Assign[c, p, k] <= Open[k];
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AMPL Model
s.t. SendEnoughTrucksToEachConsolToCarryCube
{k in CONSOLS}:
3000*Trucks[k] >=
sum{c in CUSTS, p in PRODS}
Cubes[p]*Demand[c, p]*Assign[c, p, k];
s.t. SendEnoughTrucksToEachConsolToCarryWeight
{k in CONSOLS}:
30000*Trucks[k] >=
sum{c in CUSTS, p in PRODS}
Weight[p]*Demand[c, p]*Assign[c, p, k];
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AMPL Model
s.t. MeetServiceRequirementAtOpenConsols
{k in CONSOLS}:
Trucks[k] >= 112*Open[k];
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Question 4
• In our discussions on Location problems, we
observed that locating a facility at the “center of
gravity” of a set of customers (average the x and y
coordinates of the customers) does NOT minimize
the sum of the Euclidean Distances to those
customers.
• Does locating a facility at the “center of gravity” of
a set of customers minimize the sum of the
distances to those customers under the Manhattan
Metric, where the distance between two points (x,
y) and (x’, y’) is
|x – x’| + |y – y’|?
No!
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Question 4
• If you answered “Yes” to part A, provide a brief
argument supporting your conclusion. If you
answered “No” to part A, provide an example
showing the center of gravity is not the best
location.
• Counterexample?
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Questions?
Good Luck!
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