Projects Summarized
John H. Vande Vate
Spring, 2008
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Projects Review
• GE: Look over the models to be sure you
understand how they handled the fact that
deliveries can be either early or late and these
involve different costs
• Sikorsky: Look over the model to be sure you
understand the difficulties of modeling the
flow of transport assets (helicopters) required
to support the flow of cargo (passengers).
Equipment balance is a major area of concern
in every mode of transportation.
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Projects Summarized
• CARE: Forecasting offers the
opportunity for planning. What we didn’t
quite get to – and it’s not surprising,
because things take time in this kind of
organization – is the segmentation of
response: what demand must be met
immediately, what can wait.
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Projects Summarized
• Alternative Apparel and BMW: Be sure you
understand the computation of safety stock in
these contexts especially the computation of
safety stock in terms of time, i.e., days of
supply.
• DIN/ABC and Disney: These are real cases of
the material from part 1 of the course. Be sure
you understand why inventory is less relevant
for DIN/ABC and how variability complicates
the idealized setting we considered in class.
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Projects Summarized
• Coca Cola and J&J projects: Be sure you
understand the “news vendor” nature of the
economics involved in committing to
“contract” supply vs “spot” supply. J&J takes
this one step further to illustrate the meaning
of collaboration in the supply chain – sharing
risk to mutual benefit. Be sure you understand
the value of strengthening your supply base
and how collaboration can help do that.
• More on the “news vendor” nature of Coca
Cola to follow
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Projects Summarized
• Intel: Recall we started with a Financial
perspective. The three drivers are: Growth,
Profitability and Return on Assets. This case
illustrates the tensions among these drivers.
Growth via the low cost pc product puts
pressure on margins and return on assets.
• New models of VMI
• Improving Return on Assets (ATM Capacity)
via a combination of MTS and MTO
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Coca Cola Brazil Reprised
John H. Vande Vate
Spring, 2008
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The Setting
• The bottlers deliver KO products to
– Large format stores (modern) and
– Small format stores (traditional)
from distribution centers across Brazil using a
combination of owned or leased trucks and trucks
provided by 3rd parties as needed.
• Different distribution centers are operated by
different bottlers. Some are owned by CCE, some
are franchises
• Each distribution center has its own fleet of trucks
(the dcs are hundreds of miles apart and moving
vehicles from one state in Brazil to another has
significant tax implications)
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The Problem
• Recommend an appropriate fleet
composition at each DC, i.e.,
• What numbers and types of vehicles
should each DC purchase or lease?
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Considerations
• Delivery requirements vary from day to day in
terms of both
– The number of stops required
– The number of cases to be delivered
• Different types of vehicles have different
capabilities, e.g., an investment in small trucks
yields more capacity to make stops than the same
investment in large trucks. The large trucks yield
more capacity to deliver cases.
• Delivering to small format stores requires
experience. KO will not allow a third party to
handle this portion of its business.
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Trade-offs
• This is the typical trade-off of capital
costs and operating costs
• Owned or leased vehicles are generally
less expensive to operate than 3rd party
vehicles, but the more vehicles we own
the lower the utilization on these
vehicles
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Current Situation (Roughly)
• KO’s current approach is clever, but faces three
challenges:
– Data requirements: The current approach employs a
daily history of stops and cases delivered from the DC
over the course of a year or more to capture the
variability. That’s cumbersome
– Computational requirements: The quantity of data and
the level of detail in these models precludes relying on
readily available tools like Solver. KO resorts to
heuristics
– The current tool does not recommend a fleet
composition. It just indicates the total capacity of the
fleet.
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General Approach
• Step 1: Develop a distribution of daily demand
in terms of stops and cases.
– The first observation here is that given a list of daily
demands, the order in which those days occur has no
effect on the answer. For a given fleet composition,
the utilization will be the same regardless of the
order.
– There is an exception to this observation: If the
highest demand days occur together, it probably
makes sense to develop isolate that peak season and
develop a separate business strategy for dealing with
it. This is exactly what KO does during the peak
season in December.
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General Approach
• Use the distribution rather than a detailed “simulation”
of demand. This offers several advantages:
– If we observe that demands at different DCs follow standard
distributions, we can quickly adjust our representations of the
variability in demand using appropriate summary statistics
rather than extensive and detailed data. For example, if daily
demand for cases is normally distributed, we can simply work
with the mean and standard deviation in daily demand rather
than long lists of 365 days of demand.
– Standard distributions also simplify the computations without
unduly compromising the fidelity of the analysis, e.g., we can
rely on standard functions like NormDist rather than counting
days with demands less than a given quantity, etc.
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General Approach
• Employ a “News Vendor” type analysis
to make the appropriate trade-offs.
– The news vendor gives a simple critical
ratio and is easy to work with
– We will need to extend the basic approach
to address
• The two dimensions of demand: stops and cases
• Traditional vs Modern customers
• The differences among the vehicle types
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Two dimensions of demand
• Let’s assume for the time being that we have a pdf
for the number of stops and number of cases on a
given day. This looks something like
• Prob{demand for cases S< C
and demand for stops <
C
S on a given day} =
f (c, s)dcds

0
0
• We can build f from historical data and then fit that
data to a standard distribution.
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Two Dimensions of Demand
• To keep things simple, let’s ignore for
now the Traditional vs Modern issue and
simply work with the pdf f.
• We’ll also assume for the time being that
we only have one type of vehicle in our
fleet – it can make s* stops per day and
deliver c* cases per day.
• How many of these vehicles should we
own?
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A News Vendor Approach
• We’ll need to understand the economics of the
decision
• For owned or leased vehicles we have
– F_owned: the amortized fixed costs of owning a
vehicle on a per day basis. This includes the cost of
money, insurance, etc. that we incur even if we
don’t use the vehicle
– V_owned: the variable cost of operating one of our
vehicles to its capacity on a given day. This
includes the mileage (fuel,…) and time (driver pay,
…) related costs of operating the vehicle.
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A News Vendor Approach
• For 3rd party vehicles we have
– F_3pl: the amortized fixed costs of owning
vehiclecosts.
on a perThe
day basis.
This includes
Noa fixed
3pl builds
these
the cost of money, insurance, etc. that we
into
the
variable
price
we
pay
incur even if we don’t use the vehicle
– V_3pl: the variable cost of hiring a 3pl
vehicle for a day.
– We assume F_owned+V_owned < V_3pl
Otherwise, there would be no point in owning any
vehicles
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News Vendor
• The risk we incur from the last vehicle is
the fixed cost of owning it whenever
demand is less than the capacity of our
fleet so
• F_owned * P
• We have to do some work to determine
the prob. P that demand is less than the
capacity of n vehicles.
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The Probability
ns* nc*
 
• P=
0
0
f (c, s)dc ds
This is the probability n vehicles is enough to
meet demand both for cases and for stops.
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News Vendor
• If we have n vehicles in our fleet, the
incremental value of an additional
vehicle is:
– It saves us from hiring a 3pl vehicle
whenever demand exceeds the capacity of
our fleet. So
(V_3pl – F_owned – V_owned)*(1-P)
– We have to do some work to determine the
prob. 1-P that demand exceeds the capacity
of n vehicles.
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The Critical Ratio
• So we want P to be the critical ratio
V_3pl – V_owned – F_owned
V_3pl – V_owned
• Given this critical ratio , compute the correct
fleet size by finding n so that
ns* nc*
 
0
0
f (c, s)dcds  
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Traditional vs Modern
• This actually turns out to be quite simple. The key
observation is that KO’s fleet must be large enough to
meet (all but possibly a specified small fraction of)
traditional demand.
• Thus the maximum anticipated demand traditional
customers imposes lower bounds on the daily
capacity of KO’s fleet
– Smin: a lower bound on the number of stops the fleet can
handle each day
– Cmin: a lower bound on the number of cases the fleet can
handle each day
• We only need to consider demand in excess of these
quantities to address the question of how much
additional capacity, above these minima KO’s fleet
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should have.
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2 dimensions of demand and
2 types of customers
• What if we have one kind of vehicle, but must
consider both traditional and modern
customers?
• First, Smin and Cmin impose a lower bound on
the number of vehicles in our fleet n_min =
max{Smin/s*, Cmin/c*}
• Now, use the news vendor to compute the
critical ratio  and from it the ideal fleet size
n*.
• If n* is too small to meet the demands of our
traditional customers, our fleet should have
n_min vehicles. Otherwise, our fleet should be
of size n*.
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Many Vehicle Types
• Assume the 3pl has a single type of vehicle
with capacities c* and s*, which makes sense
since modern customers are pretty
homogeneous (this just simplifies things a little)
• We have several types of vehicles
characterized by:
• s(i): stop capacity
• c(i): case capacity
• F(i): Fixed cost
• V(i): Variable cost
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An LP
• Consider the problem
Min f(i) n(i) + v(i) u(i) +V_3pl u_3pl
s.t.
s(i) u(i) + s* u_3pl ≥ stops
c(i) u(i) + c* u_3pl ≥ cases
n(i) u(i)
≥0
Here, the variables are
n(i) is the number of vehicle of type i in the fleet
u(i) is the number of vehicles of type i we would use
u_3pl is the number of the 3pl’s vehicles we would use.
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A Parametric LP
Given a fleet composition in terms of (n(1), n(2),
…) essentially the same LP tells us how best to
use it and what cost we incur as a function of the
demand (stops and cases)
Cost(stops, cases; n) = f(i) n(i) +
Min v(i) u(i) +V_3pl u_3pl
s.t. s(i) u(i) + s* u_3pl ≥ stops
c(i) u(i) + c* u_3pl ≥ cases
u(i)
≤ n(i)
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The Bases
• For each choice of demands (stops, cases) an
optimal solution to this problem will occur at a
basic feasible solution.
• Basic feasible solutions to this problem have at
most two basic variables (there are two
“structural constraints”), but we must also
specify for each non-basic u(i) whether it is nonbasic at 0 or non-basic at its upper bound n(i).
• In other words, we use none or all of the vehicles
of each type in our fleet except for at most two
types. If we use 3pl capacity then only at most
one type can be partially utilized.
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Bases
Example Basis
u(1) and u_3pl are basic rest non-basic at upper
bound:
s(1) u(1) + u_3pl = stops - (s(i) u(i): i=2,…)
c(1) u(1) = cases - (c(i) u(i): i = 2, …)
We also set each
u(i) = n(i)
(except for u(1))
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Bases
• So the cost is a linear function of demands
(stops, cases) and of the fleet composition so
long as the basis is optimal (which is so long as
it is feasible)
• E.g.
s(1) u(1) + u_3pl = stops - (s(i) u(i): i=2,…)
c(1) u(1) = cases - (c(i) u(i): i = 2, …)
Means
u(1) = [cases - (c(i) n(i): i=2, 3, …)]/c(1)
surplus = s(1) [cases - (c(i) n(i):i = 2, 3, …)]/c(1)
+(s(i) n(i): i = 2, 3, …)- stops
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So
• Cost(s, c; n) is a piecewise linear function of
s, c and n the fleet composition and the
function is linear on the regions in which a
basis is optimal
• So Cost(s, c; n)/n(i) is constant say f(i) +
k_b on the region in which a basis B_b is
optimal.
• So [Cost(n)]/n(i) = f(i) + (k_b * P_b)
where P_b is the probability demand (s, c) is in the
region where basis B_b is optimal.
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Finding P
• So, we compute the terms c_b and solve
the system
• f(i) + (k_b * P_b) = 0 for each vehicle type
• To obtain the probabilities P_b
• Then find the fleet composition to match
these probabilities (it might not be integral,
but we can round)
• That’s just 5 constraints!
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Wrapping it up
• If the recommended fleet composition is
not sufficient to meet the demands of
our traditional customers, then we use
the partial derivatives to move to a
minimum cost fleet that does.
• This requires a bit more elaboration, but
could be accomplished with a
Lagrangean approach.
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