11

Projects Summarized

John H. Vande Vate

Spring, 2008

22

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.

33

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.

44

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.

55

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

66

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

77

Coca Cola Brazil Reprised

John H. Vande Vate

Spring, 2008

88

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)

99

The Problem

• Recommend an appropriate fleet composition at each DC, i.e.,

• What numbers and types of vehicles should each DC purchase or lease?

1010

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.

1111

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

1212

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.

1313

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.

1414

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.

1515

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

1616

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 < C and demand for stops < S on a given day} =

• We can build f from historical data and then fit that data to a standard distribution.

dsdcscfS C

0 0),(

1717

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?

1818

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.

1919

A News Vendor Approach

• For 3rd party vehicles we have– F_3pl: 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_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

No fixed costs. The 3pl builds these into the variable price we pay

2020

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.

2121

The Probability

• P =

This is the probability n vehicles is enough to meet demand both for cases and for stops.

dsdcscfns nc

*

0

*

0),(

2222

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.

2323

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

dsdcscfns nc*

0

*

0),(

2424

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 should have.

2525

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*.

2626

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

2727

An LP

• Consider the problemMin f(i) n(i) + v(i) u(i) +V_3pl u_3pls.t. s(i) u(i) + s* u_3pl ≥ stops c(i) u(i) + c* u_3pl ≥ cases n(i) - u(i) ≥ 0Here, the variables aren(i) is the number of vehicle of type i in the fleetu(i) is the number of vehicles of type i we would useu_3pl is the number of the 3pl’s vehicles we would use.

2828

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)

2929

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 non-basic 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.

3030

Bases

Example Basisu(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))

3131

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

3232

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.

3333

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!

3434

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.