Diffusion model with capacity constraints

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Re: Diffusion model with capacity constraints

Postby John Morecroft » Tue Mar 27, 2012 8:03 am

It's interesting that you mention a variant of the Bass model that includes delivery delay feedback. Delivery delay is a practical measure of dynamic imbalance between supply and demand. It is commonly used in SD models as a mediating variable between the firm and the market to capture the effect of capacity constraints. The formulation of delivery delay feedback from SMBD Chapter 7 is as follows:

Customer Orders = Sales Force * Normal Sales Force Productivity * Effect of Delivery Delay on Orders {systems per month}
Effect of Delivery Delay on Orders = GRAPH(Delivery Delay Recognized)
(0.00, 1.00), (1.00, 0.97), (2.00, 0.87), (3.00, 0.73), (4.00, 0.53), (5.00, 0.38), (6.00, 0.25), (7.00, 0.15), (8.00, 0.08), (9.00, 0.03), (10.0, 0.02)

Importantly the full formulation extends to encompass the concept 'Delivery Delay Recognized' which is an information smoothing process explained as follows:

Delivery delay recognised represents customers’ perception of delivery delay. It takes time for customers to build an impression of delivery delay from their own recent experience of deliveries and from rumours circulating in the industry. The natural formulation is information smoothing. The equation is written as a SMTH1 function of delivery delay indicated, where the time constant of the smoothing process is called the ‘time to recognize delivery delay’ and is set at ten months. Think about this formulation for a moment in terms of the Baker criterion – what do customers know and when do they know it? What they know and use as the basis for ordering is delivery delay recognized. Customers form this impression by averaging, over a period of ten months, their day-to-day experience of ‘delivery delay indicated’. This measure of delivery delay is the actual time it takes the factory to fill orders, defined as the ratio of the order backlog to the order fill rate. A brief spate of late deliveries, lasting just a few weeks, will do little harm to demand and customers will continue to order in the belief that normal delivery will be resumed. But if the factory is slow to deliver for months on end then customers begin to think that is the norm. As a result some will stop ordering and place their orders with rivals instead.

Delivery Delay Recognized = SMTH1 (Delivery Delay Indicated , Time to Recognize Delivery Delay, 2) {months}
Time to Recognize Delivery Delay = 10 {months}
Delivery Delay Indicated = Order Backlog / Order Fill Rate {months}

The formulations above usually work quite well for a manufacturing business. But in a service business delivery delay is not necessarily a meaningful concept. A different approach is needed to capture the effect on demand of capacity constraints. I will give an example in my next post.
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Re: Diffusion model with capacity constraints

Postby Thomas Fiddaman » Tue Mar 27, 2012 9:02 am

Christian, I was surprised by your comment that the Sterman et al. (Behavioral Analysis of Learning Curve Strategy) model used only delivery delay feedback, so I looked at the model again, and you're right. Also, the delivery delay feedback is a competitive issue (affecting firm choice) but does not affect the total market. I guess I've seen other variants of the model (perhaps the B&B Enterprises simulator) that did include it. I think the standard Bass model must have been used in the paper for simplicity of exposition.

A simple modification to the model implements a capacity constraint on diffusion.

Code: Select all
Adoption Rate dM=Potential Adopters N*(Constant Propensity to Adopt alpha+Propensity to Adopt from WOM beta
                       *Users/Population POP)
Users=SUM(Installed Base I[Firm!])/Units per Household mu

'Users' consists of adopters who actually have received the product, so unlike adopters, it lags due to the shipping capacity constraint. The number of 'Adopters' who are philosophically convinced that they'd like to have the product will then exceed the number of 'Users' who actually have it. Substituting 'Users' for 'Adopters' in the 'Adoption Rate' equation amounts to assuming that philosophical adopters aren't producers of word-of-mouth; that only users who actually have the product to show off are convincing to potential customers.

In more complex variants, e.g. with outflows from adopters, you might need to track users explicitly as a level, with households transitioning from potential->adopter->user, in order to prevent situations in which users > adopters from occurring.

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Re: Diffusion model with capacity constraints

Postby Christian Weitert » Tue Mar 27, 2012 6:24 pm


you have actually described two solutions that are very close to approaches I have found during my Literature review so far. Both are not using SD, but I think transforming their equations into a model should be pretty straightforward. The first part of your answer is pretty close to an article by Kumar titled Diffusion of Innovations under Supply Constraints. I have not tried to implement it yet, but I am a little sceptical that the adoption rate might exceed capacity at some point. I ll look at it and post the findings.
The second solution sounds promising too. (and is very close to a marketing science article: Jain 1991 - Innovation Diffusion in the Presence of Supply Restrictions). I ll try to model this as well. My concerns about this solution are, that the rate from adopter to users will be governed by the actual delivery rate and there fore the rate as well as the level will be mere copies of the sales rate and cumulated sales. Additionally there will be a problem if the product has a limited time of use and replacement sales from already adopters occur.

Thanks for all the input so far.

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Re: Diffusion model with capacity constraints

Postby John Morecroft » Wed Mar 28, 2012 8:36 am

Tom’s distinction between users and adopters is an interesting way to introduce a supply constraint within the framework of a word-of-mouth diffusion process. I can imagine the formulation working well at the industry level where a chronic constraint on shipments serves to slow the growth of interest. However, at the firm level, interest that is not satisfied could damage the firm’s ‘ability-to-supply reputation’ and cause customers to switch to rivals or simply lose interest. Delivery delay feedback is one way to formulate the effect.

In the market growth model ‘excess’ delivery delay arises from a coordination problem: the difficulty of coordinating capital investment and sales effort in a new product market. In the wake of rising sales, delivery delay increases and delivery standards fall. Promising growth in early years is transformed into premature stagnation and decline. Yet the model assumes market size is very large relative to production capacity of the supplying firm. So the ultimate decline of sales is not due to a market limit per se but rather to cognitive limits (of normally competent managers) that, in this particular case, lead to overinvestment in sales force relative to production capacity. Only a fraction of the potential market is exploited and growth strategy fails as sales unexpectedly collapse.

I find these kind of coordination problems fascinating and it seems to me that credible formulations of capacity constraints on diffusion, market growth and market development require careful consideration of how firms’ key resources (or asset stocks) are coordinated with each other and with growth of demand.

Consider the growth of People Express and its subsequent demise. In the CD folder for Chapter 6 of SMBD is a smallish model that investigates this phenomenon. It is not the model that lies behind Sterman’s People Express Flight Simulator, though it draws on the same Harvard case for its conceptualisation. Here is a clear example of word-of-mouth growth constrained by capacity. The capacity constraint operates through service reputation which itself depends on the degree of coordination achieved in the growth of planes, staff and passengers. I explain more in a lecture originally delivered at WPI in 2009. The lecture is now on YouTube and can be found by Googling ‘System Dynamics and Strategy’.

The upshot is that there more to modelling capacity constraints on growth and market development then one might first think. Indeed there are quite a few academics working on this topic as it affects market development of electric vehicles, solar panels, liquid soap and even boil-in-the-bag rice. One thing is certain. A simple formulation of a capacity constraint, like the logic switch below, misses the richness of the underlying coordination problem.
Customer Orders = MIN (Sales Force * Normal Sales Force Productivity, Production Capacity)
A moment’s reflection reveals this is not a good formulation. It fails the Baker criterion because it implies that customers know the factory’s production capacity and ignore the efforts of the sales force the moment the limit of production capacity is reached.
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Re: Diffusion model with capacity constraints

Postby James Thompson » Wed Apr 04, 2012 11:54 pm

It's probably too late for this to be of help for this question, but I can offer a tiny model (19 equations) that simulates diffusion, capacity constraint, and efficiency gained from experience. It runs in any Vensim including PLE. The Forum doesn't permit files to be uploaded with a dot mdl extension, so please email me at gmsjpt@nus.edu.sg and I will send it to you.
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Re: Diffusion model with capacity constraints

Postby Jean-Jacques Lauble » Thu Apr 05, 2012 7:19 am

Hi Jim

To post files with a mdl extension I generally zip it and files with a zip extension are accepted.

You can too change simply the extension of the file into an extension that works like zip, upload it and the person who downloads the file will have to change the extension back to mdl.


Jean-Jacques Laublé
Jean-Jacques Lauble

Re: Diffusion model with capacity constraints

Postby James Thompson » Sun Apr 08, 2012 9:31 pm

Thanks, JJL. The model is posted as diffusion with capacity constraint and efficiency.zip.
diffusion with capacity constraint and efficiency.zip
Diffusion is simplified to ‘uptake generating more uptake’.
Learning is linked to patients receiving services as a proxy for employee experience.
The capacity constraint is exogenous.
(1.71 KiB) Downloaded 105 times
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Re: Diffusion model with capacity constraints

Postby Robert Eberlein » Mon Apr 09, 2012 7:03 am

I put back the ability to add models of various sorts. Apparently this got knocked out during one of the forum software updates.
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