This paper presents the use and validation of a generalized learning curve in the economies of scale purchasing experience. The model, based on Wright’s curve, incorporates two extra degrees of freedom to accommodate initial purchases of multiple (instead of single) units and a finite asymptotic price at high volumes. The study shows that each time the part purchase quantity is doubled, the price is reduced either by a constant percentage (a learning rate) or by an approach to an asymptotic plateau rate indicating a point of diminishing returns. Supplier price quotations at multiple purchase quantities were obtained for a pool of 17 critical parts. The data were fitted with the generalized learning curve by the method of least squares regression. The regressed learning rate, first unit price, and the asymptotic price can be used to infer supplier pricing strategies. Coupled with a “should-cost” analysis based on estimates of standard time and material, a system cost reduction task was carried out by the supply chain organization.

One of the challenges in a supply chain function is to carry out win-win negotiations with suppliers for part quality, cost, and delivery. Part quality is essential and should not be negotiable, provided that the specification is sound. A well-written specification with measurable attributes and well-defined statistical requirements will provide a precise quality foundation. Just-in-time (JIT) part delivery schedule would be ideal. However, some inventories of parts may help to ensure the continuity of supplies during abnormal circumstances. In the case of critical parts, the company may want to source them from more than one supplier. Tradeoff analysis must be conducted judiciously with multiple supplier sourcing strategy since it usually works against the part cost.

Here is a brief description of a cost reduction strategy used in a fuel cell company. A Wright-based learning curve was used to determine the economic order quantity and to aid in the negotiation of the lowest possible, but fair, part price. A win-win negotiation strategy can build upon openness and trust. However, doing the homework upfront will go a long way in the negotiation process. Competitive benchmarking of suppliers is an important step to narrow down the list of potential suppliers. Should-cost analysis should be carried out on 20% of parts that make up 80% of the system cost. The analysis takes into account the production method, the amount of material used, the various processing times, and the estimated profits. While this analysis may provide a good understanding of the variable costs, the fixed costs may vary widely among suppliers. A typical classification between variable and fixed costs is shown in Table

Cost classification.

Cost item | Category | Description |

| ||

Direct material cost | Variable | The materials (parts, and subsystems) that go into the final product |

Direct labor cost | Variable | Constant productivity of people directly involved in production |

Energy | Variable | Used primarily for production |

SAG | Variable | Selling, administrative, and general costs |

Commissions | Variable | Usually proportional to the quantity of goods sold |

Shipping | Variable | Sometimes, a portion with the supplier and a portion with the buyer |

R&D | Fixed | Research and development tasks |

Fixed cost | Fixed | Amortization of capital goods, rent, depreciation of buildings and equipment, taxes on real estate, insurance, salaries of top management and operating personnel |

Discretionary fixed | Fixed | Advertising, public relations, management development programs |

IP acquisition cost | Quasi-fixed | Intellectual property design cost, licensing agreement, and royalty payment |

Manufacturing overhead cost | Quasi-fixed | Maintenance and repairs on production equipment, heat and light, and depreciation |

Indirect labor | Quasi-fixed | Discrete jump will happen when the overall scale of production drastically changes. |

Profits | Quasi-fixed | Depending on pricing strategies |

Not all variable costs have exactly the same meaning. Some variable costs behave in a true variable or proportionately variable fashion. Other variable costs behave in a step-variable fashion. The same can be said about the fixed and quasifixed costs. For simplicity, part costs shall be classified as either variable or fixed from here on in this paper.

The objectives of this paper are organized primarily as follows:

Learning curves have been around since the early 1900s and have been used to model productivity or efficiency improvement based on learning experience. Some forty different equations on published learning curves were reviewed by Thurstone [

The hyperbolic model fits well with his empirical experiment. He concluded that typewriter learning increases with practice and attains an upper limit quickly. The meaning for the rate of learning in his model is simply a dimensional model coefficient and is not nearly as meaningful as the dimensionless rate in Wright’s [

Another common form to capture the learning effect is the exponential decay model. In their modeling work of a production-inventory study, Das et al. [

Wright studied the variation of cost with quantity since 1922 and described a basic theory for obtaining cost estimates based on repetitive production of airplane assemblies. His curve was derived empirically based on previous production experience. Although Wright did not name his curve and the word “learning” never appeared in his paper, somehow his curve became widely known as the learning curve.

The theory of learning is simple. It is recognized that repetition of the same operation results in less time or effort expended on that operation. For the Wright learning curve, the underlying hypothesis is that the direct labor man-hours necessary to complete a unit of production will decrease by a constant percentage each time the production quantity is doubled. Wright’s learning curve is a simple power model and can be interpreted as

Typical learning rates range between 70 and 95% for manufacturing and procurement activities. Accordingly, each time the volume doubles, the cost is reduced by 30 to 5% (i.e., 100% minus the learning rate). As the production quantity (

In light of the law of diminishing returns, a factor of incompressibility was added to the power model by DeJong [

In the case where the incompressibility factor is zero, DeJong’s model reverts back to the Wright learning curve. DeJong found that the incompressibility factor was about 0.25 in assembly operations, and it increased (less learning) as tasks became more uniform. In his study, the exponent of reduction stayed at 0.32 corresponding to a learning rate of 80%.

Over the years, variants of Wright’s learning curve and applications have been reported widely in many industries and disciplines. Here are examples on the selected few of them. Globerson and Gold. [

Kara and Kayis [

Nemet [

After 76 years of examination since Wright [

The power curve model of

The equation based on this foundational concept is simply

The activity level “

The general form of the equation with “

To generalize by letting

Equation (

In a more conventional form, the generalized learning curve becomes

The generalized learning curve, (

In a similar effort as documented by Sinclair [

It was prudent to work with existing suppliers on critical parts and, if necessary, to develop low cost suppliers on noncritical parts. Suppliers were requested to provide quotations on part pricing at three or four quantities of purchase. Concurrently, should-cost analysis of parts was conducted to somewhat validate the quotations. An experienced supply chain group would determine the price versus volume data and would fit them to the generalized learning curve model by regression using the least square fit of the percent errors. In this way, negotiations would be based on analytical insights, and cost-pricing relationships could be established.

The coefficients of each part model provided a “directional” estimate of the fixed cost

If the part was a COTS, we further estimated the percentage of our purchase with respect to the production volume of the supplier to understand leverage. Custom designed parts are unique and have huge learning potentials. Based on the previous comparisons and part classifications, a pricing strategy for each part was relegated to one of the four likely standards: cost-plus pricing, target return pricing, value-based pricing, and psychological pricing. Supplier visits and negotiations were conducted with a majority of the existing/new and with/without contract in place suppliers.

The generalized learning curve model was coded in an MS Excel spreadsheet. The built-in “solver” function was used to perform a least square regression fit. Three examples have been selected for discussions. Figure

Model summary for filter.

The inputs are the supplier quotations at three volume-cost pairs. In this case, the initial volume (cell C3) is 100 pieces of a filter. The primary outputs (the initial guess box) are the learning rate, model-calculated initial cost, and the terminal cost. It should be noted that the learning rate (cell C4) applies only to the learnable portion of the cost. The learnable portion does not include the terminal cost, which can be finite or zero.

The calculated pseudolearning rates (column H), which are based on the ratio of absolute costs, trend to a plateau effect at higher volumes. By definition, the learning rate is supposed to stay constant at each doubling of volume. However, with a nonzero terminal cost, the prefix “pseudo” is added to represent a learning rate that is changing with each volume doubling. The average for the pseudo-learning rate (H17) calculated within the quoted volume range may be used for comparison in the absence of the constant learning rate. The backward projected cost for the first unit (cell E12) is also calculated using the learning curve model.

With only three volume-cost input pairs, the regression may give a perfect fit as in this case. The supplier quotation price (or purchaser cost) versus the model cost and the general learning curve shape are plotted in Figure

Graph of quotation price versus learning model price for filter.

The least square regression fit algorithm is quite robust using just the default regression fit parameters. Convergence to the best fit may take up to three iterations by inputting initial guess with common sense that the learning rate is between 50 and 99%, and the initial cost is higher than the terminal cost. These three values from the initial guess will be written over automatically after each iteration. Most of the time, convergence takes only one iteration.

The second example is an air compressor part shown in Figures

Model summary for air compressor.

Graph of quotation price versus learning model price for air compressor.

The last example is a pressure sensor shown in Figures

Model summary for pressure sensor.

Graph of quotation price versus learning model price for pressure sensor.

Of the 17 regression analyses, 10 parts have a zero terminal cost. Their learning rates are shown in Table

Learning rate of parts with no plateau.

Item no. | Purchase part name | Learning rate |
---|---|---|

1 | Pressure sensor | 84% |

2 | Air compressor | 84% |

3 | Current sensor | 88% |

4 | Gas sensor | 90% |

5 | Ultracapacitor module | 93% |

6 | Inductor | 94% |

7 | Ultracapacitor assembly | 94% |

8 | Air mass flow sensor Assembly | 96% |

9 | NiMH battery (supplier A) | 98% |

10 | High power NiMH battery | 98% |

| ||

Average | 92% |

Average pseudo learning rate of parts with plateau.

Item no. | Purchase part name | Learning rate |
---|---|---|

1 | T-fitting assembly | 74% |

2 | Filter | 77% |

3 | Radiator fan | 92% |

4 | Li-ion battery | 92% |

5 | Air filter | 93% |

6 | Tracking regulator | 95% |

7 | NiMH battery (supplier B) | 99% |

It is evident that the learning curve is a helpful tool for part purchasing negotiations. In most cases, the curve fits well with the supplier pricing quotations, assuming the fixed and variable costs dominated. In the case where the fit is not as smooth, it could be caused by the quasifixed cost or step variable cost in the supplier pricing analysis.

The generalized learning curve provides two extra degrees of freedom to handle the initial purchase quantity variation and the law of diminishing return issue. The regression gives the first unit cost and the terminal cost, which can be viewed as the total (fixed plus the variable) cost, and the variable cost, respectively. Normally, it is not a good idea to extrapolate to the two ends significantly beyond the supplier quotation range. This concern is supported by 10 out of 17 parts that have a zero terminal value, a surrogate for the variable cost. Nevertheless, the insights extracted are valuable.

Early on with the company, almost all supplier quotations were single point in nature for JIT purchases. Price elasticity of parts was not apparent, which made negotiations contentious and difficult. By doing the essential cost and elasticity study upfront, negotiations can become win-win for both parties. The supplier can win by openly discussing cost saving measures and potential business opportunities, such as gainsharing.

The practice of lean manufacturing has been proven to be a crucial competitive advantage. The lean supply chain is a newer concept. Companies need to share information and coordinate demand forecasts, production planning, and inventory replenishment with suppliers. Progressive companies strive to build a highly collaborative business environment and may want to consult with their suppliers on business best practices.

Additionally, lean means doing one’s own homework and minimizing the nonvalue added burdens on the suppliers. Very often and truly believing “the customer is always right,” companies can place unreasonable demands on suppliers resulting in unnecessary (throw away) activities. The erosion of supplier profit margins may negate any potential for cost reduction negotiations.

With a “lean supply chain” approach, like the work documented in this paper, promise for a win-win approach in purchasing relationships is clearly feasible. One supplier was willing to drop the price of a part by 50% with a small packaging change. Many suppliers reduced their price to be on par with a should-cost analysis and/or with published learning rates. Yet, there were some suppliers who would not budge due partially to being the only game in town, financial demands (returns), or corporate culture. Overall, a 20% system cost reduction, purely through “purchasing,” was achieved for a fuel cell product in production.