Management And Accounting Web

Provided by James R. Martin, Ph.D., CMA

Professor Emeritus, University of South Florida

**What is Constrained Optimization?**

Constrained optimization models are based on a set of
underlying assumptions. The main assumption is that most, if not all, of the various
constraints in the model are static. The general idea is to find the optimum
solution given a set of static constraints. Most of the models in accounting, finance,
management, economics and quantitative methods fall into this category and can
be criticized from the continuous improvement perspective (e.g., Deming's
theory, JIT, TOC, ABM, etc.).^{1} Some of these models include the following:

1. ** The theoretical microeconomic non-linear
cost-volume-profit model.** This is perhaps one of the earliest constrained
optimization models. In this model the capacity of the firm and resulting fixed
costs are constant or static, but unit sales prices and unit variable costs are
allowed to vary. Sales prices reflect the law of demand, i.e., consumers are
willing and able to buy more at lower prices than higher prices. Unit variable
costs change as a result of chances in productivity, i.e., output per input. The
optimum profit is obtained where production and sales are where marginal revenue
is equal to marginal cost. This produces the greatest difference between total
revenue and total costs. The theoretical model is illustrated on the left-hand
side of Exhibit 1.

2. **The conventional linear cost-volume-profit model** (derived from the microeconomic model) provides another example. In this model,
the sales price and unit variable costs are assumed to be constant or static.
The constant sales price reflects a horizontal demand curve, while the constant
unit variable costs reflects an underlying assumption of constant productivity
and input prices, e.g. materials, labor and indirect inputs. Optimum profit is
obtained where the company produces at capacity. The simplified linear model is illustrated on the right-hand
side of Exhibit 1.

3. **The standard cost variance analysis model** provides
another example, although this model is merely an extension or variation of the
linear cost-volume-profit model. Standard input prices and quantities allowed
per output are set to reflect some acceptable level of performance. This level
could be the optimum level, but the standards are more meaningful if they are
set at the mean of the possible outcomes when the system is stable. This model
falls into the constrained optimization category because it assumes a static
system with a static set of constraints. Standard sales prices,
standard variable costs per unit, (i.e., standard input prices of direct
materials, direct labor , indirect resources and productivity), budgeted fixed
costs and sales mix are all assumed to be constant as far as the standards are
concerned. Some of the problems with this model are illustrated in Exhibit 2
which focuses on factory overhead. Overhead variance analysis does not even
isolate the price and quantity variances for indirect resources. In addition,
the model ignores many of the cost drivers of indirect resource costs and
ignores the concept of system variability identified as randomness in Exhibit 2.
Therefore the standard cost model is of little help in identifying potential
improvements.

4. ** The economic order quantity and economic batch size models** provide
other examples of constrained optimization techniques. For example, in the
economic order quantity (EOQ) model carrying costs are equated with ordering
costs to find the optimum order quantity as illustrated in Exhibit 3. Carrying
costs increase for larger order quantities because of increases in the costs of
materials storage, handling, obsolescence, insurance and other cost related to
carrying a larger inventory. Ordering costs decrease because there are fewer
orders. Supposedly, the two cost functions intersect which produces a U-shaped
total cost curve as illustrated in Exhibit 3. Of course the intersection of the
two curves identifies the EOQ as indicated above. However, the main problem with
the EOQ model is that it conflicts with the idea of continuously finding ways to
reduce inventory to a minimum. In other words, do not accept the constraints and
attempt to optimize. Instead find ways to continuously reduce or remove the
constraints. For example, using JIT purchasing concepts, ordering costs are
reduced by allowing the vendor to have access to the buyerâ€™s production
schedule. This means that the vendor performs part of the purchasing function
for the buyer.

Example: Assume demand per period is Di = 112,500 units, ordering costs per
purchase order is P = $100, carrying cost per unit per period i is C = $10. Then
the EOQ is found by equating ordering costs with carrying costs as follows:
(112,500)(100)/Q = (Q)(10)/2. Then 11,250,000/Q = Q5 and 2,250,000 = Q^{2}.
Take the square root and the EOQ = 1,500 units.

5. ** The quality cost conformance model** provides another example of a
constrained optimization approach. In this model the economic conformance level
(ECL) is obtained where prevention and appraisal costs are equal to external and
internal failure costs. Prevention and appraisal costs increase as the level of
conformance quality increases. Conformance quality refers to conformance to
specifications as opposed to design quality. Failure costs are expected to
decrease as the level of conformance quality increases. Therefore, the total
costs associated with conformance quality will be U-shaped as indicated in
Exhibit 4. The optimization concept and related calculations are essentially the
same as in the EOQ model. Prevention costs include quality engineering, training
and related supervision costs. Appraisal costs include inspection, testing and
supervision related to these activities. Internal failure costs include
spoilage, scrap, rework and the associated downtime costs, while external
failure costs include warranty costs and the costs of lost customers.

**The ECL model is associated with Juran**. Deming and others, such as Crosby (*Quality
Is Free: The Art of Making Quality Certain*) view the calculation of the ECL as a waste of
time. From this perspective, the main problem with the ECL methodology is that
the model is likely to be mis-specified by underestimating or ignoring the costs
associated with lost customers. A revised model with the "quality is
free" perspective is provided in Exhibit 5. This is a long run view where
the lost sales dollars resulting from past failures are included in external
failure costs. Of course, lost sales dollars are unknown amounts, but there is
adequate reason in most industries to believe that they represent substantial
amounts, perhaps so large that the two curves never intersect.

**Quality Models Compared**

Two quality models have appeared in the accounting literature in recent years. Juran's quality cost conformance model is associated with the zero defect philosophy shown on the left side in the illustration below. Juran's model includes a target value and tolerance, or specification limits, for the variation that occurs in a parameter or characteristic (X). In Juran's model, no loss occurs if the value of X is within the specification limits, i.e., it is considered acceptable. If the value is outside the limits it is considered unacceptable or a defect and becomes either scrap, spoilage or rework.

Deming, on the other hand, was associated with the robust quality philosophy based on Taguchi's loss function shown in the center of the illustration and combined with a distribution of X on the right hand side. Taguchi and Deming believed that some loss occurs for the manufacturer, the customer, or society when the value of X is not on target. In the graphic illustration above, the distribution of X is drawn so that it appears that the mean of X is on the target value, but of course this is not usually the case. The idea in the robust quality philosophy is to continuously improve the process by moving the mean value of the parameter closer to the target value and by reducing the amount of variation in the parameter.

6.** Relevant cost (incremental, differential or cost-benefit) models**
such as special order pricing decisions, product mix
decisions, make or buy (out-sourcing) decisions, to process joint
products further decisions and similar short term decisions all fall into
the constrained optimization category because a static environment is typically
assumed. Some relevant cost problems, such as the product mix decision model,
use techniques such as linear programming because there are a large number of
constraints that must be considered simultaneously.

7. ** The conventional capital budgeting investment decision model**. This
model is essentially a long run relevant cost model that emphasizes the
discounted cash flow methodology, i.e., net present value and internal rate of
return approaches. It falls into the constrained optimization category because a
static environment is typically assumed. The investment management concept adds
the moving baseline approach to the model to make it more dynamic.

8. ** The statistical process control (SPC) model** might also be included
in the constrained optimization category from the perspective that it measures
the variability within a stable or static system. However, sample means that
fall outside the control limits indicate that the system is no longer stable.
Changes in the system that cause changes in the mean or changes in the range are
also reflected on the control charts. This adds a dynamic element to the SPC methodology
that makes it useful as a tool for measuring improvements in the system. A system improvement represents a
change that either improves the mean outcome or reduces the variability within
the system. For the illustration in Exhibit 7, a change in the production
process that reduced the mean drilling time or reduced the variability in
drilling time would represent a system improvement.

9. ** Converting static models into dynamic models**. In defense of the
constrained optimization techniques, one can argue that all of the methods
discussed above can be converted into dynamic, rather than static models by
adding sensitivity analysis to the model. Sensitivity analysis involves testing how
sensitive the solution is to changes in the constraints associated with the
model. For example, sensitivity analysis is frequently used in product mix
decisions based on linear programming. One can also argue that ignoring
potential improvements in the system when using a model does not represent a
flaw in the model, but instead indicates a myopic misuse of the model. A
rebuttal is that most users may ignore the sensitivity issue because it is not a
formal part of the model.

___________________________________________

^{1} Martin, J. R. 1994. A controversial-issues approach to enhance management accounting education. *Journal of Accounting Education
*12(1): 59-75. (Summary).

**Other related summaries**:

Albright, T. L. and H. Roth. 1992. The measurement of quality
costs: An alternative paradigm. *Accounting Horizons* (June): 15-27. (Summary).

Albright, T. L. and H. P. Roth. 1994. Managing quality through the quality loss function. *Journal of Cost Management*
(Winter): 20-37. (Summary).

Anderson, S. W. and K. Sedatole. 1998. Designing quality into products: The use of accounting data in new product development. *Accounting
Horizons* (September): 213-233. (Summary).

Deming, W. E. 1993. *The New Economics For Industry, Government & Education*. Massachusetts Institute of
Technology Center for Advanced Engineering Study. (Summary).

Goldratt, E. M. 1990. *What is this thing called Theory of Constraints*. New York: North River Press.
(Summary).

Johnson, H. T. 1989. Professors, customers, and value: bringing a global perspective to management accounting education. *Proceedings
of the Third Annual Management Accounting Symposium*. Sarasota: American
Accounting Association: 7-20. (Summary).

Johnson, H. T. 1992. *Relevance Regained: From
Top-Down Control to Bottom-up Empowerment*. The Free Press. (Summary).