INTERFACES
Vol. I I , No. 6, December 1981
Copyright © 1981, The Institute of Management Sciences
0092-2102/81/1106/0048$01.25
MATCHING SUPPLIES TO SAVE LIVES:
LINEAR PROGRAMMING THE
PRODUCTION OF HEART VALVES
Said S. Hilal and Warren Erikson
American Edwards Laboratories, Division of American Hospital Supply Corporation, P.O.
Box 11150, Santa Ana, California 92711
ABSTRACT. This paper describes the application of a linear programming model at
American Edwards Laboratories and the resulting improved productivity in biological heart
valve production. The valves are bioprostheses manufactured from porcine hearts and used
for human implantation. Since valves demanded by the human population have a different
size distribution than valves supplied by pigs, the result has been a supply and demand
mismatch that generated hundreds of thousands of dollars of unwanted inventory. The
linear programming model was utilized to determine the combination of available suppliers
that would provide the best match for the demand distribution. As a result, annual savings
exceeded $1,500,000, and both availability and manufacturing control were improved.
In late 1958, a retired engineer by the name of Lowell Edwards approached a
cardiovascular surgeon with a concept for an artificial heart. The surgeon. Dr. Albert
Starr, convinced Edwards to try instead for a mechanical heart valve. Together, the
two men came up with the first mechanical heart valve, which signaled the start of a
new life to tens of thousands of heart patients. This valve is still considered the
standard to which other prostheses are compared.
The company founded to manufacture the valve was another success story. The
entrepreneurial venture that became Edwards Laboratories was established in April,
1961 and quickly became a pioneer and innovator in heart valves, vascular surgery,
and critical care catheterization and monitoring. In 1966 Edwards Laboratories was
acquired by American Hospital Supply Corporation, and the name later became
American Edwards Laboratories.
By the late 196O's new valve designs were already competing with the StarrEdwards Ball Valve. Shiley introduced the tilting disc valve; Hancock Laboratories
introduced the porcine valve, a bioprosthesis manufactured from the natural heart
valve of pigs. Both designs were first pursued by American Edwards but were
introduced to the market by former employees of Edwards. By the time American
Edwards introduced the Carpentier-Edwards porcine valve, both Shiley and Hancock
were established competitors in the valve market.
By 1980, American Edwards had grown into a medium-size company with
1,600 employees and over 240,000 square feet of manufacturing space. A reorganization in mid-1980 facilitated the establishment of three business centers within
American Edwards to serVe the cardiovascular, critical care, and noninvasive market
segments. The Cardiovascular Business Center product lines include both mechanical and tissue valves as well as vascular catheters. This market segment is charac-
PROGRAMMING, LINEAR, APPLICATIONS; HEALTH CARE, TREATMENT
48
INTERFACES December 1981
terized by the need for continuous and costly research and developmental work,
stringent regulatory requirements, and a generally cautious acceptance of new concepts and products by surgeons.
From an operational standpoint, the company's highest priorities were defined
as increased productivity and efficient management of inventory. Due to the nature of
the product line in the Cardiovascular Business Center, the two objectives had to be
achieved while highest quality and continued availability of the products were maintained.
The need for biological components for the porcine valve posed a unique and
challenging obstacle to the operational objectives. The work described in this paper
is a simple but innovative approach that helped overcome the many difficulties
characterizing the nature of the product. The approach helped increase productivity
while contributing to quality, and reduced inventory while increasing availability. In
the process, annual cost was reduced by close to $1.5 million, thus freeing revenues
for new innovative medical products.
THE PROBLEM
Improved productivity is a continuing effort at American Edwards Laboratories.
However, the porcine valve area was faced with diminishing returns on new productivity improvements. Employee skills and motivation were both at very high levels
and automation had limited application in a production process that used biological
components as raw materials. Additionally, previous efforts had indicated that low
production yields and high levels of biological component inventory were inherent
characteristics ofthe product. Nevertheless, yields and inventories appeared to be the
two areas where the greatest productivity gain could be attained. A small improvement in yield could reduce cost significantly.
Due to the nature of the product and its use, product availability was of
paramount importance; a safety stock equivalent to six to twelve weeks of demand
was maintained on hand in the form of finished goods. This policy ruled out the
application of stockout cost/carrying cost optimization. Efforts to reduce the biological component inventory were futile because shortages in some valve sizes necessitated increased production on all sizes. The dilemma becomes apparent when the
reader recognizes that pig valves could not be procured by specific size. Actually,
sizes are not determined until the biological component is processed.
The size of a raw pig heart is highly variable. The breed of pig, age when
slaughtered, and feed mix, to name a few, all result in variations in the physiology of
the heart. The result is that each supplier may have a distinctly different size distribution of pig heart valves.
The immeasurability ofthe biological valve until the valve is totally processed is
perhaps the most vexing aspect ofthe problem. Raw heart vendors supply a distribution that usually results in six to eight valve sizes. Clearly, units of unwanted size are
procured and processed with units of desirable sizes. The result is a spiraling mismatch between the size distribution of biological components in inventory and that of
market demand.
An early attempt to attain a better match of the size distributions of incoming
valves and the demand by tracking and adjusting the vendor base resulted in a very
marginal improvement. A subsequent attempt employed a two-pronged approach:
• Computerization of vendor distributions and yield records
• Utilization of porcine specialists to work with vendors to ensure distribution
stability and proper procedures.
INTERFACES December 1981
49
This attempt successfully cut the scrap rate in half.
Search for additional improvement continued. In early 1980, despite the giant
inroads made, inventory of biological components was still climbing at close to
$80,000 per month and shortage on two sizes was looming. Although excess was
accepted as part of the nature of the product, concern was mounting with regard to
the potential shortages.
Management intensified efforts to identify additional managerial tools in order
to:
• Predict shortages over the next fifteen months if the current procurement
policies were maintained
• Predict the annual excess inventory cost if the current procurement policies
were maintained
• Develop a method to accurately predict shortages and excess inventory that
would result from new procurement policies
• Develop a method to react more effectively to changes in either supply
availability or market demand.
In addition to the above, valve design changes scheduled for 1980 and 1981
were to change the demand size distribution. The Director of Manufacturing expressed concern about the impact of such changes on excess and shortage. Preliminary analyses indicated that there would be no additional shortages if existing purchasing policies were continued, but excess inventory associated with the modified
designs would probably exceed $1,500,000. The director's concern was justifiable;
one manufacturer had already opted to pull out of the porcine valve market when
faced with the same problem.
THE SOLUTION
The solution to the problem utilizes a relatively simple linear programming
model to determine the optimum quantity of hearts to be ordered from selected
vendors. The output of the model provides a minimum cost procurement plan that
guides inventory toward the desired level for every valve size, thus reducing inventory excess and shortage. The first step involves gathering and analyzing recent data
on procurement, processing and inventory ctsts, heart valve sales, raw heart supply
by vendor, and vendor heart valve size distributions and yields.
The early version of the model did not include the option of eliminating vendors. Constraints were set to maintain a safety stock of at least three months' supply
of all sizes of biological components. Results indicated a slowdown in the rate of
increase in inventory levels. Net savinge from implementing the solution were estimated at $500,000 per year. Results were presented to management and accepted
with cautious optimism. Approval was given for the implementation of the linear
programming model for procurement decisions beginning May 1, 1980.
The May results were very successful. Processed heart valves were within 6% of
the distribution projected by the model. Subsequent analysis indicated that much of
this deviation was caused by raw hearts that were procured prior to May 1, but were
still in process.
The success of the May run justified a more ambitious plan that reduced safety
stock to two months' supply. Further, the option of eliminating unfavorable vendors
was incorporated into the model.
This latter change was implemented by first setting the lower bounds to zero for
all vendors. The model was run and management decided which ofthe lowest volume
50
INTERFACES December 1981
vendors would be dropped. Unfavorable vendors were removed from the data base,
the lower bounds for the remaining vendors were reset, and the model was run again.
The linear programming model answered the following questions:
• Which suppliers should be tapped for raw hearts?
• How many raw hearts should be purchased from a selected supplier?
Input data needed to answer these questions included:
Supplier size distributions of raw heart valves
Supplier cost and yield
Market sales forecasts
Current inventory status
Total inventory reduction goals
Minimum inventory levels by size
Minimum monthly purchases of raw hearts necessary to retain a supplier
Maximum number of raw hearts that a supplier could ship.
The linear programming model was incorporated into the procurement/
production/inventory system.
A secondary, but still important, use of the linear programming model is to
assess the effects of changes in raw heart valve characteristics. Valves obtained from
suppliers are analyzed to detect any shifts in expected size distributions. If changes
occur due to errors on the part of the supplier, the supplier is contacted by porcine
specialists who will assist the supplier in correcting the problem. If changes are
permanent, the data base is changed to refiect the new supplier characteristics. The
model is then used to evaluate the impact of the permanent change.
Inventory reduction goals are not addressed directly by the linear programming
model. Instead, the model is used to find the minimum procurement cost solution.
The lowest procurement cost solution will, naturally, be one that relies heavily on
existing excess inventory. Each of these solutions is then analyzed further to highlight its effect on employee work hours and the stability of orders to each supplier.
Fluctuations in the production labor force were to be avoided, even at the risk of
causing limited increases in procurement cost, in order to maintain morale and
minimize training. Similarly, large fiuctuations in orders from vendors were to be
avoided in order to maintain a goodwill relationship with vendors. Although constraints of this type can be formulated, management did not feel that enough was
known about the nature of "smoothing functions" to warrant such inclusion in the
model. Finally, a single procurement plan and its resultant set of supplier orders is
selected and implemented.
Two major points should be interjected here:
• Data gathering, as described, is not cheap.
• The linear programming model runs are only a minor part of the total effort.
To elaborate: extensive data on all incoming valves, the valve source, and its final
status, have to be collected in order for the model to work. Reasons for rejection,
vendor yields, and size distributions are also important. However, all these requirements are dictated by the nature of the product and the need for traceability for all
valves and must be collected regardless of the needs of the LP model. The added cost
of the model is, therefore, minimal.
December 1981
51
V
Model Formulation
Successful applications of OR/MS in business frequently involve a compromise
between what is possible and what is useful. Too often, the formulation ofa problem
expands in complexity because each new feature helps to explain some element ofthe
situation being modeled. This process is reinforced by those who argue that a particular model is incorrect because it does not include some specific feature. The unfortunate result is that there are too many problems that remain unsolved because the
"correct" formulation is too difficult to solve.
Some of the possible features, or models that might have been (but were not)
included:
Probabilistic models were not used, even though the basic problem involves the
matching of input and output distributions. Fortunately, in this problem, the nature of
the product caused the distributions to fall neatly into a histogram structure that could
be handled easily by the linear programming model.
Fuzzy constraints could probably have been applicable, since inventory goals
and valve sizes were not absolutely fixed, but no payoff could be seen for this
increase in complexity.
An integer programming model could have been used, but was not. This is
probably the item of greatest potential controversy, because this model commits the
sin of being run as a regular LP problem and the output is then rounded to obtain
integer results. This was believed to be justified because the constraint coefficient
and the right-hand-side values were not absolutely fixed. Further, the nature of the
vendor's size distributions made it unlikely that any strange-shaped feasible regions
would cause bothersome results. (It is possible, of course, to create a set of data to
prove this wrong, but no such problems have occurred in practice.)
0-1 variables could have been used to handle the vendors who might or might
not serve as suppliers in any given month. The problem is related to vendors'
unwillingness to deal with low volume orders. Inclusion of lower bound constraints
in all runs would force every supplier to be included every month. Naturally, this
distracts from the purpose ofthe effort. The approach taken was simple. The first run
was always made without the lower bounds. Lower bounds for the remaining
suppliers were then reinstated and the model was run again. The resultant optimal
solution was then used for the procurement plan.
The problem could also have been formulated as a multiperiodproblem, but was
not, because management wanted to make the final decision regarding time frames
for lowering inventory and phasing in new product configurations.
Disadvantages of the Simple Model
In some cases, the simple model and the procedures that were used might solve
for a local optimum and miss the real global optimum for the problem (considered in
this case a very low priority problem).
Advantages of the Simple Model
• Easier to run (any computer and almost any LP code can be used)
• Faster
• Lower computer costs
• Lower data entry cost
• Easier to explain
• Lower chance of formulation errors.
52
INTERFACES December 1981
Implementation
A key element to the success of any new managerial tool is its acceptance by the
intended user. In general, user acceptance is a function ofthe user's understanding of
the tool to the terms and definitions involved. For that reason the LP model was
tailored to the production environment, and the effort involved the production team
members, who participated in the identification and definition of all pertinent terms
and characteristics.
The initial reaction of most team members was a predictable one. To most, a
computer was a tool that did not belong in the process of selecting porcine heart
valves. The biggest challenge faced by the model advocate was to demonstrate to
team members that the valve size distribution data reduced the problem to a simple
mathematical relationship. Once that point was accepted, the team began to consider
the proposed approach.
The approach was never presented or seen as a one-man plan; the production
team was kept informed, involved, and interested in the process. Since the defmitions of terms emanated from the eventual users of the model, input and output thus
used terms that were already familiar to the team. Ten months following implementation an impressive record was being made: savings and inventory reduction goals
were met or exceeded every month.
The advantages of implementing the program were outstanding and noticeable.
One disadvantage did surface, however. Porcine specialists who, prior to the program, went out and worked with vendors and adjusted the distributions on a manual
basis were very demoralized by what they conceived to be a take-over by the computer. Understandably, the specialists felt discouraged by better computer results and
threatened by the misconception that their jobs were about to be phased out. For a
few weeks the attitude was beginning to impact the overall performance of the plan
and the team. The specialists' concern was totally inaccurate, however; their jobs
were about to be redefined and reemphasized. The need for their manipulation ofthe
procurement plan would no longer be necessary, but their ability to train the
operators at the slaughterhouses in order to get a well-isolated, well-trimmed heart
continued to be essential. Within the first three months of plan implementation, the
Production Control Manager was able to convey that need and to successfully
motivate the porcine specialists.
Results
Savings resulting from the optimization approach were outstanding. Excess that
formerly ran at $77,000 per month was limited to less than $10,000 per month during
each of the first three months of model application. Later adjustments to the model
eliminated new excess and depleted old excess at the rate of $11,000 per month. The
annual reduction in excess was slightly over a million dollars. In May, 1980, inventory at the biological component level was $1,230,000 and was forecast to increase to
approximately $2,770,000 by the end of 1981. Upon implementation ofthe optimization approach, the inventory started to decline and projections for the end of 1981
showed the biological component inventory level at approximately $849,000. This
constitutes an inventory reduction of $1,921,000. Annual savings on reduced excess
and carrying cost exceeds $1,476,000.
The above savings are expected to continue for at least three years, after which a
modified product may be introduced. The new product will still be of biological
origin and will continue to use the optimization program. The savings, however, may
be different.
INTERFACES December 1981
53
Aside from the dollar savings that the program brought in, American Edwards
managers have enjoyed increased ability to forecast and control. As many production
control managers will recall, inventory reduction carries with it risks of shortages and
back orders. In that light, the optimization program described in this paper is unique
in that it insures reduction in inventory levels and concurrent elimination of shortages. Potential shortages were predicted nine months ahead of time of occurrence and
proper corrective measures were implemented without need for overtime, additional
resources, or added expense.
The implementation of the model resulted in a reduction of required raw heart
valves. This, in turn, allowed the production group to concentrate on only a few
vendors and to give the appropriate level of training and attention to each remaining
vendor. In time, the added training resulted in related improvements in yields and
quality.
An additional unexpected benefit was also realized in the process. When demand for a certain size increased by an incremental amount, the cost of procurement
(including cost of excess) increased also. This increase in procurement cost per unit
of additional demand is the marginal cost of the last unit. Use of the LP model
showed that the marginal cost of one specific size was six times the standard cost.
The LP model can be used to adjust prices to levels that more accurately reflect cost
of excess.
Another benefit is the ability to project the impact of a design change on product
cost. A new design proposed for early 1982 would have shifted the required sizes
upward. Demand by size was, therefore, adjusted to reflect such a shift and the
optimization program was run. Procurement costs, due to additional excess, skyrocketed; the actual cost ofthe new valve would have been twice the present cost. Under
previous conditions, such an increase in procurement cost would have gone unnoticed until full production was implemented. As a result of the findings, a better
vendor mix was determined 18 months ahead of the potential crisis.
Additional benefits are hard to quantify. To some patients, the most important
benefit will be the immediate availability ofthe valve they require in order to enjoy a
healthy life. To others, the benefits may be more abstract. A profitable American
Edwards will invest in new products and new research that will filter back to the
public in the form of improved medical care. Better inventory control and increased
productivity has, in the past, helped American Edwards achieve such goals. This
program will contribute to that capability in the future.
OTHER APPLICATIONS
The authors see two broad areas where the results of this work might be of
benefit to others:
• Problems that involve matching of probability distributions
• Integer programming problems in general.
Matching of probability distributions is a relatively common problem in business. The type of problem tackled at American Edwards exists whenever a supply
distribution has to match a demand distribution (and individual items cannot be
acquired separately). The approach applies to any process that uses biological or
organic items as raw material, but can also apply to more traditional manufacturing
situations. For example, boring of cylinders and machining of pistons in automotive
engine production results in two distributions that must be matched. The type of
54
INTERFACES December 1981
model used at American Edwards can be applied to any such problem.
Integer programming is almost always introduced to new students with an
example that has a sharply spiked feasible region. The result of rounding the linear
programming solution is then demonstrated as infeasible and far from the real integer
optimum. One result of this exercise is a growing army of OR/MS analysts who use
increasingly larger amounts of computer time to solve integer problems.
Another result is an even larger army of OR/MS analysts who will not attempt to
solve any problem that resembles an integer programming problem because of their
awareness of the great expenses of correct solutions and the great embarrassment that
results from rounding a regular LP solution.
The unfortunate aspect of this scenario is that in many situations the rounded LP
solution may not be only feasible, it may be optimal. Rounding is almost always
acceptable if either the constraint coefficients or the right-hand sides are not absolutely fixed. Both of these situations existed in the American Edwards case. The
probabilistic nature of the valve sizes in each batch of hearts resulted in constraint
coefficients that are not fixed exactly. At the same time, management goals of having
inventory set at, say, a two-month supply, was not fixed exactly. Management
demonstrated no concern if a desired inventory level of 100 valves ended up at 98 or
103.
The only caution the authors suggest is the necessity of conducting a simple
sensitivity analysis to ensure that the acceptable variations in the coefficients or the
right-hand sides will permit at least one of those rounded solutions to be feasible.
Purists will also note that there may well be a global optimum that is better than
the rounded solution. If a problem is one where such a condition is significant, then
integer programming analysis is called for. For many applications, however, a quick
and easy improvement is more important than the ability to prove that an optimum
has been obtained.
ACKNOWLEDGEMENT
The authors and American Edwards Laboratories recognize and appreciate the
considerable contributitn of the xenograft production team to this effort. Special
thanks are due Robert Cooper, Marilyn Howell, Robert Larson, John Hendrick, Stan
Komatsu, and Chris Serocke. The authors wish to express their sincere appreciation
to Dr. Michael Estes, General Manager, for his support and interest.
INTERFACES December 1981
55
APPENDIX A
Model Structure
In financial planning, stocks and bonds of various degrees of risk can be combined in specific amounts to form a portfolio of assets with the desired risk and return
characteristics. This same concept was used to match the supply distribution of raw
heart valves to demand. Thus, by ordering certain quantities from specific suppliers,
the total amount procured can be made to conform at least closely to the desired size
distribution. An example may best illustrate the approach. Assume a situation where
a company is buying from two suppliers, A and B, who could supply three sizes, 1,
2, and 3, with the following historical size distribution:
Vendor
A
B
Size
1
.30
.10
2
.50
.60
3
.20
.30
Assume also that the total purchasing and processing costs for size 1, 2, and 3 are
$10, $14, and $12, respectively. The objective function would be:
Minimize C = 12.4X,, + 13.0^6.
If the demands for the three sizes were, respectively, 100, 300, and 250 units, the
demand constraints would be as follows:
.3^^ + .1X6^ 100,
.5X,, + .6Xh ^ 300,
.2Xa + .3^6 « 250.
Since vendors have a maximum amount (//,) that cannot be exceeded and a minimum
amount (L,) below which they will not be bothered, upper and lower bounding
constraints are also needed:
Obviously this is a standard optimization program requiring linear programming.
The solution for two or twenty vendors, and three or thirty sizes, is basically the
same; the difficulty is not in expanding the number of vendors or sizes. Rather, it is
in deciding on definitions and time horizons for demand, and handling the integer
and probabilistic elements of the problem.
56
INTERFACES December 1981