Benchmarking Warehousing and Distribution Operations

Transcription

1 Benchmarking Warehousing and Distribution Operations ISyE 6202 Fall Introduction Benchmarking warehousing and distribution operations is an appropriate and popular first step in re-engineering warehousing operations. Within the warehousing and distribution industry benchmarking has traditionally been focused on comparing quantitative performance measures, such as operating cost typically measured as warehousing and/or distribution cost as percent of sales; operating productivity typically measured in units (lines, orders, cases, pieces, pallets, pounds, etc.) handled per person-hour; and response time and shipping accuracy. These operating cost and productivity performance measures are easy to compute and comprehend, and have become the industry standards. Statistics describing such performance measures are available for a number of key industries from the Food Distributor s International Association, the Warehousing Engineering Research Council, the National Association of Service Merchandisers, and a number of consulting firms. The existing paradigm for quantitative benchmarking is ratio-based; that is, the performance measures are always constructed as the ratio of an output dimension to an input dimension. Consequently, while simple, they can be seriously misleading. For example, warehousing and distribution cost as a percentage of sales varies directly and widely with product pricing and sales volume, aspects of the operation that are usually outside the control of warehouse and distribution management. In addition, cost as a percentage of sales figures vary widely across industries. As another example, units handled per person-hour ignores the capital investment in material handling and storage systems, which often substitutes for investment in labor. Furthermore, not all units are equal in the amount of handling time required per unit. For example, the popular case quantity shipped per person-hour is often used to compare facilities that perform a combination of broken case, full case and pallet picking, which is inappropriate. Regardless of their true efficiency, facilities that perform primarily broken case picking, which requires the time-consuming task of handling individual units within each case, will be rated highly inefficient, while facilities that perform primarily full pallet picking, in which many cases are handled with one simple pallet move, will be rated highly efficient. A new benchmarking approach is sorely needed one that simultaneously considers several dimensions of performance and makes it possible to compare warehouses across a broader spectrum. In this chapter, we will describe 1

2 a system-based approach to measure operational efficiency. At the heart of the new approach is a model of a warehouse that explicitly accounts for and integrates the many dimensions of critical resources and required work found in any warehouse. The outline of this chapter is as follows. Section 2 explains the key concepts underlying the system-based approach. Section 3 presents a prototype warehouse system model, and Section 4 describes the linear programming model used to compute efficiency. Section 5 describes an extension of the prototype warehouse system model that was used to analyze the warehouse industry, and summarizes the findings reported in [6]. 2 A Systems-Based Approach to Measuring Efficiency Our goal is to assess the operational efficiency of warehouse A. Assume that the only 2 relevant inputs used are labor and the capital expenditure on storage and handling equipment, and the only relevant output is total lines shipped. In the past year, warehouse A shipped 1.6 million lines and used 100 thousand hours of labor (= 50 full-time equivalents at 2000 hours per year) to perform all activities related to receiving and shipping operations. Its replacement cost of the storage and handling equipment, hereafter referred to as the capital input, is estimated to be 1 million dollars. Based on this data how efficient is warehouse A? Without any other data the only possible way to quantitatively assess efficiency is via the production function approach. Suppose we knew that a warehouse that uses K units of capital and L units of labor can process a maximum of Y = Φ(K, L) lines. Given the production function Φ we could measure output efficiency of warehouse A as the ratio 1.6/Φ(1, 100). For example, if Φ(1, 100) = 2.0, then the ratio would be In percentage terms, we might say that warehouse A is 80% efficient in that it should be able to produce (0.20/0.80)100 = 25% more output given its current levels of inputs. While not immediately obvious it is also possible to use the production function to assess input efficiency, which asks the following question: If our goal is to produce the same level of output that we currently are producing, by how much can we decrease all inputs in percentage terms? Using the production function we can compute input efficiency in the following way: Find the smallest value of θ for which Φ(θK, θl) Y. Since the parameter θ multiplies the arguments of the function Φ, this calculation, while conceptually simple, cannot be found simply by forming a ratio as we did in the output efficiency case. The production function approach requires an analytical expression, 2

3 which has to be found by some means. In very simple manufacturing settings or engineering processes (e.g. chemicals), such functions may be known to a desired degree of accuracy. If this is not the case, then the next step is to collect data from representative firms, posit a functional form for the production function with unknown values of the parameters, and find the best choice for the parameters so that the fitted form matches the data as closely as possible. This parametric approach almost always involves some kind of regression. The 2 main problems with the parametric approach are that it requires a specific functional form and that it is only applicable in cases where one can aggregate the various outputs of the firm into a single output measure. We shall now describe a nonparametric approach that requires no a priori functional form and is applicable for many outputs. In effect, the approach uses the data to implicitly estimate the multi-output analog to the production function, and the computations (for most models) involve the use of linear programming, which is now widely used and readily available. 2.1 Nonparametric Approach The nonparametric approach we use to measure operational efficiency is best explained via an example. Suppose we have collected data on warehouse B, just down the road from warehouse A. 1 In the past year, it used 200 thousand hours of labor and 2 million in capital to ship 3.2 million lines. In symbolic terms, we shall represent this transformation as Recall that warehouse A may be represented as B : (2, 200) 3.2. (1) A : (1, 100) 1.6. (2) Obviously, the numbers for B are twice that of A. We shall assume that operations may be scaled in which case B and A are equivalent, and we have no more information as to whether A is inefficient or efficient. Now suppose we add warehouse C to our dataset, represented symbolically as C : (0.9, 90) 2.0. (3) How can we compare C to A? First, we scale C downwards by 1.6/2.0 = 0.80 so that it has the same output as A, i.e., we represent it as C : (0.72, 72) 1.6. (4) 1 Warehouse B executes similar operations as warehouse A, but is in a different industry, so they were happy to participate in the comparison. 3

4 Now we can easily see that C reveals A to be 72% efficient. To illustrate one more issue here, let us suppose that warehouse D has been added to our dataset, represented symbolically as D : (0.24, 20) (5) Now scale D upwards by the ratio 1.6/0.64 = 2.5 so that it may be represented symbolically as D : (0.60, 50) 1.6. (6) With respect to the capital input, D has revealed A to be 60% efficient, whereas with respect to the labor input it has revealed A to be 50% efficient. To be conservative, we shall assess A as 60% efficient. In this way, we can reduce all inputs by at least 40% and still achieve the same level of output. In most situations, what we have just described would still be inadequate, since the data would not be so friendly. However, if we add one more assumption to our scaling hypothesis, then we will have a powerful methodology for comparison. To illustrate, suppose our dataset consisted of the following 3 warehouses E : (0.76, 80) 1.2 (7) F : (3.60, 192) 3.6 G : (2.32, 150) 2.4 respectively, and we seek to measure the input efficiency of G. (Please refer to Figure 1.) First, we scale firms E and F so that each firm produces 2.4, the same output as G: E : (1.52, 160) 2.4 (8) F : (2.40, 128) 2.4. A one-to-one comparison of firm E to G or firm F to G will not reveal any inefficiency in G, since in each case G is using less of one of the inputs. Suppose in addition to the scaling we further extrapolate the dataset by forming (hypothetical) composite warehouses that result from weighted averages of warehouses in the dataset. (Technically, we are convexifying the data.) As an example, consider a 30%-70% mixing of E and F, which we shall denote as H : (2.136, 137.6) 2.4. (9) Now comparing H to G we see that G is approximately 92% efficient. In fact, the 30%-70% mixture (to the nearest percent) turns out to be the best mixture if our goal is to reveal G as inefficient as possible. The above examples hopefully illustrate the type of quantitative approach to measuring input efficiency. For the case of several inputs and 4

5 outputs, the problem of finding the best mixture to find the lowest efficiency can be formulated as a mathematical programming problem. While the computations are relatively easy, the results are meaningful only in so far as the model of inputs and outputs that describe a warehouse facility are meaningful, and that a large enough sample of accurate data can be obtained. The modeling aspect turns out to be most difficult, as we shall see, since a variety of proxies must be invented given the availability of data and the scope of the system being represented. The next section builds such a systems view of a warehouse. A more extensive systems view will be presented in Section 5. 3 A Prototype Warehouse System Model The warehouse is viewed as a system that uses input to produce output. In the prototype model the input categories are labor and equipment, and the output proxies the required movement of material. 3.1 Labor Input The index constructed to represent the labor input is the sum of the direct and indirect labor hours expended performing all operations necessary for receiving, putaway, storing, order picking and shipping. Hours spent on maintenance, supervision and management constitute the category of indirect hours. Direct and indirect hours devoted to specific requirements not generally common to all warehouses, such as specialized inspections, security, customer satisfaction, traffic and personnel are excluded. 3.2 Material Handling and Storage Equipment Input The index constructed to represent the investment in storage and material handling equipment is the summation of the number of units of each equipment used by the warehouse weighted by the average replacement cost per unit of equipment. Replacement cost is used in lieu of book value since what companies pay for similar equipment and how they determine their book values varies widely. The categories of equipment are Vehicles, Storage Systems and Conveyor Systems. The (1991) replacement cost numbers used are shown in Tables 1, 2 and 3 in the Appendix. (The cost of conventional storage systems such as shelving and pallet rack should be included if their costs are significant in relation to these other capital investments.) Since the focus is on operational efficiency within the warehouse, any rental or depreciation for the building is excluded. 5

6 3.3 Movement Output The picking/shipping workload is driven by the number of orders and the number of lines on those orders. The orders and lines should be further broken down into broken case, full case and pallet picking, since each type of picking determines a different type of workload. So that we may interpret the approach graphically we shall proxy the workload associated with the movement of material by the single index of total lines shipped. 4 Computational Model to Measure Operating Efficiency Let T LH i and INV EST i denote the total labor hours and capital investment inputs, and let LINES i denote the total lines shipped as the output for warehouse i in the database. The linear programming formulation used to compute the operating efficiency θ for reference warehouse O is as follows: subject to: min θ N λ i T LH i θ T LH O (10) j=1 N λ i INV EST i θ INV EST O (11) j=1 N λ i LINES i LINES O (12) j=1 λ i 0 (13) The solution λ and θ to the linear program has the following interpretation. There exists a (hypothetical) composite warehouse formed as a weighted average of the existing warehouses using the components of λ as the weights. This composite warehouse has 2 important attributes: It produces at least as much output (total lines shipped) as the reference warehouse; and consumes no more than 100 θ % of the labor and capital investment than the reference warehouse. From this perspective the reference warehouse can theoretically scale its labor and capital investment by 100 θ % amount and still achieve its current levels of output. The linear programming model presented here is a form of 6

7 Data Envelopment Analysis (DEA), first introduced in the late 1970 s in [3]. 2 For a 2-input, 1-output model such as the one presented here the efficiency scores obtained from the linear programs may be graphically represented, as follows. Consider the sample data (obtained from [6]) shown in Table 4. As discussed in Section 2, first divide each warehouse s 2 inputs by its output to obtain normalized warehouses, all of whom have unit output. Since all the outputs are the same, we only need to concentrate on how each warehouse consumes the two inputs. Figure 2a plots the normalized capital and labor inputs for each warehouse in the sample. The next step is to construct the collection of all capital-labor pairs that can achieve unit output. If we had data on all warehouses we would know this set exactly, and simply plot it as in Figure 2a. As discussed in Section 2, we approximate the true set via extrapolation of the original (normalized) data by adding to the points in Figure 2a all possible weighted averages (or convex combinations) of these points, and including all points that lie above, since we assume that inputs are freely disposable. That is, if a labor-capital pair (K 1, L 1 ) can achieve unit output, and (K 2, L 2 ) (K 1, L 1 ), then we assume that (K 2, L 2 ) can also achieve unit output. The resulting set is depicted in Figure 2b. Notice that the set in Figure 2b is completely determined by its efficient frontier, which is the darkened piecewise-linear boundary. Every point on the efficient frontier will have an efficiency score (its θ ) of 1.0. Every point lying above the efficient frontier will have an efficiency score less than 1.0. Each inefficient warehouse s efficiency score can be determined graphically by forming the line segment joining the origin (the point (0, 0)) to the point (K 0, L 0 ) describing the warehouse, and determining the point of intersection (K0, L 0 ) of the line segment with the efficient frontier. The point of intersection (K0, L 0 ) is simply a scaled version of (K 0, L 0 ), and so may be expressed as (θ K 0, θ L 0 ) for a unique value of θ. The value of θ is the warehouse s input efficiency. To determine θ one simply measures the distance along the line segment joining the origin to (K0, L 0 ) and dividing this distance by the distance along the line segment joining the origin to (K 0, L 0 ). Figure 3 illustrates this graphical approach for one of the warehouses. 3 5 Analysis of the Warehouse Industry Once the efficiency scores have been calculated it is then possible to determine if high efficiency is correlated with unique design parameters, and if certain industry practices outperform others. Specific questions include: Do larger warehouses perform more efficiently? 2 For further discussion and interpretation of DEA, consult the monographs [2], [4] and [7]. 3 When there are more dimensions it will not be possible to graphically depict the whole efficient frontier, though it is possible to represent the relevant 2-dimensional portion for computational purposes. See [8]. 7

8 Do capital-intensive warehouses perform more efficiently? Do non-union facilities outperform their union counterparts? With the advent of Quick Reponse and Micro-Marketing strategies, warehouse managers have seen a vast increase in the number of sku s, and smaller, more frequent deliveries of both inbound and outbound traffic. Consequently, facilities are having to perform more work in less time. Answers to these questions would point the way to the optimum warehouse design. We now summarize an analysis of the Warehouse Industry reported in [6]. We describe the database, extensions to the prototype warehouse system model, and finally the conclusions of the study. 5.1 The Georgia Tech Database The necessary data, feedback and site visits were obtained from the Material Handling Research Center (MHRC) at the Georgia Institute of Technology (now called the Logistics Institute), a consortium of over 20 firms that either have warehouses, or sell material handling equipment, provide information systems or consulting services. Over an initial period of 1 year a variety of models were developed and tested that were used to finalize and encapsulate the data requirements into a data survey [5]. Over the next 3 years data was collected for a total of 57 warehousing and distribution facilities located throughout the United States, a third of whom belonged to companies outside the MHRC. The participating companies range in size in annual sales from $500 million to over $50 billion. The range of industries represented was broad, including service parts, electronics assembly, health care, photographic supplies, and food products. The range of products shipped included auto parts, dental and office supplies, electronics, fine papers, hardware, health care, industrial packaging, mail order apparel, office machines, photographic supplies, and wholesale drugs. Almost all of the participating facilities received and stored product from their vendors and then shipped product to their retail customers. A few facilities were central distribution centers that re-shipped product received from manufacturers to the smaller warehouse facilities that served their local retail markets. 5.2 Extensions to the Prototype Warehouse System Model Since the amount of space affects the consumption of capital and labor, an input, termed space, was added. It was proxied by the square feet associated with the receiving, storage and shipping operations. On the output side, the single output of total lines shipped was disaggregated into its 3 components of broken case, full case and pallet lines. Two additional outputs, termed accumulation and storage, were also added. Each is described below. 8

9 Investment in material handling equipment (conveyors, sortation) may be required to accumulate orders due to the necessity to perform picking operations within a short time window. For example, in a walk-and-pick system for broken case picking, the picker places the respective quantities in containers such as totes staged on movable carts that are pushed by the picker to the respective locations. When a batch of lines has been picked, the order picker may be asked to drop the tote onto a conveyor that routes the tote to shipping, which must then accumulate the customer order from the totes arriving from the various zones. This task is often achieved using bar-coding and (expensive) sortation equipment. Alternatively, the order picker may be asked to pass the tote manually or via a conveyor to the next zone for picking, in which case the order is being progressively assembled, so that no sortation equipment is required downstream in shipping. However, the progressive assembly strategy is more labor-intensive. In either case extensive accumulation requirements will result in a significant increase in the capital and labor inputs. The index proposed to measure the cost to accumulate lines picked into orders shipped is the difference between the lines picked and the orders shipped. (Since every order has at least one line this number is nonnegative.) Note that when there are only single-line orders the accumulation index, justifiably so, is zero. As the lines per order increases (keeping the number of orders constant) the accumulation index grows, too, as it should. Note also that proportional increases in lines and orders results in a proportional increase in the index, in sharp contrast to the conventional industry standard of the lines per order ratio. The storage output index is designed to measure the cost to store the inventory in the warehouse. A distribution center that must store large quantities of slower-moving, larger-sized parts will have a large footprint, which necessitates a considerable investment in conveyors and vehicles to move material. For such a facility the investment in material handling equipment is largely due to its storage requirement. Unfortunately, financial measures of inventory, while readily available, are not directly related to operations requirements. Here is how this cost was proxied. Each sku that was picked primarily in broken case quantities was allocated 1 square foot on the floor ; each pallet rack location was allocated 25 square feet of floor space; and the actual square feet of floor storage was also recorded. (The pallet rack and floor storage areas are used to handle the case and pallet quantities.) The next step aggregated the 3 different amounts of allocated floor space into a single proxy for storage, as follows. Order picking constitutes the dominant portion of a warehouse facility s operating cost, and traveling to and from storage dictates a sizeable portion of that cost. Given that the average length of an optimal tour that visits points located randomly in a rectangular region is proportional to the square root of the area traveled ([1]), the index constructed was a weighted average of the square root of the broken case, pallet rack and floor storage square 9

10 footage estimations. The weights were determined by the frequency of visitation to these areas, with frequency being proxided by the lines shipped in the respective categories. The justification for the weighting system is that a facility that has a large physical requirement for pallet storage, for example, but only visits that area infrequently should receive less credit for storage requirement than a facility with a smaller physical requirement for pallet storage that is visited frequently. To summarize the model development, the storage index was calculated by the following formula: π ( # broken case SKUs + (1 π) 5 # pallet locations + ) floor storage sq. ft. where π denotes the proportion of lines picked broken case. 5.3 Modeling Issues With regard to the systems model just described, the following caveats are in order. 1. Cubic feet in the facility was not proposed because not all of the clear height can or should be used within a facility. For example, the receiving and shipping operations typically do not require the full use of the clear height. As another example, consider pallet storage. Pallets can be stored either on the floor, typically stacked as high as the material can support, or in different types of racking systems (e.g. single-deep, double-deep, drive-through, deep-lane) in order to use the cube of a facility. When several pallets of an item can be stacked, or when there is sufficiently high turnaround, it will be cheaper to store these pallets on the floor, instead of pallet rack. For these reasons multiplying the clear height by the square footage could be a less accurate proxy for space than simply using square feet. 2. The volume of material moved was not proxied. Comparing units shipped or determining an appropriate aggregator is fraught with problems, since units vary in size, weight, and handling characteristics. Just as important, the workload required to pick the units of an item is largely dictated by the need to visit the location, not the amount of units picked, which is especially true for broken case picking. For example, the time to pick 2 or 3 pens out of case is essentially the same. 3. The role of the Warehouse Management System (WMS) software, which essentially controls the flow of material, and keeps statistics such as work requirements, labor productivity and equipment utilization, was not proxied. Many of the best strategies for efficiency cannot be implemented without support from the WMS. The participating companies all had comparable WMS systems, so no further attention was given to developing a suitable proxy for the the quantity of such 10

11 input. Note that it is the quality and type of services provided by the WMS, not its purchase cost, that determines the relevant input. 4. Use of replacement cost could underestimate a firm s efficiency, if that firm experienced an initial high period of demand, which may have justified a design decision of purchasing expensive equipment, but then saw that demand subsequently decline. 5. A variety of useful submodels are obtained by constraining the choices of the λ i s. For example, it is possible to eliminate the assumption that warehouses may be scaled. 5.4 Industry Analysis The raw data from the study is reproduced in Table 4 in the Appendix. All data, except for storage, are measured in units of thousands. SF i, A i, S i, BCL i, F CL i and P L i denote, respectively, the square footage, accumulation, storage and broken case, full case and pallet lines for warehouse i. Several submodels were run as described in the previous paragraph. Three factors were analyzed: workforce demographics (union versus non-union), warehouse size (measured by square footage), level of automation (measured by the level of investment). Typically, large sized or highly automated firms were less efficient than smaller sized or less automated firms. However, highly automated smaller sized firms tended to be less efficient than highly automated large sized firms. We examine each factor below. Workforce Demographics Unionization was associated with higher operating efficiency than nonunion facilities. This result may seem somewhat surprising, since it is generally perceived that union operations tend to be less efficient. All union operations that were highly rated paced the flow of work by using such strategies as time standards and incentives to motivate high productivity; on-line communication via light aids and/or radio frequency terminals to establish continuous, real-time monitoring of transactions; and progressive order assembly or automated storage/retrieval systems to pace the work flow and facilitate supervision. 11

12 Warehouse Size Statistical analysis provided very little support for the idea of technical scale efficiencies in warehousing and distribution operations. (Small-sized warehouses were under 175,000 square feet and large warehouses were over 500,000 square feet.) In large facilities, long travel distances, poor work flow visibility, and difficult communication and supervision appear to offset any improvements brought on by increased order volumes, high levels of mechanization, and information system enhancements. It is important to note that a distribution network consisting of larger but fewer facilities may result in lower aggregate inventory in the supply chain, which may offset potential losses in operating efficiency. Level of Automation To determine how investments in material handling systems correlate with warehouse efficiency, the level of investment was used as a proxy for the level of material handling systems automation. Low-investment warehouses used less than $650K in storage and handling equipment and highinvestment warehouses used more than $4M. A low investment per square foot warehouse had less than $3 per square foot, and a high investment per square foot warehouse had more than $12 per square foot. A low investment per person warehouse had less than $10,000 per person, and a high investment per person warehouse had more than $35,000 per person. (Keep in mind that these figures represent 1991 costs.) Generally speaking, statistical analysis demonstrated that low investment warehouses outperformed high investment warehouses. The negative effects of high automation were partially mitigated by size. That is, The large sized facilities with high levels of automation tended to be less inefficient than small sized facilities with high levels of automation. lack of adequate system maintenance, 12

13 inappropriate selection of system types, and the difficulty of reconfiguring highly automated systems to meet changing business requirements are the likely causes of the inefficiencies in facilities with high levels of investment. There were several highly efficient facilities with high levels of automation. These facilities generally made very targeted use of the automation, and made every effort to simplify and streamline the material flow, which is the key to success in conventional warehouse operations. 13

14 6 Appendix Table 1: Vehicle Replacement Cost Type Unit Cost Pallet Trucks $10,000 Walkie Stackers 8,000 Sit-down Counterbalance 30,000 Stand-up Counterbalance 30,000 Straddle Trucks 35,000 Straddle Reach Trucks 45,000 Side-loader Trucks 65,000 Turret Trucks 80,000 Hybrid Trucks 100,000 Pallet ASRS Machines 150,000 Rail-guided Order Pickers 30,000 Wire-guided Order Pickers 35,000 AGVs 80,000 Table 2: Storage Systems Replacement Cost Type Unit Cost Person-aboard ASRS Aisles $ 110,000 Horizontal Carousels 40,000 Vertical Carousels 65,000 Miniload ASRS Aisles 125,000 A-Frame Dispensers 750 Table 3: Conveyor Systems Replacement Cost Type Cost per Linear Foot Non-Powered Roller $ 50 Powered Roller 200 Powered Belt 100 Skate Wheel 25 Tow Line 100 Tilt-Tray Sorter 1,000 Pallet Conveyor 1,000 14

15 Warehouse i T LH i INV i L i BCL i F CL i P L i S i A i SF i Union Y Y N Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y N N N N N N N N N N Y N N N N N N N N N N Y N Y N Y N Y N N N N N N N Table 4: Georgia Tech Database 15

16 References [1] Beardwood, J., J.H. Halton and J. Hammersley, The Shortest Path Through Many Points, Proceedings of the Cambridge Philosophical Society, 55 (1959), [2] Charnes, A., W.W. Cooper, A.Y. Lewin, and L.M. Seiford, Data Envelopment Analysis: Theory, Methodology, and Application, Kluwer Academic Publishers, Boston, [3] Charnes, A., W.W. Cooper, and E. Rhodes, Measuring the Efficiency of Decision Making Units, European Journal of Operations Research, 2 (1978), [4] Fare, R., S. Grosskopf and C.A.K. Lovell, Production Frontiers, Cambridge University Press, Cambridge, [5] Frazelle, E.H. and S.T. Hackman, The Warehouse Performance Index: A Single-Point Metric for Benchmarking Warehouse Performance, Material Handling Research Center Technical Report TR-93-14, Georgia Institute of Technology, Atlanta, GA. [6] Hackman, S.T., Frazelle, E.H., Griffin, P.M., Griffin, S.O. and Vlatsa, D.A. Benchmarking Warehousing and Distribution Operations: An Input-Output Approach, Journal of Productivity Analysis, Vol. 16, pp , [7] Fried, H.O., C.A.K. Lovell and S.S. Schmidt, The Measurement of Productive Efficiency: Techniques and Applications, Oxford University Press, New York, [8] Hackman, S.T., L.K. Platzman and U. Passy, Explicit Representation of the Two-Dimensional Section of the Production Possiblity Set, Journal of Productivity Analysis, Vol. 5, 1994, pp