A risk management model for FMS selection decisions: A multiple-criteria decision-making approach

Computers in Industry 23 (1993) 99-108 Elsevier

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A risk management model for FMS selection decisions: A multiple-criteria decision-making approach Markku Kuula Helsinki School of Economics, Department of Management Science, Helsinki, Finland Most manufacturing system selection decisions are made under risk. To cope with such risks, many manufacturing companies invest in complex and flexible manufacturing systems (FMS) which allow them to adjust production to unforeseen developments. In this paper a multiple-criteria decision-making model is introduced to aid m a n a g e m e n t in selecting the most appropriate technology and design for a flexible manufacturing system under simultaneous consideration of a number of possible market scenarios. This study is part of a larger project which tests different decision support methods for FMS selection.

Keywords: FMS; Risk management; Multicriteria

1. Introduction

Flexible manufacturing systems (FMS) have extensively been studied over the past fifteen years. An FMS is an integrated manufacturing system that consists of one or several work stations linked by a computerized inventory system, making it possible for jobs to follow diverse routes through the production system. An advantage of an FMS is that it can simultaneously meet several goals: small batch sizes, high quality standards and efficiency of the production process. Both the industrial and the academic community [1-3] have been interested in the design of flexible manufacturing systems. Correspondence to: Markku Kuula, Helsinki School of Economics, Department of M a n a g e m e n t Science, Runeberginkatu 14-16, SF-00100 Helsinki, Finland.

Gupta and Goyal [4] in their survey article have provided a number of different definitions for "flexibility" in manufacturing. They have pointed out that flexibility is a property of the system that indicates the system's potential behavior, rather than its performance. Furthermore, they have found out that in many definitions flexibility is not a self-contained concept. It should be used together with other production objects, such as products, volume and quantity. Ranta and Alabian [5], Ranta [6], Kuula and Stam [7] and further Stam and Kuula [8] have developed multiple-criteria optimization models for FMS selection problems. In their model formulations they have used flexibility criteria that are based on different definitions of flexibility. In this paper we adopt an alternative view to manufacturing flexibility. Because the flexibility indicates the system's potential behavior in unforeseen developments, we ask the decisionmaker (DM) to make subjective predictions for some (say five) economic situations (scenarios), for example, about the future prices and demands for each product. The scenarios aim to approximate all possible future development paths or risk situations. Ater such scenarios have been defined, the DM can optimize, by using an interactive multiple-criteria optimization method, his utility subject to the production restrictions of the FMS configuration. The utility function is assumed quasi-concave. For example (see Fig. 1), if we have two scenarios and one judgement

0166-3615/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved

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:;Of2a:;r

~ 'omputers m Industo,

A robust alternative

÷ ÷.

. =,,,' +

* nondominated + dominated

.,I÷

÷

÷ ÷

+

÷

~r

÷ +

÷ 4Profit for scenario

1

Fig. 1. An example of a decision problem under risk.

criterion, the DM may select a policy which is robust with respect to the scenarios. While selecting a robust alternative the DM enters his risk preferences and subjective probabilities for the scenarios. Typically, FMS investment decisions are made in companies at the highest managerial level. They are strategic decisions and require diverse complex analysis. In previous studies on FMS selection a decision support framework has been suggested that consists of two separate phases [7,8]. In these papers it is assumed that management has already made the decision to switch their production system to an FMS. It is further assumed that management has general information about the different available FMS designs. Because the initial number of available alternatives may be large, phase one is a pre-screening procedure to narrow the list of candidate FMS configurations. Management uses only rough estimates of revenues, costs and other benefits. It is possible to use both quantitative and qualitative information. Management can also simultane-

. . . .

M a r k k a K u u l a is a senior researcher in the D e p a r t m e n t of M a n a g e m e n t Systems at the Helsinki School of Economics and Business Administration. He received his Master's degree from the Helsinki School of Economics and Business Administration. He is currently preparing his doctoral dissertation on multiple-criteria decision aids for the strategic management of flexible manufacturing. His research interests include flexible I ................ m a n u f a c t u r i n g systems, multiplecriterion decision making and negotiation support systems. He has published articles in several international journals.

ously use " h a r d " (possible to give numerical values) and "soft" (not possible to give numerical values) criteria to evaluate all possible FMS alternatives, In this phase the Analytic Hierarchy Process [9,10] is used. This study concentrates on the second phase and assumes that the pre-screcning phase is already completed. Only a small number of candidate FMS configurations are investigated in more detail in the second phase. Detailed quantitative information is required in this phase. A separate risk management model is formulated for each remaining FMS configuration. We will explore the performance tradeoffs between the different scenarios of each remaining FMS configuration, subject to the physical limitations and performance characteristics of the design. The model formulation is modified from Refs. [5-8]. Model formulations used in these studies were deterministic, whereas the model formulation used in this study is stochastic. However, we use the same data, based on a real case from a Finnish metal product company, as was used in the above studies. We use the IFPS optimizer [11,12] to illustrate our model. Many previous approaches to assist the DM involved in an FMS selection problem exist. These include simulation models (for example, [13]) and analytical models (for example, [8,14,15]). Although the simulation models are very useful in evaluating new FMS designs, the analytical models provide more insight into thc process [1]. Our approach belongs to the category of analytical models. In comparison with other analytical models developed for FMS selection problems, our approach is unique. First of all it is stochastic, whereas most of the other analytical models are deterministic. The stochastic models developed for this particular problem (for example, [15]) require the DM to determine his risk preferences precisely. This is not the case in our approach. The only assumption needed is that the (DM's underlying) utility function is quasiconcave. The remainder of the paper is organized as follows. In next section, we show how to solve, by multiple-criteria optimization methods, decision problems that involve risk. Then the case background is discussed, and a mathematical programming model formulation of our example case is described. In Section 4 we apply our model to a

Computers in Industry

M. Kuula / Risk management model

specific case problem. We conclude the paper with some final remarks.

Choose x

2. Risk management using multiple-criteria optimization

101

Choose y

Obeerve scenario

• ..he

Fig. 2. The two-stage decision problem.

Let us assume a decision situation where a DM is preparing (at time 0) production plans for two periods. The first period is fully deterministic. The product prices, costs, demand, etc. are known. The second period is stochastic. At the beginning of the planning horizon, the DM has for the second period K alternative scenarios for price, demand etc. estimates. He also knows that the production program chosen in period one affects the demand of product i in period two. Therefore the DM has to consider both periods simultaneously. Traditionally, this two-stage problem is divided into K + 1 separate modules (one module per scenario plus one module for the first period). The modules related to the second period are linked together with the module for the first period by some of the constraints. Let x i denote the first-stage variables (i = 1. . . . . n 1) and Ykl the second-stage variables (l = 1..... r/z) for scenario k (k = 1 , . . . , K ) . When the probabilities pk for the different scenarios (states of nature) are known, an optimal decision rule under an expected "profit" criterion can be solved with the following LP model [16]:

Only the objective function changes. Let U be the utility function for profit. The objective function for the expected utility maximization is then given as follows:

m a x ~ - , P k U ( X l . . . . . xn,,Yll . . . . . Yx,,,).

(4)

k

In the two-stage formulation, we solve production programs simultaneously for the first period and for all alternative scenarios related to the second period. The DM first implements the production program x, chosen by the optimization model, for period one (see Fig. 2). Then after the first period he observes the scenario realized, and therefore chooses the production program y for period two. In our multicriteria approach [17,18] we assume that the DM's utility function is explicitly unknown. Therefore, profits for the K different scenarios are considered as separate objectives and the problem will be solved by using an interactive multiple-criteria decision-making procedure. The formulation for this model is: max fl

= Ecixi-}-

i

Eellyll, l

subject to (5)

for j = 1,2 . . . . . e

Y'~aijx i = bj

(2)

i

Y~.dk,jXi+ ~ a k t j Y k t = b k j i

forj=g+

1. . . . . G

l

k = 1.....

(3)

where c i and ckt are the profit coefficients, aii and aktj the structural coefficients, and bj and bkj the right-hand-side coefficients for the first and second stage, respectively. Coefficients dki j connect the first-stage variables to the secondstage. All the second-stage coefficients CkZ, akt j and bkj can be scenario dependent. Similarly the expected utility maximization leads to a linearly constrainted NLP formulation.

maxfK = Y~.cix i + ~..CKtY m, i

l

subject to (2) and (3) Now, the DM accounts for risk aversion and for the probabilities of different scenarios subjectively. He may also test different decision rules (e.g. minmax, maxmax, etc.) by using the same restriction set. Note that the expected utility criterion with a strictly quasi-concave utility function leads to a single Pareto optimal point in the objective space. Our multicriteria approach allows the DM to study the Pareto frontier carefully before making such a final choice. In the example of Fig. 3 we have presented feasible

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f2

thermore, we have one objective function associated with each scenario. Next we describe the model in detail.

3.1. Time The total time Tjk that machine j is used per period is calculated as follows:

Ti~= E ( C + t i j ) b i ~ z , i ~ i Fig. 3. Production alternatives in the objective space (fk is the objective function of scenario k).

solutions in the objective space in a three-scenario case.

3. Case background and model formulation The FMS configuration used in our study is drawn from a Finnish metal product company [5,6]. The formulation of the model as a twoperiod multicriteria mathematical programming problem has been modified from the deterministic one-period models used in [6-8]. Suppose that this FMS configuration consists of four machines j, which are used to produce thirteen different parts i, and the decision problem is to determine an optimal production program for this particular configuration. Let k = 0 refer to the first period, and k > 0 to the second period in scenario k. The decision variables in our model formulation are b~k, the bach size, and V~k, the number of batches produced per period, for each i and k. in this linear model the batch sizes are fixed: b~ = 5 for all i and k. We also use some auxiliary variables in our model formulation. The model consists of four fundamental constraints: (1) cost, (2) time, (3) marketing and (4) definitional constraints. The cost and time constraints are related to scarce resources. The marketing constraints limit production via the demands of different scenarios. The contribution to profit and volume constraints are definitional. As mentioned above our model consists of two periods. The first period is fully deterministic (master problem). The second period is stochastic and consists of five different scenarios (subproblems). The subproblems are connected to the master problem only through marketing constraints. Fur-

( j = I . . . . m) (k : 0 . . . . . K),

(6)

where T/j is the actual unit tooling time of part i on machine j, and tij the unit overhead time of part i on machine j. The overhead time consists of changing, checking, repairing and waiting times. The tooling times and overhead times are identical for both periods. The technical nonavailability times (disturbance and service times) are estimated by using recent empirical case studies [19-21]. In these studies the technical nonavailability time Tdjk is a function of part complexity, number of batches of each part, size and complexity of software used and personnel training. In our illustration we use the same coefficients for all machines and time periods (thereby following Stam and Kuuta [8] and Kuula and Stam [7]). However, in reality, for example the part complexity factors will be machine-dependent. The technical nonavailability time is given by Tdjl~ = d g E g i

i + d b Y'~ t:ik + d s S -. i

dpL PL

( j = 1 . . . . ,m)

(7)

( / , = 0 .... ,K), where gi is a measure of complexity of part i, S is complexity of the software needed and PL is the number of employees trained per period; the coefficients dg (nonavailability due to part complexity), d b (nonavailability due to batch changes), d s (nonavailability due to software size and complexity) and dpL (nonavailability due to personnel training) are positive scaling constants. The maximum TjkMAx and minimum TjkMIN times (minutes) that machine j can operate per period requires TjkMI N ~ Tjk + Tdj k < TjkMAX

(J = 1,... ,m)

(k = 0 . . . . . K ) . (8)

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M. Kuula / Risk management model

In addition to the nonavailability and normal production times, we also account for the batch change times tbi , for each product i. The total batch change time per period is

Tbk = ~tbiVik

(k = 0 . . . . . K ) .

(9)

i

The utilization time (minutes) for all machines combined per period is limited by TkMIN and TkMAX as follows: (k = 0 . . . . . K ) ,

(10)

where T k = E~ Tjk is the total operation time of all machines and Tdk= S, jTdj k is the total nonavailable (disturbance) time of all machines.

3.2. Costs Let CMk be the machine costs, CLk the tool costs, Cpk the parts pallet costs, Cs~ the software costs, CTk the transportation costs and Co~ all other costs. The total costs Ck per period are then defined as C k = CMk 4- CLk 4- Cpk 4- Csk 4- CTk 4- Cok

(k=0 ..... K).

=

EejMj J

(12)

The tool costs per period CLk are a function of part complexity, gi, and the number of tools L~ needed to produce part i. The tool costs can be expressed as i

EL i (k=0 ..... K),

(13)

i

where qg is the tool cost of part complexity gi and q~ is the tool cost per tool needed. The parts pallet costs Cpk depend on part complexity, batch size and number of batches produced:

Cpk=Pg•gi

+pbY'~bik +pv~_,Vik

i

(k = 0,...,K),

i

i

i

(15)

where Sg are software costs of part complexity, s are software costs of the total number of batches produced, s v are software costs of the number of batches produced, s I are software costs of the number of tools needed, and s e are software costs of machine efficiency. The other costs Cok consist of training costs CTR k and residual costs CRES~. The training costs are a function of PL, employees to be trained per period, and of CpL, the training costs per employee per period. In this notation we have Cok = CTR k + CRESk = CpLPL + CRESk

(16)

(11)

( k = 0 .... , K ) .

CLk=qgEgi+q,

Csk = s g E g i + (s +Sv) EVik + s , E L i + s e E e j

(k=0 ..... K).

We assume that the machine prices are dependent on the relative efficiency ej of machine j. For each machine j, the direct investment cost per period is Mj. The machine costs CMk consist of direct investment costs adjusted for efficiency:

CMk

ity, Pb is the parts pallet cost of batch size and Pv the parts pallet cost of number of batches produced. The software costs Csk depend on part complexity, number of batches produced, number of tools needed to produce part i and efficiency of machine j:

( k = O .... ,K),

TkM1N ~ T k + Talk 4- Tbk ~ TkMAX

103

i

i

(14)

where pg is the parts pallet cost of part complex-

3.3. Contribution to profit Define the contributions to profit R k as Rk = E(Pik--VCi)Uikbik i

(k =0 ..... K),

(17)

where Pik is the price of product i for scenario k and vc i is the variable material cost of product i.

3. 4. Marketing constraints The overall economic situation limits production. For the first period, minimum and maximum levels (V/0MIN and ~0MAX) for production, for each product i, require V/0MIN ~ biovio <~ Vi0MAX.

(18)

For the second period, we assume that such sales limits are stochastic. The maximum production level of product i depends on the production volume in the first period and on a sales change quantity chik. For each scenario k, the marketing limit in the second period is stated as

bikUik <~bioUio 4- Chik

( k = 1 . . . . . K ).

(19)

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Table 1 Products i, maximum ~0MAX and m i n i m u m VioMiN production limits, part complexity g~, tooling Ti~ and overhead tz~ times, (for machines j), batch change times Tbi and n u m b e r of tools needed in production L, i 1 2 3 4 5 6 7 8 9 10 11 12 13

V/OMIy (parts)

ViOMAX (parts)

gi

Til

Ill

TI2

li2

L3

li~

Tt4

ti4

?bz

L

(facets)

(min)

(min)

(min)

(min)

(min)

(min)

(min)

(min)

(min)

(tools)

500 2000 1500 1500 1000 100 200 3000 3000 1500 200 150 100

850 3000 2400 2400 1500 360 360 4100 4100 2800 360 350 360

4 2 3 4 4 6 8 2 2 3 9 10 10

20 12 20 20 40 20 40 12 12 12 48 60 0

2.0 1.6 2.0 2.0 1.2 1.6 2.0 1.6 1.6 0.8 4.0 5.0 0.0

20 6 14 20 10 20 40 6 6 8 60 45 40

2.0 1.2 2.0 2.0 1.2 2.0 2.4 1.2 1.2 (}.8 4.(} 5.0 5.0

20 6 14 20 it) 4{) 60 6 6 8 60 45 60

2.0 1.2 2.0 2.0 1.2 2.0 2.4 1.2 1.2 (I.8 4.0 5.0 5.5

8 4 8 8 8 20 40 4 4 8 80 80 50

4.0 2,0 4.0 4.0 4.(} 4.8 6.0 2.0 2.0 2.0 6.0 6.0 8.0

4.(} 2.1} 4.0 -1.0 4.0 4.8 6.0 2.0 25) 2.{1 6.0 6.{! S.0

50 50 50 50 50 50 50 50 50 50 100 100 100

3. 5. The decision problem For each scenario, we define one objective:

fk = ( R o - C o ) + (1 + i ) ( R k - C~) (k = 1 , . . . , K ) ,

(20)

where i is a discounting factor. Table 2 Symbols, dimensions and values for disturbance coefficients dr,, dg, d s, dpL, S and PL, and constraints ~kMAX, cost coefficients Sg, s, s v, s e, s~, pg, Pb, Pv, qg, q~, /14:. and c m, and efficiency coefficients ej. db dg d~ dpL S PL

Tjk MAX Sg s sv s¢ s1 pg Pb p~ Mj qg qE ¢PL e1 e2 e3 e4

3 min/batch 40 m i n / f a c e t 0.05 m i n / l i n e 3 1,000,000 lines 100 hr 316,800 min 500 S / f a c e t 10 S / b a t c h 20 S / b a t c h 2 $ min/mm 300 S/tool 10 S / f a c e t 3 S / b a t c h size 200 S / b a t c h 100 $ m i n / m m 500 S / f a c e t 10 S/tool 100 $ / h r 3,000 m m / m i n 3,000 m m / m i n 3,000 m m / m i n 6,000 m m / m i n

In our risk management model the DM aims to maximize the contribution to profit, but he does not know for sure which scenario will occur. Therefore, taking into account his subjective views on the likelihood of different scenarios, as well as his preference for risk, he aims to balance the different scenarios in his mind. We may interpret this as an implicit maximization of the expected utility criterion.

4. Illustration As mentioned above, our example is based on a real case. The data collected from a Finnish metal product company [5] are confidential. For that reason the name of the company is not disclosed. In our illustration we use the interactive financial planning system I F P S / P l u s and IFPS/Optimum software packages together. These packages run on a minicomputer (HP9000). We solve the multicriteria problem by the reference point method, as suggested by Kallio ct al. [22]. For a detailed discussion of various multicriteria approaches, see, for instance Refs. [2326]. See also Refs. [27-29] for an overview of reference point methods. For the reference point method we modify our M O L P model formulation. Thereby we introduce an auxiliary decision variable y, a set of extra constraints and a single objective function. The extra constraints restrict the value of y as follows:

Y>G--fk

(k=l

. . . . . K),

(21)

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105

Table 3 T h e product prices Pik ($) in different scenarios k and the variable material costs vci ($) Product prices

Product

Variable cost VCi

Pik

1 2 3 4 5 6 7 8 9 10 11 12 13

k=l

k=2

k=3

k=4

k=5

50.0 50.0 50.0 50.0 50.0 200.0 200.0 50.0 50.0 50.0 200.0 200.0 200.0

45.0 37.5 37.5 37.5 45.0 70.0 70.0 47.5 47.5 37.5 214.0 220.0 220.0

47.5 40.0 40.0 40.0 47.5 85.0 85.0 49.0 49.0 40.0 206.0 210.0 210.0

52.5 57.5 57.5 57.5 52.5 206.0 206.0 52.5 52.5 57.5 85.0 85.0 85.0

55.0 62.5 62.5 62.5 55.0 214.(I 214.0 55./) 55.0 62.5 70.0 70.0 70.0

20.0 10.0 15.0 20.0 25.0 30.(I 60.0 20.0 3(I.0 10.0 50.0 50.0 50.0

Table 4 M a x i m u m increase from period o n e to period two in the annual sales for different scenarios Product

k = 1

k = 2

k = 3

k = 4

k = 5

l 2 3 4 5 6 7 8 9 10 11 12 13

100 250 250 200 125 25 25 350 350 200 150 25 20

75 200 150 150 100 - 75 - 25 200 200 200 - 25 75 50

85 200 175 175 110 - 25 - 10 250 250 200 - 10 50 20

100 250 250 200 125 50 50 350 350 200 200 - 25 - 50

11)(I 25(I 25(I 200 125 75 75 350 35(I 21)(I 225 - 50 - 8(I

where r~ is the reference value of profit for scenario k. The reference values r k are obtained from the D M during the interactive process. They represent the wishes of the D M regarding the objective function values. The scalar objective function to be minimized is

y-eEL

(k=l ..... K),

(22)

k

where e is a small multiplier (e.g. e < 0.001), to guarantee nondominated optimal solutions [23]. In our example both periods are divided into several subperiods. Period one is four years long and period two six years long. All the monetary values for costs and revenues used represent real

values in US dollars. Tables 1 to 4 contain the data for our illustration. In Table 1 the first column provides an index i for the parts, followed by complexity coefficients g,, the maximum and minimum production boundaries for each part in period o n e V/0mi n and V,0....... machining and overhead times for each part and machine T,~ and tij, batch change times Tb, and the number of tools needed in production Li. Table 2 provides data for time and cost constraints as follows: the coefficients db, dg, d s and dpL and the constants S and PL for the technical nonavailability equation (7); the maximum limit ~kMAX for the machine time equation (8); the coefficients Sg, s, s,,, s e and s~ for the software cost equation (15), the coefficients pg, Pb and Pv

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Table 5 Utopian values (in 1000 $)

Table 6 Reference points (in 1000 $) for initial scanning

Scenario k

Utopian value for fk

1 2 3 4 5

2,538 1,928 2,047 2,537 2,737

for the parts pallet cost equation (14), the coefficients ej and the constants Mj for the machine cost equation (12), the coefficients qg and qj for the tool cost equation (13) and the coefficient CpL for the other cost equation (16). Table 3 provides the product prices p~ for different scenarios and the variable material costs vc~. The product prices are identical for the first period and scenario one. We assume that in scenario one the market situation is stable. In

Ref. value

Reference point l

2

3

4

rI r. r~ r~ r~

2538 1928 21147 2537 2737

2738 1928 2047 2537 2737

2538 212~ 2247 2537 2737

2538 1928 2047 2737 2937

scenarios two and three the sales forecasts show that the prices will decrease, except for products 11, 12 and 13. In scenarios four and five the economy is booming and the product prices will go up, except for the product prices I1, 12, and 13. The sales change factors chik ~or (14) are given in Table 4. They are based on the DM's subjective estimates about the competition in the field and they determine how much the sales of prod-

Table 7 The results of the interactive decision process: criterion values J~, machine times the first period

Tik

and Tk and production volumes b,l ,~ t,i for

Reference point I

2

3

4

5

Profit (1000 $) f~ f2 f3 f4 f5

2513 1898 2018 2507 2707

2538 1910 2029 2486 267t)

2495 1928 2047 2396 2584

2529 1879 1998 2537 2737

2513 i 897 2018 25(77 27118

Machine times (1000 rain/year) T~k T2k Z~k T4k Tk

317 224 242 208 1093

317 230 251 218 1118

317 229 246 222 1116

317 224 245 208 1097

~,17 224 242 208 !093

First period production volumes (part/year) part 1 part 2 part 3 part 4 part 5 part 6 part 7 part 8 part 9 part 10 part 11 part 12 part 13

500 3000 1500 1500 1000 360 231 3000 3000 2800 253 150 300

50/) 2191 15011 15111t 1000 360 360 3000 3000 2800 360 150 300

500 2000 151117 1500 1000 360 200 3000 3000 2800 360 293 300

500 2803 15{)1/ 1500 1000 36/1 360 3000 3000 280[) 200 150 300

500 31100 1500 1500 10IX) 360 232 3000 3000 2800 252 t50 3170

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M. Kuula / Risk management model

the situation where scenarios 2 and 3 are more likely than the other scenarios and reference point 4 refers to the situation where scenarios 4 and 5 are more likely than the others. The results of these analyses are collected in Table 7, where the objective function values for each scenario, the work loads for each machine and for the whole production line, and production volumes for each part i are given. Suppose that after this preliminary analysis, the DM thinks that scenarios 1, 3 and 4 are more likely than scenario 2 and 5. Therefore, he chooses next the following reference point number 5: r k = (2638, 1928, 2147, 2637, 2737). The solution corresponding to reference point 5 is also presented in Table 7. Suppose that the DM is satisfied with the solution related to reference point 5. Its production program for the first period is in Table 7, and for the second period the production programs together with machine times corresponding to the five scenarios are shown in Table 8. It should be clear, that only the production plan for the first period will be implemented. The role of

uct i can increase during the second period compared to the first period. Next we demonstrate the interactive process using I F P S / P L U S and I F P S / O p t i m u m packages. First we solve the problem five times to derive the utopian values for each scenario (Table 5). The DM is in a better position to compare solutions with each other, when he has information about the best possible outcomes (utopian values) for each scenario. The DM may also use the utopian values when selecting the new reference points. Suppose that the DM first wishes to get an overall picture of the effects of the different scenarios on the production program for the first period. For that purpose, he solves the optimization model four times by using the reference values given in Table 6. These reference points may be interpreted as follows: The first reference point corresponds to the situation where the DM believes that each scenario is equally likely. The second reference point corresponds to the situation where the first scenario is more likely than the others. Similarly, reference point 3 refers to

Table 8 Second-period solution corresponding to reference point 5 Scenario 1

2

3

4

5

317 246 265 229 1162

317 227 240 209 1097

317 227 241 208 1098

317 240 259 218 1140

317 218 237 185 1062

600 3250 1750 1700 63 385 257 3350 3350 2800 253 150 300

575 3200 1650 1650 729 285 0 3200 3200 3000 227 225 350

585 3200 1675 1675 658 335 0 3250 3250 3000 242 200 320

600 3250 1750 1700 238 410 282 3350 3350 3000 452 0 250

600 3250 1750 1700 769 435 307 335(/ 3350 3000 0 0 220

Machine time (1000 min/year) T1k

T2k T~k T4k T~

Production volumes (parts/year) part 1 part 2 part 3 part 4 part 5 part 6 part 7 part 8 part 9 part 10 part 11 part 12 part 13

107

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(.MRs & FOF

the second period in the model is only to account for future consequences of the "here-and-now" decision [16].

5. Conclusions In this paper we introduced a user-friendly interactive decision support system for FMS selection problems under risk. First we presented the mathematical model formulation for our example configuration and then illustrated how the D M can use our support system to explore various tradeoffs between different scenarios. This way, the D M can get a better understanding of the risks involved in the FMS selection problem.

References [1] J.A. Buzacott and D.D. Yao, "Flexible manufacturing systems: A review of analytical models", Manage. Sci., Vol. 32, 1986, pp. 890-905. [2] R. Jaikumar, "Postindustrial manufacturing", Hart,. Bus. Ret,., Vol. 6, 1986, pp. 69-79. [3] J. Ranta, K.Koskinen and M. Ollus, "Flexible production automation and computer integrated manufacturing: Recent trends in Finland", Computers in Industry, Vol. 11, No. 1, 1988, pp. 55-76. [4] Y.P. Gupta and S. Goyal, "Flexibility of manufacturing systems: Concepts and measures", Eur. Z Oper. Res.. Vol. 43, 1989, pp. 119-135. [5] J. Ranta and A. Alabian, "Interactive analysis of FMS productivity and flexibility", Working Paper WP-88-ll98, International Institute for Applied Systems Analysis, Laxenburg, Austria, 1988. [6] J. Ranta, "FMS investment as a multiobjective decisionmaking problem", Unpublished Manuscript, International Institute for Applied Systems Analysis, Laxenburg, Austria, 1989. [7] M. Kuula and A. Stare, "A nonlinear multicriteria model for strategic FMS selection decisions", WP-89-62, International Institute for Applied Systems Analysis, Laxenburg, Austria, 1989. [8] A. Stam and M. Kuula, "Selecting a flexible manufacturing system using multiple criteria analysis", Int. J. Prod. Res., Vol. 29, No. 4, 1991, pp. 803-820. [9] T.L. Saaty, The Analytic Hierarchy Process, McGraw-Hilt, New York, 1980. [10] T.L. Saaty, Decision Making for Leaders--The Analytic Hierarchy Process for Decision in Complex World, University of Pittsburgh Press, Pittsburgh, PA, 1987. [11] IFPS/Optimum Users Manual, Release 4.0, Execucom Systems Corporation, Austin, TX, 1989. [12] IFPS/Plus Users Manual, Release 4.0, Execucom Systems Corporation, Austin, TX, 1989.

Computers in lndm~tO, [13] I. Sabuncuoglu and D.L. Hommertzheim, "'Expert simulation systems--Recent developments and applications in flexible manufacturing systems', Compur Ind. Eng., Vol. 16, No. 4, 1989, pp. 575-585. [14] G.W. Hintz and H.-J. Zimmermann, "'A method to control flexible manufacturing systems", Eur..1. Oper. Res.. Vol. 41, 1989, pp. 321-334 [15] GJ. Miltenburg and I. Krinsky, 'Evaluating flexible manufacturing systems", liE Trans., Vol. 19, No. 2, 1987, pp. 222-233. [16] G.B. Dantzig, Linear Programming and Extensions, Princeton University Press, Princeton, NJ, 1963. [17] M. Kallio, "On applied systems analysis", Finn. J. Bus. Econ., Vol. 2, 1981, pp. t57-167 (in Finnish),. [18] G. Klein, H. Moskowitz and A. Ravinran, "Interactive multiobjective optimization under uncertainty", Manage. Sck, Vol. 36, 1990, pp. 58--75. [19] R. Kuivanen, J. Lepist6 and R. Tiusanen, "'Disturbances in flexible manufacturing", 3rd IFAC / IFIP / lEA / IFORS Conf on Man-Machine Systems, June 14-16, 1988, Oulu, Finland. [20] T. Lakso, "The influence of FMS-technology on the efficiency of NC-controlled machine tools". Publication #50, Tampere University of Technology, Tampere, Finland, 1988. [21] L. Norros, K. Toikka and R. Hy6tytiiinen, "FMS design from the point of view of implementation results of a case study", 3rd I F A C / I F I P / I E A / I F O R S Conf on Man-Machine Systems, June 14-16, 1988, Oulu, Finland. [22] M. Kallio, A. Lewandowski and W. Orhard-Hays, "An implementation of the reference point approach for multiobjective optimization", in: G.B. Dantzig, A.H. Dempstcr and M.J. Kaltio (eds.), Large-Scale Linear Programruing, (Proc. IIASA Workshop, 2-6 June 1980), IIASA Collaborative Proceedings Series, C'P-81-SI, Volume 2, International Institute for Applied Systems Analysis. Laxenburg, Austria, 1981. [23] D. Bell, R. Keeney and H. Raiffa (eds.), Conflicting Objectives in Decisions, IIASA International Series on Applied Systems Analysis, Wiley, New York, 1977. [24] M. Starr and M. Zeleny (eds.), Multiple Criteria Decision Making, TIMS Studies in the Management Sciences, Vol. 6, 1977. [25] A.P. Wierzbicki, "'A methodological guide to muttiobjecrive decision making", WP-79-122. International Institute for Applied Systems Analysis, Laxenburg, Austria, 1979. [26] R.E. Steuer, Multiple Criteria Optimization: Theory, Computation and Application, Wiley, New York. 1986. [27] A.P. Wierzbicki, "The use of reference objectives in multiobjective optimization", in: G. Fandel and 'IF. Gal (eds.), Multiple Criteria Decision Making; 7~'ory and Application, Springer-Verlag, Berlin, 1980, pp. 468-486. [28] A.P. Wierzbicki, "A Mathematical basis for satisficing decision making", Math. Modelling,,. Vol. 3, 1982, pp. 391-405. [29] A. Lewandowski and A.P. Wierzbicki, " Theory, software and testing examples in decision support systems", WP88-071, International Institute for Applied Systems Analysis, Laxenburg, Austria, 1988.