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Multiple machine replacement analysis
~~g~neer~ng Costs and Preduciien Ece~omics, 20
265
( 1990) 265-275
Elsevier
MULTIPLE MACHINE
REPLACEMENT ANALYSIS
Lawrence C. Leung Department
of industrial and Management
Systems, Universiry of South Florida, Tampa, FL 33620,
and J.M.A.
U.S.A.
Tanchoco
School of industrial Engineering, furdue Un~vers~ry, West Lafayette, IN 47907,
U.S.A.
ABSTRACT
Historically, the study of economic equipment replacement is primarily limited to that of a single machining system. The replacement situation whereby the machines under consideration are part ofa large integrated system has received little attention. There are two major dif~cult~es in analyzing such a problem: the in-
teractive nature of an integrated system and the combinatorial nature of replacement alternatives. This paper attempts to first view the produ~tion system as a network of centers, and then approach the multiple equipment replacement decision as a &on~guration selection problem. An elaborate example is included.
1. INTRODUCTION
to a multi-period model. The model is applicable to replacement situations with multiple defenders and multiple challengers, within an integrated system and over a finite planning horizon. Essentially, the integrated system is constructed as a network of centers through which multiple commodities flow. The replacement problem is viewed as a configuration selection problem which assesses ‘if and when’ a certain piece or pieces of equipment should be installed in a given configuration. This work also attempts to integrate vital factors in capital investment, production, and product market within the multi-machine replacement environment. The proposed model structure is a multi-stage problem with a set of multi-commodity flow subproblems at each stage. The subproblems are formulated as linear programs which are then nested into the multi-stage problem formulated as a dynamic program.
Traditional economic equipment replacement research involves the evaluation of either a single machine or a single machining system. Works on this topic include Alchian [ I 1, Terborgh [ 2 1, Bellman [ 3 1, Dreyfus [4 1, Smith [5], Eilon et al. [6], Sethi and Morton [7], Sethi and Chand [ 8 1, Chand and Sethi [ 9 1, Oakford et al. [ IO], Bean et al. [ Ill, Leung and Tanchoco [ 121 and Tanchoco and Leung [ 13 1. While these works help depict the economic nature of replacement decisions, their single-machine environment remains an unrealistic one in today’s complex production environment. Several authors have addressed the issue of machine replacement within an integrated system: Hannssmann [ 14 ] , Ray [ 15 ] and Leung and Tanchoco [ 16 1. Their works, however, omitted many of the important factors in the multiple machine environment. This paper extends the single-period model in [ 161 0167-188X/90/$03.50
0 1990-Elsevier
Science Publishers B.V.
266 2. ISSUES ON MULTIPLE REPLACEMENT
MACHINE
The manufacturing environment for this replacement problem is one of multi-machine producing multi-product; each individual machine is capable of performing many different operations, and each part may have alternate routes through the system. The machines are assumed to utilize multiple inputs, and their consumption rates per machine are known; the input consumption rate varies proportionately with the output level. The sequence of operations per part type is predetermined, and the price-volume relationship of the part is described by price breaks. There are two major categories of issues regarding multiple machine replacement within this environment. The first is with respect to the effects on the entire system as one or more of its components are altered. It is related to the interactive nature of a system and involves considerations such as production level, product mix, inputs requirement, etc. The second is with respect to the possibilities of machinemix arising from various replacement alternatives. It is related to the combinatorial nature of multiple machine replacement and addresses replacement possibilities ranging from one-for-one to several-for-several. Interactions
and parts assignment
The interactive issues have been examined in [ 161. For completeness, the key proposition thereof is stated in brief as follows. It is proposed that the desirability of a local replacement situation needs to be examined at the system level, thereby incorporating all possible interactions; that an optimal assignment of parts to machines within the system implicitly accounts for all the interactive factors; and that the optimal parts assignment process should be based on operating profit maximization. The part assignment model is shown in Table 1 with its variables defined in Table la.
TABLE 1 The assignment model
=
Qo C Y,,,,, .T
and all Y, > 0
TABLE la Definitions of variables for the assignment model quantity of part p which flows from machine s to machine m for the k-th operation of the part. distance travelled from the s-th machine to the mth machine. utilization rate of the material handling system. cost per unit distance travelled. processing time required by the k-th operation of part p on machine m. cost per unit machine time for the m-th machine. operation set required by part p. machine set which can perform operation k. capacity of the material handling system. capacity of the m-th machine. amount available for the I-th input. inputs per unit machine time to the m-th machine. unit cost of the I-th input. price-volume relationship of part p, represented as price-breaks. is the output level of part p. denotes n( Q,), the unit selling price of part p for output level Q, k= l,..., k, m= l,..., 51, p= l,..., fi, I= l,..., 7:
267 In essence, this model consists of a series of multi-commodity flow problems with each representing a price-break situation. And for each price-break, the maximum operating profit is determined while subject to flow logic and inputs restrictions including the material handling capacity. The overall maximum operating profit across price-breaks would indicate the corresponding optimal production volume per part type as well as the optimal parts assignment. The optimal operating profit is then adjusted according to all the capital costs of the system in order to determine the system’s net profit. The optimal annual net profit of a system scenario as derived in [ 161 is expressed as: Net profit = optimal operating profit-tax - oppo~unity
payment
cost
where OP is the optimal operating profit of the system obtained from the parts assignment model, M is the total number of machines in the system, Di is the depreciation value of the i;th machine for the year (i= 1,._,M), p is the tax rate, M”,is the bookvalue of the i-th machine at the beginning of the year (i= 1,...,M), and Yis the opportunity cost interest rate. Possible system configurations planning horizon
over the
Since a replacement situation can give rise to many system scenarios, the above net profit determination would have to be repeated for all scenarios. And since a replacement scheme is but a sequence of system scenarios, the economic desirability of such a scheme can be evaluated according to the discounted values of the system scenarios within the scheme. Here, we attempt to describe all possible systems scenarios - given a set of challengers and
defenders - for each year in the entire planning horizon. We assume that there is only one layout for a machine-mix of unique vintages. While it is possible that a machine-mix can be laid out in many different ways and that the optimal layout problem can in fact be incorporated within the replacement problem, this however would require all layout combinations per machine-mix for all the machinemixes within the entire planning horizon to be considered. That, we believe, is a research topic in its own right and is left for future research. For the remainder of this paper, we will define a machine-mix of unique vintages as a configuration whose layout is unique. We also assume that (i) the total number of machines in the manufacturing system cannot exceed the original number of machines in the system, (ii > once a challenger is introduced into the system it will not be considered further as a challenger, (iii) when a new machine is acquired it will not become a replacement candidate, (iv) all machines to be replaced are known and are ready to be replaced at the beginning of the planning horizon, and (v) each of the challengers is capable of replacing each of the defenders as well as all of the defenders. At the beginning of the planning horizon and each subsequent year, a decision is to be made on which system configuration to select. This keep-or-replace process throughout the horizon forms a tree structure, with each node representing a system con~guration and each linking arc indicating a precedence reiationship between configurations, i.e., certain configurations can only precede or be followed by a specific set of configurations. An expression for the total number of possible system configurations for a single period was given in [ 161:
where n is the total number of challengers and
268 d is the total number of defenders. Since this expression is only applicable to a single-period situation, i.e., it makes no distinction between con~gurations of the same machine-mix but of different vintage type, modification is required for the multi-period case. Let n, be the accumulated total number of challengers through year t, i.e., n,=tn,; the simple multiple is due to the assumption that the challengers each year are of the same type as those in the beginning of the planning horizon. The purpose of defining an accumulated total is to separately account for the configurations which consist of challengers of the same type but of different vintage. For now define TC: as follows:
(2b) (status quo), n,< d
This expression, however, is still incorrect for it includes certain con~gurations which consist of challengers of the same type (although of different vintage). This violates the premise that once a challenger type is introduced into the system the same type will not be further considered through the remainder of the horizon. The total number of such challenger combinations is n, C j&k for both n1> d and n, -c d TABLE 2 Machine in the horizon and their identification Machine description
Year of make
Defender machine A Defender machine B Existing machine C Existing machine D
-
2 3 4 5
Challenger Challenger Challenger Challenger Challenger Challenger Challenger Challenger
1 1 2 2 3 3 4 4
6 7 8 9 10 11 12 13
X Y X Y X Y X Y
Label as M/C
and will be adjusted in the total configuration computation. TCt, the total number of configurations (nodes in the replacement tree) in year t, can be written as: 6. I TC,=TC;-n,
c il k=”
(fc)
To illustrate, consider a manufacturing system of four machines A, B, C and D, where only TABLE 3 System configurations
1
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41
per year and their representations
Year 1
Year 2
(2,3,4,5) (63,425) (7,3,4,5) (2,6,4,5) (2,7,4,5) (6,435) (7,435 ) (67,435 1
12,3,4,51 (6.3,4,5) (7,3,4,5) (8,3,4,5) (9,3,4,5) (2,6.4,5)
C&7,45) (SM~ 1 (2,9,4,51 (6,435) (7,4.5 1
(8/G 1 (9,4,5) (f&7,4,5) t&9,4,5) (7,8,4,5) f 8.97435)
Year 3
(2,3,4S) f&3,4,5) (7,3,4,5) (8,3,4,5) (9,3,4,5) (10,3,4,5) (11,3,4,5) (2,6.4,5 1
P-,7,4,5) 1X8,4,51 f&9,4,5 1 (2,10,4,5) (2,11,4,5) (6,4,5) (7,4.5) (8,4,5) (9,4,5) (10,4,5)
(11,4,5) (t&7,4,5) (6,9,4.5) (6,11,4,5) (7,8,4>5 1 (7,10,4,5) (x,9,4,5 1 (8,11,4,5) (9,10,4,5) (10,L1,4,5)
Year 4
(2,X4,5 1 (6,X4,5) (7,3.4,5) (8,374s) (9,3.4,5) ( 10,3,4,5 1 (I 1,3.4,5) ( 12.3,4,5) ( 13,3,4.5) (w4.5) (2,7,4,5) (2,8,4,5)
12,9,451 (2.10,4,5) (23 1,4,5) (2,12,4S) (2,13,4,5) (6,455) (7,4,5) (8345) (9,4.5) (10,4,5) (11,4,5)
(12,4,5) ( 13,4,5 1 (6,7,4S 1 (6,9,4S 1 (6,11,4,5) (6.13.4,s) (7,8.4,5) (7,10,4,5) (7,12,4,5) (8,9,4,5) (8,11,4,5) (8,13,4,5) (9,10,4,5) (9,12,4,5) (10,11,4,5) (l&13,4,5) (11,12,4.5) (12,13,4,5)
269 A and B are under consideration for replacement, and there are two challenger types: X and Y. Now let the planning horizon be four years. The machines which will be involved in the analysis are shown in Table 2; identification numbers are assigned to facilitate the presentation. To calculate the total number of possi-
n/c
A
ble configurations in each year, eqns. (2b) and (2~) are used. There are eight, seventeen, twenty-eight, and forty-one configurations for year one, two, three and four, respectively, (see Table 3). Though not shown here, there exist implicit precedence relationships amongst these configurations. For instance, contiguration five in year two (9,3,4,5) implies that defender A is replaced by challenger X at year two and therefore can only be preceded by configuration one of year one (2,3,4,5 ). However it could precede both configurations five and twenty-seven of year three; the former configuration implies no replacement activity whereas the latter implies replacing defender B by challenger Y at year three.
M/C B
LOAD
UNLOAD
STATION
STATION
c n/cc
Fig. 1.
3. OPTIMAL
SEQUENCE OF SYSTEM CONFIGURATIONS OVER TYEARS
TABLE 4
This section presents a model which would optimally select the sequence of system configurations over the planning horizon; the optimality criterion is the maximization of the after-tax terminal wealth of the production system. The model is a forward recursive Dynamic Programming formulation. The use of Dynamic Programming in single machine replacement analysis is well documented [ 3,4,10,11,13 ] - largely due to the problem’s
The distance matrix for the machine Unloading area
Loading area 68 M/CA 50 M/CB 14 M/CC 110 M/CD 74
M/C A
M/C B
M/C C
M/C D
26 0 92 68 32
62 44 0 104 68
86 68 32 0 92
122 104 68 44 0
TABLE 5 Respective input consumption
rates by machine and unit costs of inputs
Direct labor (min.)
Indirect labor (min.)
Energy 20 kW M/C (kWh)
Maintenance (min.)
M/C A M/C B M/CC M/CD M/C X M/C Y
0.5 0.6 0.55 0.65 0.4 0.3
0.3 1.0 0.7 0.4 0.3 0.25
0.6 0.8 0.7 0.65 0.4 0.25
0.1 0.2 0.15 0.1 0.1 0.05
Unit cost
$25/hr. ($0.4167/min.)
$35/hr. ($0,5833/min.)
$4/hr. ($0.25/min.) ($O.ZO/kW/hr.)
$15/hr. ($0.25/min.)
270 sequential structure. In the case of multiple machine replacement, the problem structure remains the same; the difference lies in defining the state variables (the yearly system configurations) and the recursive relationships (additional precedence conditions). This model will consider the traditional time-dynamic factors such as time-value-of-money, deterioration and obsolescence, changes in machine values, depreciation, and changes in values of both initial inputs and final outputs (products). All assumptions from the prior sections remain valid. In addition it is assumed that there will be no inventory carryover from year to year, and that the sale of the incumbent machines takes place at the beginning of the year.
figuration (i,j,t ), its net profit value can be explicitly determined. That is, NP,, = - wi,t(r)+ ( 1-PI (OF’,, -Di,r)
(3a)
It is obtained as follows (see also eqn. ( 1) ): maximum operating profit OPijf minus aggregate machine depreciation D,, minus opportunity Cost Wijf( Y) minus tax OUtflOW (OP,, - D,,)p. Due to the assumption that machine market value is equal to book value, i.e., D,,= Wu,_ 1- Wijt, the expression is reduced to eqn. (3a). Starting
condition
Let_& be an optimal value function which is TABLE 6
Definitions
(Lit)
wjt wijt
Dijt T
Machine times per operation per part type per machine (in min. )
A triple denoting the i-th machine-mix of the j-th vintage type operating in year t, thus defining a system configuration in year t. Maximum operating profit of the system configuration in year t characterized by (i,j,t). Aggregate machine value at year t for configuration ( i,j, t ). Aggregate depreciation charge at year t for configuration ( i,j, t ) . The planning horizon.
Backward
recursive
dynamic
programming
formulation
The determination of the optimal system configuration sequence can be viewed as a multi-stage problem with sequential decisions. The output of each decision stage is the system configuration which in turn becomes the input to the next decision stage. The decision at each stage is to select a system configuration; the stage return is the net profit, in year T dollars, contributed by the system configuration selected at that stage. Note that for a given con-
Operation No.
Machine L/U
Part One 1 2 3 4 5 6 Part Two 1 2 3 4 5 6 7 Part Three 1 2 3 4 5 6 7
a 9 10
A
B
C
D
X
Y
la.5 -
10.0
0
15 -
12.5 25 16.5 -
12.5 10.0 10.0 14.0 -
-
23.5 _
14.5 10.5 _
0
33.2 15.0 14.5 _
41.9 40.0 20.0 _
40.0 35.0 -
36.8 38.4 15.0 -
40.0 30.0 30.0 30.0 -
35.0 35.0 15.0 15.0 -
34.5 10.0 30.6 20.0 5.0 -
40.0 15.0 10.0 17.0 -
34.8 25.0 9.4 -
30.0 30.0 10.0 15.0 25.0 a.0 15.0 15.0 5.0 -
25.0 25.0 10.0 a.0 20.0
-
36.3 32.0 10.0 0
20.0 la.0 5.8 -
‘I-” = not able to perform.
a
7.0 16.0 12.0 -
a.0 -
271 defined as the optimal future worth (in year T dollars) for system configuration (i,j,t) from year t to year T, given that the optimal replacement strategy is pursued during this period of time. At the first stage, t= T, the optimal value functions for the set of possible system configurations can be expressed as: .L,T= NP,,T
v(~,~,~)~{J*)
(3b)
where {JT} is the set of possible system configuration in year T. Equation (3b) is simply the set profit value at year T, for that is the optimal value at year T.
Recursive relationship
For each stage t where t= 1,2,...T- 1, the optimal value function for configuration (i,j,t) is j;,=NP,,z( 1+~)~-‘+max v(i,j,t)E{JI}
v(i’,j’J+
&+I
,.&,,~,,+I1
(3c)
1)E{J)t+,),
where {Jt} is defined as the set of possible configurations in year t, and {Jt+ ,} is defined as the set of new machine configurations in year t+ 1, resulting from replacement activities on configuration (i, j,t ). Equation ( 3c) states that the optimal value function J;j.t (for t=2,..., T- 1) is simply the net profit value of (i,j,t )
TABLE I Yearly deterioration
and obsolescence rates for respective machines
A. Machine deterioration per machine each year (I) Percentage increase in parts operating time: Machine 2 3 4 5 Year Year Year Year
1. 2. 3. 4.
10 10 10 10
15 15 15 15
10 10 10 10
10 10 10 10
6
7
8
9
10
11
12
13
10 10 10 10
10 10 10 10
10 10 10
10 10 10
10 10
10 10
10
10
(II) Percentage increase in input consumption (for all inputs) Machine 2 3 4
5
6
7
8
9
10
11
12
13
Year Year Year Year
10 10 10 10
10 10 10 10
10 10 10 10
10 10 10
10 10 10
10 10
10 10
10
10
10
11
12
13
5
5 -
5
5
10
11
12
13
5
5 5
5
1. 2. 3. 4.
10 10 10 10
15 15 15 15
10 10 10 10
rates:
B. Machine obsolescence per challenger each year (I) Percentage decrease in parts operating time: Machine 6 7 8 9 Year Year Year Year
1. 2. 3. 4.
5
5 5 -
-
5 -
-
(II) Percentage decrease in input consumption rates: (for all inputs) Machine 6 I 8 9 Year Year Year Year
1. 2. 3. 4.
5
5 5
-
5
272 in year T dollars plus the maximum of the optimal values from possible configurations in year t + 1. Here, f;i,+ 1 represents optimal value of a ‘keep’ decision andf;, ,js,I+ 1 represents each of the ‘replace’ decisions. Stopping conditions
When thef-values at year 1 are determined, max {fjl} Vi,j is then selected. This value is the optimal future wealth at year T when the optimal replacement strategy is pursued at the beginning of the planning horizon. By backtracking, the sequence of configurations which leads to this maximum f-value can be identified. TABLE 8 Upper limits of resources Inputs availability Direct labor = 5 workers/shift x 8 hr./shift x 2 shift/ dayx 5 day/weekx 60 min. x 50 week/ year = 1200 000 min. Indirect labor = 5 workers/shift x 8 hr./shift x 2 shift/ day x 5 day/weekx 60 min. x 50 week/ year = 1200 000 min. Energy = 200 000 kWh weekx 50 weeks = 10 000 000 kWh Maintenance = 5 workers/shift x 8 hr./shift x 1 shift/ dayx 5 dayfweekx 60 min. X 50 week/ year = 600 000 min. Machine capacity (50 week/yearx 5 day/week x 16 hr./ dayx60 min./hr.) Machine A = 240 000 min. Machine B = 240 000 min. Machine C = 240 000 min. Machine D = 240 000 min. Machine X = 240 000 min. Machine Y = 240 000 min. Capacity of the materials handling system AGV = 120 ft./min. x 50 week/yearx 5 day/ (capacity) weeks 16 hr./dayX60 min./hr. = 28 800 000 ft. Cost per foot travelled by AGV = $0. I
From the sequence of configurations, the optimal output schedule can also be obtained. 4. AN ILLUSTRATIVE
EXAMPLE
The previous four-machine system with two challengers is used in this illustrative example. The layout for this system is shown in Fig. 1; the distance between machines is shown in Table 4 and the material handling system is an Automated Guided Vehicle System ( AGVS ) . There are four major inputs to each of the four machines: (a) direct labor, (b) indirect labor, (c ) energy, and (d) maintenance. The respective consumption rates and the unit cost of respective inputs are listed in Table 5. This manufacturing system produces three part types. The machining times for each operation per part type for each incumbent machine and for each challenger are listed in Table 6. The rates TABLE 9 Demand profile for years I, 2,3 and 4 (A ) Part type one Production volume
Selling price/
year
year. year. year. year.
1 2 3 4
1000-1400 200 250 300 350
1401-2400 180 230 280 330
2401-3800 160 210 260 310
(B) Part type two Production volume 1200-1600 Selling price/ year
year. year. year. year.
I 250 2 300 3 350 4 400
1601-2200 225 275 325 375
2201-3000 200 250 300 350
(C) Part type three Production volume
Selling price/ year
year. year. year. year.
1 2 3 4
1500-2000 400 450 500 550
2001-2700 350 400 450 500
2701-3500 300 350 400 450
273 TABLE 10 Machine values over the entire planning horizon (all beginning-of-year Machine
2
3
6
7
Year Year Year Year Year
132 300 84 000 42 000 0 0
189 000 126 000 63 000 0 0
500 000 425 000 315 000 210 000 105 000
600 510 378 252 126
1. 2. 3. 4. 5.
8 000 000 000 000 000
TABLE 11 Year
Configuration
1 2 3 4
5 7 24 31
No.
Content (2,7,4,5) (2,7,4,5) (7,10,4,5) (7,10,4,5)
TABLE 12 Optimal price-volume Part type
Price
combination Volume
Year 1: Configuration (2,7,4,5) 1 180 2400 2 250 1600 3 400 2000 Year 2: Configuration (2,7,4,5) 1 250 1400 2 300 1600 3 450 2000 Year 3: Configuration (7,10,4,5) 1 260 3800 2 350 1600 3 500 2000 Year 4: Configuration (7,10,4,5) 1 330 2400 2 400 1600 3 550 2000
of deterioration and obsolescence for all machine types are listed in Table 7; the effect of machine deterioration is assumed to increase both machining times and input consumption rate. In this example problem, deterioration
550 467 346 231
values) 9
000 500 500 000
660 561 415 277
000 000 800 200
10
11
12
13
605 000 514 000 381 150
726 000 617 100 456 750
665 500 565 675
797 500 677 875
and obsolescence rates are assumed to be uniform. The capacity limits for all resources and the AGV’s unit distance cost are listed in Table 8. The unit cost of indirect labor, direct labor, energy and maintenance are expected to have a yearly increase of 6%, 8%, 7% and 7%, respectively; for the AGV unit cost, the yearly increase is 10%. The price-volume relationships of the three part types for each of the next four years are shown in Table 9; the product market is assumed to be stable with steady increase in selling prices. All machine values over the entire planning horizon are shown in Table 10; the 1982 ACRS schedule is used for depreciation values. Lastly, the amount of resources are assumed identical each year and a time-valueof-money of 10% is assumed. The optimal
replacement
sequence
The maximum optimal value function at the beginning of the planning horizon is found to be $2,789,991 which belongs to configuration 5 of that year. Tracing the configurations sequence which leads to this maximumf-values, the resulting replacement policy is as stated in Table 11. That is, in the beginning of year one, defender B (machine 3 ) should be replaced by challenger Y (machine 7 ). This system configuration would be used for two years. Then, in the beginning of year three, replace defender A (machine 2 ) by challenger X (machine 10); this system configuration would be used till the end of the planning horizon. Lastly, the set of optimal price-volume relationships over the
274 TABLE 13 Part assignment for optimal replacement sequence Operation
Machine label
Year 1 L/UL
Year 2 2
7
4
Part typeI 0 2400 1 2 3 4 5 6 2400
2400 0 -
2400 0 -1902498 1902 1902 0 -
Parttype2 0 1600
-
-
I
-
- 1600
2 3 4 S 6 I
1600
0 1600 0 67 _-0 0 -
Part t.vpe 3 0 2000 1 2 3 4 5 6 7 8 9 10 2000
1581 419 1581 419 - 2000 02000 -0 -0 2000 2000 2000 -
5
Year 3
L/UL
2
I
4
498 498 -
1400 1400
1400 0 -
1400 1322 1322 -
0 1400 0 -
-
-
1600
-
-
-
-
1600
-
-
-
-
-
-
I533 0 0 -
1600 1600 1600 -
1600
1600 0 0 -
1600 1600 1600 -
-
- 1600 0 0 0 -
1600 0 0 -
- 2000 -1533467 0 0 1533 467 0 - 2000 0 2000 - 2000 0 0 2000 0 2000 0 2000 2000 - 2000 -
5
L/UL
- 3800 0 78 78 -3800
Year 4 IO
7
4
5
L/UL
10
3800 0 0 0 -
0 1382 1382 -
3800 3800 0 -
0 2418 2418 -
2400 2400
2400 0 0 563 -
-
1600 1600 0 0 -
0 0 0 0 -
1600 535 535 -
1065 1065 1600 -
2000 53 53 53 53 53 -2000 2000 2000 2000 2000 -
1947 1947 1947 1947 1947 0 -
0 0 0 0 -
0 0 0 -
-
-
-
1600
1600
0
-
-
-
1600
1600 0 0 -
0 0 0 -
1600 616 616 -
984 984 1600 -
1600
- 2000 0 0 0 -20u0 0 2000 0 - 2000
186 186 186 186 186 2000 2000 2000 2000 -
1814 I814 1814 1814 1814 0 -
0 0 0 0 -
0 0 0 -
entire horizon is shown in Table 12, and their corresponding part assignment is shown in Table 13. 5. CONCLUSION A multi-machine replacement model has been presented. The analysis is oriented towards the many issues relevant to the planning of capital equipment within an integrated system. The model also integrates vital elements on capital investment, production, and product market into the multi-machine replacement environment. Based on this model, sensitivity analysis can be performed on the
7
4
S
0 2400 - 2400 563 0 0 -
0 1837 1837 -
critical parameters. For example, one can examine the equipment acquisition situation with different product demand profiles or even with the introduction of new products. Similarly, the effects of changes in input prices on the system design can also be addressed. Implicitly, this replacement model not only can approach the flexibility of an equipment within a system, but also the flexibility of the entire system with respect to the demand placed on it both internally and externally. Further, the determination of the system’s input-output structure directly defines the capital intensiveness of the production system. While this model is believed to be robust, work is far from complete.
275
The dimensionality of the problem size could present problems in the solution procedure, hence certain bounds or heuristic rules ought to be developed to facilitate the computation process. Also, the linear input-output structure remains a restrictive assumption and extension to nonlinear structures should be studied.
8
9
10
11
12
REFERENCES Alchian, A.A., 1952. Economic replacement policy. Publ. R-224, Santa Monica, CA, The Rand Corporation. Terborgh, G., 1949. Dynamic Equipment Policy, McGraw-Hill, New York. Bellman, R., 1955. Equipment replacement policy. SIAM J. Appl. Math., 3(3): 133-136. Dreyfus, Stuart E., 1957. A generalized equipment replacement study. Santa Monica, CA, Rand Corporation, March, p. 1039. Smith, V.L., 1966. Investment and Production. Harvard University Press, Cambridge, MA. Eilon, S., Ring, J.R. and Hutchison, D.E., 1966. A study of equipment replacement. Oper. Res. Q., 17( 1): 5971. Sethi, S. and Morton, T., 1972. A mixed optimization technique for the generalized machine replacement problem. Nav. Res. Logis. Q., 19( 3): 471-481.
13
14 15
16
17
Sethi, S. and Chand, C., 1979. Planning horizon procedures for machine replacement models. Manage. Sci., 25(2): 140-151. Chand, C. and Sethi, S., 1982. Planning horizon procedure for machine replacement models with several possible alternatives. Nav. Res. Logis. Q., 29(3): 483-493. Oakford, R.V., Lohmann, J.R. and Salazar, A.S., 1989. A dynamic replacement economy decision model. IIE Trans., 16( 1): 65-72. Bean, J.C., Lohmann, J.R. and Smith, R.L., 1985. A dynamic infinite horizon replacement economy decision model. Eng. Economist, 30(2): 99-120. Leung, L.C. and Tanchoco, J.M.A., 1984. After-tax equipment replacement analysis using discrete transform. IIETrans., 16(2): 152-158. Tanchoco, J.M.A. and Leung, L.C., 1987. An input-output model for equipment replacement decisions. Eng. Costs Prod. Econ., 11(2): 69-78. Hannssmann, F.S., 1968. Operations Research Techniques for Capital Investment. Wiley, New York. Ray, Thomas R., 197 1.A System Approach to Replacement. Ph.D. Thesis, Dept. Ind. Eng. Oper. Res., Virginia Polytechnic Institute and State University. Leung, L.C. and Tanchoco, J.M.A., 1987. Multiple equipment replacement within an integrated system framework. Eng. Economist, 32 (2): 89-114. Verheyen, P.A., 1978. Economic interpretation of models of the replacement of machines. Eur. J. Oper. Res., 3: 150-156.
(Received February 29, January 23, 1990)
1988;
accepred
in revised form
265
( 1990) 265-275
Elsevier
MULTIPLE MACHINE
REPLACEMENT ANALYSIS
Lawrence C. Leung Department
of industrial and Management
Systems, Universiry of South Florida, Tampa, FL 33620,
and J.M.A.
U.S.A.
Tanchoco
School of industrial Engineering, furdue Un~vers~ry, West Lafayette, IN 47907,
U.S.A.
ABSTRACT
Historically, the study of economic equipment replacement is primarily limited to that of a single machining system. The replacement situation whereby the machines under consideration are part ofa large integrated system has received little attention. There are two major dif~cult~es in analyzing such a problem: the in-
teractive nature of an integrated system and the combinatorial nature of replacement alternatives. This paper attempts to first view the produ~tion system as a network of centers, and then approach the multiple equipment replacement decision as a &on~guration selection problem. An elaborate example is included.
1. INTRODUCTION
to a multi-period model. The model is applicable to replacement situations with multiple defenders and multiple challengers, within an integrated system and over a finite planning horizon. Essentially, the integrated system is constructed as a network of centers through which multiple commodities flow. The replacement problem is viewed as a configuration selection problem which assesses ‘if and when’ a certain piece or pieces of equipment should be installed in a given configuration. This work also attempts to integrate vital factors in capital investment, production, and product market within the multi-machine replacement environment. The proposed model structure is a multi-stage problem with a set of multi-commodity flow subproblems at each stage. The subproblems are formulated as linear programs which are then nested into the multi-stage problem formulated as a dynamic program.
Traditional economic equipment replacement research involves the evaluation of either a single machine or a single machining system. Works on this topic include Alchian [ I 1, Terborgh [ 2 1, Bellman [ 3 1, Dreyfus [4 1, Smith [5], Eilon et al. [6], Sethi and Morton [7], Sethi and Chand [ 8 1, Chand and Sethi [ 9 1, Oakford et al. [ IO], Bean et al. [ Ill, Leung and Tanchoco [ 121 and Tanchoco and Leung [ 13 1. While these works help depict the economic nature of replacement decisions, their single-machine environment remains an unrealistic one in today’s complex production environment. Several authors have addressed the issue of machine replacement within an integrated system: Hannssmann [ 14 ] , Ray [ 15 ] and Leung and Tanchoco [ 16 1. Their works, however, omitted many of the important factors in the multiple machine environment. This paper extends the single-period model in [ 161 0167-188X/90/$03.50
0 1990-Elsevier
Science Publishers B.V.
266 2. ISSUES ON MULTIPLE REPLACEMENT
MACHINE
The manufacturing environment for this replacement problem is one of multi-machine producing multi-product; each individual machine is capable of performing many different operations, and each part may have alternate routes through the system. The machines are assumed to utilize multiple inputs, and their consumption rates per machine are known; the input consumption rate varies proportionately with the output level. The sequence of operations per part type is predetermined, and the price-volume relationship of the part is described by price breaks. There are two major categories of issues regarding multiple machine replacement within this environment. The first is with respect to the effects on the entire system as one or more of its components are altered. It is related to the interactive nature of a system and involves considerations such as production level, product mix, inputs requirement, etc. The second is with respect to the possibilities of machinemix arising from various replacement alternatives. It is related to the combinatorial nature of multiple machine replacement and addresses replacement possibilities ranging from one-for-one to several-for-several. Interactions
and parts assignment
The interactive issues have been examined in [ 161. For completeness, the key proposition thereof is stated in brief as follows. It is proposed that the desirability of a local replacement situation needs to be examined at the system level, thereby incorporating all possible interactions; that an optimal assignment of parts to machines within the system implicitly accounts for all the interactive factors; and that the optimal parts assignment process should be based on operating profit maximization. The part assignment model is shown in Table 1 with its variables defined in Table la.
TABLE 1 The assignment model
=
Qo C Y,,,,, .T
and all Y, > 0
TABLE la Definitions of variables for the assignment model quantity of part p which flows from machine s to machine m for the k-th operation of the part. distance travelled from the s-th machine to the mth machine. utilization rate of the material handling system. cost per unit distance travelled. processing time required by the k-th operation of part p on machine m. cost per unit machine time for the m-th machine. operation set required by part p. machine set which can perform operation k. capacity of the material handling system. capacity of the m-th machine. amount available for the I-th input. inputs per unit machine time to the m-th machine. unit cost of the I-th input. price-volume relationship of part p, represented as price-breaks. is the output level of part p. denotes n( Q,), the unit selling price of part p for output level Q, k= l,..., k, m= l,..., 51, p= l,..., fi, I= l,..., 7:
267 In essence, this model consists of a series of multi-commodity flow problems with each representing a price-break situation. And for each price-break, the maximum operating profit is determined while subject to flow logic and inputs restrictions including the material handling capacity. The overall maximum operating profit across price-breaks would indicate the corresponding optimal production volume per part type as well as the optimal parts assignment. The optimal operating profit is then adjusted according to all the capital costs of the system in order to determine the system’s net profit. The optimal annual net profit of a system scenario as derived in [ 161 is expressed as: Net profit = optimal operating profit-tax - oppo~unity
payment
cost
where OP is the optimal operating profit of the system obtained from the parts assignment model, M is the total number of machines in the system, Di is the depreciation value of the i;th machine for the year (i= 1,._,M), p is the tax rate, M”,is the bookvalue of the i-th machine at the beginning of the year (i= 1,...,M), and Yis the opportunity cost interest rate. Possible system configurations planning horizon
over the
Since a replacement situation can give rise to many system scenarios, the above net profit determination would have to be repeated for all scenarios. And since a replacement scheme is but a sequence of system scenarios, the economic desirability of such a scheme can be evaluated according to the discounted values of the system scenarios within the scheme. Here, we attempt to describe all possible systems scenarios - given a set of challengers and
defenders - for each year in the entire planning horizon. We assume that there is only one layout for a machine-mix of unique vintages. While it is possible that a machine-mix can be laid out in many different ways and that the optimal layout problem can in fact be incorporated within the replacement problem, this however would require all layout combinations per machine-mix for all the machinemixes within the entire planning horizon to be considered. That, we believe, is a research topic in its own right and is left for future research. For the remainder of this paper, we will define a machine-mix of unique vintages as a configuration whose layout is unique. We also assume that (i) the total number of machines in the manufacturing system cannot exceed the original number of machines in the system, (ii > once a challenger is introduced into the system it will not be considered further as a challenger, (iii) when a new machine is acquired it will not become a replacement candidate, (iv) all machines to be replaced are known and are ready to be replaced at the beginning of the planning horizon, and (v) each of the challengers is capable of replacing each of the defenders as well as all of the defenders. At the beginning of the planning horizon and each subsequent year, a decision is to be made on which system configuration to select. This keep-or-replace process throughout the horizon forms a tree structure, with each node representing a system con~guration and each linking arc indicating a precedence reiationship between configurations, i.e., certain configurations can only precede or be followed by a specific set of configurations. An expression for the total number of possible system configurations for a single period was given in [ 161:
where n is the total number of challengers and
268 d is the total number of defenders. Since this expression is only applicable to a single-period situation, i.e., it makes no distinction between con~gurations of the same machine-mix but of different vintage type, modification is required for the multi-period case. Let n, be the accumulated total number of challengers through year t, i.e., n,=tn,; the simple multiple is due to the assumption that the challengers each year are of the same type as those in the beginning of the planning horizon. The purpose of defining an accumulated total is to separately account for the configurations which consist of challengers of the same type but of different vintage. For now define TC: as follows:
(2b) (status quo), n,< d
This expression, however, is still incorrect for it includes certain con~gurations which consist of challengers of the same type (although of different vintage). This violates the premise that once a challenger type is introduced into the system the same type will not be further considered through the remainder of the horizon. The total number of such challenger combinations is n, C j&k for both n1> d and n, -c d TABLE 2 Machine in the horizon and their identification Machine description
Year of make
Defender machine A Defender machine B Existing machine C Existing machine D
-
2 3 4 5
Challenger Challenger Challenger Challenger Challenger Challenger Challenger Challenger
1 1 2 2 3 3 4 4
6 7 8 9 10 11 12 13
X Y X Y X Y X Y
Label as M/C
and will be adjusted in the total configuration computation. TCt, the total number of configurations (nodes in the replacement tree) in year t, can be written as: 6. I TC,=TC;-n,
c il k=”
(fc)
To illustrate, consider a manufacturing system of four machines A, B, C and D, where only TABLE 3 System configurations
1
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41
per year and their representations
Year 1
Year 2
(2,3,4,5) (63,425) (7,3,4,5) (2,6,4,5) (2,7,4,5) (6,435) (7,435 ) (67,435 1
12,3,4,51 (6.3,4,5) (7,3,4,5) (8,3,4,5) (9,3,4,5) (2,6.4,5)
C&7,45) (SM~ 1 (2,9,4,51 (6,435) (7,4.5 1
(8/G 1 (9,4,5) (f&7,4,5) t&9,4,5) (7,8,4,5) f 8.97435)
Year 3
(2,3,4S) f&3,4,5) (7,3,4,5) (8,3,4,5) (9,3,4,5) (10,3,4,5) (11,3,4,5) (2,6.4,5 1
P-,7,4,5) 1X8,4,51 f&9,4,5 1 (2,10,4,5) (2,11,4,5) (6,4,5) (7,4.5) (8,4,5) (9,4,5) (10,4,5)
(11,4,5) (t&7,4,5) (6,9,4.5) (6,11,4,5) (7,8,4>5 1 (7,10,4,5) (x,9,4,5 1 (8,11,4,5) (9,10,4,5) (10,L1,4,5)
Year 4
(2,X4,5 1 (6,X4,5) (7,3.4,5) (8,374s) (9,3.4,5) ( 10,3,4,5 1 (I 1,3.4,5) ( 12.3,4,5) ( 13,3,4.5) (w4.5) (2,7,4,5) (2,8,4,5)
12,9,451 (2.10,4,5) (23 1,4,5) (2,12,4S) (2,13,4,5) (6,455) (7,4,5) (8345) (9,4.5) (10,4,5) (11,4,5)
(12,4,5) ( 13,4,5 1 (6,7,4S 1 (6,9,4S 1 (6,11,4,5) (6.13.4,s) (7,8.4,5) (7,10,4,5) (7,12,4,5) (8,9,4,5) (8,11,4,5) (8,13,4,5) (9,10,4,5) (9,12,4,5) (10,11,4,5) (l&13,4,5) (11,12,4.5) (12,13,4,5)
269 A and B are under consideration for replacement, and there are two challenger types: X and Y. Now let the planning horizon be four years. The machines which will be involved in the analysis are shown in Table 2; identification numbers are assigned to facilitate the presentation. To calculate the total number of possi-
n/c
A
ble configurations in each year, eqns. (2b) and (2~) are used. There are eight, seventeen, twenty-eight, and forty-one configurations for year one, two, three and four, respectively, (see Table 3). Though not shown here, there exist implicit precedence relationships amongst these configurations. For instance, contiguration five in year two (9,3,4,5) implies that defender A is replaced by challenger X at year two and therefore can only be preceded by configuration one of year one (2,3,4,5 ). However it could precede both configurations five and twenty-seven of year three; the former configuration implies no replacement activity whereas the latter implies replacing defender B by challenger Y at year three.
M/C B
LOAD
UNLOAD
STATION
STATION
c n/cc
Fig. 1.
3. OPTIMAL
SEQUENCE OF SYSTEM CONFIGURATIONS OVER TYEARS
TABLE 4
This section presents a model which would optimally select the sequence of system configurations over the planning horizon; the optimality criterion is the maximization of the after-tax terminal wealth of the production system. The model is a forward recursive Dynamic Programming formulation. The use of Dynamic Programming in single machine replacement analysis is well documented [ 3,4,10,11,13 ] - largely due to the problem’s
The distance matrix for the machine Unloading area
Loading area 68 M/CA 50 M/CB 14 M/CC 110 M/CD 74
M/C A
M/C B
M/C C
M/C D
26 0 92 68 32
62 44 0 104 68
86 68 32 0 92
122 104 68 44 0
TABLE 5 Respective input consumption
rates by machine and unit costs of inputs
Direct labor (min.)
Indirect labor (min.)
Energy 20 kW M/C (kWh)
Maintenance (min.)
M/C A M/C B M/CC M/CD M/C X M/C Y
0.5 0.6 0.55 0.65 0.4 0.3
0.3 1.0 0.7 0.4 0.3 0.25
0.6 0.8 0.7 0.65 0.4 0.25
0.1 0.2 0.15 0.1 0.1 0.05
Unit cost
$25/hr. ($0.4167/min.)
$35/hr. ($0,5833/min.)
$4/hr. ($0.25/min.) ($O.ZO/kW/hr.)
$15/hr. ($0.25/min.)
270 sequential structure. In the case of multiple machine replacement, the problem structure remains the same; the difference lies in defining the state variables (the yearly system configurations) and the recursive relationships (additional precedence conditions). This model will consider the traditional time-dynamic factors such as time-value-of-money, deterioration and obsolescence, changes in machine values, depreciation, and changes in values of both initial inputs and final outputs (products). All assumptions from the prior sections remain valid. In addition it is assumed that there will be no inventory carryover from year to year, and that the sale of the incumbent machines takes place at the beginning of the year.
figuration (i,j,t ), its net profit value can be explicitly determined. That is, NP,, = - wi,t(r)+ ( 1-PI (OF’,, -Di,r)
(3a)
It is obtained as follows (see also eqn. ( 1) ): maximum operating profit OPijf minus aggregate machine depreciation D,, minus opportunity Cost Wijf( Y) minus tax OUtflOW (OP,, - D,,)p. Due to the assumption that machine market value is equal to book value, i.e., D,,= Wu,_ 1- Wijt, the expression is reduced to eqn. (3a). Starting
condition
Let_& be an optimal value function which is TABLE 6
Definitions
(Lit)
wjt wijt
Dijt T
Machine times per operation per part type per machine (in min. )
A triple denoting the i-th machine-mix of the j-th vintage type operating in year t, thus defining a system configuration in year t. Maximum operating profit of the system configuration in year t characterized by (i,j,t). Aggregate machine value at year t for configuration ( i,j, t ). Aggregate depreciation charge at year t for configuration ( i,j, t ) . The planning horizon.
Backward
recursive
dynamic
programming
formulation
The determination of the optimal system configuration sequence can be viewed as a multi-stage problem with sequential decisions. The output of each decision stage is the system configuration which in turn becomes the input to the next decision stage. The decision at each stage is to select a system configuration; the stage return is the net profit, in year T dollars, contributed by the system configuration selected at that stage. Note that for a given con-
Operation No.
Machine L/U
Part One 1 2 3 4 5 6 Part Two 1 2 3 4 5 6 7 Part Three 1 2 3 4 5 6 7
a 9 10
A
B
C
D
X
Y
la.5 -
10.0
0
15 -
12.5 25 16.5 -
12.5 10.0 10.0 14.0 -
-
23.5 _
14.5 10.5 _
0
33.2 15.0 14.5 _
41.9 40.0 20.0 _
40.0 35.0 -
36.8 38.4 15.0 -
40.0 30.0 30.0 30.0 -
35.0 35.0 15.0 15.0 -
34.5 10.0 30.6 20.0 5.0 -
40.0 15.0 10.0 17.0 -
34.8 25.0 9.4 -
30.0 30.0 10.0 15.0 25.0 a.0 15.0 15.0 5.0 -
25.0 25.0 10.0 a.0 20.0
-
36.3 32.0 10.0 0
20.0 la.0 5.8 -
‘I-” = not able to perform.
a
7.0 16.0 12.0 -
a.0 -
271 defined as the optimal future worth (in year T dollars) for system configuration (i,j,t) from year t to year T, given that the optimal replacement strategy is pursued during this period of time. At the first stage, t= T, the optimal value functions for the set of possible system configurations can be expressed as: .L,T= NP,,T
v(~,~,~)~{J*)
(3b)
where {JT} is the set of possible system configuration in year T. Equation (3b) is simply the set profit value at year T, for that is the optimal value at year T.
Recursive relationship
For each stage t where t= 1,2,...T- 1, the optimal value function for configuration (i,j,t) is j;,=NP,,z( 1+~)~-‘+max v(i,j,t)E{JI}
v(i’,j’J+
&+I
,.&,,~,,+I1
(3c)
1)E{J)t+,),
where {Jt} is defined as the set of possible configurations in year t, and {Jt+ ,} is defined as the set of new machine configurations in year t+ 1, resulting from replacement activities on configuration (i, j,t ). Equation ( 3c) states that the optimal value function J;j.t (for t=2,..., T- 1) is simply the net profit value of (i,j,t )
TABLE I Yearly deterioration
and obsolescence rates for respective machines
A. Machine deterioration per machine each year (I) Percentage increase in parts operating time: Machine 2 3 4 5 Year Year Year Year
1. 2. 3. 4.
10 10 10 10
15 15 15 15
10 10 10 10
10 10 10 10
6
7
8
9
10
11
12
13
10 10 10 10
10 10 10 10
10 10 10
10 10 10
10 10
10 10
10
10
(II) Percentage increase in input consumption (for all inputs) Machine 2 3 4
5
6
7
8
9
10
11
12
13
Year Year Year Year
10 10 10 10
10 10 10 10
10 10 10 10
10 10 10
10 10 10
10 10
10 10
10
10
10
11
12
13
5
5 -
5
5
10
11
12
13
5
5 5
5
1. 2. 3. 4.
10 10 10 10
15 15 15 15
10 10 10 10
rates:
B. Machine obsolescence per challenger each year (I) Percentage decrease in parts operating time: Machine 6 7 8 9 Year Year Year Year
1. 2. 3. 4.
5
5 5 -
-
5 -
-
(II) Percentage decrease in input consumption rates: (for all inputs) Machine 6 I 8 9 Year Year Year Year
1. 2. 3. 4.
5
5 5
-
5
272 in year T dollars plus the maximum of the optimal values from possible configurations in year t + 1. Here, f;i,+ 1 represents optimal value of a ‘keep’ decision andf;, ,js,I+ 1 represents each of the ‘replace’ decisions. Stopping conditions
When thef-values at year 1 are determined, max {fjl} Vi,j is then selected. This value is the optimal future wealth at year T when the optimal replacement strategy is pursued at the beginning of the planning horizon. By backtracking, the sequence of configurations which leads to this maximum f-value can be identified. TABLE 8 Upper limits of resources Inputs availability Direct labor = 5 workers/shift x 8 hr./shift x 2 shift/ dayx 5 day/weekx 60 min. x 50 week/ year = 1200 000 min. Indirect labor = 5 workers/shift x 8 hr./shift x 2 shift/ day x 5 day/weekx 60 min. x 50 week/ year = 1200 000 min. Energy = 200 000 kWh weekx 50 weeks = 10 000 000 kWh Maintenance = 5 workers/shift x 8 hr./shift x 1 shift/ dayx 5 dayfweekx 60 min. X 50 week/ year = 600 000 min. Machine capacity (50 week/yearx 5 day/week x 16 hr./ dayx60 min./hr.) Machine A = 240 000 min. Machine B = 240 000 min. Machine C = 240 000 min. Machine D = 240 000 min. Machine X = 240 000 min. Machine Y = 240 000 min. Capacity of the materials handling system AGV = 120 ft./min. x 50 week/yearx 5 day/ (capacity) weeks 16 hr./dayX60 min./hr. = 28 800 000 ft. Cost per foot travelled by AGV = $0. I
From the sequence of configurations, the optimal output schedule can also be obtained. 4. AN ILLUSTRATIVE
EXAMPLE
The previous four-machine system with two challengers is used in this illustrative example. The layout for this system is shown in Fig. 1; the distance between machines is shown in Table 4 and the material handling system is an Automated Guided Vehicle System ( AGVS ) . There are four major inputs to each of the four machines: (a) direct labor, (b) indirect labor, (c ) energy, and (d) maintenance. The respective consumption rates and the unit cost of respective inputs are listed in Table 5. This manufacturing system produces three part types. The machining times for each operation per part type for each incumbent machine and for each challenger are listed in Table 6. The rates TABLE 9 Demand profile for years I, 2,3 and 4 (A ) Part type one Production volume
Selling price/
year
year. year. year. year.
1 2 3 4
1000-1400 200 250 300 350
1401-2400 180 230 280 330
2401-3800 160 210 260 310
(B) Part type two Production volume 1200-1600 Selling price/ year
year. year. year. year.
I 250 2 300 3 350 4 400
1601-2200 225 275 325 375
2201-3000 200 250 300 350
(C) Part type three Production volume
Selling price/ year
year. year. year. year.
1 2 3 4
1500-2000 400 450 500 550
2001-2700 350 400 450 500
2701-3500 300 350 400 450
273 TABLE 10 Machine values over the entire planning horizon (all beginning-of-year Machine
2
3
6
7
Year Year Year Year Year
132 300 84 000 42 000 0 0
189 000 126 000 63 000 0 0
500 000 425 000 315 000 210 000 105 000
600 510 378 252 126
1. 2. 3. 4. 5.
8 000 000 000 000 000
TABLE 11 Year
Configuration
1 2 3 4
5 7 24 31
No.
Content (2,7,4,5) (2,7,4,5) (7,10,4,5) (7,10,4,5)
TABLE 12 Optimal price-volume Part type
Price
combination Volume
Year 1: Configuration (2,7,4,5) 1 180 2400 2 250 1600 3 400 2000 Year 2: Configuration (2,7,4,5) 1 250 1400 2 300 1600 3 450 2000 Year 3: Configuration (7,10,4,5) 1 260 3800 2 350 1600 3 500 2000 Year 4: Configuration (7,10,4,5) 1 330 2400 2 400 1600 3 550 2000
of deterioration and obsolescence for all machine types are listed in Table 7; the effect of machine deterioration is assumed to increase both machining times and input consumption rate. In this example problem, deterioration
550 467 346 231
values) 9
000 500 500 000
660 561 415 277
000 000 800 200
10
11
12
13
605 000 514 000 381 150
726 000 617 100 456 750
665 500 565 675
797 500 677 875
and obsolescence rates are assumed to be uniform. The capacity limits for all resources and the AGV’s unit distance cost are listed in Table 8. The unit cost of indirect labor, direct labor, energy and maintenance are expected to have a yearly increase of 6%, 8%, 7% and 7%, respectively; for the AGV unit cost, the yearly increase is 10%. The price-volume relationships of the three part types for each of the next four years are shown in Table 9; the product market is assumed to be stable with steady increase in selling prices. All machine values over the entire planning horizon are shown in Table 10; the 1982 ACRS schedule is used for depreciation values. Lastly, the amount of resources are assumed identical each year and a time-valueof-money of 10% is assumed. The optimal
replacement
sequence
The maximum optimal value function at the beginning of the planning horizon is found to be $2,789,991 which belongs to configuration 5 of that year. Tracing the configurations sequence which leads to this maximumf-values, the resulting replacement policy is as stated in Table 11. That is, in the beginning of year one, defender B (machine 3 ) should be replaced by challenger Y (machine 7 ). This system configuration would be used for two years. Then, in the beginning of year three, replace defender A (machine 2 ) by challenger X (machine 10); this system configuration would be used till the end of the planning horizon. Lastly, the set of optimal price-volume relationships over the
274 TABLE 13 Part assignment for optimal replacement sequence Operation
Machine label
Year 1 L/UL
Year 2 2
7
4
Part typeI 0 2400 1 2 3 4 5 6 2400
2400 0 -
2400 0 -1902498 1902 1902 0 -
Parttype2 0 1600
-
-
I
-
- 1600
2 3 4 S 6 I
1600
0 1600 0 67 _-0 0 -
Part t.vpe 3 0 2000 1 2 3 4 5 6 7 8 9 10 2000
1581 419 1581 419 - 2000 02000 -0 -0 2000 2000 2000 -
5
Year 3
L/UL
2
I
4
498 498 -
1400 1400
1400 0 -
1400 1322 1322 -
0 1400 0 -
-
-
1600
-
-
-
-
1600
-
-
-
-
-
-
I533 0 0 -
1600 1600 1600 -
1600
1600 0 0 -
1600 1600 1600 -
-
- 1600 0 0 0 -
1600 0 0 -
- 2000 -1533467 0 0 1533 467 0 - 2000 0 2000 - 2000 0 0 2000 0 2000 0 2000 2000 - 2000 -
5
L/UL
- 3800 0 78 78 -3800
Year 4 IO
7
4
5
L/UL
10
3800 0 0 0 -
0 1382 1382 -
3800 3800 0 -
0 2418 2418 -
2400 2400
2400 0 0 563 -
-
1600 1600 0 0 -
0 0 0 0 -
1600 535 535 -
1065 1065 1600 -
2000 53 53 53 53 53 -2000 2000 2000 2000 2000 -
1947 1947 1947 1947 1947 0 -
0 0 0 0 -
0 0 0 -
-
-
-
1600
1600
0
-
-
-
1600
1600 0 0 -
0 0 0 -
1600 616 616 -
984 984 1600 -
1600
- 2000 0 0 0 -20u0 0 2000 0 - 2000
186 186 186 186 186 2000 2000 2000 2000 -
1814 I814 1814 1814 1814 0 -
0 0 0 0 -
0 0 0 -
entire horizon is shown in Table 12, and their corresponding part assignment is shown in Table 13. 5. CONCLUSION A multi-machine replacement model has been presented. The analysis is oriented towards the many issues relevant to the planning of capital equipment within an integrated system. The model also integrates vital elements on capital investment, production, and product market into the multi-machine replacement environment. Based on this model, sensitivity analysis can be performed on the
7
4
S
0 2400 - 2400 563 0 0 -
0 1837 1837 -
critical parameters. For example, one can examine the equipment acquisition situation with different product demand profiles or even with the introduction of new products. Similarly, the effects of changes in input prices on the system design can also be addressed. Implicitly, this replacement model not only can approach the flexibility of an equipment within a system, but also the flexibility of the entire system with respect to the demand placed on it both internally and externally. Further, the determination of the system’s input-output structure directly defines the capital intensiveness of the production system. While this model is believed to be robust, work is far from complete.
275
The dimensionality of the problem size could present problems in the solution procedure, hence certain bounds or heuristic rules ought to be developed to facilitate the computation process. Also, the linear input-output structure remains a restrictive assumption and extension to nonlinear structures should be studied.
8
9
10
11
12
REFERENCES Alchian, A.A., 1952. Economic replacement policy. Publ. R-224, Santa Monica, CA, The Rand Corporation. Terborgh, G., 1949. Dynamic Equipment Policy, McGraw-Hill, New York. Bellman, R., 1955. Equipment replacement policy. SIAM J. Appl. Math., 3(3): 133-136. Dreyfus, Stuart E., 1957. A generalized equipment replacement study. Santa Monica, CA, Rand Corporation, March, p. 1039. Smith, V.L., 1966. Investment and Production. Harvard University Press, Cambridge, MA. Eilon, S., Ring, J.R. and Hutchison, D.E., 1966. A study of equipment replacement. Oper. Res. Q., 17( 1): 5971. Sethi, S. and Morton, T., 1972. A mixed optimization technique for the generalized machine replacement problem. Nav. Res. Logis. Q., 19( 3): 471-481.
13
14 15
16
17
Sethi, S. and Chand, C., 1979. Planning horizon procedures for machine replacement models. Manage. Sci., 25(2): 140-151. Chand, C. and Sethi, S., 1982. Planning horizon procedure for machine replacement models with several possible alternatives. Nav. Res. Logis. Q., 29(3): 483-493. Oakford, R.V., Lohmann, J.R. and Salazar, A.S., 1989. A dynamic replacement economy decision model. IIE Trans., 16( 1): 65-72. Bean, J.C., Lohmann, J.R. and Smith, R.L., 1985. A dynamic infinite horizon replacement economy decision model. Eng. Economist, 30(2): 99-120. Leung, L.C. and Tanchoco, J.M.A., 1984. After-tax equipment replacement analysis using discrete transform. IIETrans., 16(2): 152-158. Tanchoco, J.M.A. and Leung, L.C., 1987. An input-output model for equipment replacement decisions. Eng. Costs Prod. Econ., 11(2): 69-78. Hannssmann, F.S., 1968. Operations Research Techniques for Capital Investment. Wiley, New York. Ray, Thomas R., 197 1.A System Approach to Replacement. Ph.D. Thesis, Dept. Ind. Eng. Oper. Res., Virginia Polytechnic Institute and State University. Leung, L.C. and Tanchoco, J.M.A., 1987. Multiple equipment replacement within an integrated system framework. Eng. Economist, 32 (2): 89-114. Verheyen, P.A., 1978. Economic interpretation of models of the replacement of machines. Eur. J. Oper. Res., 3: 150-156.
(Received February 29, January 23, 1990)
1988;
accepred
in revised form