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The series–parallel replacement problem
Robotics and Computer Integrated Manufacturing 18 (2002) 215–221
The series–parallel replacement problem J.C. Hartman*, J. Ban Department of Industrial and Systems Engineering, Lehigh University, Bethlehem, PA, USA
Abstract Traditional equipment replacement models focus on single machine problems. However, the capacity of many production facilities is defined by multiple, heterogeneous machines. In this situation, optimal replacement (and expansion) decisions must consider all machines and their integration simultaneously as they define system capacity and are therefore economically interdependent. We model a multiple machine replacement problem that is characterized as a parallel flow shop environment. Work flows through processes in a facility according to a predetermined processing order for the product(s). For a given process, numerous machines, which may differ according to type (different manufacturers), capacity, and/or age, operate in parallel. A series of these processes defines a line and in our analysis, a plant is comprised of multiple, parallel lines. In this preliminary investigation, we present an integer programming formulation to determine optimal purchase, salvage, utilization and storage decisions for each asset over a finite horizon. We illustrate that this model is difficult to solve. We provide valid inequalities to improve the lower bound provided by the linear programming relaxation and a dynamic programming approach to provide initial upper bounds. r 2002 Elsevier Science Ltd. All rights reserved. Keywords: Equipment replacement; Discrete optimization
1. Introduction Traditional replacement analysis models focus on the evaluation of the keep or replace option for each period within the planning horizon for a single machine [1]. The literature in this area, termed serial replacement, is rich. Research has focused on several motivations for replacement, including deterioration, capacity expansion and technological change. Solutions generally include the use of dynamic programming under various assumptions concerning costs, forecast horizons or the number of asset types. For more on this literature, see, for example, Bean et al. [2]. This analysis is appropriate if looking to replace a single machine, an entire facility or multiple independent machines, but when investigating the option(s) of machine replacement within an integrated system, it is necessary to capture the effects of the replacement decision(s) on all of the components of the system in the model [3]. In general, there are two types of replacement models which consider multiple assets that interact: parallel and series. In parallel replacement analysis, it is assumed that assets are economically interdependent and operate in parallel. Conditions which lead to *Corresponding author.
economic interdependence among assets include economies of scale in asset purchases, capital budgeting constraints and/or service constraints on multiple assets. Series replacement models may also have economic interdependencies, but they also have operational dependencies as the machines operate in series. The distinction between these two types of models is that in parallel models, the capacity of the system is simply the sum of the capacities of the individual assets. However, in the series configuration, the minimum capacity asset, or set of assets, in the series defines the capacity of the system. Multiple asset replacement analysis, whether parallel or series, is a difficult problem due to the combinatorial nature of replacing components of groups of assets [4]. To obtain a complete analysis of the series replacement problem, the interactive nature of the integrated system, as well as the combinatorial nature of the replacement alternatives, must be incorporated [3]. When compared to serial replacement analysis, there is little work in multiple machine replacement analysis. Tanchoco and Leung [5], Leung and Tanchoco [3,6], Lotfi and Suresh [7], Suresh [8,9] and Stinson and Khumawala [10] provide various approaches to integrated settings where machines operate in series. In the area of parallel replacement analysis, Jones et al. [11],
0736-5845/02/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 7 3 6 - 5 8 4 5 ( 0 2 ) 0 0 0 1 2 - 1
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J.C. Hartman, J. Ban / Robotics and Computer Integrated Manufacturing 18 (2002) 215–221
Chen [12], Rajagopalan [13], Hartman [14], Hartman and Lohmann [15], and Karabakal et al. [16] analyze various cases of economic interdependence among assets, including economies of scale, demand and budget constraints. In this paper, we analyze a multiple machine problem in which assets work in both parallel and series, as described in the following section.
2. Problem environment and notation The system analyzed in this multiple machine replacement problem can be characterized as a parallel flow shop environment. Work flows through processes in a facility according to a predetermined processing order for the product(s). For a given process, numerous machines, which may differ according to type (different manufacturers), capacity, and/or age, operate in parallel. A series of these processes defines a line and in our analysis, a plant is comprised of multiple, parallel lines. From a replacement analysis point of view, this may be defined as a parallel replacement problem at the process level and a series replacement problem at the line level. However, as we allow for the movement of machines between lines, the decisions are interrelated. Additionally, as plant capacity is defined according to both processes and lines, machine replacement decisions are economically interdependent. We present an integer programming formulation to determine optimal purchase, salvage, utilization and storage decisions for each asset in each period t over a finite horizon t ¼ 0; 1; y; T: Utilization and storage decisions refer to an asset either being ‘‘online’’ or ‘‘offline’’ for a given period. If an asset is online, it provides capacity to the system for that given period. Define the number of production lines as l ¼ 1; 2; y; L and the number of processes in each line as k ¼ 1; 2; y; K: For a given process k; there are a number of asset types, termed challengers, j ¼ 1; 2; y; J k that may vary in age i ¼ 0; 1; y; N jk : An asset is defined by its age i; type j and process k in a given line l during time period t: The capacity of a machine in period t is ajkl it : This allows for the modeling of technological change, as newer assets may have greater capacity when compared to older machines, and deterioration, as older assets may lose capacity over time due to wear, increased maintenance and/or higher failure rates. This parameter may also be used to model yields in a process. An asset that is online from the end of period t to the end of period t þ 1 is denoted as Ojkl it : Assets may also be taken offline for preventive maintenance or shutting down excess capacity. While offline, assets may also be moved from one line to another line. As this is not a trivial process, it is assumed that it takes one period to move and integrate the machinery into the new line.
Finally, taking assets offline allows for the modeling of redundant, backup systems. Although not explicitly considered in this paper, requiring a minimum asset availability for a given process would determine a lower bound on the number of assets needed in storage in0 case of failure. We denote an asset that is offline as Iitjkll ; as it is placed in inventory. If l ¼ l 0 ; then the asset is placed in inventory, else the asset is moved from line l to l 0 : This allows an additional cost to be charged for moving an asset. Assets owned at time zero are denoted as Iitjkl as it is assumed that they are currently in a line. The capacity of a process in a given line is defined as the sum of the individual machine capacities operating in parallel. Define PCtkl as the operating capacity of process k in line l from the end of period t to the end of period t þ 1: Mathematically, it is PCtkl
¼
jk J k NX 1 X
j¼1
jkl ajkl it Oit ; 8k ¼ 1; 2; :::; K;
i¼0
l ¼ 1; 2; :::; L; t ¼ 0; 1; :::; T 1:
ð1Þ
Note that assets in storage or those being moved do not contribute to capacity in that period. As the processes in a given line operate in series, as illustrated in Fig. 1, the capacity of the line is equal to the minimum process capacity of all processes in the line. Define the capacity of line l from the end of period t to the end of period t þ 1 as LCtl ; or: LCtl ¼ mink¼1;2;:::;K PCtkl ; 8l ¼ 1; 2; :::; L; t ¼ 0; 1; :::; T 1:
ð2Þ
Thus, the maximum output from a line with two processes is equal to the minimum capacity of the two individual processes. A number of these lines may operate in parallel, as shown in Fig. 2. Defining TCt as the total capacity of the plant from the end of period t to the end of period t þ 1; it is the sum of the individual line capacities, or: TCt ¼
L X
LCtl 8t ¼ 0; 1; :::; T 1:
ð3Þ
l¼1
In addition to keeping assets online or taking them offline, we determine asset purchase decisions at the end of each period t ¼ 0; 1; :::; T 1 ðBjkl it Þ and salvage decisions at the end of each period t; where t ¼ 0; 1; :::; T ðSitjkl Þ: An asset must be salvaged on or before it reaches its maximum physical life N jk and all assets
Process 1
Process 2
Process K
Fig. 1. Line comprised of processes in a series with parallel machines in each process.
J.C. Hartman, J. Ban / Robotics and Computer Integrated Manufacturing 18 (2002) 215–221
Line 1 Line 2
storage costs, less salvage values over a finite horizon. The objective function is given as follows. " jk J X K X L T 1 NX 1 X T 1 X X jkl jkl pjkl B þ cjkl min t t it Oit j¼1 k¼1 l¼1
…
Fig. 2. Facility comprised of multiple, parallel lines.
are salvaged at time T: For simplicity, we assume that only new assets may be purchased and thus reduce the purchase variables to Bjkl t ; as i 0: It should be noted that purchases are not restricted for just replacement as additional assets may be purchased to create more capacity. This environment describes many semiconductormanufacturing facilities currently in operation, although individual asset or process capacities may have to be altered in order to accommodate re-entrant flow. Due to the high capital cost of tooling, prudent asset replacement policies are imperative to profitable operations. We define this problem as SPRP, for series–parallel replacement, as machines operate both in parallel and series. The formulation for SPRP is defined in the next section.
3. Integer programming formulation of SPRP An integer programming formulation for SPRP is presented here using the variables defined in the previous section. The following costs associated with each of these decision variables and parameters are defined as follows: ft is the discounted fixed cost for the purchase of any asset(s) at time t; pjkl t is the discounted unit purchase cost for a new asset of type j for process k in line l purchased at the end of period t; cjkl it is the discounted operating and maintenance cost for an i-period old asset of type j for process k in line l in use from the end of period t to 0 t þ 1; hjkll is the discounted holding cost for an i-period it old asset of type j for process k in line l in storage from the end of period t to t þ 1 if l ¼ l 0 ; else, discounted cost to move an asset from line l to line l 0 at the end of period t such that it is operable at time t+1; rjkl is the it discounted salvage value revenue for an i-period old asset of type j for process k in line l sold at the end of period t; dt is the demand from end of period t to the end of period t þ 1; and bt is the budget for asset purchases at time t: The objective is to minimize all discounted costs, including asset purchase, operating, maintenance and
t¼0
i¼0
jk
þ
LineL
217
NX 1 X L X T 1
t¼0
#
jk
0 jkll 0 hjkll it Iit
i¼0 l 0 ¼1 t¼0
N X T X
jkl rjkl it Sit
i¼0 t¼0
þ
T 1 X
ft Zt
t¼0
The following constraints represent the capacity, budgetary and balance constraints of the system. The balance constraints refer to the fact that the number of assets must balance from period to period. Each of these constraints must hold for each asset type j of each process k in K and line l in L: Purchased assets must be placed immediately into service or storage, represented by constraint (4): jkl jkl Bjkl t Oit Iit ¼ 0 8j; 8k; 8l; i ¼ 0;
8t ¼ 0; 1; :::; T 1:
ð4Þ
Assets that are already owned at time zero may either be retained for service in line l; stored, moved to another line, or immediately salvaged, as in constraint (5): Ojkl i0 þ
L X
0
jkl Ii0jkll þ Si0 ¼ Ii0jkl ;
l 0 ¼1
8j; 8k; 8l; i ¼ 1; 2; :::; N jk ; t ¼ 0:
ð5Þ
If any assets owned at time zero have reached their maximum service life, they must be immediately salvaged, as in constraint (6): jkl jkl SN 8j; 8k; 8l: jk 0 ¼ IN jk 0
ð6Þ
Assets that are utilized, moved or stored in one period are either salvaged at the end of the period or retained for the following period where they are again utilized, stored or moved to another line. This is captured in constraint (7): Ojkl i1;t1 þ
L X
0
jkl l Ii1;t1 Ojkl it
l 0 ¼1
L X
0
jkll Ii1;t1 Sitjkl ¼ 0;
l 0 ¼1 jk
8j; 8k; 8l; i ¼ 1; 2; :::; N 1; t ¼ 0; 1; :::; T 1:
ð7Þ
However, if an asset that is utilized, moved or stored reaches its maximum service life, it must be salvaged, as in constraint (8): Ojkl þ N jk 1;t1
L X
0
l jkl INjkljk 1;t1 SN jk ;t ¼ 0;
l 0 ¼1
8j; 8k; 8l; t ¼ 0; 1; :::; T 1:
ð8Þ
After the final period of the finite horizon, all assets are salvaged, as in (9): Ojkl i1;T1 þ
L X
0
jkl l jkl Ii1;T1 Si;T ¼ 0;
l 0 ¼1
8j; 8k; 8l; i ¼ 1; 2; :::; N jk :
ð9Þ
J.C. Hartman, J. Ban / Robotics and Computer Integrated Manufacturing 18 (2002) 215–221
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From these balance constraints, the number of assets in operation defines the operating capacity of the plant. With the variables Ojkl it defining the number of assets in operation for process k in line l, the capacity of the process is given in (10): PCtkl
¼
jk J k NX 1 X
j¼1
jkl ajkl it Oit ;
i¼0
8k ¼ 1; 2; :::; K; l ¼ 1; 2; :::; L; t ¼ 0; 1; :::; T 1:
ð10Þ
As the line capacity is equivalent to the minimum process capacity in the line, the process capacities define upper bounds on the line capacity, as in (11): LCtl pPCtkl 8k; 8l; 8t ¼ 0; 1; :::; T 1:
ð11Þ
Clearly, the capacity of the line is restricted to the minimum process capacity in the line. The total capacity of the plant is equivalent to the sum of the line capacities. This value must be at least as great as the required capacity, as given in (12): L X
LCtl Xdt 8t ¼ 0; 1; :::; T 1:
ð12Þ
4.1. Lower bound procedure A number of valid inequalities were derived in order to improve the lower bound provided by the linear programming relaxation of SPRP. Note that these were derived under the assumptions that (1) the value of a was decreasing in age i for constant j to represent the deterioration of an asset but increasing in j for constant i to represent technological change and (2) demand was constant over the horizon. For this experiment, we analyzed the problem without budgeting constraints. Thus, constraint (13) had to be replaced with J X K X L X
Bjkl t pMZt ; 8t ¼ 0; 1; :::; T 1;
j¼1 k¼1 l¼1
where M takes on a value that is sufficiently large so as not to cut off feasible solutions, but small enough to drive Z towards a value of one when purchases are made. This constraint was replaced by the two following constraints: & ’ dt jkl ð14Þ Bt p jkl Zt 8l; k; j; t: aN jk ;tþN jk
l¼1
The final constraint is the capital budgeting constraint, which restrains purchases in a given period. In constraint (13), the Zt variable which forces the fixed charge for any asset purchases in period t is included on the right hand side: J X K X L X
jkl pjkl t Bt pðbt ft ÞZt
XX
j¼1 k¼1 l¼1
8t ¼ 0; 1; :::; T 1:
Constraint (14) states that the maximum number of assets of type j for process k in line l that can be purchased at time t is equal to demand divided by the minimum capacity for the asset over its life (due to deterioration), rounded (up) to the nearest integer. A similar substitution was made for purchases within a process:
ð13Þ
In addition to these constraints and upper bounds, all variables are general integer variables with the exception that the Zt variables are binary.
4. Initial solution investigations In an attempt to understand the difficulty of solving SPRP, we tested a number of different problems without the budgeting constraints with the use of the integer programming solver CPLEX [17]. For problems with two lines, two processes and two machine types for each process, the standard branch and bound solver of CPLEX could not solve 10- or 20-period problems that were generated. The problems were either too large in that there was not enough memory to store all of the information or the solution took excessively long. The next sections describe methods that we employed to improve both lower bounds for the problem.
j
l
& Bjkl t p
dt minj ajkl N jk ;tþN jk
’ Zt 8k; t:
With demand being constant and assets deteriorating after they are purchased, it should be clear that each sale of an asset leads to the purchase of another asset (or assets). Thus, salvages could be substituted for purchases in (14) and (15). Assets owned in inventory at time zero may also be sold, representing an upper bound that can be used to move the Z value to one, as jkl Siþt;t pIi0jkl Zt 8l; k; j; t ¼ 0; 1; ::; N jk :
ð16Þ
Constraint (16) has proved extremely effective for solving parallel replacement problems [18]. As the assets in this system deteriorate, it should be clear that if demand is being met exactly in one period, then an asset purchase must occur in the following period for feasibility. This notion leads to the following two constraints: X PCtkl dtþ1 þ Ztþ1 X1; l
8k; t ¼ 0; 1; :::; N jk 2:
ð17Þ
J.C. Hartman, J. Ban / Robotics and Computer Integrated Manufacturing 18 (2002) 215–221
Constraint (17) states that if the total process capacity is equal to demand in a period, then a purchase must occur (making Z ¼ 1). This is made more explicit in (18): jk 2 X X NX
l
j
0
jkl jkl l ajkl iþ1tþ1 ðOit þ Iit Þ þ dtþ1 Ztþ1 Xdtþ1
i¼0
8k; t ¼ 0; 1; :::; N jk 2:
ð18Þ
Constraint (18) examines all assets in a process for period t that will be available in period t þ 1 calculates their capacity. If it falls below demand, the Z value must be positive for feasibility. 4.2. Initial computational testing To test the upper and lower bounding procedures, we generated 10 test problems with two lines (L ¼ 2), two processes (K ¼ 2) and two machine types for each process (J ¼ 4). We generated five test problems with T ¼ 10 and five with T ¼ 20: All costs and number of initial assets were generated randomly. For each generated problem, we solved the following problems to obtain bounds: LB1: Linear programming relaxation of SPRP with constraints (14) and (15). LB2: Linear programming relaxation of SPRP with all valid inequalities. BIN: LB2 with Z variables set to be binary. INT: LB2 with Z variables binary and all other variables integer (upper bound). Table 1 provides the solution statistics for solving the 10 problem instances with the four solution procedures outlined above. Objective function values are provided along with the total number of Simplex iterations performed and branch and bound nodes visited. The total solution time (in seconds) is also provided. For the integer programming problems, the solution procedure was halted after 30 min and the best solution found was reinstated. The solution gap provided is the difference between the upper bound (provided by the integer solution) and each lower bound, divided by the upper bound. All solutions were found on a 300 Mhz personal computer with 126 MB of RAM. The results given in Table 1 illustrate that this research is preliminary. The valid inequalities provide roughly a 5–10% improvement on the gap over the original linear programming formulation. This also helps with the solution of the binary problem, which has produced good bounds for these test problems. However, it is clear that these bounds will not be as tight for larger problems. Thus, more work is needed to develop stronger lower bounds.
219
Current and future research is investigating different ways in which to approach this problem. Computationally attractive solution approaches in the parallel replacement analysis literature (see [12,14]) suggest that decomposing the problem according to processes might be logical. However, the parallel replacement analysis literature generally assumes that the parameter value a ¼ 1: As noted earlier, the value of a varying from one, as it allows for technological change, capacity variation and deterioration, is what makes the SPRP problem hard. Thus, another approach would be to dualize this constraint and use Lagrangean relaxation procedures to improve the lower bounds. A second problem deals with improving the upper bound, which is discussed in the following section.
5. Realistic problem Although the problems tested in the previous section proved to be quite difficult, we also examined a larger, more realistic problem. The system was defined by two lines with four processes each and two machine types available for each process. The problem was solved over a 40 period horizon. The same solution procedures presented in the previous section were utilized on this problem instance. As shown in Table 2, the solution gaps produced are not very encouraging. Both the BIN and INT solutions were terminated after 30 min of processing time. This instance highlights the need to produce better upper bounds, in addition to work on improving lower bounds. We have devised a dynamic programming approach to aid in tackling the upper bounding problem. Determining asset replacement decisions for this example scenario is not trivial. Given two lines, four processes and two machine types per process, the number of combinations of asset replacements is 82 ¼ 64 for a given line in the system, assuming only one machine is purchased for a given process. Thus, after the first period, there are 64 possible asset states. This grows to 642 in the second period. Obviously, this rate of growth is too large for an optimal analysis over a long time horizon. We limit the search through two methods. First, we fix the periods in which a purchase can be made (much like the approach of Chand et al. [19]). This comes from the solution of BIN as all Z variables are integers. Given these periods where purchases (and thus sales) can occur, we sell assets that cannot be retained until the next period of sales and then evaluate sales of assets according to oldest assets (as in the Older Cluster Replacement Rule of Jones et al. [11]). While this still leads to a large number of combinatorial decisions in each period, it is drastically smaller than searching the entire state space. With this approach, the problem was
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J.C. Hartman, J. Ban / Robotics and Computer Integrated Manufacturing 18 (2002) 215–221
Table 1 Bounds for the 10 generated test problems Problem
Bound
1
LB1 LB2 BIN INT
Gap (%)
Simplex
B&B
Time
35136 38640 50936 52571
33.16 26.50 3.11 F
159 242 1789 475190
F F
0.1 0.98 3.2 1800
LB1 LB2 BIN INT
41714 45219 57514 59139
29.46 23.54 2.75 F
174 213 1632 455301
LB1 LB2 BIN INT
31635 38859 49848 51413
38.47 24.42 3.04 F
200 236 2573 369250
LB1 LB2 BIN INT
42495 46649 58714 61397
30.79 24.02 4.37 F
220 207 2189 587640
F F
LB1 LB2 BIN INT
32596 38450 56209 57774
43.58 33.45 2.71 F
209 185 2179 92186
F F
6
LB1 LB2 BIN INT
81816 85466 103460 109754
25.46 22.13 5.73 F
7
LB1 LB2 BIN INT
71989 78988 106263 111454
8
LB1 LB2 BIN INT
9
10
2
3
4
5
Objective
21 103409 F F 21 107040 F F 54 120025
42 85623
0.1 0.15 2.0 1800 0.12 0.19 3.7 1800 0.14 0.19 3.7 1800
38 21030
0.14 0.21 4.1 1800
452 592 35361 548064
F F 1008 57256
0.26 0.56 74 1800
35.41 29.13 4.66 F
510 924 37269 352847
F F 994 38612
0.35 1.0 100 1800
60990 68302 103948 109955
44.53 37.88 5.46 F
501 1283 92464 489510
F F 2009 16531
0.44 1.8 290 1800
LB1 LB2 BIN INT
59171 67236 107064 119205
50.36 43.60 10.18 F
550 894 102974 394019
F F 2842 14587
0.55 1.3 410 1800
LB1 LB2 BIN INT
57654 65717 105535 116363
50.45 43.52 9.31 F
523 829 93640 451178
F F 2111 16637
0.52 1.2 420 1800
Table 2 Solutions for larger problem instance Bound
Objective
Gap (%)
Simplex
B&B
Time
LB1 LB2 BIN INT
230549 240201 290781 440551
47.68 45.48 34 F
8417 20232 67222 43675
F F 660 1422
5.2 92 1800 1800
J.C. Hartman, J. Ban / Robotics and Computer Integrated Manufacturing 18 (2002) 215–221
solved with an optimal solution value of 377,918, a 14.2% improvement over the feasible solution provided by CPLEX after 30 min. This results in a new gap of 23.06%. While this gap is not acceptable for such a high cost decisions, it does provide promise that it can be refined to find better solutions. It should be noted that it took this approach just under 10 min to solve.
6. Conclusions and future research This paper has presented an integer programming formulation for a series–parallel replacement problem (SPRP). The model determines optimal purchase, salvage, operating and storage decisions for assets in a parallel flow-shop configuration. That is, assets work in parallel in a given process and a number of processes work in series for a given line. There are generally multiple lines for a given plant. Although investigations are preliminary, we illustrated that solving SPRP is extremely difficult. While many integer programming formulations are difficult, SPRP is difficult because of its ability to model deterioration and technological change through changes in system capacity. We provided a dynamic programming approach based on previous results in the literature in order to develop upper bounds for the problem. This approach still needs further testing. Also, we provided valid inequalities to improve the lower bound provided by the linear programming relaxation of SPRP.
Acknowledgements This research was sponsored in part by NSF grants DMI-9713690 and DMI-9984891.
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The series–parallel replacement problem J.C. Hartman*, J. Ban Department of Industrial and Systems Engineering, Lehigh University, Bethlehem, PA, USA
Abstract Traditional equipment replacement models focus on single machine problems. However, the capacity of many production facilities is defined by multiple, heterogeneous machines. In this situation, optimal replacement (and expansion) decisions must consider all machines and their integration simultaneously as they define system capacity and are therefore economically interdependent. We model a multiple machine replacement problem that is characterized as a parallel flow shop environment. Work flows through processes in a facility according to a predetermined processing order for the product(s). For a given process, numerous machines, which may differ according to type (different manufacturers), capacity, and/or age, operate in parallel. A series of these processes defines a line and in our analysis, a plant is comprised of multiple, parallel lines. In this preliminary investigation, we present an integer programming formulation to determine optimal purchase, salvage, utilization and storage decisions for each asset over a finite horizon. We illustrate that this model is difficult to solve. We provide valid inequalities to improve the lower bound provided by the linear programming relaxation and a dynamic programming approach to provide initial upper bounds. r 2002 Elsevier Science Ltd. All rights reserved. Keywords: Equipment replacement; Discrete optimization
1. Introduction Traditional replacement analysis models focus on the evaluation of the keep or replace option for each period within the planning horizon for a single machine [1]. The literature in this area, termed serial replacement, is rich. Research has focused on several motivations for replacement, including deterioration, capacity expansion and technological change. Solutions generally include the use of dynamic programming under various assumptions concerning costs, forecast horizons or the number of asset types. For more on this literature, see, for example, Bean et al. [2]. This analysis is appropriate if looking to replace a single machine, an entire facility or multiple independent machines, but when investigating the option(s) of machine replacement within an integrated system, it is necessary to capture the effects of the replacement decision(s) on all of the components of the system in the model [3]. In general, there are two types of replacement models which consider multiple assets that interact: parallel and series. In parallel replacement analysis, it is assumed that assets are economically interdependent and operate in parallel. Conditions which lead to *Corresponding author.
economic interdependence among assets include economies of scale in asset purchases, capital budgeting constraints and/or service constraints on multiple assets. Series replacement models may also have economic interdependencies, but they also have operational dependencies as the machines operate in series. The distinction between these two types of models is that in parallel models, the capacity of the system is simply the sum of the capacities of the individual assets. However, in the series configuration, the minimum capacity asset, or set of assets, in the series defines the capacity of the system. Multiple asset replacement analysis, whether parallel or series, is a difficult problem due to the combinatorial nature of replacing components of groups of assets [4]. To obtain a complete analysis of the series replacement problem, the interactive nature of the integrated system, as well as the combinatorial nature of the replacement alternatives, must be incorporated [3]. When compared to serial replacement analysis, there is little work in multiple machine replacement analysis. Tanchoco and Leung [5], Leung and Tanchoco [3,6], Lotfi and Suresh [7], Suresh [8,9] and Stinson and Khumawala [10] provide various approaches to integrated settings where machines operate in series. In the area of parallel replacement analysis, Jones et al. [11],
0736-5845/02/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 7 3 6 - 5 8 4 5 ( 0 2 ) 0 0 0 1 2 - 1
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Chen [12], Rajagopalan [13], Hartman [14], Hartman and Lohmann [15], and Karabakal et al. [16] analyze various cases of economic interdependence among assets, including economies of scale, demand and budget constraints. In this paper, we analyze a multiple machine problem in which assets work in both parallel and series, as described in the following section.
2. Problem environment and notation The system analyzed in this multiple machine replacement problem can be characterized as a parallel flow shop environment. Work flows through processes in a facility according to a predetermined processing order for the product(s). For a given process, numerous machines, which may differ according to type (different manufacturers), capacity, and/or age, operate in parallel. A series of these processes defines a line and in our analysis, a plant is comprised of multiple, parallel lines. From a replacement analysis point of view, this may be defined as a parallel replacement problem at the process level and a series replacement problem at the line level. However, as we allow for the movement of machines between lines, the decisions are interrelated. Additionally, as plant capacity is defined according to both processes and lines, machine replacement decisions are economically interdependent. We present an integer programming formulation to determine optimal purchase, salvage, utilization and storage decisions for each asset in each period t over a finite horizon t ¼ 0; 1; y; T: Utilization and storage decisions refer to an asset either being ‘‘online’’ or ‘‘offline’’ for a given period. If an asset is online, it provides capacity to the system for that given period. Define the number of production lines as l ¼ 1; 2; y; L and the number of processes in each line as k ¼ 1; 2; y; K: For a given process k; there are a number of asset types, termed challengers, j ¼ 1; 2; y; J k that may vary in age i ¼ 0; 1; y; N jk : An asset is defined by its age i; type j and process k in a given line l during time period t: The capacity of a machine in period t is ajkl it : This allows for the modeling of technological change, as newer assets may have greater capacity when compared to older machines, and deterioration, as older assets may lose capacity over time due to wear, increased maintenance and/or higher failure rates. This parameter may also be used to model yields in a process. An asset that is online from the end of period t to the end of period t þ 1 is denoted as Ojkl it : Assets may also be taken offline for preventive maintenance or shutting down excess capacity. While offline, assets may also be moved from one line to another line. As this is not a trivial process, it is assumed that it takes one period to move and integrate the machinery into the new line.
Finally, taking assets offline allows for the modeling of redundant, backup systems. Although not explicitly considered in this paper, requiring a minimum asset availability for a given process would determine a lower bound on the number of assets needed in storage in0 case of failure. We denote an asset that is offline as Iitjkll ; as it is placed in inventory. If l ¼ l 0 ; then the asset is placed in inventory, else the asset is moved from line l to l 0 : This allows an additional cost to be charged for moving an asset. Assets owned at time zero are denoted as Iitjkl as it is assumed that they are currently in a line. The capacity of a process in a given line is defined as the sum of the individual machine capacities operating in parallel. Define PCtkl as the operating capacity of process k in line l from the end of period t to the end of period t þ 1: Mathematically, it is PCtkl
¼
jk J k NX 1 X
j¼1
jkl ajkl it Oit ; 8k ¼ 1; 2; :::; K;
i¼0
l ¼ 1; 2; :::; L; t ¼ 0; 1; :::; T 1:
ð1Þ
Note that assets in storage or those being moved do not contribute to capacity in that period. As the processes in a given line operate in series, as illustrated in Fig. 1, the capacity of the line is equal to the minimum process capacity of all processes in the line. Define the capacity of line l from the end of period t to the end of period t þ 1 as LCtl ; or: LCtl ¼ mink¼1;2;:::;K PCtkl ; 8l ¼ 1; 2; :::; L; t ¼ 0; 1; :::; T 1:
ð2Þ
Thus, the maximum output from a line with two processes is equal to the minimum capacity of the two individual processes. A number of these lines may operate in parallel, as shown in Fig. 2. Defining TCt as the total capacity of the plant from the end of period t to the end of period t þ 1; it is the sum of the individual line capacities, or: TCt ¼
L X
LCtl 8t ¼ 0; 1; :::; T 1:
ð3Þ
l¼1
In addition to keeping assets online or taking them offline, we determine asset purchase decisions at the end of each period t ¼ 0; 1; :::; T 1 ðBjkl it Þ and salvage decisions at the end of each period t; where t ¼ 0; 1; :::; T ðSitjkl Þ: An asset must be salvaged on or before it reaches its maximum physical life N jk and all assets
Process 1
Process 2
Process K
Fig. 1. Line comprised of processes in a series with parallel machines in each process.
J.C. Hartman, J. Ban / Robotics and Computer Integrated Manufacturing 18 (2002) 215–221
Line 1 Line 2
storage costs, less salvage values over a finite horizon. The objective function is given as follows. " jk J X K X L T 1 NX 1 X T 1 X X jkl jkl pjkl B þ cjkl min t t it Oit j¼1 k¼1 l¼1
…
Fig. 2. Facility comprised of multiple, parallel lines.
are salvaged at time T: For simplicity, we assume that only new assets may be purchased and thus reduce the purchase variables to Bjkl t ; as i 0: It should be noted that purchases are not restricted for just replacement as additional assets may be purchased to create more capacity. This environment describes many semiconductormanufacturing facilities currently in operation, although individual asset or process capacities may have to be altered in order to accommodate re-entrant flow. Due to the high capital cost of tooling, prudent asset replacement policies are imperative to profitable operations. We define this problem as SPRP, for series–parallel replacement, as machines operate both in parallel and series. The formulation for SPRP is defined in the next section.
3. Integer programming formulation of SPRP An integer programming formulation for SPRP is presented here using the variables defined in the previous section. The following costs associated with each of these decision variables and parameters are defined as follows: ft is the discounted fixed cost for the purchase of any asset(s) at time t; pjkl t is the discounted unit purchase cost for a new asset of type j for process k in line l purchased at the end of period t; cjkl it is the discounted operating and maintenance cost for an i-period old asset of type j for process k in line l in use from the end of period t to 0 t þ 1; hjkll is the discounted holding cost for an i-period it old asset of type j for process k in line l in storage from the end of period t to t þ 1 if l ¼ l 0 ; else, discounted cost to move an asset from line l to line l 0 at the end of period t such that it is operable at time t+1; rjkl is the it discounted salvage value revenue for an i-period old asset of type j for process k in line l sold at the end of period t; dt is the demand from end of period t to the end of period t þ 1; and bt is the budget for asset purchases at time t: The objective is to minimize all discounted costs, including asset purchase, operating, maintenance and
t¼0
i¼0
jk
þ
LineL
217
NX 1 X L X T 1
t¼0
#
jk
0 jkll 0 hjkll it Iit
i¼0 l 0 ¼1 t¼0
N X T X
jkl rjkl it Sit
i¼0 t¼0
þ
T 1 X
ft Zt
t¼0
The following constraints represent the capacity, budgetary and balance constraints of the system. The balance constraints refer to the fact that the number of assets must balance from period to period. Each of these constraints must hold for each asset type j of each process k in K and line l in L: Purchased assets must be placed immediately into service or storage, represented by constraint (4): jkl jkl Bjkl t Oit Iit ¼ 0 8j; 8k; 8l; i ¼ 0;
8t ¼ 0; 1; :::; T 1:
ð4Þ
Assets that are already owned at time zero may either be retained for service in line l; stored, moved to another line, or immediately salvaged, as in constraint (5): Ojkl i0 þ
L X
0
jkl Ii0jkll þ Si0 ¼ Ii0jkl ;
l 0 ¼1
8j; 8k; 8l; i ¼ 1; 2; :::; N jk ; t ¼ 0:
ð5Þ
If any assets owned at time zero have reached their maximum service life, they must be immediately salvaged, as in constraint (6): jkl jkl SN 8j; 8k; 8l: jk 0 ¼ IN jk 0
ð6Þ
Assets that are utilized, moved or stored in one period are either salvaged at the end of the period or retained for the following period where they are again utilized, stored or moved to another line. This is captured in constraint (7): Ojkl i1;t1 þ
L X
0
jkl l Ii1;t1 Ojkl it
l 0 ¼1
L X
0
jkll Ii1;t1 Sitjkl ¼ 0;
l 0 ¼1 jk
8j; 8k; 8l; i ¼ 1; 2; :::; N 1; t ¼ 0; 1; :::; T 1:
ð7Þ
However, if an asset that is utilized, moved or stored reaches its maximum service life, it must be salvaged, as in constraint (8): Ojkl þ N jk 1;t1
L X
0
l jkl INjkljk 1;t1 SN jk ;t ¼ 0;
l 0 ¼1
8j; 8k; 8l; t ¼ 0; 1; :::; T 1:
ð8Þ
After the final period of the finite horizon, all assets are salvaged, as in (9): Ojkl i1;T1 þ
L X
0
jkl l jkl Ii1;T1 Si;T ¼ 0;
l 0 ¼1
8j; 8k; 8l; i ¼ 1; 2; :::; N jk :
ð9Þ
J.C. Hartman, J. Ban / Robotics and Computer Integrated Manufacturing 18 (2002) 215–221
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From these balance constraints, the number of assets in operation defines the operating capacity of the plant. With the variables Ojkl it defining the number of assets in operation for process k in line l, the capacity of the process is given in (10): PCtkl
¼
jk J k NX 1 X
j¼1
jkl ajkl it Oit ;
i¼0
8k ¼ 1; 2; :::; K; l ¼ 1; 2; :::; L; t ¼ 0; 1; :::; T 1:
ð10Þ
As the line capacity is equivalent to the minimum process capacity in the line, the process capacities define upper bounds on the line capacity, as in (11): LCtl pPCtkl 8k; 8l; 8t ¼ 0; 1; :::; T 1:
ð11Þ
Clearly, the capacity of the line is restricted to the minimum process capacity in the line. The total capacity of the plant is equivalent to the sum of the line capacities. This value must be at least as great as the required capacity, as given in (12): L X
LCtl Xdt 8t ¼ 0; 1; :::; T 1:
ð12Þ
4.1. Lower bound procedure A number of valid inequalities were derived in order to improve the lower bound provided by the linear programming relaxation of SPRP. Note that these were derived under the assumptions that (1) the value of a was decreasing in age i for constant j to represent the deterioration of an asset but increasing in j for constant i to represent technological change and (2) demand was constant over the horizon. For this experiment, we analyzed the problem without budgeting constraints. Thus, constraint (13) had to be replaced with J X K X L X
Bjkl t pMZt ; 8t ¼ 0; 1; :::; T 1;
j¼1 k¼1 l¼1
where M takes on a value that is sufficiently large so as not to cut off feasible solutions, but small enough to drive Z towards a value of one when purchases are made. This constraint was replaced by the two following constraints: & ’ dt jkl ð14Þ Bt p jkl Zt 8l; k; j; t: aN jk ;tþN jk
l¼1
The final constraint is the capital budgeting constraint, which restrains purchases in a given period. In constraint (13), the Zt variable which forces the fixed charge for any asset purchases in period t is included on the right hand side: J X K X L X
jkl pjkl t Bt pðbt ft ÞZt
XX
j¼1 k¼1 l¼1
8t ¼ 0; 1; :::; T 1:
Constraint (14) states that the maximum number of assets of type j for process k in line l that can be purchased at time t is equal to demand divided by the minimum capacity for the asset over its life (due to deterioration), rounded (up) to the nearest integer. A similar substitution was made for purchases within a process:
ð13Þ
In addition to these constraints and upper bounds, all variables are general integer variables with the exception that the Zt variables are binary.
4. Initial solution investigations In an attempt to understand the difficulty of solving SPRP, we tested a number of different problems without the budgeting constraints with the use of the integer programming solver CPLEX [17]. For problems with two lines, two processes and two machine types for each process, the standard branch and bound solver of CPLEX could not solve 10- or 20-period problems that were generated. The problems were either too large in that there was not enough memory to store all of the information or the solution took excessively long. The next sections describe methods that we employed to improve both lower bounds for the problem.
j
l
& Bjkl t p
dt minj ajkl N jk ;tþN jk
’ Zt 8k; t:
With demand being constant and assets deteriorating after they are purchased, it should be clear that each sale of an asset leads to the purchase of another asset (or assets). Thus, salvages could be substituted for purchases in (14) and (15). Assets owned in inventory at time zero may also be sold, representing an upper bound that can be used to move the Z value to one, as jkl Siþt;t pIi0jkl Zt 8l; k; j; t ¼ 0; 1; ::; N jk :
ð16Þ
Constraint (16) has proved extremely effective for solving parallel replacement problems [18]. As the assets in this system deteriorate, it should be clear that if demand is being met exactly in one period, then an asset purchase must occur in the following period for feasibility. This notion leads to the following two constraints: X PCtkl dtþ1 þ Ztþ1 X1; l
8k; t ¼ 0; 1; :::; N jk 2:
ð17Þ
J.C. Hartman, J. Ban / Robotics and Computer Integrated Manufacturing 18 (2002) 215–221
Constraint (17) states that if the total process capacity is equal to demand in a period, then a purchase must occur (making Z ¼ 1). This is made more explicit in (18): jk 2 X X NX
l
j
0
jkl jkl l ajkl iþ1tþ1 ðOit þ Iit Þ þ dtþ1 Ztþ1 Xdtþ1
i¼0
8k; t ¼ 0; 1; :::; N jk 2:
ð18Þ
Constraint (18) examines all assets in a process for period t that will be available in period t þ 1 calculates their capacity. If it falls below demand, the Z value must be positive for feasibility. 4.2. Initial computational testing To test the upper and lower bounding procedures, we generated 10 test problems with two lines (L ¼ 2), two processes (K ¼ 2) and two machine types for each process (J ¼ 4). We generated five test problems with T ¼ 10 and five with T ¼ 20: All costs and number of initial assets were generated randomly. For each generated problem, we solved the following problems to obtain bounds: LB1: Linear programming relaxation of SPRP with constraints (14) and (15). LB2: Linear programming relaxation of SPRP with all valid inequalities. BIN: LB2 with Z variables set to be binary. INT: LB2 with Z variables binary and all other variables integer (upper bound). Table 1 provides the solution statistics for solving the 10 problem instances with the four solution procedures outlined above. Objective function values are provided along with the total number of Simplex iterations performed and branch and bound nodes visited. The total solution time (in seconds) is also provided. For the integer programming problems, the solution procedure was halted after 30 min and the best solution found was reinstated. The solution gap provided is the difference between the upper bound (provided by the integer solution) and each lower bound, divided by the upper bound. All solutions were found on a 300 Mhz personal computer with 126 MB of RAM. The results given in Table 1 illustrate that this research is preliminary. The valid inequalities provide roughly a 5–10% improvement on the gap over the original linear programming formulation. This also helps with the solution of the binary problem, which has produced good bounds for these test problems. However, it is clear that these bounds will not be as tight for larger problems. Thus, more work is needed to develop stronger lower bounds.
219
Current and future research is investigating different ways in which to approach this problem. Computationally attractive solution approaches in the parallel replacement analysis literature (see [12,14]) suggest that decomposing the problem according to processes might be logical. However, the parallel replacement analysis literature generally assumes that the parameter value a ¼ 1: As noted earlier, the value of a varying from one, as it allows for technological change, capacity variation and deterioration, is what makes the SPRP problem hard. Thus, another approach would be to dualize this constraint and use Lagrangean relaxation procedures to improve the lower bounds. A second problem deals with improving the upper bound, which is discussed in the following section.
5. Realistic problem Although the problems tested in the previous section proved to be quite difficult, we also examined a larger, more realistic problem. The system was defined by two lines with four processes each and two machine types available for each process. The problem was solved over a 40 period horizon. The same solution procedures presented in the previous section were utilized on this problem instance. As shown in Table 2, the solution gaps produced are not very encouraging. Both the BIN and INT solutions were terminated after 30 min of processing time. This instance highlights the need to produce better upper bounds, in addition to work on improving lower bounds. We have devised a dynamic programming approach to aid in tackling the upper bounding problem. Determining asset replacement decisions for this example scenario is not trivial. Given two lines, four processes and two machine types per process, the number of combinations of asset replacements is 82 ¼ 64 for a given line in the system, assuming only one machine is purchased for a given process. Thus, after the first period, there are 64 possible asset states. This grows to 642 in the second period. Obviously, this rate of growth is too large for an optimal analysis over a long time horizon. We limit the search through two methods. First, we fix the periods in which a purchase can be made (much like the approach of Chand et al. [19]). This comes from the solution of BIN as all Z variables are integers. Given these periods where purchases (and thus sales) can occur, we sell assets that cannot be retained until the next period of sales and then evaluate sales of assets according to oldest assets (as in the Older Cluster Replacement Rule of Jones et al. [11]). While this still leads to a large number of combinatorial decisions in each period, it is drastically smaller than searching the entire state space. With this approach, the problem was
220
J.C. Hartman, J. Ban / Robotics and Computer Integrated Manufacturing 18 (2002) 215–221
Table 1 Bounds for the 10 generated test problems Problem
Bound
1
LB1 LB2 BIN INT
Gap (%)
Simplex
B&B
Time
35136 38640 50936 52571
33.16 26.50 3.11 F
159 242 1789 475190
F F
0.1 0.98 3.2 1800
LB1 LB2 BIN INT
41714 45219 57514 59139
29.46 23.54 2.75 F
174 213 1632 455301
LB1 LB2 BIN INT
31635 38859 49848 51413
38.47 24.42 3.04 F
200 236 2573 369250
LB1 LB2 BIN INT
42495 46649 58714 61397
30.79 24.02 4.37 F
220 207 2189 587640
F F
LB1 LB2 BIN INT
32596 38450 56209 57774
43.58 33.45 2.71 F
209 185 2179 92186
F F
6
LB1 LB2 BIN INT
81816 85466 103460 109754
25.46 22.13 5.73 F
7
LB1 LB2 BIN INT
71989 78988 106263 111454
8
LB1 LB2 BIN INT
9
10
2
3
4
5
Objective
21 103409 F F 21 107040 F F 54 120025
42 85623
0.1 0.15 2.0 1800 0.12 0.19 3.7 1800 0.14 0.19 3.7 1800
38 21030
0.14 0.21 4.1 1800
452 592 35361 548064
F F 1008 57256
0.26 0.56 74 1800
35.41 29.13 4.66 F
510 924 37269 352847
F F 994 38612
0.35 1.0 100 1800
60990 68302 103948 109955
44.53 37.88 5.46 F
501 1283 92464 489510
F F 2009 16531
0.44 1.8 290 1800
LB1 LB2 BIN INT
59171 67236 107064 119205
50.36 43.60 10.18 F
550 894 102974 394019
F F 2842 14587
0.55 1.3 410 1800
LB1 LB2 BIN INT
57654 65717 105535 116363
50.45 43.52 9.31 F
523 829 93640 451178
F F 2111 16637
0.52 1.2 420 1800
Table 2 Solutions for larger problem instance Bound
Objective
Gap (%)
Simplex
B&B
Time
LB1 LB2 BIN INT
230549 240201 290781 440551
47.68 45.48 34 F
8417 20232 67222 43675
F F 660 1422
5.2 92 1800 1800
J.C. Hartman, J. Ban / Robotics and Computer Integrated Manufacturing 18 (2002) 215–221
solved with an optimal solution value of 377,918, a 14.2% improvement over the feasible solution provided by CPLEX after 30 min. This results in a new gap of 23.06%. While this gap is not acceptable for such a high cost decisions, it does provide promise that it can be refined to find better solutions. It should be noted that it took this approach just under 10 min to solve.
6. Conclusions and future research This paper has presented an integer programming formulation for a series–parallel replacement problem (SPRP). The model determines optimal purchase, salvage, operating and storage decisions for assets in a parallel flow-shop configuration. That is, assets work in parallel in a given process and a number of processes work in series for a given line. There are generally multiple lines for a given plant. Although investigations are preliminary, we illustrated that solving SPRP is extremely difficult. While many integer programming formulations are difficult, SPRP is difficult because of its ability to model deterioration and technological change through changes in system capacity. We provided a dynamic programming approach based on previous results in the literature in order to develop upper bounds for the problem. This approach still needs further testing. Also, we provided valid inequalities to improve the lower bound provided by the linear programming relaxation of SPRP.
Acknowledgements This research was sponsored in part by NSF grants DMI-9713690 and DMI-9984891.
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