- Email: [email protected]
A rigorous comparative study of temporal versus spatial Lagrangean decomposition in production planning problems
20th European Symposium on Computer Aided Process Engineering – ESCAPE20 S. Pierucci and G. Buzzi Ferraris (Editors) © 2010 Elsevier B.V. All rights reserved.
A rigorous comparative study of temporal versus spatial Lagrangean decomposition in production planning problems Sebastian Terrazas-Moreno,a Ignacio E. Grossmann,a Philipp Trotterb a
Carnegie Mellon University, 5000 Forbes Ave, Pittsburgh PA 15232, USA, [email protected] b RWTH Aachen University, Templergraben 55, 52056 Aachen, Germany, [email protected]
Abstract Temporal and spatial Lagrangean decompositions are alternatives for solving largescale planning problems. In this paper we compare the strength of the bounds provided by both decompositions on the full space optimal solution. We also use the economic interpretation of the Lagrange multipliers to speed the convergence to the optimal dual solution. Keywords: Production planning, Lagrangean relaxation, temporal decomposition, spatial decomposition, mixed-integer programming.
1. Introduction The optimal planning of a network of manufacturing sites and markets is a complex problem. It involves assigning which products to manufacture in each site, how much to ship to each market and how much to keep in inventory to satisfy future demand. Each site has different production capacities and operating costs, while demand for products varies significantly across markets. Production and distribution planning is concerned with mid to long-term decisions usually involving several months, adding a temporal dimension to the spatial distribution given by the multi-site network. In addition, the production of each product can involve a setup or cleaning time that in some cases is dependent on the sequence of production. When setups and sequence-dependent transitions are included, the optimal planning problem becomes a mixed integer linear programming (MILP) problem. The computational expense of solving large-scale MILPs of this type can be decreased by using decomposition techniques. This paper presents temporal and spatial Lagrangean decompositions that allow the independent solution of time periods, production sites, and markets. The importance of choosing the best between alternative decomposition strategies such as temporal and spatial is discussed in Gupta and Maranas (1999). Although it has been reported that temporal decomposition provides a tighter bound on the full space solution and has faster dual convergence (Jackson and Grossmann, 2003), there is no rigorous generalization of the observed result. One objective of this paper is to compare the bounds obtained through Lagrangean temporal and spatial decompositions for a class of MILPs derived from the lot-sizing problem with setup times. The second objective is to use the economic interpretation of the Lagrange multipliers to provide a reduced dual search space and accelerate the convergence of the optimal multipliers.
S. Terrazas-Moreno et al.
2. Problem Statement There is a set of products that are manufactured in several production sites and shipped to a set of markets where they are sold. Let I , S , and M be the sets of products, production sites, and markets, respectively. There is a finite time horizon divided into a set T of time periods of length Lt . Figure 1 shows the multi-period, multi-site network structure. Month 1
Month 2
Month n -1
Month n
Multi-site Network Market 1 Production Site 1
Market 2
Production Site n-1
Market 3 Production Site 2
Production Site n
Figure 1. Network of production sites and markets for a multi-period planning problem
At each market m M and time period t T there is a forecasted demand d im,t for product i I . The production at each site s S incurs in a manufacturing cost D i,s t and the shipment from any site s to any market m involves a transportation cost J is,t,m . Each site has a limited production rate (production per unit time) T is . Every product has a setup time STi s and setup cost SCis . The products have a sale price of E i,mt at market m . The excess production can be stored as inventory at the production sites, where the inventory holding costs are G i,st . All products that are sent to a market are sold, but the sales cannot be greater than the demand d im,t . The planning problem is to determine the production in each site, the inventory levels, and the amounts of products shipped to each market during each time period, in order to maximize the profit. Profit is defined as sales minus production, shipment, inventory, and setup costs. We assume that the size of the problem prohibits its direct solution and we consider that temporal and spatial lagrangrean decomposition techniques are alternatives to overcome this challenge. The objective of this work is to rigorously compare the strength of both decompositions. We also illustrate the use of the economic interpretation of the Lagrange multipliers involved in the decomposition schemes to provide a reduced search space for the optimal Lagrangean dual. We illustrate our analysis using a planning problem with setup times but no sequence-dependent changeovers. The results will be extended to a more general model with sequence-dependent changeovers in a full length paper.
3. Illustrative Production planning model The planning problem can be formulated as the following MILP model.
A rigurous comparative study of temporal versus spatial Lagrangean decomposition in production planning problems
¦ ¦ ¦ ¦ E im,t sl is,t,m
max S
sS tT iI mM D is,t xis,t G is,t inv is,t tT iI ss
¦¦ ¦
SC is stp is,t ¦ ¦
¦ ¦ J is,t,m f is,t,m
(1a)
tT iI mM sS
s.t. s
¦f
inv i ,t 1 xis,t
s ,m i ,t
invis,t
¦ ST
xis,t d T is,t Lt stpis,t
¦
T is xis,t
iI
(1b)
s
s i stp i ,t
i I , t T , s S
(1c)
d Lt
t T , s S
(1d)
iI
f i ,st,m
¦
i I , t T , s S
mM
slis,t,m
slis,t,m
d
i I , t T , s S , m M (1e)
d im,t
i I , t T , m M
(1f)
i I , t T , s S
(1g)
sS
s
invis,t
inv i ,t
f is,t,m
s ,m i ,t
f
i I , t T , s S , m M (1h)
invis d inviUP ,s
i I , t T , s S
(1i)
s inv i
i I , t T , s S
(1j)
d inviUP ,s
f i s,t,m d f iUP , s ,m
i I , t T , m M , s S (1k)
s ,m
i I , t T , m M , s S (1l)
f i ,t d f iUP , s ,m I u M u|S|x T
f , f , sl
I u S uT
; inv, inv, x
; stp ^0,1`
I u S uT
(1m)
The variable sl is,t,m represents the sales of product i in market m at time t ; the superscript s indicates the production site of origin. The variable invis,t is the level of inventory, xis,t is the amount produced of product i in site s during time t , and f i ,st,m corresponds to the shipments between s and m . The setup variable stp is,t takes the value of one when a product i is manufactured in site s during time t . Equation (1a) represents the maximization of profit. Equation (1b) is the mass balance of each product at each site and time period. Equation (1c) is a setup constraint where T is,t Lt is a valid upper bound for the production of each product. In (1d) the summation of production, setup, and transition times should be less than the length of the time period. Equations (1e) and (1f) are the market constraints. Constraints (1g) and (1h) set the duplicated variables inv,inv and f, f to be equal. Constraints (1i) - (1l) contain upper bounds for inventory and transportation levels. Finally constraint (1m) indicates the domains of all variables.
S. Terrazas-Moreno et al.
4. Temporal and spatial Lagrangean duals The problem can be made decomposable into time periods dualizing constraint (1g). The objective function takes the following form: max S t
¦ ¦ ¦ ¦ E im,t sl is,t,m ¦ ¦ ¦ D is,t xis,t G is,t inv is,t SC is stp is,t
sS tT iI mM
¦¦ ¦ ¦
J is,t,m f is,t,m
tT iI ss
tT iI mM sS
(2a)
s
¦ ¦ ¦ Ot is,t (inv is,t inv i,t )
sS iI tT
On the other hand, the model can be decomposed into individual sites and markets by dualizing constraint (1h), and using the following objective function: max S s
¦ ¦ ¦ ¦ E im,t sl is,t,m ¦ ¦ ¦ D is,t x is,t G is,t inv is,t SC is stp is,t
sS tT iI mM
¦¦ ¦ ¦
J is,,tm f is,t,m
tT iI mM sS
tT iI ss
¦¦ ¦
(3a)
s,m
¦ Os is,,tm ( f i,st,m f i,t )
tT iI mM sS
Problems (2) and (3) can be completed by adding equation (2b)-(2f), (2h)-(2m) and (3b)-(3g), (3i)-(3m) identical to the corresponding constrains in problem (1). Let F , FT , and FS be the feasible regions of problems (1), (2), and (3). The temporal and spatial duals are defined as: Dt
min
max
St
(4)
Ds
min
max
Ss
(5)
Ot ( sl , x, f , f , stp ,inv,inv )FT Os ( sl , x, f , f ,stp ,inv ,inv )FS
The well known result of Geoffrion (1974) establishes the following equivalences:
^(sl, x, f , f , stp, inv, inv) : (sl, x, f , f , stp, inv, inv) Co(FT ^(sl, x, f , f , stp, inv, inv) : (sl, x, f , f , stp, inv, inv) Co(FS
T S D
t
D
s
^max S : (sl, x, f , f , stp, inv, inv) T ` ^max S : (sl, x, f , f , stp, inv, inv) S `
^
LP
) (inv, inv) : inv
LP
) (f, f ): f
^
f
``
inv (6)
``
(7) (8) (9)
where F LP refers to the linear programming relaxation of F . The main result of our analysis is that in general D t d D s . The remainder of this section is devoted to presenting an outline of the derivation of this result. The complete proof will be available in a full length paper. Definition pF proj sl ,stp F
^(sl, stp) : (sl, x, f , f , stp, inv, inv) F `.
A rigurous comparative study of temporal versus spatial Lagrangean decomposition in production planning problems Key Assumption The larger the feasible region in the space of product sales ( sl ), the higher the attainable profit. Theorem: For the mixed integer planning problem presented in this work, temporal decomposition provides a tighter upper bound than spatial decomposition ( D t d D s ). Outline of the proof:
By Fourier-Motzkin elimination, we can obtain the projection of FT LP and FS LP onto the ( sl , stp ) space. The projections show that pFT LP pFS LP , which implies
p Co( FT LP ) p Co( FS LP ) .
satisfies f
f ,
we
p Co( FT ) {(inv, inv) : inv
By
showing
establish
p Co( FT LP ) p Co( FS LP ) {( f , f ) : f LP
that
every
the
point
following
in
Co( FT LP )
relationship:
f } . Furthermore, we can conclude that LP
inv} p Co( FT ) p Co( FS LP ) {( f , f ) : f f } . This result and equations (6) and (7) can be used to show that pT pS . Given that a larger feasible region in the product sales space leads to a higher profit, Dt d D s Ƒ
5. Economic interpretation of Lagrange multipliers. Solving the temporal and spatial Lagrangean duals (equations (4) and (5)) requires finding the optimal Lagrange multipliers Ȝt and Ȝs . We propose a method that exploits the economic interpretation of these multipliers to reduce their search space. For more details we refer the reader to Trotter (2009), available by request from the authors. The main idea is that the dual multipliers correspond to the transfer prices between time periods (for temporal decomposition) or between markets and sites (for spatial decomposition). The transfer prices depend on the set of active constraints. The sales are limited either by production capacity (equation 1d) or by forecast (equation 1f). Our approach for exploiting this interpretation can be outlined as follows. The first step is to solve the linear relaxation of the MILP problem (1) and obtain the values of the optimal dual variables of constraints (1g) for temporal decomposition or (1h) for spatial decomposition. The relaxed MILP solution can overestimate the production capacity since the setup variables can take fractional values. For this reason, constraint (1d) can become active in the MILP solution when it is inactive in the solution to the linear relaxation. However, if constraint (1d) is active in the relaxed solution it is also active in the MILP solution. In a second step, this property of the active set and the values of the dual variables are used to obtain rigorous bounds on the multipliers Ȝt and Ȝs .
6. Computational results We ran an instance of the planning problem (1) that consists of three time periods, markets, production sites, and products. This example has two objectives. The first is to illustrate the difference in the bounds on the full space solution obtained by using temporal and spatial Lagrangean decompositions. The second is to show the effect of using the economic interpretation of the Lagrange multipliers on the computational effort required to find the dual solutions. We solve the Lagrangean dual using a generic implementation of the cutting plane algorithm (Kelley, 1960). We choose this algorithm since it has a rigorous convergence criterion. Other subgradient-based algorithms can be
S. Terrazas-Moreno et al.
much faster and useful in practice but lack the convergence properties of the cutting plane method. The full space model of this example consists of 27 binary variables, 298 continuous variables, and 253 constraints. The full space and decomposed problems were solved using the MILP and LP solver CPLEX 11.2.0 in GAMS 22.9. In Table 1 we show the optimal solutions of the full space problem and both duals. As shown in the table, the bound provided by temporal decomposition is tighter than the one provided by spatial decomposition. Also, the temporal dual requires fewer iterations to converge to its optimal solution than the spatial dual. Note that we are not concerned with comparing CPU times in this example; we assume that decomposition techniques are required as would be the case in larger problem instances. We leave such examples for a full length paper. Table 1. Full space, temporal dual, and spatial dual solutions.
Full space solution* Temporal dual solution Temporal dual solution with economic interpretation of multipliers Spatial dual solution Spatial dual solution with economic interpretation of multipliers
Optimal Profit 41.6 41.8 41.8
Cutting plane iterations** 467 28
42.0 42.3
5000§ 1627
* 0% optimality gap in Branch and Bound algorithm ** 1% tolerance for convergence of cutting plane algorithm § Maximum number of iterations reached with 1.9% GAP
7. Conclusions A multi-site, multi-period production planning problem based on lot-sizing models with setup times was presented in this paper. We assumed that the solution of the resulting mixed-integer linear programming (MILP) formulation requires decomposition techniques. Our work presents a rigorous comparison of the bounds on the optimal solution obtained by temporal and spatial Lagrangean decompositions. Temporal decomposition was shown to be the preferred scheme. We also proposed a procedure based on the economic interpretation of the multipliers for reducing their search space in order to accelerate the convergence of the Lagrangean dual. The results confirm that temporal decomposition provides a tighter dual bound and that less computational effort is required when the economic interpretation of the multipliers is exploited.
References Maranas C. D. and Gupta A.,1999, A Hierachical Lagrangean Relaxation Procedure for Solving Midterm Planning Problems, Ind. Eng. Chem. Res., Vol 38, No 5 pp 1937 – 1947. Jackson. J. R. and Grossmann I. E., 2003, Temporal Decomposition Scheme for Nonlinear Multisite Production Planning and Distribution Models, Ind. Eng. Chem. Res., Vol 42, No 13 pp 3045 – 3055. Geoffrion A. M., 1974, Lagrangean Relaxation for Integer Programming, Math. Prog. Studies, Vol 2, pp 82 – 114. Trotter P. A., 2009, Economic Interpretation of Lagrange Multipliers in Lagrangean Decomposition of a Planning Problem , Technical Report, RWTH Aachen University. Kelley J. E., 1960, The Cutting-Plane Method for Solving Convex Programs, Journal of the SIAM, Vol 8, pp 703 – 712.
A rigorous comparative study of temporal versus spatial Lagrangean decomposition in production planning problems Sebastian Terrazas-Moreno,a Ignacio E. Grossmann,a Philipp Trotterb a
Carnegie Mellon University, 5000 Forbes Ave, Pittsburgh PA 15232, USA, [email protected] b RWTH Aachen University, Templergraben 55, 52056 Aachen, Germany, [email protected]
Abstract Temporal and spatial Lagrangean decompositions are alternatives for solving largescale planning problems. In this paper we compare the strength of the bounds provided by both decompositions on the full space optimal solution. We also use the economic interpretation of the Lagrange multipliers to speed the convergence to the optimal dual solution. Keywords: Production planning, Lagrangean relaxation, temporal decomposition, spatial decomposition, mixed-integer programming.
1. Introduction The optimal planning of a network of manufacturing sites and markets is a complex problem. It involves assigning which products to manufacture in each site, how much to ship to each market and how much to keep in inventory to satisfy future demand. Each site has different production capacities and operating costs, while demand for products varies significantly across markets. Production and distribution planning is concerned with mid to long-term decisions usually involving several months, adding a temporal dimension to the spatial distribution given by the multi-site network. In addition, the production of each product can involve a setup or cleaning time that in some cases is dependent on the sequence of production. When setups and sequence-dependent transitions are included, the optimal planning problem becomes a mixed integer linear programming (MILP) problem. The computational expense of solving large-scale MILPs of this type can be decreased by using decomposition techniques. This paper presents temporal and spatial Lagrangean decompositions that allow the independent solution of time periods, production sites, and markets. The importance of choosing the best between alternative decomposition strategies such as temporal and spatial is discussed in Gupta and Maranas (1999). Although it has been reported that temporal decomposition provides a tighter bound on the full space solution and has faster dual convergence (Jackson and Grossmann, 2003), there is no rigorous generalization of the observed result. One objective of this paper is to compare the bounds obtained through Lagrangean temporal and spatial decompositions for a class of MILPs derived from the lot-sizing problem with setup times. The second objective is to use the economic interpretation of the Lagrange multipliers to provide a reduced dual search space and accelerate the convergence of the optimal multipliers.
S. Terrazas-Moreno et al.
2. Problem Statement There is a set of products that are manufactured in several production sites and shipped to a set of markets where they are sold. Let I , S , and M be the sets of products, production sites, and markets, respectively. There is a finite time horizon divided into a set T of time periods of length Lt . Figure 1 shows the multi-period, multi-site network structure. Month 1
Month 2
Month n -1
Month n
Multi-site Network Market 1 Production Site 1
Market 2
Production Site n-1
Market 3 Production Site 2
Production Site n
Figure 1. Network of production sites and markets for a multi-period planning problem
At each market m M and time period t T there is a forecasted demand d im,t for product i I . The production at each site s S incurs in a manufacturing cost D i,s t and the shipment from any site s to any market m involves a transportation cost J is,t,m . Each site has a limited production rate (production per unit time) T is . Every product has a setup time STi s and setup cost SCis . The products have a sale price of E i,mt at market m . The excess production can be stored as inventory at the production sites, where the inventory holding costs are G i,st . All products that are sent to a market are sold, but the sales cannot be greater than the demand d im,t . The planning problem is to determine the production in each site, the inventory levels, and the amounts of products shipped to each market during each time period, in order to maximize the profit. Profit is defined as sales minus production, shipment, inventory, and setup costs. We assume that the size of the problem prohibits its direct solution and we consider that temporal and spatial lagrangrean decomposition techniques are alternatives to overcome this challenge. The objective of this work is to rigorously compare the strength of both decompositions. We also illustrate the use of the economic interpretation of the Lagrange multipliers involved in the decomposition schemes to provide a reduced search space for the optimal Lagrangean dual. We illustrate our analysis using a planning problem with setup times but no sequence-dependent changeovers. The results will be extended to a more general model with sequence-dependent changeovers in a full length paper.
3. Illustrative Production planning model The planning problem can be formulated as the following MILP model.
A rigurous comparative study of temporal versus spatial Lagrangean decomposition in production planning problems
¦ ¦ ¦ ¦ E im,t sl is,t,m
max S
sS tT iI mM D is,t xis,t G is,t inv is,t tT iI ss
¦¦ ¦
SC is stp is,t ¦ ¦
¦ ¦ J is,t,m f is,t,m
(1a)
tT iI mM sS
s.t. s
¦f
inv i ,t 1 xis,t
s ,m i ,t
invis,t
¦ ST
xis,t d T is,t Lt stpis,t
¦
T is xis,t
iI
(1b)
s
s i stp i ,t
i I , t T , s S
(1c)
d Lt
t T , s S
(1d)
iI
f i ,st,m
¦
i I , t T , s S
mM
slis,t,m
slis,t,m
d
i I , t T , s S , m M (1e)
d im,t
i I , t T , m M
(1f)
i I , t T , s S
(1g)
sS
s
invis,t
inv i ,t
f is,t,m
s ,m i ,t
f
i I , t T , s S , m M (1h)
invis d inviUP ,s
i I , t T , s S
(1i)
s inv i
i I , t T , s S
(1j)
d inviUP ,s
f i s,t,m d f iUP , s ,m
i I , t T , m M , s S (1k)
s ,m
i I , t T , m M , s S (1l)
f i ,t d f iUP , s ,m I u M u|S|x T
f , f , sl
I u S uT
; inv, inv, x
; stp ^0,1`
I u S uT
(1m)
The variable sl is,t,m represents the sales of product i in market m at time t ; the superscript s indicates the production site of origin. The variable invis,t is the level of inventory, xis,t is the amount produced of product i in site s during time t , and f i ,st,m corresponds to the shipments between s and m . The setup variable stp is,t takes the value of one when a product i is manufactured in site s during time t . Equation (1a) represents the maximization of profit. Equation (1b) is the mass balance of each product at each site and time period. Equation (1c) is a setup constraint where T is,t Lt is a valid upper bound for the production of each product. In (1d) the summation of production, setup, and transition times should be less than the length of the time period. Equations (1e) and (1f) are the market constraints. Constraints (1g) and (1h) set the duplicated variables inv,inv and f, f to be equal. Constraints (1i) - (1l) contain upper bounds for inventory and transportation levels. Finally constraint (1m) indicates the domains of all variables.
S. Terrazas-Moreno et al.
4. Temporal and spatial Lagrangean duals The problem can be made decomposable into time periods dualizing constraint (1g). The objective function takes the following form: max S t
¦ ¦ ¦ ¦ E im,t sl is,t,m ¦ ¦ ¦ D is,t xis,t G is,t inv is,t SC is stp is,t
sS tT iI mM
¦¦ ¦ ¦
J is,t,m f is,t,m
tT iI ss
tT iI mM sS
(2a)
s
¦ ¦ ¦ Ot is,t (inv is,t inv i,t )
sS iI tT
On the other hand, the model can be decomposed into individual sites and markets by dualizing constraint (1h), and using the following objective function: max S s
¦ ¦ ¦ ¦ E im,t sl is,t,m ¦ ¦ ¦ D is,t x is,t G is,t inv is,t SC is stp is,t
sS tT iI mM
¦¦ ¦ ¦
J is,,tm f is,t,m
tT iI mM sS
tT iI ss
¦¦ ¦
(3a)
s,m
¦ Os is,,tm ( f i,st,m f i,t )
tT iI mM sS
Problems (2) and (3) can be completed by adding equation (2b)-(2f), (2h)-(2m) and (3b)-(3g), (3i)-(3m) identical to the corresponding constrains in problem (1). Let F , FT , and FS be the feasible regions of problems (1), (2), and (3). The temporal and spatial duals are defined as: Dt
min
max
St
(4)
Ds
min
max
Ss
(5)
Ot ( sl , x, f , f , stp ,inv,inv )FT Os ( sl , x, f , f ,stp ,inv ,inv )FS
The well known result of Geoffrion (1974) establishes the following equivalences:
^(sl, x, f , f , stp, inv, inv) : (sl, x, f , f , stp, inv, inv) Co(FT ^(sl, x, f , f , stp, inv, inv) : (sl, x, f , f , stp, inv, inv) Co(FS
T S D
t
D
s
^max S : (sl, x, f , f , stp, inv, inv) T ` ^max S : (sl, x, f , f , stp, inv, inv) S `
^
LP
) (inv, inv) : inv
LP
) (f, f ): f
^
f
``
inv (6)
``
(7) (8) (9)
where F LP refers to the linear programming relaxation of F . The main result of our analysis is that in general D t d D s . The remainder of this section is devoted to presenting an outline of the derivation of this result. The complete proof will be available in a full length paper. Definition pF proj sl ,stp F
^(sl, stp) : (sl, x, f , f , stp, inv, inv) F `.
A rigurous comparative study of temporal versus spatial Lagrangean decomposition in production planning problems Key Assumption The larger the feasible region in the space of product sales ( sl ), the higher the attainable profit. Theorem: For the mixed integer planning problem presented in this work, temporal decomposition provides a tighter upper bound than spatial decomposition ( D t d D s ). Outline of the proof:
By Fourier-Motzkin elimination, we can obtain the projection of FT LP and FS LP onto the ( sl , stp ) space. The projections show that pFT LP pFS LP , which implies
p Co( FT LP ) p Co( FS LP ) .
satisfies f
f ,
we
p Co( FT ) {(inv, inv) : inv
By
showing
establish
p Co( FT LP ) p Co( FS LP ) {( f , f ) : f LP
that
every
the
point
following
in
Co( FT LP )
relationship:
f } . Furthermore, we can conclude that LP
inv} p Co( FT ) p Co( FS LP ) {( f , f ) : f f } . This result and equations (6) and (7) can be used to show that pT pS . Given that a larger feasible region in the product sales space leads to a higher profit, Dt d D s Ƒ
5. Economic interpretation of Lagrange multipliers. Solving the temporal and spatial Lagrangean duals (equations (4) and (5)) requires finding the optimal Lagrange multipliers Ȝt and Ȝs . We propose a method that exploits the economic interpretation of these multipliers to reduce their search space. For more details we refer the reader to Trotter (2009), available by request from the authors. The main idea is that the dual multipliers correspond to the transfer prices between time periods (for temporal decomposition) or between markets and sites (for spatial decomposition). The transfer prices depend on the set of active constraints. The sales are limited either by production capacity (equation 1d) or by forecast (equation 1f). Our approach for exploiting this interpretation can be outlined as follows. The first step is to solve the linear relaxation of the MILP problem (1) and obtain the values of the optimal dual variables of constraints (1g) for temporal decomposition or (1h) for spatial decomposition. The relaxed MILP solution can overestimate the production capacity since the setup variables can take fractional values. For this reason, constraint (1d) can become active in the MILP solution when it is inactive in the solution to the linear relaxation. However, if constraint (1d) is active in the relaxed solution it is also active in the MILP solution. In a second step, this property of the active set and the values of the dual variables are used to obtain rigorous bounds on the multipliers Ȝt and Ȝs .
6. Computational results We ran an instance of the planning problem (1) that consists of three time periods, markets, production sites, and products. This example has two objectives. The first is to illustrate the difference in the bounds on the full space solution obtained by using temporal and spatial Lagrangean decompositions. The second is to show the effect of using the economic interpretation of the Lagrange multipliers on the computational effort required to find the dual solutions. We solve the Lagrangean dual using a generic implementation of the cutting plane algorithm (Kelley, 1960). We choose this algorithm since it has a rigorous convergence criterion. Other subgradient-based algorithms can be
S. Terrazas-Moreno et al.
much faster and useful in practice but lack the convergence properties of the cutting plane method. The full space model of this example consists of 27 binary variables, 298 continuous variables, and 253 constraints. The full space and decomposed problems were solved using the MILP and LP solver CPLEX 11.2.0 in GAMS 22.9. In Table 1 we show the optimal solutions of the full space problem and both duals. As shown in the table, the bound provided by temporal decomposition is tighter than the one provided by spatial decomposition. Also, the temporal dual requires fewer iterations to converge to its optimal solution than the spatial dual. Note that we are not concerned with comparing CPU times in this example; we assume that decomposition techniques are required as would be the case in larger problem instances. We leave such examples for a full length paper. Table 1. Full space, temporal dual, and spatial dual solutions.
Full space solution* Temporal dual solution Temporal dual solution with economic interpretation of multipliers Spatial dual solution Spatial dual solution with economic interpretation of multipliers
Optimal Profit 41.6 41.8 41.8
Cutting plane iterations** 467 28
42.0 42.3
5000§ 1627
* 0% optimality gap in Branch and Bound algorithm ** 1% tolerance for convergence of cutting plane algorithm § Maximum number of iterations reached with 1.9% GAP
7. Conclusions A multi-site, multi-period production planning problem based on lot-sizing models with setup times was presented in this paper. We assumed that the solution of the resulting mixed-integer linear programming (MILP) formulation requires decomposition techniques. Our work presents a rigorous comparison of the bounds on the optimal solution obtained by temporal and spatial Lagrangean decompositions. Temporal decomposition was shown to be the preferred scheme. We also proposed a procedure based on the economic interpretation of the multipliers for reducing their search space in order to accelerate the convergence of the Lagrangean dual. The results confirm that temporal decomposition provides a tighter dual bound and that less computational effort is required when the economic interpretation of the multipliers is exploited.
References Maranas C. D. and Gupta A.,1999, A Hierachical Lagrangean Relaxation Procedure for Solving Midterm Planning Problems, Ind. Eng. Chem. Res., Vol 38, No 5 pp 1937 – 1947. Jackson. J. R. and Grossmann I. E., 2003, Temporal Decomposition Scheme for Nonlinear Multisite Production Planning and Distribution Models, Ind. Eng. Chem. Res., Vol 42, No 13 pp 3045 – 3055. Geoffrion A. M., 1974, Lagrangean Relaxation for Integer Programming, Math. Prog. Studies, Vol 2, pp 82 – 114. Trotter P. A., 2009, Economic Interpretation of Lagrange Multipliers in Lagrangean Decomposition of a Planning Problem , Technical Report, RWTH Aachen University. Kelley J. E., 1960, The Cutting-Plane Method for Solving Convex Programs, Journal of the SIAM, Vol 8, pp 703 – 712.