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Successive smoothing algorithm for solving large-scale optimization models with fixed cost

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Abstract

This study addresses the solution of large-scale, non-convex optimization problems with fixed and linear variable costs in the objective function and a set of linear constraints. A successive smoothing algorithm (SSA) is developed to solve a non-convex optimization problem by solving a sequence of approximated convex problems. The performance of the SSA is tested on a series of randomly generated problems. The computation time and the solution quality obtained by the SSA are compared to a mixed integer linear programming (MILP) solver (CPLEX) over a wide variety of randomly generated problems. The results indicate that the SSA performs consistently well and produces high-quality near optimal solutions using substantially shorter time than the MILP solver. The SSA is also applied to solving a real-world problem related to regional biofuel development. The model is developed for a “system of systems” that consists of refineries, transportation, agriculture, water resources and crops and energy market systems, resulting in a large-scale optimization problem. Based on both the hypothetical problems and the real-world application, it is found that the SSA has considerable advantage over the MILP solver in terms of computation time and accuracy, especially when solving large-scale optimization problems.

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Abbreviations

\(T_{t,p}^{r,s}\) :

Transport of product \(p\) from node \(r\) to node \(s\) in year \(t\), mass per year

\(N^r \) :

Set of nodes adjacent and connected to node \(r\)

\(D_{t,p}^{r,0} \) :

Product \(p\) at node \(r\) sold to external markets, or dumped, in year \(t\), mass per year

\(S_{t,p}^r \) :

Supply of product \(p\) at node \(r \)in year \(t\), mass per year

\(D_{t,p}^r \) :

Demand for product \(p\) from refinery operations at node \(r \) in year \(t\), mass per year

\(u^{r,s}\) :

Free flow travel time from \(r\) to \(s\), hours

\(\alpha _{\mathrm{time},p} \) :

Cost per unit travel time per unit mass, $ per hour per unit mass

\(K_{t,\mathrm{transport}} \) :

Total cost of transportation in year \(t\), $ per year

\(\alpha _{t,p}^r\) :

Price of product \(p \)in outside markets in year \(t\) at transport node \(r\), $ per unit mass

\(\beta _{\mathrm{ethacorn}} \) :

Mass of ethanol per unit mass of corn

\(\beta _{\mathrm{ethamisc}} \) :

Mass of ethanol per unit mass of miscanthus

\(\beta _{\mathrm{ethacstov}} \) :

Mass of ethanol per unit mass of corn stover

\(\beta _{\mathrm{ddgscorn}} \) :

Mass of DDGS per unit mass of corn

\(C_{t,\mathrm{cellref}}^r \) :

Biomass ethanol refinery capacity at node \(r\) in year \(t\), mass of ethanol

\(C_{t,\mathrm{cornref}}^r \) :

Corn ethanol refinery capacity at node \(r\) in year \(t\), mass of ethanol

\(\Delta \hbox {Cap}_{\mathrm{max}} \) :

Maximum increment in capacity, mass of ethanol

\(Z_{t,\mathrm{cornref}}^r \) :

1 if there is an increase in refinery capacity, 0 otherwise

\(Z_{t,\mathrm{cellref}}^r \) :

1 if there is an increase in refinery capacity, 0 otherwise

\(\hbox {Cap}_{\mathrm{max}} \) :

Maximum capacity, mass of ethanol

\({\textit{ZP}}_{t,\mathrm{cornref}}^r \) :

1 if there is refinery, 0 otherwise

\({\textit{ZP}}_{t,\mathrm{cellref}}^r \) :

1 if there is refinery, 0 otherwise

\(C_{fixed}^{\mathrm{opr}} ,C_{fixed}^{\exp } \) :

Fixed cost for operation and refinery expansion, respectively

\(C_{PUC}^{\mathrm{opr}} ,C_{PUC}^{\exp } \) :

Per unit cost for operation and refinery expansion, respectively

\(X_{t,\mathrm{corn}}^l \) :

Fraction of land parcel \(l\) in corn in year \(t\)

\(X_{t,\mathrm{soyb}}^l \) :

Fraction of land parcel \(l\) in soybean in year \(t\)

\(X_{t,\mathrm{cstov}}^l \) :

Fraction of land parcel \(l\) in corn in year \(t\) where corn stover is also harvested

\(X_{t,\mathrm{misc},a}^l \) :

Fraction of land parcel \(l\) in miscanthus \(a\) years after establishment in year \(t\)

\(A^{l}\) :

Total agriculture area of land parcel \(l\)

\(y_{\mathrm{corn}}^l \) :

Per unit area yield of corn in land parcel \(l\)

\(y_{\mathrm{cstov}}^l \) :

Per unit area yield of corn stover in land parcel \(l\)

\(y_{\mathrm{misc},a}^l \) :

Per unit area yield of miscanthus \(a\) years after establishment in land parcel \(l\)

\({\textit{NP}}^{l}\) :

Set of transportation nodes neighboring land parcel \(l\)

\(Y_{t,q}^l \) :

Crop \(q\) from land parcel \(l\) in year \(t\), mass per year

\(L,U\) :

Lower and upper bounds for the crops ratios

\(X_{\max ,\mathrm{misc}} \) :

Maximum fraction of miscunthus

\({\textit{TL}}_{t,q}^{l,r} \) :

Transport of crop \(q\) from land parcel \(l\) to node \(r\) in year \(t\), mass per year

\({\textit{NP}}^{r}\) :

Set of land parcels neighboring transport node \(r\)

\(\alpha _{t,\mathrm{corn}}^l \) :

Cost of producing corn in land parcel \(l\) in year \(t\), $/unit area

\(\alpha _{t,\mathrm{cstov}}^l \) :

Cost of producing corn stover in land parcel \(l\) in year \(t\), $/unit area

\(\alpha _{t,\mathrm{soyb}}^l \) :

Cost of producing soybean in land parcel \(l\) in year \(t\), $/unit area

\(\alpha _{t,\mathrm{misc},a}^l \) :

Cost of producing miscanthus \(a\) years, in land parcel \(l\) year \(t\), $/unit area

\({\textit{AT}}^l \) :

Total land area of land parcel \(l\) including both agricultural and non-agricultural lands

\(x^{w,l} \) :

Fraction of subwatershed \(w\) overlapping land parcel \(l\)

\({\textit{QO}}_{t,m}^w \) :

Outflow from subwatershed \(w\) month \(m\), in year \(t\), volume of water per unit time

\(X_{t,m,c}^w \) :

Fraction of subwatershed \(w\) in land cover \(c\) in month \(m\), in year \(t\)

\({\textit{WY}}_{m,c}^w \) :

Per unit area streamflow contribution of subwatershed \(w\) in crop \(c\) in month \(m\)

\(A^w \) :

Total area of subwatershed \(w\)

\({\textit{QR}}_{t,m}^w \) :

Withdrawal from subwatershed \(w\) by refineries in month \(m\), in year \(t\)

\(\hbox {US}_w \) :

Set of subwatershed immediately upstream of subwatershed \(w\)

\(\beta _{\mathrm{watmiscref}} \) :

Water consumed per unit mass of miscanthus ethanol

\(\beta _{\mathrm{watcstovref}} \) :

Water consumed per unit mass of corn stover

\(\beta _{\mathrm{watcornref}} \) :

Water consumed per unit mass of corn

\(\hbox {R}_w \) :

Set of refineries withdrawing water from subwatershed \(w\)

\({\textit{QO}}_{\mathrm{min}}^w \) :

Minimum flow allowed

\({\textit{NO}}_{t,m}^w \) :

Mass of nitrate leaving subwatershed \(w\) in month \(m\), in year \(t\)

\({\textit{NY}}_{m,c}^w \) :

Per unit area nitrate contribution of subwatershed \(w\) in crop \(c\) in month \(m\)

\({\textit{NO}}_{m-\mathrm{max}}^w \) :

Monthly maximum nitrate load allowed

\({\textit{NO}}_{y-\mathrm{max}}^w \) :

Yearly maximum nitrate load allowed

\(i\) :

Discount rate

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Author information

Authors and Affiliations

  1. Faculty of Management, Department of Natural Resources and Environmental Management, University of Haifa, Haifa, Israel

    Mashor Housh

  2. Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, 205 N Mathews Ave, Urbana, IL, 61801, USA

    Ximing Cai

Authors
  1. Mashor Housh
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  2. Ximing Cai
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Corresponding author

Correspondence to Ximing Cai.

Appendix 1: Formulation of the model for biofuel development

Appendix 1: Formulation of the model for biofuel development

Objective function:

$$\begin{aligned} \max B=\sum _t \frac{G_{t,\mathrm{sale}} -\left( K_{t,\mathrm{transport}} +K_{t,\mathrm{refinery}} + K_{t,\mathrm{crop}} + K_{t,\mathrm{localtransport}} \right) }{(1+i)^{t-1}} \end{aligned}$$
(14)

The objective of the model [Eq. (14)] is to maximize the total net profit of the biofuel systems, including the profit of selling the products, cost of producing raw material, costs of refinery operations, capital cost of refineries, and the transportation costs of raw materials and end products.

The constraints include the relationships that link different components of the system and environmental and operational constrains.

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Housh, M., Cai, X. Successive smoothing algorithm for solving large-scale optimization models with fixed cost. Ann Oper Res 229, 475–500 (2015). https://doi.org/10.1007/s10479-015-1795-7

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  • DOI: https://doi.org/10.1007/s10479-015-1795-7

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