1 Introduction

Environmental protection and resource recycling continue to challenge municipal solid waste (MSW) managers. The challenges are being complicated with the rapid socio-economic development associated with increasing waste-generation rates and decreasing waste disposal capacities (Maqsood and Huang 2003). Due to the scarcity of land near urban centers and growing opposition from the public with regard to landfill disposal, many communities are putting their efforts toward waste diversion through an integrated solid waste management approach (Huang et al. 2001). Driven by the difficulty of sitting landfills and the recognition that the present levels of land resource consumption are not sustainable, most industrialized jurisdictions have adopted policies for waste diversion to extend the lives of landfills. Waste to energy (WTE) and composting facilities are considered an integrated management system. Optimal schemes for effective utilization of waste-management facilities may vary among different periods due to temporal variations between waste generation rates and available system capacities (Baetz 1990). Thus, identification of desirable expansion schemes and waste flow allocation plans are important aspects in a long-term solid waste management plan.

During the past few decades, the mixed-integer linear programming (MILP) method has been widely used for solving the above problems (Baetz 1990; Huang et al. 1997; Glen 2003; Emam 2006; Chen 2007). Since conventional MILP methods cannot reasonably address the complex uncertainties, a number of interval-parameter MILP (IMILP) methods have been developed and applied in MSW management and planning (Chi and Huang 1998; Huang et al. 1997, 1993a, b). For example, Huang (1993a, b) presented interactive interval mixed integer linear programming (IMILP) approaches and applied them to various practical problems. Chi and Huang (1998) applied the IMILP model to the City of Regina for the long-term planning of integrated solid waste management system under uncertainty. Wu et al. (2006) proposed an inexact nonlinear programming (INLP) model with a nonlinear objective function and linear constraints. The INLP model was applied to a waste management system with economies-of-scale (EOS) effects on system costs for the planning of waste flow allocation and facility operation. These models are useful in dealing with the uncertainties in parameters expressed as intervals and/or the nonlinearity resulted from the economies-of-scale (EOS) effects on the costs of operation and transportation. However, they cannot reflect probabilistic distribution information of a constraint’s right-hand side.

More recently, stochastic programming methods were incorporated into the IMILP framework to address the random and interval information in MSW management problems (Huang et al. 2001; Maqsood and Huang 2003; Maqsood et al. 2004; Li et al. 2006a, b, c; Li and Huang 2006a, b). For instance, Maqsood et al. (2004) developed an inexact two-stage mixed integer linear programming model for waste management under uncertainty. Li et al. (2006a, b) presented an interval-parameter two-stage stochastic integer programming model for environmental systems planning under uncertainty and an inexact two-stage mixed integer linear programming method for solid waste management in the City of Regina. Li et al. (2006b) provided an interval fuzzy two-stage stochastic mixed-integer linear programming for environmental management and planning. Li et al. (2006c) proposed an interval-parameter two-stage chance-constrained mixed integer linear programming method for MSW management under uncertainty. Li and Huang (2006a) applied an interval-stochastic programming model for supporting the analysis of various policy scenarios associated with different levels of economic penalties. These approaches are effective in handling optimization problems where an analysis of policy scenarios is desired and the right-hand side coefficients are random with known probability distributions.

A remarkable limitation of the aforementioned methods is their incapability in reflecting the parameters’ dynamic feature though they may be useful in dealing with a wide range of programming problems. Yet in reality, the parameters may be more complex, which can hardly be expressed as ordinary interval- and/or stochastic-valued parameters. For example, a crisp interval parameters (R) can be defined as [a, b] where a and b are both constants, representing the lower and upper bounds of R, respectively (Moore and Yang 1959). If R varies with the change of its independent variables (say, time), it becomes no longer crisp. Under this situation, a concept of functional interval can be proposed to describe this type of parameter uncertainty (He and Huang 2004; He et al. 2007). If R is a function of time (s), then it can be represented as a functional interval [a(s), b(s)], where a(s) and b(s) are lower- and upper bounded functions of time s. When parameters in the programming problems are expressed as functional intervals, the dynamic feature, i.e. variations of the parameters with time, can be reflected. Note that each constraint with functional interval parameters implies that a set of infinite constraints will be produced as it must be satisfied at any time. This can be named as an inexact semi-infinite programming problem. Presently, there have been a number of studies focused on development and application of the semi-infinite programming (SIP) methods (López and Still 2007; Gómez et al. 2005; Vaz et al. 2004; León et al. 2000, 2004; Geletu et al. 2004; Goberna and López 2002; Stein and Still 2002; Žakovi and Rustem 2002; Wang and Kuo 1999; Fang et al. 1999; Lin et al. 1998). However, none of the previous studies focused on simultaneously deal with problems with parameters expressed as functional intervals and stochastic parameters, which results in the need of developing a new method for overcome this challenge.

One potential approach is to incorporate the stochastic programming, interval programming (IP), integer programming and semi-infinite programming methods into a general optimization framework. Thus, this study aims to develop an inexact stochastic mix-integer linear semi-infinite programming (ISMISIP) method for solid waste management under uncertainties. The specified tasks entails: (1) handling uncertainties in the constraints’ left-hand sides presented as intervals and functional intervals and those in the right-hand sides as intervals, functional intervals and probability distributions, (2) applying the developed method to a case study for MSW management and (3) comparing the ISMISIP with interval-parameter mixed integer linear semi-infinite programming (IMISIP) to demonstrate the advantages of the developed method over conversional approaches when dealing with various uncertainties.

2 Inexact chance-constrained mixed integer linear semi-infinite programming

When coefficients in a programming problem are allowed to be functional intervals, an interval linear semi-infinite programming (ILSIP) problem can be formulated as follows (He et al. 2007):

$$ {\hbox{Max}}\quad f^{\pm} = C^{\pm} X^{\pm} $$
(1a)

subject to:

$$ A^{\pm} (s)X^{\pm} \leq B^{\pm} (s),\quad s = [{\rm sl, su}] $$
(1b)
$$ X^{\pm} \geq 0 $$
(1c)

where A ± (s) ∈ {R ±}m × n, B ± (s) ∈ {R ±}m × 1, C ± ∈ {R ±}1 × n and X ± ∈ {R ±}n × 1; (R ±) denotes the sets with elements being intervals or functional intervals; s is independent variable, ranging between [sl, su]. If some elements in B can be expressed as probability distribution functions, a chance-constrained programming (CCP) problem can be introduced as follows:

$$ \hbox{Max} \quad CX $$
(2a)

subject to:

$$ A (t)X \leq B (t) $$
(2b)
$$ X \geq 0 $$
(2c)

where A(t) and B(t) are sets with random elements defined on a probability space T, t ∈ T. The CCP problem can be solved by converting it to a deterministic version through: (1) fixing a certain level of probability p i ∈ [0,1] for each constraint i, and (2) imposing the condition that the constraint is satisfied with at least a probability of 1 − p i . The feasible solution set is thus subject to the following constraints:

$$\Pr {\left[ {\left\{{t\vert A_{i} (t)X \leq b_{i} (t)} \right\}} \right]} \geq 1 - p_{i},\quad A_{i} (t) \in A(t), \quad b_{i} (t) \in B(t),\,i = 1, 2,\ldots, m $$
(3)

Constraint (3) is generally nonlinear. When the left-hand side coefficients (a ij ) are deterministic and the right-hand side constraints (b i ) are random, it can be transformed to an equivalent linear constraint by introducing a probability of violating constraint (p i ). Thus, constraint (3) can be converted to (Loucks et al. 1981)

$$ A_{i} X \leq b_{i} (t)^{(p_{i})},\quad \forall i $$
(4)

where \(b_{i} (t)^{{(p_{i})}} = F^{{- 1}}_{i} (p_{i}),\) in which F i (b i ) is the cumulative distribution function of b i and p i is the probability of violating the ith constraint. Equation (4) is a linear constraint that can only reflect the case where A is deterministic. If both A and B are uncertain, the constraints become more complicated (Ellis 1991; Infanger 1993; Zare et al. 1995). To address the uncertainties of functional interval and stochastic parameters, an inexact stochastic mixed integer linear semi-infinite programming (ISMISIP) problem is formulated based on the ILSIP and CCP methods:

$$ {\hbox{Max}}\,f^{\pm} = C^{\pm} X^{\pm} $$
(5a)

subject to:

$$ A_{i}^{\pm} (s)X^{\pm} \leq b_{i} (t)^{(p_{i})},\quad s = [{\rm sl, su}],\quad i \in M, \, i\neq r $$
(5b)
$$ A_{r}^{\pm} (s)X^{\pm} \leq B_{r}^{\pm} (s),\quad s = [{\rm sl, su}], \quad r \in M, r\neq i $$
(5c)
$$ X^{\pm} \geq 0 $$
(5d)

Letting c ± j be the jth element of C ± and a ± ij (s i ) be the ith row and jth line element of A ±(s). Corresponding to each s value, there will be a set of intervals for each functional interval. The x ± j is the jth element of X ± and \( b_{i} (t)^{{(p_{i})}} \) represents the corresponding values given the cumulative distribution function of b i and the probability of violating constraint i(p i ). The above model can be solved by converting it into two submodels (He et al. 2007):

Submodel 1:

$$ {\hbox{Max}}f^{+} = {\sum\limits_{j = 1}^k {c_{j}^{+} x_{j}^{+}}} + {\sum\limits_{j = k + 1}^n {c_{j}^{+} x_{j}^{-}}} $$
(6a)

subject to:

$$ {\sum\limits_{j = 1}^k {\vert a_{{ij}} {\hbox{(s)}} \vert ^{-} {\hbox{Sign}}(a_{ij}^{-})x_{j}^{+}}} + {\sum\limits_{j = k + 1}^n {\vert a_{{ij}} {\hbox{(s)}} \vert^{+} {\hbox{Sign}}(a_{ij}^{+})x_{j}^{-}}} \leq b_{i} (t)^{{(p_{i})}},\quad \forall i,i \ne r $$
(6b)
$$ {\sum\limits_{j = 1}^k {\vert a_{{rj}} {\hbox{(s)}} \vert^{-} {\hbox{Sign}}(a_{rj}^{-})x^{+}_{j}}} + {\sum\limits_{j = k + 1}^n {\vert a_{{rj}} {\hbox{(s)}} \vert^{+} {\hbox{Sign}}(a_{rj}^{+})x_{j}^{-}}} \leq b_{r} (s)^{+},\quad \forall r,r \ne i $$
(6c)
$$ x_{j}^{+} \geq 0, \quad x_{j}^{-} \geq 0,\quad \forall j $$
(6d)

Submodel 2:

$$ {\hbox{Max}}\,f^{-} = {\sum\limits_{j = 1}^k {c_{j}^{-} x^{-}}} + {\sum\limits_{j = k + 1}^n {c_{j}^{-} x^{+}}} $$
(7a)

subject to:

$$ {\sum\limits_{j = 1}^k {\vert a_{{ij}} {\hbox{(s)}} \vert^{+} {\hbox{Sign}}(a_{ij}^{+})x_{j}^{-}}} + {\sum\limits_{j = k + 1}^n {\vert a_{{ij}} {\hbox{(s)}} \vert^{-} {\hbox{Sign}}(a_{ij}^{-})x_{j}^{+}}} \leq b_{i} (t)^{{(p_{i})}} \quad \forall i,i \ne r $$
(7b)
$$ {\sum\limits_{j = 1}^k {\vert a_{{rj}} {\hbox{(s)}} \vert^{+} {\hbox{Sign}}(a^{+}_{{rj}})x_{j}^{-}}} + {\sum\limits_{j = k + 1}^n {\vert a_{{rj}} {\hbox{(s)}} \vert^{-} {\hbox{Sign}}(a_{rj}^{-})x_{j}^{+}}} \leq b_{r} (s)^{-} \quad \forall r,r \ne i $$
(7c)
$$ x_{j}^{+} \geq 0,\, x_{j}^{-} \geq 0,\quad \forall j $$
(7d)
$$ x_{j}^{-} \leq x^{+}_{{j_{\rm opt}}},\quad j = 1, 2, \ldots, k $$
(7e)
$$ x_{j}^{+} \geq x^{-}_{{j_{\rm opt}}},\quad j = k + 1, k + 2, \ldots, n $$
(7f)

where x + j_opt and x − j_opt are solutions of models (6) and (7), respectively. The sign(·) is a signal function, which is defined as

$$ {\rm Sign}(x^{\pm}) = \left\{ {\begin{array}{*{20}c} 1 & (x^{\pm} \geq 0)\\ - 1 & (x^{\pm} < 0) \end{array}}\right. $$
(8)

By combining solutions to models (6) and (7), the optimal interval solution of the ISMISIP can be represented as:

$$ f_{\rm opt}^{\pm} =\left [f_{\rm opt}^{-}, f_{\rm opt}^{+}\right ], $$
(9)
$$ x_{j{\,{\rm opt}}}^{\pm} =\left [x_{j\,{\rm opt}}^{-},x_{j\,{\rm opt}}^{+}\right ], \quad \forall j $$
(10)

3 Application

3.1 Overview of the study system

The developed ISMISIP method is applied to a hypothetical MSW management planning system. The system consists of three cities (Fig. 1), where three waste disposal facilities are available, including one landfill, one waste-to-energy (WTE) facility and one composting facility. The planning horizon is divided into three time periods, with each period having 5 years. At the beginning of the planning horizon, capacities of the landfill, WTE and composting facilities are [3.4, 4.6] × 106 tonnes, [100, 120] and [200, 232] tonnes/day, respectively. In addition, the WTE facility generates residues of approximately 30% (on a mass basis) for the incoming waste stream and is transported to the landfill directly. The revenue from the WTE facility is 15–25 $/tonne. The composting facility generates approximately 15% of residue (on a mass basis) from the incoming waste stream and is transported directly to the landfill. Revenue from the composting facility is between 10 and 15 $/tonne.

Fig. 1
figure 1

Hypothetical study municipalities and waste management facilities

Full size image

Waste transporting and operating costs in the three periods vary temporally and spatially, which are assumed to be intervals (Table 1). Since the capacity of each facility may be insufficient to deal with the incoming waste flow, expansions are optional. At the same time, three types of expansion resolutions for the WTE and composting facilities can be made, with different facilities having different expansion capacities and costs. Table 2 shows the expansion options and the corresponding costs for the three facilities. Table 3 gives the cumulative distribution functions for the existing capacities of three facilities: landfill, WTE and composting.

Table 1 Costs for waste transportation and treatment
Full size table
Table 2 Capacity expansion options and expansion costs for facilities (s ∈ [0, 5])
Full size table
Table 3 Distribution information of TL (106 tonnes), TW (tonnes/day), and TC (tonnes/day)
Full size table

The MSW generation rates, varying among different cities and different periods, are assumed to be functional intervals (Table 4), with their lower- and upper-bounds being functions of an independent variable (time, ranging from 0 to 5). The capacity expansion options for landfill are also assumed to be functional intervals associated with time (Table 2). Actually, they may be affected by many other factors related to various social, economic and human activities in a waste-collection district. However it is difficult to quantify these complex relations by simply fixing their lower and upper bounds. If the lower and upper bounds of these parameters are regarded as functions of time, it would be more robust in representing not only the uncertain but also the dynamic features of these factors.

Table 4 Waste generation rate in each city during three periods WG± jk (s)
Full size table

The functional intervals can be obtained through the time series and multivariate analysis methods (Gupta et al. 2006; Kowalski et al. 2004; Panagopoulos et al. 2002; Strait 1994). Figure 2 shows an illustration of a functional interval. For each of the possible s value, there will be a set of intervals corresponding to it; for example, in period 1, when s is 1.2, WG ± 2k equals [348.36, 402.36], while it becomes [348.45, 402.45] when s is 1.5. It is observed that prolongation of the time can increase the uncertainty of the parameter. Figure 3 shows the relationship between probability density functions and functional interval parameters. If the parameter is an interval, then there is only one probability density function with a determinate lower bound (a) and an upper one (b). If the parameter is a functional interval, then there is a group of probability density functions with an interval lower bound [a(s)−, a(s)+] and upper bound [b(s)−, b(s)+]. Each line represents a random event sampled at time s.

Fig. 2
figure 2

A illustration of functional interval

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Fig. 3
figure 3

Relationship of probability density functions and functional interval parameters

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3.2 Modeling formulation

When considering a regional MSW management system with limited resources of human power, waste-disposal capacity, waste-generation rates and operating fund, a waste manager encounters the problem of how to allocate the MSW from the cities to the disposal facilities over a planning horizon. This problem can be formulated as an ISMISIP model with the objective of minimizing the expected value of net system cost. The model should take into account the following issues:

  1. 1.

    selecting the preferred capacity expansion schemes for waste management facilities during different periods;

  2. 2.

    allocating waste flows under different management conditions, in order to minimize the net system cost;

  3. 3.

    incorporating the independent variable (time) into the planning problem with the least system disruption (Wang and Kuo 1999).

Under this consideration, the model (9) is formulated as follows:

  1. 1.

    Objective function (9a) = (a) + (b) + (c) − (d) + (e).

    1. (a)

      Waste collection and transportation costs

      $$ 365 \times 5 \times {\sum\limits_{i = 1}^3 {\sum\limits_{j = 1}^3 {\sum\limits_{k = 1}^3 {{\rm TR}^{\pm}_{{ijk}} \cdot x_{ijk}^{\pm}}}}} $$
    2. (b)

      Facility operating costs

      $$ 365 \times 5 \times {\sum\limits_{i = 1}^3 {\sum\limits_{j = 1}^3 {\sum\limits_{k = 1}^3 {{\rm OP}^{\pm}_{{jk}} \cdot x_{ijk}^{\pm}}}}} $$
    3. (c)

      Facility capital costs

      $$ {\sum\limits_{l = 1}^3 {{\sum\limits_{k = 1}^3 {({\rm FTW}_{lk}^{\pm} \cdot {\rm YW}^{\pm}_{{lk}}}}}} + {\rm FTC}_{lk}^{\pm} \cdot {\rm YC}_{lk}^{\pm}) + {\sum\limits_{k = 1}^3 {{\rm FLC}_{k}^{\pm} \cdot Z_{k}^{\pm}}} $$
    4. (d)

      Revenues from WTE and composting facilities

      $$ {\sum\limits_{i = 1}^3 {{\sum\limits_{j = 1}^2 {{\sum\limits_{k = 1}^3 {{\rm RE}_{jk}^{\pm} \cdot x_{ijk}^{\pm}}}}}}} $$
    5. (e)

      Residual costs

      $$ {\sum\limits_{i = 1}^3 {{\sum\limits_{j = 1}^2 {{\sum\limits_{k = 1}^3 {{\rm RF}_{j} \cdot \left({\rm FT}^{\pm}_{{jk}} + {\rm OP}_{3k}^{\pm}\right) \cdot x_{ijk}^{\pm}}}}}}} $$
  2. (2)

    Capacity limitation constraints (9b)–(9d).

    $$ {\sum\limits_{i = 1}^3 {x_{{ij{k}^{\prime}}}^{\pm} - {\sum\limits_{l = 1}^3 {\sum\limits_{k = 1}^{{k}^{\prime}} {{\rm YW}_{lk}^{\pm} \cdot \Updelta {\rm TW}_{l}^{\pm}}}}}} \leq {\rm TW}_{j}^{p_{\rm TW}} \quad(j = 1;\; {k}^{\prime}= 1, 2, 3) $$
    (9b)
    $$ {\sum\limits_{i = 1}^3 {x_{ij{k}^{\prime}}^{\pm} - {\sum\limits_{l = 1}^3 {\sum\limits_{k = 1}^{{k}^{\prime}} {{\rm YC}_{lk}^{\pm} \cdot \Updelta {\rm TC}_{l}^{\pm}}}}}} \leq {\rm TC}_{j}^{p_{\rm TC}}\quad (j = 2;\; {k}^{\prime}= 1, 2, 3) $$
    (9c)
    $$ 365 \times 5 \times {\sum\limits_{i = 1}^3 {\sum\limits_{k = 1}^{{k}^{\prime}} \left({x_{i3k}^{\pm}} + {\rm RF} \cdot {\sum\limits_{j = 1}^2 {x_{ijk}^{\pm}}}\right)}} - \Updelta {\rm TL}^{\pm} (s) \cdot {\sum\limits_{k = 1}^{{k}^{\prime}} {Z_{k}^{\pm}}} \leq {\rm TL}^{{p_{{\rm TL}}}} \quad({k}^{\prime}= 1, 2, 3) $$
    (9d)
  3. (3)

    Mass balance constraints (9e)

    $$ {\sum\limits_{j = 1}^3 {x_{ijk}^{\pm}}} \geq WG_{ik}^{\pm} (s)\quad (i = 1, 2, 3; k = 1, 2, 3) $$
    (9e)
  4. (4)

    Technical constraints (9f–9o)

    $$ {\sum\limits_{l = 1}^3 {YW_{lk}^{\pm}}} \leq 1\quad (k = 1, 2, 3) $$
    (9f)
    $$ {\sum\limits_{l = 1}^3 {YC_{lk}^{\pm}}} \leq 1 \quad(k = 1, 2, 3) $$
    (9g)
    $$ {\sum\limits_{k = 1}^3 {Z_{k}^{\pm}}} \leq 1 $$
    (9h)
    $$ {\rm YW}_{lk}^{\pm} \leq 1\quad ({\rm YW}_{lk}^{\pm}\,\hbox{are integer numbers},\; l = 1, 2, 3;\; k = 1, 2, 3) $$
    (9i)
    $$ {\rm YC}_{lk}^{\pm} \leq 1\quad ({\rm YC}_{lk}^{\pm} \,\hbox{are integer numbers},\; l = 1, 2, 3;\; k = 1, 2, 3) $$
    (9j)
    $$ Z_{k}^{\pm} \leq 1\quad (Z_{k}^{\pm} \,\hbox{are integer numbers},\; k = 1, 2, 3) $$
    (9k)
    $$ x_{ijk}^{\pm} \geq 0\quad \forall i, j, k $$
    (9l)
    $$ {\rm YW}_{lk}^{\pm} \geq 0\quad \forall l, k $$
    (9m)
    $$ {\rm YC}_{lk}^{\pm} \geq 0\quad \forall l,k $$
    (9n)
    $$ Z_{k}^{\pm} \geq 0\quad \forall k $$
    (9o)

where s is an independent variable representing time with the range of [0, 5]. Model (9) can be converted into the following two submodels (He et al. 2007)

Submodel 1:

$$ \begin{aligned} \hbox{Min} f^{-} & = 365 \times 5 \times \left[{\sum\limits_{i = 1}^3 {{\sum\limits_{j = 1}^3 {{\sum\limits_{k = 1}^3 {({\rm TR}_{ijk}^{-} + {\rm OP}_{jk}^{-})}}}}}} \cdot x_{ijk}^{-} + {\sum\limits_{i = 1}^3 {{\sum\limits_{j = 1}^2 {{\sum\limits_{k = 1}^3 {({\rm RF}_{j} \cdot ({\rm FT}_{jk}^{-} + {\rm OP}_{3k}^{-}) - {\rm RE}_{jk}^{+}) \cdot x_{ijk}^{-}}}}}}}\right]\\ \quad & + {\sum\limits_{l = 1}^3 {{\sum\limits_{k = 1}^3 {({\rm FTW}_{lk}^{-} \cdot {\rm YW}_{lk}^{-} + {\rm FTC}_{lk}^{\pm} \cdot {\rm YC}_{lk}^{\pm})}}}} + {\sum\limits_{k = 1}^3 {{\rm FLC}_{k}^{-} \cdot Z_{k}^{-}}} \end{aligned} $$
(10a)

subject to:

$$ {\sum\limits_{i = 1}^3 {x_{ij{k}^{\prime}}^{-} - {\sum\limits_{l = 1}^3 {{\sum\limits_{k = 1}^{{k}^{\prime}} {YW_{lk}^{-} \cdot \Updelta{\rm TW}_{l}^{+}}}}}}}(s) \leq {\rm TW}_{j}^{p_{\rm TW}} \quad(j = 1; \;{k}^{\prime}= 1, 2, 3) $$
(10b)
$$ {\sum\limits_{i = 1}^3 {x_{ij{k}^{\prime}}^{-} - {\sum\limits_{l = 1}^3 {{\sum\limits_{k = 1}^{{k}^{\prime}} {{\rm YC}_{lk}^{-} \cdot \Updelta {\rm TC}_{l}^{+}}}}}}}(s) \leq {\rm TC}_{j}^{p_{\rm TC}} \quad(j = 2;\; {k}^{\prime}= 1, 2, 3) $$
(10c)
$$ 365 \times 5 \times {\sum\limits_{i = 1}^3 {{\sum\limits_{k = 1}^{{k}^{\prime}} {(x_{i3k}^{-}}}}} + {\rm RF} \cdot {\sum\limits_{j = 1}^2 {x_{ijk}^{-}}}) - \Updelta {\rm TL}^{+} (s) \cdot {\sum\limits_{k = 1}^{{k}^{\prime}} {Z_{k}^{-}}} \leq {\rm TL}^{{(p_{{\rm TL}})}} ({k}^{\prime}= 1, 2, 3) $$
(10d)
$$ {\sum\limits_{j = 1}^3 {x_{ijk}^{-}}} \geq WG_{ik}^{-} (s) \quad (i = 1, 2, 3;\; k = 1, 2, 3) $$
(10e)
$$ {\sum\limits_{l = 1}^3 {YW_{lk}^{-}}} \leq 1 \quad (k = 1, 2, 3) $$
(10f)
$$ {\sum\limits_{l = 1}^3 {YC_{lk}^{-}}} \leq 1\quad (k = 1, 2, 3) $$
(10g)
$$ {\sum\limits_{k = 1}^3 {Z_{k}^{-}}} \leq 1 $$
(10h)
$$ YW_{lk}^{-} \leq 1 \quad (YW_{lk}^{\pm}\,\hbox{are integer numbers}, l = 1, 2, 3;\; k = 1, 2, 3) $$
(10i)
$$ {\rm YC}_{lk}^{-} \leq 1\quad (YC_{lk}^{\pm} \,\hbox{are integer numbers}, l = 1, 2, 3;\; k = 1, 2, 3) $$
(10j)
$$ Z_{k}^{-} \leq 1\quad (Z_{k}^{\pm} \,\hbox{are integer numbers}, k = 1, 2, 3) $$
(10k)
$$ x_{ijk}^{-} \geq 0,\quad \forall i,j,k $$
(10l)
$$ {\rm YW}_{lk}^{-} \geq 0\quad \forall l,k $$
(10m)
$$ {\rm YC}_{lk}^{-} \geq 0\quad \forall l,k $$
(10n)
$$ Z_{k}^{-} \geq 0\quad \forall k $$
(10o)

Submodel 2:

$$ \begin{aligned} \hbox{Min} f^{+} &= 365 \times 5 \times \left[{\sum\limits_{i = 1}^3 {{\sum\limits_{j = 1}^3 {{\sum\limits_{k = 1}^3 {({\rm TR}_{ijk}^{+} + {\rm OP}_{jk}^{+})}}}}}} \cdot x_{ijk}^{+} + {\sum\limits_{i = 1}^3 {{\sum\limits_{j = 1}^2 {{\sum\limits_{k = 1}^3 {({\rm RF}_{j} \cdot ({\rm FT}_{jk}^{+} + {\rm OP}_{3k}^{+}) - {\rm RE}_{jk}^{-}) \cdot x_{ijk}^{+}}}}}}}\right]\\ \quad & + {\sum\limits_{l = 1}^3 {{\sum\limits_{k = 1}^3 {({\rm FTW}_{lk}^{+} \cdot {\rm YW}_{lk}^{+} + {\rm FTC}_{lk}^{+} \cdot {YC}_{lk}^{+})}}}} + {\sum\limits_{k = 1}^3 {{\rm FLC}_{k}^{+} \cdot Z_{k}^{+}}} \end{aligned} $$
(11a)

subject to:

$$ {\sum\limits_{i = 1}^3 {x_{ij{k}^{\prime}}^{+} - {\sum\limits_{l = 1}^3 {{\sum\limits_{k = 1}^{{k}^{\prime}} {{\rm YW}_{lk}^{+} \cdot \Updelta {\rm TW}_{l}^{-}}}}}}}(s) \leq {\rm TW}_{j}^{p_{\rm TW}} \quad (j = 1;\; {k}^{\prime}= 1, 2, 3) $$
(11b)
$$ {\sum\limits_{i = 1}^3 {x_{ij{k}^{\prime}}^{+} - {\sum\limits_{l = 1}^3 {{\sum\limits_{k = 1}^{{k}^{\prime}} {{\rm YC}_{lk}^{+} \cdot \Updelta {\rm TC}_{l}^{-}}}}}}}(s) \leq {\rm TC}_{j}^{p_{\rm TC}} \quad (j = 2; \;{k}^{\prime}= 1, 2, 3) $$
(11c)
$$ 365 \times 5 \times {\sum\limits_{i = 1}^3 {{\sum\limits_{k = 1}^{{k}^{\prime}} {(x_{i3k}^{+}}}}} + {\rm RF} \cdot {\sum\limits_{j = 1}^2 {x_{ijk}^{+}}}) - \Updelta {\rm TL}^{-} (s) \cdot {\sum\limits_{k = 1}^{{k}^{\prime}} {Z_{k}^{+}}} \leq {\rm TL}^{{(p_{{\rm TL}})}} \quad ({k}^{\prime}= 1, 2, 3) $$
(11d)
$$ {\sum\limits_{j = 1}^3 {x_{ijk}^{+}}} \geq WG_{ik}^{+} (s),\quad (i = 1, 2, 3;\; k = 1, 2, 3) $$
(11e)
$$ {\sum\limits_{l = 1}^3 {{\rm YW}_{lk}^{+}}} \leq 1\quad (k = 1, 2, 3) $$
(11f)
$$ {\sum\limits_{l = 1}^3 {{\rm YC}_{lk}^{+}}} \leq 1 \quad (k = 1, 2, 3) $$
(11g)
$$ {\sum\limits_{k = 1}^3 {Z_{k}^{+}}} \leq 1 $$
(11h)
$$ {\rm YW}_{lk}^{+} \leq 1 \quad({\rm YW}_{lk}^{\pm} \,\hbox{are integer numbers}, l = 1, 2, 3;\; k = 1, 2, 3) $$
(11i)
$$ {\rm YC}_{lk}^{+} \leq 1\quad ({\rm YC}_{lk}^{\pm}\,\hbox{are integer numbers}, l = 1, 2, 3;\; k = 1, 2, 3) $$
(11j)
$$ Z_{k}^{+} \leq 1,\quad (Z_{k}^{\pm} \,\hbox{are integer numbers}, k = 1, 2, 3) $$
(11k)
$$ 0 \leq x_{ijk}^{+} \leq x_{ijk,{\rm opt}}^{-}, \quad \forall i,j,k $$
(11l)
$$ YW_{lk}^{+} \geq 0\quad \forall l,k $$
(11m)
$$ YC_{lk}^{+} \geq 0\quad \forall l,k $$
(11n)
$$ Z_{k}^{+} \geq 0\quad \forall k $$
(11o)

The ISMISIP programming model allows the coefficients to be expressed as probability distribution function, interval, and functional intervals. To handle the stochastic coefficients, a chance-constrained programming approach is applied. The inexact programming and semi-infinite programming methods are used for dealing with interval and functional interval coefficients, respectively. In addition, binary decision variables are integrated to identify whether the expansions would be required or not, thus a mixed integer programming problem is formulated.

3.3 Results analysis

In this case two scenarios are taken into account. In scenario 1, expansions of the WTE and composting facilities are considered; in scenario 2, expansions of the landfill, WTE and composting facilities are permitted. After solving models (10) and (11), the solutions are acquired as shown in Tables 5, 6, 7, 8, 9, 10 and 11. The solutions are provided under different levels of risk of violating the constraints. The integer solutions are listed in Table 11 (those equal to zero are not presented).

Table 5 Waste flow quantity from city i to facility j in period k and the system cost under p i  = 0.01
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Table 6 Waste flow quantity from city i to facility j in period k and the system cost under p i = 0.05
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Table 7 Waste flow quantity from city i to facility j in period k and the system cost under p i  = 0.1
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Table 8 Waste flow quantity from city i to facility j in period k and the system cost under p i  = 0.25
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Table 9 Waste flow quantity from City i to facility j in period k and the system cost under different p i levels in scenario 1
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Table 10 Waste flow quantity from city i to facility j in period k and the system cost under different p i level in scenario 2
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Table 11 Integer solutions for facility expansion
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Tables 5, 6, 7 and 8 give the solutions obtained from the ISMISIP model. Tables 9, 10 indicate the optimal waste flows under different p i levels (in scenarios 1 and 2). It is indicated from Table 9 that, under scenario 1, the waste flow allocation patterns vary between different p i levels due to the temporal and spatial variations of waste management conditions over the planning horizon under given uncertainty inputs. The cost would be [328, 558], [321, 535], [318, 523] and [314, 514] under scenario 1 when p i  = 0.01, 0.05, 0.1 and 0.25, respectively (Table 9). This indicates that the willingness to accept high level of probability of violating constraints would guarantee a low system cost but a high risk of violating the allowable criterion; and a strong desire to acquire low risk of violating the criterion would run into a high system cost. Waste from city 1 should be shipped to either WTE facility or the landfill. Only under p i = 0.01 during period 1, waste from city 1 would be shipped to the landfill less often than other p i levels and some of the waste would be shipped to a composting facility but under other p i levels no waste would be shipped to the composting facility. The solutions of x ± 211 to x ± 223 = 0 (waste flow from the city 2 to WTE facility and composting facility during three periods) indicate that those from city 2 should be transported to landfill. However, no waste from city 2 would be transported to the WTE and composting facilities. This might be due to the high disposal cost at the WTE and composting facilities, such that the majority of the generated waste would be allocated to the landfill. Those from city 3 should be shipped half to the composting facility, with the remainder would be delivered to the WTE facility. However, no waste from city 3 would be transported to the landfill. Under scenario 2, waste from cities 1 and 2 should be shipped to the landfill and waste from city 3 should be mainly transported to the WTE and composting facilities.

During period 1, most waste generated by city 1 should be transported to the landfill in scenario 1 with a value within the range [206, 226] tonnes/day under p i = 0.1. A small amount of waste should be shipped to the WTE facility with a value of 8 tonnes/day under p i = 0.1. All the waste should be shipped to the landfill under scenario 2 with a value between [214, 234] tonnes/day under p i = 0.1. Waste from city 2 would be mainly transported to the landfill. For example, waste flowing into the landfill from city 2 is 363 tonnes/day under p i = 0.1. City 3 would contribute most of the waste to the composting facility in both scenarios but only a small amount of waste should be shipped to the WTE facility with a value between [0, 28] tonnes/day in scenario 1 under p i  = 0.01. City 3 would contribute approximately 90 percent of the waste flow into the composting facility and 10% into the WTE facility under scenario 1.

In period 2, the optimal waste flow patterns in cities 1 and 3 would be different from those in period 1. Approximately 65% of the waste from city 1 would be transferred to the composting facility in scenario 1, with the remainder would be sent to the WTE facility. City 3 would contribute approximately 60% of the waste flowing into the composting and 40% into the WTE facility in scenario 1 under p i = 0.01. Under scenario 2, waste from city 1 would be shipped to the landfill. City 3 would contribute 70% of the waste flowing into the composting facility and 30% into the WTE facility in scenario 2. All the waste from city 2 would be transported to the landfill in both scenarios.

In period 3, city 1 would contribute most of the waste to the landfill under both scenarios. When p i = 0.25, the transported waste would be in the range of [269, 273] and [269, 281] tonnes/day under scenarios 1 and 2, respectively. The majority of the waste from city 2 would be shipped into the landfill, with the amount of flow between [422, 432] tonnes/day under both scenarios when p i  = 0.25. In addition, most of the waste generated by city 3 would be shipped to the composting facility. The amounts would be between 178 and 354 tonnes/day under both scenarios. In summary, most waste to the WTE facility would be from city 1, the waste to the landfill would be from city 2 and the waste to the composting facility would be from city 3.

Figure 7 displays the relationships between p i levels and cost under the two scenarios. It is also indicated that scenario 2 corresponds to a lower system cost than scenario 1. For example, the lower bound for the cost in scenario 2 would be 3.2 × 106 dollars, which is less than that in scenario 1, 3.28 × 106 dollars. The upper bound costs would be 5.47 × 106 and 5.58 × 106 dollars under the two scenarios, respectively. Therefore, to ensure a low system cost, scenario 2 is preferred to scenario 1 if the landfill expansion is permitted.

Table 11 shows the solutions of facility expansion obtained from the ISMISIP model. It can be found that under scenario 1, when p i = 0.05, the WTE facility should be expanded once in period 1 and a further expansion is desired in period 2; however, since sufficient capacities have been developed in the previous periods, no expansion should be carried out in period 3; under p i  = 0.01, 0.1 and 0.25, the WTE facility should be expanded once in period 2 and a further expansion is desired in period 3; however, no expansion should be carried out in period 1. In comparison, the composting facility should be expanded once in period 1 when p i = 0.01, while no expansion should be carried out in this period when p i = 0.05, 0.1 and 0.25. No expansion should be carried out in period 2 under all p i levels. The composting facility should be expanded once in period 3 when p i = 0.05, 0.1 and 0.25; however, no expansion should be carried out in period 3 when p i = 0.01.

Under scenario 2, because the landfill expansion is available, the expansion plan is different from that under scenario 1. The WTE facility should be expanded once in period 3 when p i = 0.01; but no expansion should be carried out in periods 1 and 2 when p i = 0.01. The WTE facility should be expanded once in period 2 when p i = 0.05, 0.1 and 0.25; but no expansion should be carried out in periods 1 and 3. In contrast, the composting facility should be expanded once in period 3 under all p i levels; but no expansion should be carried out in periods 1 and 2. In addition, the landfill should be expanded, with the possible lowest expansion capability of 1.548 × 106 tonnes (when the variable s is equal to 0) and the highest capability of 2.501 × 106 tonnes (when the variable s is equal to 5). The landfill should be expanded once in period 1 when p i = 0.01 and 0.05; but no expansion should be carried out in periods 2 and 3, since sufficient capacities have been developed in the previous periods, but the landfill should be expanded once in period 3 when p i = 0.1  and 0.25, because no expansion was carried out in periods 1 and 2.

Some detailed expansion schemes for the facilities under different expansion options, p i levels and scenarios are illustrated in Figs.  4, 5, 6. Figure 4 shows the expansion schemes for the WTE facility under scenario 1 when p i = 0.25. It is indicated that when p i = 0.25 in period 1 (under scenario 1), the WTE facility should not be expanded (i.e. YW11 = 0). In comparison, the WTE facility should be expanded with an increment of 100 tonnes per day in period 2 (i.e. YW12 = 1). In period 3, an increment of [0, 100] tonnes per day indicates that under very optimistic settings, no expansion would be necessary; under disadvantageous conditions, an expansion of up to 100 tonnes should be implemented per day. Figure 5 exhibits the expansion planning for the composting facility under scenario 1 (when p i = 0.1). It is shown that the composting facility capacity should not be expanded between periods 1 and 2. However, its capacity should be increased by 150 tonnes per day in period 3. Figure 6 shows that, in period 1, the landfill should be expanded by [0, 2.50] × 106 tonnes, demonstrating that no expansion are necessary under very optimistic settings but an expansion of up to 2.5 × 106 tonnes need to be implemented under disadvantageous conditions (Fig. 7).

Fig. 4
figure 4

Solutions of WTE capacity expansion under scenario 1 in pi = 0.25

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Fig. 5
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Solutions of composting capacity expansion under scenario 1 in pi = 0.1

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Fig. 6
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Solutions of landfill capacity expansion under scenario 2 in pi = 0.01

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Fig. 7
figure 7

The relationship among scenario, p i , and costs

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Thus, when the decision scheme tends toward f − under advantageous system conditions, it may be appropriate not to expand the landfill; but when the scheme tends toward f + under disadvantageous conditions, it may be suitable to increase the capacity by 2.5 × 106 tonnes. The objective function value (f ± opt) of $[3.14, 5.14] × 106 provides two extremes of the net system cost under the optimal solution for the entire time horizon of 15 years. As the actual value of each variable or parameter vary between its two bounds, the system cost may change correspondingly between f − opt and f + opt with a variety of reliability levels.

The problem can also be solved using an interval-parameter mixed integer linear semi-infinite programming (IMISIP) method by letting the right-hand side coefficients in the ISMISIP model be intervals or functional intervals. Table 12 presents the solutions obtained from the ISMISIP model under different p i levels and from the IMISIP model. Since the random variable b i is represented as an interval of b ± i , its statistical information is not reflected in the IMISIP model. Therefore, only one set of interval solutions is generated without directly accounting for the probability of violating the constraints. For example, in terms of IMISIP, the solution of the waste flow from city 2 to landfill in period 2 (x ± 232) is [374, 426]; while in terms of ISMISIP, they are [380, 420], [386, 414], [389, 411] and [394, 406] under the p i levels of 0.01, 0.05, 0.1 and 0.25, respectively.

Table 12 Waste flow quantity from ISMISIP method compares with solutions from IMISIP method
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The features of the ISMISIP model can be found through the case study. Model (9) represents an ISMISIP problem where the coefficients are allowed to be functional interval, interval, and constant numbers. Compared with those inexact programming methods without introducing functional interval coefficients, the model has the following advantages that: (1) since parameters are represented as functional intervals, the parameter’s dynamic feature (i.e., the constraint should be satisfied under all possible levels within its range) can be reflected; (2) it is applicable to practical problems as the solution method does not generate more complicated intermediate models (He et al. 2007). Also, capacities of the landfill, WTE and composting facilities are treated as stochastic numbers, with the cumulative distribution functions being provided. To solve the model, the chanced-constrained programming is used to transform it into an equivalent interval semi-infinite mixed integer programming problem under a given p i level. The results show that the improved capabilities of the ISMISIP model in (1) identifying schemes regarding to the waste allocation and facility expansions with a minimized system cost, and (2) addressing tradeoffs among environmental, economic and system reliability level.

4 Conclusions

Functional intervals are presented as more extensive uncertain numbers than intervals. The numbers are described with their lower and upper bounds being functions of independent variables. Consequently, they have characteristics of both intervals and functions. By introducing functional interval and stochastic parameters into the programming problem, an inexact chance-constrained mixed integer semi-infinite programming method is developed based on interval, chance-constrained, mix integer and semi-infinite programming. The method is applied to a solid waste management system with three cities and three disposal facilities in three periods. Two scenarios are considered: one allows for expansions of the WTE and composting facilities and the other considers expansions of the WTE, composting and landfill facilities. The ISMISIP model can assist decision makers in (1) identifying schemes regarding to the waste allocation and facility expansions with a minimized system cost, and (2) addressing tradeoffs among environmental, economic and system reliability level. Though the method is applied to a MSW management problem, it could also be applicable in many other fields such as water resources management as the solution method does not generate more complicated intermediate models.

It is observed that the interval width of the system cost obtained from the IMISIP model is larger than that from the ISMISIP model. This is because that a large number of potential constraints are neglected by the IMISIP model. Thus, to ensure decisions are made with more safety, the ISMISIP model would be preferred prior to the IMISIP one under uncertainty expressed as probability distribution information. In addition, the application of the ISMISIP model to the solid waste management can be extended into many other environmental management systems to help managers make decisions under complex uncertainties.

In addition to constraints in model (9), other potentially important ones can be considered including limitations of the allowable time of waiting for services of the waste disposal, as well as the regulated air pollutants emissions and noise releasing. This is because the rise of the waiting time could affect the amount of the delivered waste in the next period, which may lead to the increase of the system cost. Implementation of practices for controlling the air pollutant and noises emissions will also lead to the growth of the system cost. These constraints were not taken into account due to two reasons. The first is that these constraints have less significant effects on the optimal system cost according to our investigations, compared with those of the facility capacity and generation rate. The second is the possible introduction of the complex nonlinear terms into the objective and constraints of the model. This issue will be discussed in future studies by transforming the nonlinear programming model into approximated linear one.