Monte Carlo bounding techniques for determining solution quality in stochastic programs

Mak, Wai-Kei; Morton, David P.; Wood, R. Kevin

doi:10.1016/s0167-6377(98)00054-6

Cited by 562 publications

(401 citation statements)

References 28 publications

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“…Table 5 shows a decrease in both these terms as the batch size m grows. In fact, it is possible to show that EU m decreases monotonically in m [17,14]. The increase in CPU times with larger batch sizes in Table 5 (and to a lesser extent in Table 4) is due, in part, to the IP (15) becoming larger.…”

Section: Procedures Mcskpmentioning

confidence: 85%

“…we see that U m is an upper bound, in expectation, on the optimal solution value z * ; see Mak et al [14]. Estimates of EU m are valuable in ascertaining the quality of a feasible candidate solutionx ∈ X.…”

Section: Skp With General Return Distributionsmentioning

confidence: 93%

“…It is clearly desirable to ascertain the quality of such a solution, and to do so we follow Mak et al [14]. This procedure consists of using the method of batch means to construct a one-sided confidence interval of the optimality gap, z * − P (rx ≥ c), by forming i.i.d.…”

Section: Skp With General Return Distributionsmentioning

confidence: 99%

“…, m; this use of common random numbers is a well-known variance reduction technique. (See, for example, Law and Kelton [13, §11.2] for a general discussion of common random numbers; for computational results in stochastic programming, see Mak et al [14].) In our current setting, common random numbers have the additional benefit of ensuring nonnegative estimates of the optimality gap, since, by construction G m ≥ 0; this could not be guaranteed if U m and L m were estimated separately with distinct random number streams.…”

Section: Skp With General Return Distributionsmentioning

confidence: 99%

“…There are two reasons for this: First, there is a contribution due toḠ(n) that originates from the inequality in (14), obtained by exchanging the optimization and expectation operators. Second, there is a contribution due to sampling error which is captured in G .…”

Section: Procedures Mcskpmentioning

confidence: 99%

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On a Stochastic Knapsack Problem and Generalizations

Morton

Wood

1998

Operations Research/Computer Science Interfaces Series

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Abstract:We consider an integer stochastic knapsack problem (SKP) where the weight of each item is deterministic, but the vector of returns for the items is random with known distribution. The objective is to maximize the probability that a total return threshold is met or exceeded. We study several solution approaches. Exact procedures, based on dynamic programming (DP) and integer programming (IP), are developed for returns that are independent normal random variables with integral means and variances. Computation indicates that the DP is significantly faster the most efficient algorithm to date. The IP is less efficient, but is applicable to more general stochastic IPs with independent normal returns. We also develop a Monte Carlo approximation procedure to solve SKPs with general distributions on the random returns. This method utilizes upper-and lower-bound estimators on the true optimal solution value 149 150 in order to construct a confidence interval on the optimality gap of a candidate solution.

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Section: Procedures Mcskpmentioning

confidence: 85%

Section: Skp With General Return Distributionsmentioning

confidence: 93%

Section: Skp With General Return Distributionsmentioning

confidence: 99%

Section: Skp With General Return Distributionsmentioning

confidence: 99%

Section: Procedures Mcskpmentioning

confidence: 99%

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On a Stochastic Knapsack Problem and Generalizations

Morton

Wood

1998

Operations Research/Computer Science Interfaces Series

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Hybrid wind‐solar grid‐connected system planning using scenario aggregation method

Abuelrub

Al‐Masri

Singh

2020

Int Trans Electr Energ Syst

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Summary Hybrid PV‐wind distribution system planning is challenging due to the uncertainties of the system components. Therefore, this paper presents a novel optimization procedure based on scenario aggregation (SA) approach for the planning of such a distribution system. The proposed procedure uses random distributions, which are built for each time step, to model the system uncertainties. System uncertainties include solar irradiance, wind velocity, system load, and energy market price. The objective function of the optimization problem measures the annual cost of the proposed system. Results of the suggested SA optimization procedure are then compared with results of two‐stage stochastic programming (SP) to verify the effectiveness of the proposed method. A case study for a grid‐connected hybrid PV‐wind distribution system in North Texas is presented to illustrate the proposed method. One‐year system components' data are collected to create the stochastic distributions needed to represent system uncertainties. The results of the SP give optimal system configuration that minimizes the system annual cost and meets reliability and load requirements of the proposed system. These results are compared with those obtained using the simpler SA approach. The case study shows that the proposed SA method gives close results compared with two‐stage SP when the uncertainty of the system is reasonable.

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Sampling Methods

Shapiro¹

2011

Wiley Encyclopedia of Operations Research and Management Science

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Monte Carlo bounding techniques for determining solution quality in stochastic programs

Cited by 562 publications

References 28 publications

On a Stochastic Knapsack Problem and Generalizations

On a Stochastic Knapsack Problem and Generalizations

Hybrid wind‐solar grid‐connected system planning using scenario aggregation method

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