Model: Facility Sizing

Description:

The facility-sizing problem is formulated as follows: \(m\) facilities are to be installed, each with capacity \(xi ≥ 0, i = 1, . . . , m\). Then the random demand \(ξi\) arrives at facility \(i\), with a known joint distribution of the random vector \(ξ = (ξ1, . . . , ξm)\).

A realization of the demand, \(ξ = (ξ1, . . . , ξm)\), is said to be satisfied by the capacity \(x\) if \(xi ≥ ξi, ∀i = 1, . . . , m\).

Sources of Randomness:

  1. random demand vector \(ξ\) follows a multivariate normal distribution and correlation coefficients \(ρi,j\) , \(i != j\) .

Model Factors:

  • \(\mu\): Mean vector of the multivariate normal distribution.

    • Default: [100, 100, 100]

  • \(\Sigma\): Variance-covariance matrix of multivariate normal distribution.

    • Default: [[2000, 1500, 500], [1500, 2000, 750], [500, 750, 2000]]

  • \(capacity\): Inventory capacities of the facilities.

    • Default: [150, 300, 400]

  • \(n_facility\): The number of facilities.

    • Default: 3

Respones:

  • \(stockout_flag\):
    0: all facilities satisfy the demand

    1: at least one of the facilities did not satisfy the demand

  • \(n_stockout\):

    the number of facilities which cannot satisfy the demand

  • \(n_cut\):

    the amount of total demand which cannot be satisfied

References:

This model is adapted from the article Rengarajan, T., & Morton, D.P. (2009). Estimating the Efficient Frontier of a Probabilistic Bicriteria Model. Proceedings of the 2009 Winter Simulation Conference. (https://www.informs-sim.org/wsc09papers/048.pdf)

Optimization Problem: Minimize Total Cost (FACSIZE-1)

Our goal is to minimize the total costs of installing capacity while keeping the probability of stocking out low.

The probability of failing to satisfy demand \(ξ = (ξ_1, . . . , ξ_m)\) is \(p(x) = P(ξ !<= x)\). Let \(epsilon ∈ [0, 1]\) be a risk-level parameter, then we obtain the probabilistic constraint:

\(P(ξ !<= x) ≤ epsilon\)

Meanwhile, the unit cost of installing facility i is \(ci\), and hence the total cost is \(\sum_{i=1}^n c_i x_i\).

Decision Variables:

  • \(capacity\)

Objectives:

Minimize the (deterministic) total cost of installing capacity.

Constraints:

1 stochastic constraint: \(P(Stockout) <= epsilon\). Box constraints: 0 < \(x_i\) < infinity for all \(i\).

Problem Factors:

  • budget: Max # of replications for a solver to take.

    • Default: 10000

  • epsilon: Maximum allowed probability of stocking out.

    • Default: 0.05

  • installation_costs: Cost to install a unit of capacity at each facility

    • Default: (1, 1, 1)

Fixed Model Factors:

None

Starting Solution:

  • capacity: (300, 300, 300)

Random Solutions:

  • Each facility’s capacity is Uniform(0, 300).

Optimal Solution:

None

Optimal Objective Function Value:

None

Optimization Problem: Maximize Service Level (FACSIZE-2)

Our goal is to maximize the probability of not stocking out subject to a budget constraint on the total cost of installing capacity.

Decision Variables:

  • \(capacity\)

Objectives:

Maximize the probability of not stocking out.

Constraints:

1 deterministic constraint: sum of facility capacity installation costs less than an installation budget. Box constraints: 0 < \(x_i\) < infinity for all \(i\).

Problem Factors:

  • budget: Max # of replications for a solver to take.

    • Default: 10000

  • installation_costs: Cost to install a unit of capacity at each facility.

    • Default: (1, 1, 1)

  • installation_budget: Total budget for installation costs.

    • Default: 500.0

Fixed Model Factors:

None

Starting Solution:

  • capacity: (100, 100, 100)

Random Solutions:

  • Use acceptance rejection to generate capacity vectors uniformly from space of vectors summing to less than installation budget.

Optimal Solution:

None

Optimal Objective Function Value:

None