facilitysizing.rst

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:

• μ: Mean vector of the multivariate normal distribution.

  • Default: [100, 100, 100]

• Σ: 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 = (ξ₁, ..., 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