ssa.py

##@file ssa.py
# @brief multi-stage (serial) safety stock allocation model
"""
Approach: use SOS2 constraints for modeling non-linear functions.

Copyright (c) by Joao Pedro PEDROSO and Mikio KUBO, 2012
"""
import math
import random

from pyscipopt import Model, quicksum

from piecewise import convex_comb_sos


def ssa(n, h, K, f, T):
    """ssa -- multi-stage (serial) safety stock allocation model
    Parameters:
        - n: number of stages
        - h[i]: inventory cost on stage i
        - K: number of linear segments
        - f: (non-linear) cost function
        - T[i]: production lead time on stage i
    Returns the model with the piecewise linear relation on added variables x, f, and z.
    """

    model = Model("safety stock allocation")

    # calculate endpoints for linear segments
    a, b = {}, {}
    for i in range(1, n + 1):
        a[i] = [k for k in range(K)]
        b[i] = [f(i, k) for k in range(K)]

    # x: net replenishment time for stage i
    # y: corresponding cost
    # s: piecewise linear segment of variable x
    x, y, s = {}, {}, {}
    L = {}  # service time of stage i
    for i in range(1, n + 1):
        x[i], y[i], s[i] = convex_comb_sos(model, a[i], b[i])
        if i == 1:
            L[i] = model.addVar(ub=0, vtype="C", name="L[%s]" % i)
        else:
            L[i] = model.addVar(vtype="C", name="L[%s]" % i)
    L[n + 1] = model.addVar(ub=0, vtype="C", name="L[%s]" % (n + 1))

    for i in range(1, n + 1):
        # net replenishment time for each stage i
        model.addCons(x[i] + L[i] == T[i] + L[i + 1])

    model.setObjective(quicksum(h[i] * y[i] for i in range(1, n + 1)), "minimize")

    model.data = x, s, L
    return model


def make_data():
    """creates example data set"""
    n = 30  # number of stages
    z = 1.65  # for 95% service level
    sigma = 100  # demand's standard deviation
    h = {}  # inventory cost
    T = {}  # production lead time
    h[n] = 1
    for i in range(n - 1, 0, -1):
        h[i] = h[i + 1] + random.randint(30, 50)
    K = 0  # number of segments (=sum of processing times)
    for i in range(1, n + 1):
        T[i] = random.randint(3, 5)  # production lead time at stage i
        K += T[i]
    return z, sigma, h, T, K, n


if __name__ == "__main__":
    random.seed(1)

    z, sigma, h, T, K, n = make_data()


    def f(i, k):
        return sigma * z * math.sqrt(k)


    model = ssa(n, h, K, f, T)
    model.optimize()

    # model.write("ssa.lp")
    x, s, L = model.data
    for i in range(1, n + 1):
        for k in range(K):
            if model.getVal(s[i][k]) >= 0.001:
                print(s[i][k].name, model.getVal(s[i][k]))
        print
    print("%10s%10s%10s%10s" % ("Period", "x", "L", "T"))
    for i in range(1, n + 1):
        print("%10s%10s%10s%10s" % (i, model.getVal(x[i]), model.getVal(L[i]), T[i]))

    print("Objective:", model.getObjVal())