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API

Documentation for ShelfSpaceAllocation.jl's interface.

Model

IO

Note

JSON stores multi-dimensional arrays as nested arrays. If we load an array a whose elements are accessed a[i, j, k], the elements of the nested array are accessed in reverse order a[k][j][i]. However, the function load_json converts the nested arrays back to multi-dimensional arrays.

We use simple plain text formats CSV and JSON for input and output of numerical values.

ShelfSpaceAllocation.Params — Method

Load sets, subsets and parameters from CSV files.

Each CSV file in product_path contains following attributes:

  • product_id
  • category_id
  • brand_id
  • width
  • height
  • depth
  • weight
  • monthly_demand
  • replenishment_interval
  • price
  • unit_margin
  • blocking_field
  • min_facing
  • max_facing
  • max_stack
  • up_down_order_criteria

Each CSV file in shelf_path contains following attributes:

  • module
  • id
  • level
  • total_width
  • total_height
  • total_length
  • product_min_unit_weight
  • product_max_unit_weight

Examples of input parameters can found inside examples/instances directory.

Arguments

  • product_path::AbstractString
  • shelf_path::AbstractString
source

Plotting

Model Reference

Sets and Subsets

  • $p∈P$ – A set of products.
  • $s∈S$ – A set of shelves.
  • $b∈B$ – A set of blocks. Blocks are an index of mutually exclusive subsets of products.
  • $m∈M$ – A set of modules. Modules are an index of a mutually exclusive subset of shelves.
  • $P_b⊆P$ – A block is a subset of products.
  • $S_m⊆S$ – A module is a subset of shelves.

Parameters

  • $N_p^{min}$, $N_p^{max}$ – The minimum and maximum number of facings for product $p$
  • $G_p$ – Unit profit of product $p$; used as shortage penalty (treated to be $\max\{0, G_p\}$
  • $R_p$ – Replenishment period of product $p$
  • $D_p$ – Demand forecast of product $p$
  • $L_p$ – Priority weight for height placement of product $p$
  • $W_p$ – Facing width of product $p$
  • $H_p$ – Height of product $p$
  • $M_p$ – Unit weight of product $p$
  • $P_{p,s}$ – Number units per facing of product $p$ on shelf $s$
  • $M_s^{min}$, $M_s^{max}$ – The minimum and maximum unit weight on shelf $s$
  • $W_s$ – Width of shelf $s$
  • $H_s$ – Height of shelf $s$
  • $L_p$ – Shelf level, counted from bottom to top
  • $SL$ – Slack, maximum difference in block starting points and between block max and min width
  • $w_1=0.5$
  • $w_2=10.0$
  • $w_3=0.1$

Objective

\[\min \left(w_1 ∑_s o_s + w_2 ∑_p G_p e_p + w_3 ∑_{p,s} L_p L_s n_{p,s}\right)\]

Basic Variables

  • $n_{p,s}$ – Number of facings of product $p$ on shelf $s$
  • $s_p$ – Amount of product $p$ sold
  • $e_p$ – Shortage of product $p$ (mismatch between demand and on-shelf inventory)
  • $o_p$ – Total empty space on shelf $s$
  • $y_p$ – $1$ if product is assigned to module $m$, $0$ otherwise

Basic Constraints

\[\begin{aligned} & n_{p,s} ∈ ℤ_{≥0}, & ∀p,s \\ & y_p ∈ \{0,1\}, & ∀p \\ & s_p ≥ 0, & ∀p \\ & e_p ≥ 0, & ∀p \\ & o_s ≥ 0, & ∀s \\ & \\ & n_{p,s}=0, & ∀p,s∣H_p > H_s \\ & n_{p,s}=0, & ∀p,s∣M_p > M_s^{max} \\ & ∑_p n_{p,s} ≥ y_{p}, & ∀p \\ & N_p^{min} y_p ≤ ∑_s n_{p,s} ≤ N_p^{max} y_p, & ∀p \\ & s_p ≤ \min\left(∑_s \frac{30}{R_p} P_{p,s} n_{p,s}, D_p\right), & ∀p \\ & s_p + e_p = D_p, & ∀p \\ & ∑_p W_p n_{p,s} + o_s = W_s, & ∀s \\ \end{aligned}\]

Block Variables

  • $z_{b,s}$ – $1$ if block is assigned on shelf $s$, otherwise $0$
  • $z_{b,s}^f$ – $1$ if shelf $s$ is the first shelf of block $b$, otherwise $0$
  • $z_{b,s}^l$ – $1$ if shelf $s$ is the last shelf of block $b$, otherwise $0$
  • $b_{b,s}$ – Width of block $b$ on shelf $s$
  • $m_{b,m}$ – Block width on module
  • $v_{b,m}$ – $1$ if block is assigned to module $m$, otherwise $0$
  • $x_{b,s}$ – Block starting location on shelf $s$
  • $x_{b,m}$ – Block starting location on module $m$
  • $w_{b,b'}$ – $1$ if block $b$ precedes block $b'$, otherwise $0$

Block Constraints

\[\begin{aligned} & b_{b,s}≥0, & ∀b,s \\ & z_{b,s}∈\{0,1\}, & ∀b,s \\ & m_{b,m}≥0, & ∀b,m \\ & z_{b,s}^f∈\{0,1\}, & ∀b,s \\ & z_{b,s}^l∈\{0,1\}, & ∀b,s \\ & x_{b,s}≥0, & ∀b,s \\ & x_{b,m}≥0, & ∀b,m \\ & w_{b,b'}∈\{0,1\}, & ∀b,b' \\ & v_{b,m}∈\{0,1\}, & ∀b,m \\ & \\ & ∑_{p∈P_b} W_p n_{p,s} ≤ b_{b,s}, & ∀s,b \\ & ∑_b b_{b,s} ≤ W_s, & ∀s \\ & b_{b,s} ≤ W_s z_{b,s}, & ∀b,s \\ & b_{b,s} ≥ m_{b,m} - W_s (1 - z_{b,s}) - SL, & ∀b,m,s∣s∈S_m \\ & b_{b,s} ≤ m_{b,m} + W_s (1 - z_{b,s}) + SL, & ∀b,m,s∣s∈S_m \\ & \\ & ∑_s z_{b,s}^f ≤ 1, & ∀b \\ & ∑_s z_{b,s}^l ≤ 1, & ∀b \\ & z_{b,s}^f = z_{b,s}, & ∀b,s=1 \\ & z_{b,s}^l = z_{b,s}, & ∀b,s=|S| \\ & z_{b,s+1}^f + z_{b,s} = z_{b,s+1} + z_{b,s}^l, & ∀b,s∣s≤|S|-1 \\ & \\ & ∑_{p∈P_b} n_{p,s} ≥ z_{b,s}, & ∀b,s \\ & n_{p,s} ≤ N_p^{max} z_{b,s}, & ∀b,p,s∣p∈P_b \\ & \\ & x_{b,s} ≤ W_s z_{b,s}, & ∀b,s \\ & x_{b,s} + b_{b,s} ≤ W_s, & ∀b,s \\ & x_{b,s} + W_s (1 - z_{b,s}) ≥ x_{b',s} + b_{b,s} - W_s (1 - w_{b,b'}), & ∀b,b',m∣b≠b' \\ & x_{b',s} + W_s (1 - z_{b', s}) ≥ x_{b,s} + b_{b,s} - W_s w_{b,b'}, & ∀b,b',m∣b≠b' \\ & x_{b,m} ≥ x_{b,s} - W_s (1 - z_{b,s}) - SL, & ∀b,m,s∣s∈S_m \\ & x_{b,m} ≤ x_{b,s} + W_s (1 - z_{b,s}) + SL, & ∀b,m,s∣s∈S_m \\ & \\ & n_{p,s} ≤ N_p^{max} v_{b,m}, & ∀p,b,m,s∣s∈S_m,p∈P_b \\ & ∑_m v_{b,m} ≤ 1, & ∀b \end{aligned}\]