# Simulated Annealing Hyper-Heuristic for a Shelf Space Allocation on Symmetrical Planograms Problem

^{*}

## Abstract

**:**

## 1. Introduction

- Dividing the shelf into the segments of variable size (which could be enlarged or reduced) and allocating the products on different horizontal shelf segments. To the best of our knowledge, this is the first model which considers vertical shelf levels (e.g., pallet, low-level, eye-level) and horizontal shelf segments (e.g., aisle, local, convenience) of flexible size for aesthetic symmetry.
- Differentiating shelf segments of a symmetrical planogram based on the customer traffic flow (e.g., segments situated near the aisle at the beginning or at the end of a planogram).
- Specifying shelf segments dedicated for the specific product types (e.g., local and convenience segments) which could be situated in arbitrary place of a planogram.
- Grouping similar products into clusters with the intention to place them one next to the other on the shelf for making the comparison of the products by customers easier.
- Considering two orientation possibilities (front or side) to allocate products on shelves.
- Incorporation of capping and nesting parameters.

- Presenting a practical retail SSAP model with above mentioned characteristics.
- Adjusting the simulated annealing (SA) algorithm for solving the retail SSAP.
- The enhancement of the well-known SA hyper-heuristic with solution improvement and product reallocation procedures which enables to get more profitable solution within the small number of iterations.

## 2. Related Works

#### 2.1. Visual Attention to Product

#### 2.2. Assortment Decisions

#### 2.3. Shelf Space Allocation Decisions

#### 2.4. Solution Approaches

## 3. Problem Model

${z}_{11}=0,{z}_{12}={s}_{i}^{l}$ | —for a pallet segment; |

${z}_{21}=0,{z}_{22}={s}_{i}^{l}$ | —for low-level segment; |

${z}_{31}=0,{z}_{32}={s}_{i}^{l}$ | —for eye-level segment; |

${z}_{41}=({v}^{lc}-1)\cdot {v}_{i}^{w},{z}_{42}={v}^{lc}\cdot {v}_{i}^{w}$ | —for local segment; |

${z}_{51}=({v}^{v}-1)\cdot {v}_{i}^{w},{z}_{52}={v}^{v}\cdot {v}_{i}^{w}$ | —for convenience segment; |

${z}_{61}={v}_{i}^{w},{z}_{62}={s}_{i}^{l}-{v}_{i}^{w}$ | —for shelf center segment; |

${z}_{71}=0,{z}_{72}={v}_{i}^{w}$ | —for first aisle segment; |

${z}_{81}={s}_{i}^{l}-{v}_{i}^{w},{z}_{82}={s}_{i}^{l}$ | —for last aisle segment. |

- representing local products,
- representing convenience products,
- grouping come products into clusters.

- the shelf constraints;
- the product constraints;
- multi-shelves constraints
- shelf segment constraints.

#### 3.1. Shelf Constraints

#### 3.2. Product Constraints

#### 3.3. Multi-Shelves Constraints

#### 3.4. Shelf Segment Constraints

#### 3.5. Relationship Constraints

#### 3.6. Decision Variables

## 4. Simulated Annealing Approach

- Initial Phase

- consider or not consider facings/cappings/nestings already on the shelf;
- allow or not allow to select the same product more than once in the next iteration;

- 2.
- Iterative Phase

- 3.
- Termination Criteria

Algorithm 1. Pseudocode of the general steps of the proposed SA algorithm |

1: Define heuristics rules $h,$$h=1,\dots ,H$. 2: Define a maximum temperature ${t}_{\mathrm{max}}=\underset{i=1,..,S}{\mathrm{min}}({s}_{i}^{l})\cdot P$. 3: Set a starting temperature $t={t}_{\mathrm{max}}$. 4: Initialise the number of iterations without improvement $w=0$. 5: Initialise the last profit ${P}_{l}=0$. 6: Initialise the initial solution $s$, the total profit is ${P}_{s}$. 7: Save current solution as the best solution $b=s$, ${P}_{b}={P}_{s}$. 8: while ($t>{t}_{\mathrm{min}}$){ 9: for ($y=1$;$y\le H$;$y++$) 10: { 11: Randomly select the heuristic rule $h=Random(1,H)$. 12: Generate a new solution $s$ based on the selected $h$, 13: $s=GenerateSolution(h)$. 14: Calculate ${P}_{s}$. 15: If (${P}_{s}>{P}_{l}$){ 16: Set $w=0$ 17: }else{ 18: Increase $w++$ 19: } 20: Set ${P}_{l}={P}_{s}$ 21: If (${P}_{s}>{P}_{b}$) 22: { 23: Save current solution as the best solution $b=s$, ${P}_{b}={P}_{s}$. 24: } 25: If ($w\ge {w}_{\mathrm{max}}$){ 26: Set $t={t}_{\mathrm{min}}$. 27: } 28: } 29: Set $\beta =\frac{{t}_{\mathrm{max}}-{t}_{\mathrm{min}}}{{t}_{\mathrm{max}}\cdot {t}_{\mathrm{min}}}\cdot K$. 30: Set $t=\mathrm{max}(0,\frac{t}{1+\beta \cdot t})$. 31:} |

Algorithm 2. Pseudocode of the iterative solution generation $GenerateSolution(h)$ |

1: $\mathrm{Procedure}GenerateSolution(h)${ 2: Generate solution $x$ increase the number of facings, cappings and nestings $k$ times in the previous solution $s$. 3: Correct (reduce) the number of facings, cappings, and nestings in the solution $x$ according to $h$, $x=UseHeuristics(h)$ with regard to all constraints, the total profit is ${P}_{x}$. 4: Improve the solution $m=Improve(x)$, the total profit is ${P}_{m}$. 5: Reallocate products in the solution $r=Reallocate(x)$, the total profit is ${P}_{r}$. 6: Set the new solution as the solution with the highest profit, 7: $x=\left\{\begin{array}{ll}m,& \mathrm{if}{P}_{m}=\mathrm{max}({P}_{m},{P}_{r})\\ r,& \mathrm{if}{P}_{r}=\mathrm{max}({P}_{m},{P}_{r})\end{array}\right\}$, ${P}_{x}=\mathrm{max}({P}_{m},{P}_{r})$ 8: $\delta ={P}_{x}-{P}_{s}$ 9: if ($\delta \ge 0$){ 10: Set solution $s=x$, ${P}_{s}={P}_{x}$. 11: } else { 12: Set $\gamma =\mathrm{exp}(\delta /t)$. 13: Generate a random $\alpha $ uniformly in range [0,1]. 14: if ($\alpha <\gamma $){ 15: Set solution $s=x$, ${P}_{s}={P}_{x}$. 16: } 17: } 18: Return $s$ 19: } |

- increase shelf ratio of the current or other shelf;
- increase shelf profit of the current or other shelf;
- reduce free shelf space of the current or other shelf.

- the product could be profitable if it is placed on the current shelf;
- the product is not profitable being placed on the current shelf;
- the product could be profitable if it is placed on another shelf;
- the product is not profitable being placed on another shelf.

## 5. Results

- Set 1—There are 10 products that must be placed on 4 shelves on the planogram. The shelf lengths in each store are 250 cm, 375 cm, 500 cm, 625 cm, 750 cm.
- Set 2—There are 20 products that must be placed on 4 shelves on the planogram. The shelf lengths in each store are 250 cm, 375 cm, 500 cm, 625 cm, 750 cm.
- Set 3—There are 30 products that must be placed on 4 shelves on the planogram. The shelf lengths in each store are 250 cm, 375 cm, 500 cm, 625 cm, 750 cm.
- Set 4—There are 40 products that must be placed on 4 shelves on the planogram. The shelf lengths in each store are 250 cm, 375 cm, 500 cm, 625 cm, 750 cm.
- Set 5—There are 50 products that must be placed on 4 shelves on the planogram. The shelf lengths in each store are 250 cm, 375 cm, 500 cm, 625 cm, 750 cm.

- The computational experiments were performed in Visual C# 2015.
- Language: Visual C# 2015
- Microsoft Visual Studio Community 2015
- Version 14.0.25431.01 Update 3
- Microsoft .NET Framework
- Version 4.6.01055
- IBM ILOG CPLEX Optimization Studio Version 12.7.1.0 was used to find the optimal or feasible solution.

- number of heuristics $H=13$;
- maximum number of iterations $K=3000$;
- maximum number of iterations without improvement ${w}_{\mathrm{max}}$ not used for 10 and 20 products;
- maximum number of iterations without improvement ${w}_{\mathrm{max}}=500$ for 30, 40, and 50 products;
- minimum temperature ${t}_{\mathrm{min}}=0.5$;
- coefficient of increase of the numbers of facings, cappings, and nestings $k=1.3$;

## 6. Conclusions

- Considering a planogram of a single product category.
- Absence of time-dependent variables in the proposed model.
- Lack of defining the product positioning for all, not only specific products.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Hübner, A. Retail Category Management: Decision Support Systems for Assortment, Shelf Space, Inventory and Price Planning; Springer: Berlin/Heidelberg, Germany, 2011; Volume 656. [Google Scholar]
- Hübner, A.; Schaal, K. A shelf-space optimisation model when demand is stochastic and space-elastic. Omega
**2017**, 68, 139–154. [Google Scholar] [CrossRef] - Bianchi-Aguiar, T.; Hübner, A.; Carravilla, M.A.; Oliveira, J.F. Retail shelf space planning problems: A comprehensive review and classification framework. Eur. J. Oper. Res.
**2021**, 289, 1–16. [Google Scholar] [CrossRef] - Buttle, F. Retail space allocation. Internat. J. Phys. Distrib. Mater. Manag.
**1984**, 14, 3–23. [Google Scholar] - Zhang, W.; Rajaram, K. Managing limited retail space for basic products: Space sharing vs. space dedication. Eur. J. Oper. Res.
**2017**, 263, 768–781. [Google Scholar] [CrossRef][Green Version] - Urban, T.L. An inventory-theoretic approach to product assortment and shelf-space allocation. J. Retail.
**1998**, 74, 15–35. [Google Scholar] [CrossRef] - Hübner, A.; Schaal, K. Effect of replenishment and backroom on retail shelf-space planning. Bus. Res.
**2017**, 10, 123–156. [Google Scholar] [CrossRef][Green Version] - Czerniachowska, K.; Hernes, M. Optimization Models for the Shelf Space Allocation Problem with Vertical Position Effects. In Proceedings of the 36th IBIMA Conference, Granada, Spain, 4–5 November 2020. [Google Scholar]
- Kim, G.; Moon, I. Integrated planning for product selection, shelf-space allocation, and replenishment decision with elasticity and positioning effects. J. Retail. Consum. Serv.
**2021**, 58, 102274. [Google Scholar] [CrossRef] - Czerniachowska, K.; Hernes, M. A genetic algorithm for the shelf-space allocation problem with vertical position effects. Mathematics
**2020**, 8, 1881. [Google Scholar] [CrossRef] - Czerniachowska, K.; Hernes, M. A heuristic approach to shelf space allocation decision support including facings, capping, and nesting. Symmetry
**2021**, 13, 314. [Google Scholar] [CrossRef] - Flores, W.; Chen, J.-C.V.; Ross, W.H. The effect of variations in banner ad, type of product, website context, and language of advertising on Internet users’ attitudes. Comput. Hum. Behav.
**2014**, 31, 37–47. [Google Scholar] [CrossRef] - Djamasbi, S.; Siegel, M.; Tullis, T. Generation Y, web design, and eye tracking. Int. J. Hum. Comput. St.
**2010**, 68, 307–323. [Google Scholar] [CrossRef][Green Version] - Huang, C.-Y.; Chou, C.-J.; Lin, P.-C. Involvement theory in constructing bloggers’ intention to purchase travel products. Tour. Manag.
**2010**, 31, 513–526. [Google Scholar] [CrossRef] - Desrochers, D.M.; Nelson, P. Adding consumer behavior insights to category management: Improving item placement decisions. J. Retail.
**2006**, 82, 357–365. [Google Scholar] [CrossRef] - Valenzuela, A.; Raghubir, P. Center of Orientation: Effect of Vertical and Horizontal Shelf Space Product Position. ACR N. Am. Adv.
**2009**, 36, 100–103. [Google Scholar] - Ngo, D.C.L.; Byrne, J.G. Application of an aesthetic evaluation model to data entry screens. Comput. Hum. Behav.
**2001**, 17, 149–185. [Google Scholar] [CrossRef] - Anic, I.-D.; Radas, S.; Lim, L.K.S. Relative effects of store traffic and customer traffic flow on shopper spending. Int. Rev. Retail. Distrib. Consum. Res.
**2010**, 20, 237–250. [Google Scholar] [CrossRef] - Daruka, G.P.; Palekar, U.S. Application of column generation techniques to retail assortment planning. In Integer Programming: Theory and Practice; CRC Press: Boca Raton, FL, USA, 2005; pp. 221–251. [Google Scholar] [CrossRef]
- Flamand, T.; Ghoniem, A.; Haouari, M.; Maddah, B. Integrated assortment planning and store-wide shelf space allocation: An optimisation-based approach. Omega
**2018**, 81, 134–149. [Google Scholar] [CrossRef] - Kök, A.G.; Fisher, M.L.; Vaidyanathan, R. Assortment planning: Review of literature and industry practice. In International Series in Operations Research and Management Science; Springer: Berlin/Heidelberg, Germany, 2015; Volume 223, pp. 175–236. [Google Scholar] [CrossRef]
- Kök, A.G.; Fisher, M.L.; Vaidyanathan, R. Assortment planning: Review of literature and industry practice. Retail supply chain management. In Retail Supply Chain Management. International Series in Operations Research and Management Science; Springer: Boston, MA, USA, 2008; Volume 122, pp. 99–153. [Google Scholar] [CrossRef]
- Irion, J.; Lu, J.C.; Al-Khayyal, F.; Tsao, Y.C. A piecewise linearisation framework for retail shelf space management models. Eur. J. Oper. Res.
**2012**, 222, 122–136. [Google Scholar] [CrossRef] - Geismar, H.N.; Dawande, M.; Murthi, B.P.S.; Sriskandarajah, C. Maximizing Revenue Through Two-Dimensional Shelf-Space Allocation. Prod. Oper. Manag.
**2015**, 24, 1148–1163. [Google Scholar] [CrossRef] - Yu, V.F.; Maglasang, R.; Tsao, Y.C. A reduced variable neighborhood search-based hyperheuristic for the shelf space allocation problem. Comput. Ind. Eng.
**2020**, 143, 106420. [Google Scholar] [CrossRef] - Silver, E.A. Operations Research and Inventory Management: A Review and Critique. Oper. Res.
**1981**, 29, 628–645. [Google Scholar] [CrossRef] - Drèze, X.; Hoch, S.J.; Purk, M.E. Shelf management and space elasticity. J. Retail.
**1994**, 70, 301–326. [Google Scholar] [CrossRef][Green Version] - Russell, R.A.; Urban, T.L. The location and allocation of products and product families on retail shelves. Ann. Oper. Res.
**2010**, 179, 131–147. [Google Scholar] [CrossRef] - Esparcia-Alcázar, A.I.; Martínez-García, A.I. Linear Shelf Space Allocation Using a Multi Objective Evolutionary Algorithm. Technical Report ITI-SAC-027. 2008, pp. 1–23. Available online: https://pdfs.semanticscholar.org/e768/f366d4c7cb4009e5d937eb177710003a38f8.pdf (accessed on 29 June 2021).
- Pinto, F.; Soares, C. Space allocation in the retail industry: A decision support system integrating evolutionary algorithms and regression models. In Proceedings of the Joint European Conference on Machine Learning and Knowledge Discovery in Databases; Springer: Berlin/Heidelberg, Germany, 2013; pp. 531–546. Available online: https://link.springer.com/chapter/10.1007/978-3-642-40994-3_34 (accessed on 29 June 2021).
- Borin, N.; Farris, P.W.; Freeland, J.R. A Model for Determining Retail Product Category Assortment and Shelf Space Allocation. Decis. Sci.
**1994**, 25, 359–384. [Google Scholar] [CrossRef] - Bai, R.; Kendall, G. An investigation of automated planograms using a simulated annealing based hyper-heuristic. Oper. Res. Comput. Sci. Interfaces Ser.
**2005**, 32, 87–108. [Google Scholar] - Van Nierop, E.; Fok, D.; Franses, P.H. Interaction between shelf layout and marketing effectiveness and its impact on optimising shelf arrangements. Mark. Sci.
**2008**, 27, 1065–1082. [Google Scholar] [CrossRef] - Bai, R.; van Woensel, T.; Kendall, G.; Burke, E.K. A new model and a hyper-heuristic approach for two-dimensional shelf space allocation. 4OR
**2013**, 11, 31–55. [Google Scholar] [CrossRef] - Hwang, H.; Choi, B.; Lee, M.J. A model for shelf space allocation and inventory control considering location and inventory level effects on demand. Int. J. Prod. Econ.
**2005**, 97, 185–195. [Google Scholar] [CrossRef] - Esparcia-Alcazar, A.I.; Lluch-Revert, L.; Sharman, K.C.; Albarracin-Guillem, J.M.; Palmer-Gato, M.E. Towards an evolutionary tool for the allocation of supermarket shelf space. In Proceedings of the eighth annual conference on genetic and evolutionary computation, Woshington, DC, USA, 8–12 July 2006; pp. 1653–1660. [Google Scholar]
- Esparcia-Alćazar, A.I.; Lluch-Revert, L.; Sharman, K.; Albarracín-Guillem, J.M.; Palmer-Gato, M.E. An Evolutionary Algorithm for the Product to Shelf Allocation Problem. In Proceedings of the 2006 IEEE International Conference on Evolutionary Computation, Vancouver, BC, Canada, 16–21 July 2006; IEEE, 2006. [Google Scholar] [CrossRef]
- Hansen, J.M.; Raut, S.; Swami, S. Retail Shelf Allocation: A Comparative Analysis of Heuristic and Meta-Heuristic Approaches. J. Retail.
**2010**, 86, 94–105. [Google Scholar] [CrossRef] - Ghazavi, E.; Lotfi, M.M. Formulation of customers’ shopping path in shelf space planning: A simulation-optimisation approach. Expert Syst. Appl.
**2016**, 55, 243–254. [Google Scholar] [CrossRef] - Ozcan, T.; Esnaf, S. A heuristic approach based on artificial bee colony algorithm for retail shelf space optimisation. In Proceedings of the 2011 IEEE Congress on Evolutionary Computation (CEC), New Orleans, LA, USA, 5–8 June 2011; pp. 95–101. [Google Scholar] [CrossRef]
- Kirkpatrick, S.; Gelatt, C.D.; Vecchi, M.P. Optimization by Simulated Annealing. Science
**1983**, 220, 671–680. [Google Scholar] [CrossRef] [PubMed] - Metropolis, N.; Rosenbluth, A.W.; Rosenbluth, M.N.; Teller, A.H.; Teller, E. Equation of state calculations by fast computing machines. J. Chem. Phys.
**1953**, 21, 1087–1092. [Google Scholar] [CrossRef][Green Version] - Otten, R.H.J.M.; van Ginneken, L.P.P.P. The Annealing Algorithm; Kluwer: Boston, MA, USA, 1989. [Google Scholar]
- Ingber, L. Simulated Annealing: Practice Versus Theory. Math. Comput. Model.
**1993**, 18, 29–57. [Google Scholar] [CrossRef][Green Version]

Parameter | Description |
---|---|

$P$ | Number of products |

$j$ | Product iterator, $j=1,\dots ,P$ |

$S$ | Number of shelves |

$i$ | Shelf iterator, $i=1,\dots ,S$ |

${V}_{i}$ | Number of shelf segments on the shelf $i$ |

$g$ | Shelf segment iterator, $g=1,\dots ,{V}_{i}$ |

$m$ | Segment index and product type index,
$m=1,\dots ,8$ $m=1$ pallet segment/product $m=2$ low-level segment/product $m=3$ eye-level segment/product $m=4$ local segment/product $m=5$ convenience segment/product $m=6$ shelf center segment/product $m=7$ first aisle segment/product $m=8$ last aisle segment/product |

$n$ | Shelf segment size index, $n=\{1,2\}$ $n=1$ left border $n=2$ right border |

$r$ | Subset index, $r=1,\dots ,3$ $r=1$ the subset A, subset before the specific segment on the shelf $r=2$ the subset C, subset inside the specific segment on the shelf $r=3$ the subset B, subset after the specific segment on the shelf |

Parameter | Description |
---|---|

Shelf parameters | |

${s}_{i}^{l}$ | Length of the shelf $i$ |

${s}_{i}^{h}$ | Height of the shelf $i$ |

${s}_{i}^{d}$ | Depth of the shelf $i$ |

${s}^{lc}$ | Shelf index where the local segment is allocated, ${s}^{lc}=\{1,\dots ,S\}$ |

${s}^{v}$ | Shelf index where convenience segment is allocated, ${s}^{v}=\{1,\dots ,S\}$ |

${s}_{mi}^{t}$ | Binary tag $m$ of the shelf $i$ ${s}_{mi}^{t}=\left\{\begin{array}{l}1,\mathrm{if}\mathrm{shelf}i\mathrm{is}\mathrm{tagged}\\ 0,\mathrm{otherwise}\end{array}\right\}$ |

Shelf segment parameters | |

${v}_{i}^{w}$ | Shelf segment width on the shelf $i$ |

${v}^{lc}$ | Local segment index on the shelf $i={s}^{lc}$, ${v}^{lc}=\{1,\dots ,{V}_{i}\}$ |

${v}^{v}$ | Convenience segment index on the shelf $i={s}^{v}$, ${v}^{v}=\{1,\dots ,{V}_{i}\}$ |

${z}_{mn}$ | Horizontal (left and right) coordinates of the segment of the type $m$ |

Product parameters | |

${p}_{j}^{w}$ | Width of the product $j$ |

${p}_{j}^{h}$ | Height of the product $j$ |

${p}_{j}^{d}$ | Depth of the product $j$ |

${p}_{j}^{s}$ | Supply limit of the product $j$ |

${p}_{j}^{u}$ | Unit profit of the product $j$ |

${p}_{j}^{n}$ | Nesting coefficient of the product $j$, ${p}_{j}^{n}<1$, or ${p}_{j}^{n}=0$ if the product cannot be nested |

${p}_{j}^{{o}_{2}}$ | Side orientation binary parameter ${p}_{j}^{{o}_{2}}=\left\{\begin{array}{l}1,\mathrm{if}\mathrm{side}\mathrm{orientation}\mathrm{is}\mathrm{available}\mathrm{for}\mathrm{product}j\\ 0,\mathrm{otherwise}\end{array}\right\}$. |

${p}_{j}^{l}$ | The cluster of the product $j$ |

${f}_{j}^{\mathrm{min}}$ | Minimum number of facings of the product $j$ |

${f}_{j}^{\mathrm{max}}$ | Maximum number of facings of the product $j$ |

${c}_{j}^{\mathrm{min}}$ | Minimum number of cappings per facings group of the product $j$ |

${c}_{j}^{\mathrm{max}}$ | Maximum number of cappings per facings group of the product $j$ |

${n}_{j}^{\mathrm{min}}$ | Minimum number of nestings of one facing of the product $j$ |

${n}_{j}^{\mathrm{max}}$ | Maximum number of nestings of one facing of the product $j$ |

${s}_{j}^{\mathrm{min}}$ | Minimum number of shelves on which the product $j$ can be allocated |

${s}_{j}^{\mathrm{max}}$ | Maximum number of shelves on which the product $j$ can be allocated |

${p}_{mj}^{t}$ | Binary tag $m$ of the product $j$ ${p}_{mj}^{t}=\left\{\begin{array}{l}1,\mathrm{if}\mathrm{product}j\mathrm{is}\mathrm{tagged}\\ 0,\mathrm{otherwise}\end{array}\right\}$ |

${b}_{mij}^{t}$ | Product to shelf compatibility tag ${b}_{mij}^{t}=\left\{\begin{array}{l}1,\mathrm{if}{s}_{mi}^{t}={p}_{mi}^{t}\\ 0,\mathrm{otherwise}\end{array}\right\},m=1$—for pallet shelf ${b}_{mij}^{t}=\left\{\begin{array}{l}1,\mathrm{if}{p}_{mj}^{t}=1\wedge {s}_{mi}^{t}={p}_{mj}^{t}\\ 0,\mathrm{if}{p}_{mj}^{t}=1\wedge {s}_{mi}^{t}\ne {p}_{mj}^{t}\\ 1,\mathrm{if}{p}_{mj}^{t}=0\end{array}\right\},m=2,\dots ,8$—for the rest not pallet shelves |

Temporary variables | |

${q}_{ir}^{ABC}$ | Occupied space temporary variable |

Parameter | Description | Formula |
---|---|---|

${x}_{ij}$ | Decision variable showing if the product $j$ is put on the shelf $i$. | ${x}_{ij}=\left\{\begin{array}{l}1,\mathrm{product}j\mathrm{is}\mathrm{put}\mathrm{to}\mathrm{the}\mathrm{shelf}i\\ 0,\mathrm{otherwise}\end{array}\right\}$ |

${f}_{ij}$ | Number of facings of the product $j$ on the shelf $i$. | ${f}_{ij}=\left\{0,1,2\dots \right\}$ |

${c}_{ij}$ | Number of cappings of the product $j$ on the shelf $i$. | ${c}_{ij}=\left\{0,1,2\dots \right\}$ |

${n}_{ij}$ | Number of nestings of the product $j$ on the shelf $i$. | ${n}_{ij}=\left\{0,1,2\dots \right\}$ |

${y}_{ij}^{{o}_{1}}$ | Decision variable showing if the product $j$ is put on the shelf $i$ on front orientation. | ${y}_{ij}^{{o}_{1}}=\left\{\begin{array}{l}1,\mathrm{if}\mathrm{product}\mathrm{is}\mathrm{on}\mathrm{front}\mathrm{orientation}\\ 0,\mathrm{otherwise}\end{array}\right\}$ |

${y}_{ij}^{{o}_{2}}$ | Decision variable showing if the product $j$ is put on the shelf $i$ on side orientation. | ${y}_{ij}^{{o}_{2}}=\left\{\begin{array}{l}1,\mathrm{if}\mathrm{product}\mathrm{is}\mathrm{on}\mathrm{side}\mathrm{orientation}\\ 0,\mathrm{otherwise}\end{array}\right\}$ |

${y}_{mijr}^{ABC}$ | Decision variable showing if the product $j$ is put on the shelf $i$ on tag $m$ and assigned to the subset $r$. | ${y}_{mijr}^{ABC}=\left\{\begin{array}{l}1,\mathrm{if}\mathrm{product}\mathrm{of}\mathrm{tag}m\mathrm{is}\\ \mathrm{assigned}\mathrm{to}\mathrm{the}\mathrm{subset}r\\ 0,\mathrm{otherwise}\end{array}\right\}$ |

Parameter | Description |
---|---|

$H$ | Number of solutions received by heuristics |

$h$ | Heuristic number, $h=1,\dots ,H$ |

$y$ | Heuristic iterator, $y=1,\dots ,H$ |

$K$ | Maximum number of iterations |

$k$ | Coefficient of increasing of the numbers of facing, cappings, and nestings |

$w$ | Number of iterations without improvement |

${w}_{\mathrm{max}}$ | Maximum number of iterations without improvement |

$\alpha $ | Solution acceptance probability |

$\beta $ | The reducing temperature coefficient |

$\gamma $ | Solution acceptance goal |

$\delta $ | Profit difference between the current and the previous solution |

${t}_{\mathrm{max}}$ | Maximum (starting) temperature |

${t}_{\mathrm{min}}$ | Minimum temperature |

$t$ | Current temperature |

${P}_{s}$ | Profit of the starting (previous) solution |

${P}_{x}$ | Profit of the current solution |

${P}_{l}$ | Profit of the last solution |

${P}_{b}$ | Profit of the best solution |

${P}_{m}$ | Profit of the improved solution |

${P}_{r}$ | Profit of the reallocated solution |

$s$ | Initial solution |

$x$ | Current solution |

$b$ | Best solution |

$m$ | Improved solution |

$r$ | Reallocated solution |

CPLEX Time Limit Equals SA Execution Time | CPLEX Time Limit Equals 5 min | ||||
---|---|---|---|---|---|

Products | Shelf Width | CPLEX Is Better | SA Is Better | CPLEX Is Better | SA Is Better |

10 | 250 | 2.91% | 2.91% | ||

375 | 4.99% | 4.99% | |||

500 | 14.32% | 10.05% | |||

625 | 2.35% | 2.35% | |||

750 | 0.96% | 0.96% | |||

20 | 250 | 14.79% | 15.55% | ||

375 | 14.92% | 14.92% | |||

500 | 12.54% | 12.54% | |||

625 | 7.99% | 8.05% | |||

750 | 14.79% | 17.16% | |||

30 | 250 | 24.01% | 24.01% | ||

375 | 24.32% | 23.72% | |||

500 | 18.28% | 17.53% | |||

625 | 6.86% | 7.45% | |||

750 | 3.27% | 3.24% | |||

40 | 250 | 14.40% | 14.40% | ||

375 | 13.08% | 13.08% | |||

500 | 12.72% | 12.29% | |||

625 | 11.03% | 11.00% | |||

750 | 13.63% | 13.59% | |||

50 | 375 | 10.31% | 10.17% | ||

500 | 11.30% | 11.30% | |||

625 | 18.16% | 17.61% | |||

750 | 1.90% | 1.23% | |||

Min | 0.96% | 2.35% | 0.96% | 2.35% | |

Avg | 11.50% | 11.26% | 11.59% | 10.69% | |

Max | 18.16% | 24.32% | 17.16% | 24.01% |

Products | Shelf Width | Time (min) | Average Time (min) | Solution Get Worse | Solution Improved |
---|---|---|---|---|---|

10 | 250 | 0.05 | 0.71 | 10/10 | |

375 | 1.72 | 8/10 | |||

500 | 0.02 | √ | 5/10 | ||

625 | 0.04 | 9/10 | |||

750 | 1.70 | 6/10 | |||

20 | 250 | 0.18 | 2.16 | 9/10 | |

375 | 1.28 | 10/10 | |||

500 | 5.69 | 9/10 | |||

625 | 3.61 | 7/10 | |||

750 | 0.03 | √ | 6/10 | ||

30 | 250 | 2.33 | 3.54 | 10/10 | |

375 | 0.40 | 10/10 | |||

500 | 0.64 | 8/10 | |||

625 | 14.11 | 8/10 | |||

750 | 0.36 | √ | 7/10 | ||

40 | 250 | 8.47 | 59.82 | 10/10 | |

375 | 71.29 | 10/10 | |||

500 | 27.43 | 10/10 | |||

625 | 103.57 | 10/10 | |||

750 | 88.36 | 10/10 | |||

50 | 375 | 12.69 | 103.60 | 10/10 | |

500 | 76.35 | 10/10 | |||

625 | 98.26 | 10/10 | |||

750 | 227.10 | 10/10 |

Parameter | Value |
---|---|

Analysis time | 0.14 s. |

Analysed variables | Products; Width; SA; CPLEX |

Number of unspecified | 0 |

Number of missing data | 0 |

Significance level | 0.05 |

Accept missing data (Durbin/Skillings-Mack) | No |

C | 600 |

A | 720 |

T1 statistic Friedman | 65.25 |

Degrees of freedom | 3 |

p-value | <0.000001 |

T2 statistic Iman-Davenport | 222.333333 |

Degrees of freedom | 3/69 |

p-value | <0.000001 |

Skillings-Mack statistic | 65.25 |

Degrees of freedom | 3 |

p-value | <0.000001 |

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## Share and Cite

**MDPI and ACS Style**

Czerniachowska, K.; Hernes, M. Simulated Annealing Hyper-Heuristic for a Shelf Space Allocation on Symmetrical Planograms Problem. *Symmetry* **2021**, *13*, 1182.
https://doi.org/10.3390/sym13071182

**AMA Style**

Czerniachowska K, Hernes M. Simulated Annealing Hyper-Heuristic for a Shelf Space Allocation on Symmetrical Planograms Problem. *Symmetry*. 2021; 13(7):1182.
https://doi.org/10.3390/sym13071182

**Chicago/Turabian Style**

Czerniachowska, Kateryna, and Marcin Hernes. 2021. "Simulated Annealing Hyper-Heuristic for a Shelf Space Allocation on Symmetrical Planograms Problem" *Symmetry* 13, no. 7: 1182.
https://doi.org/10.3390/sym13071182