Teaching–Learning Based Optimization (TLBO) with Variable Neighborhood Search to Retail Shelf-Space Allocation
Abstract
:1. Introduction
2. Related Works
3. Yang and Chen’s Model and Extant Solution Methods
- The total space allocated to all the products cannot exceed the shelf capacity of the store, as expressed in Equation (2).
- To ensure the exposure of new products or to maintain product competitiveness, there are lower and upper bounds of the amount of facings for each product, as expressed in Equation (3).
- The allocated amount of facings for each product must be a nonnegative integer, as shown in Equation (4).
- Preparatory phase: this phase checks whether the shelf capacity is smaller than the minimum space requirement, as expressed in Equation (5). If so, the operation will be terminated, meaning this problem is infeasible. If not, the unit profit of each product () will be calculated, and products will be sorted in descending order of unit profit for subsequent allocation.
- Allocation phase: based on the descending order of unit profit obtained in the previous phase, the algorithm allocates available space of shelf to the product to satisfy its minimum space demand. After the allocation, if there is no available space of shelf , the algorithm directly proceeds to the termination phase. If there is still space, the algorithm then allocates the available space of shelf to product with the highest unit profit, until all the available space of shelf is used up, but the amount of facings allocated to each products cannot exceed its upper bounds.
- Termination phase: this phase calculates the corresponding total profit for the final solution.
- Adjustment 1: this adjustment attempts to improve a solution by swapping one facing for a pair of products allocated on the same shelf. For example, a shelf is allocated with 3 beers and 2 sodas. Without violating the constraints, we replace 1 beer with 1 soda and recalculate the total profit. If a higher total profit is obtained, the solution will be updated; otherwise, the original solution will be kept.
- Adjustment 2: this adjustment attempts to improve a solution by interchanging one facing for a pair of products allocated on two different shelves. For example, there are two shelves, respectively allocated with 3 beers and 2 sodas. Without violating the constraints, we swap the display locations for 1 beer on one shelf and 1 soda on the other. If a higher total profit can be obtained after the swap, the solution will be updated; if not, the original solution will be kept.
- Adjustment 3: this is an extension of Adjustment 2. Since the length of facing varies from product to product, after the swapping of facings between the two products on two different shelves, there may be shelf space that can be reallocated to other products.
- Select the set of neighborhood structure , with = 1, 2, …, to be used in the search.
- Give the current incumbent solution .
- Enter the solution process. Set = 1 and repeat the following steps until =: (a) generate a point at random from the -th neighborhood of ; (b) apply the local search method with as the initial solution to obtain the local optimum ; and (c) if the obtained solution is better than the incumbent solution , update the solution with and continue the local search with the current neighborhood structure; otherwise, move to the next neighborhood: = + 1.
- Neighborhood structure, : Let be the incumbent solution and be the current amount of facings of the product i on the shelf k. The neighborhood structure changes the value of as follows: find the shelves with available space capacity to allocate one facing of product i, and then select one with the minimum available space to move one facing of product i. The new solution is better than X if and only if the available space of the shelf with the minimum available capacity is smaller than that of .
- Neighborhood structure, : Given that shelf and shelf are respectively allocated with facings of product and product , exchanges facing(s) of the two products. In other words, attempts to reduce the minimum available capacity of each shelf by displaying facings of product on shelf and facings of product on shelf . As in , the new solution is better than if and only if the available space of the shelf with the minimum available capacity is smaller than that of .
- Neighborhood structure, : The objective of this neighborhood structure is to optimize sales profit. It works by removing a facing of a product and replaces it with a facing of product , where and are different products. The premise is that the available capacity of the shelf after the removal of the facing of is greater than the facing of product . The new solution is better than if and only if the sales profit is greater than that of .
- Neighborhood structure, : This neighborhood structure removes a facing of product and replaces it with a facing of and with a facing of product . This neighborhood structure is based on a simple idea: The profit of two “small” products can exceed that of a “large” one. When using this structure, the new solution is better than if and only if the sales profit is greater than that of .
4. Teaching–Learning-Based Optimization
- Configure parameters: The parameters include model parameters and control parameters of the algorithm. The parameters of Yang’s SSAP model include the number of products, the number of shelves, the profit per facing of each product, the width of facing of each product, the length of all shelves, and the upper bound and lower bound of the amount of facings for each product. The control parameters of the TLBO algorithm include the population size ( and maximum number of generations ().
- Initialize the learner population: a learner , where denotes the allocated amount of facings of product on shelf . The amount of facings were generated by using Equation (6):
- Evaluate the grade of learners: By substituting the learner’s (decision variables) value into the objective function in Equation (1), we can obtain the learner’s grade (i.e., the objective value) and then select the learner with the highest grade as teacher and set it to .
- The teacher phase: Let be the mean grade of the class. In this phase, the teacher attempts to improve to his/her level (). The difference between the teacher and the learners can be expressed as in Equation (7):
- The learner phase: Learners are allowed to randomly exchange knowledge with other learners for enhancing his/her knowledge. A learner will learn new information if the other learners have more knowledge than he or she has; a learner does not learn anything new if the other learners do not have more knowledge. Let be the other randomly selected learner. The learning of from can be mathematically expressed as in (10):Similar to the teacher phase, those with a better grade between the learner and the newly generated learner will be accepted.
- Termination: Check whether the stopping condition is met (i.e., maximum number of generations is achieved). If the condition is met, the algorithm terminates and outputs the current solution, which is the optimal solution for this generation; otherwise, evaluate the grade of all learners for the next teacher phase.
5. Experimental Setting and Results
6. TLBO-VNS
7. Conclusions
Author Contributions
Funding
Conflicts of Interest
References
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References | Model Class | Solution Method |
---|---|---|
Yang and Chen [1] | MINLP/ILP | None |
Corstjens and Doyle [6] | GP | None |
Yang [8] | ILP | Specialized heuristics |
Hansen and Heinsbroek [9] | MINLP | Lagrange multiplier technique |
Hwang et al. [11] | MNILP | Gradient descent search, Genetic Algorithm (GA) |
Urban [12] | MINLP | Greedy heuristic, Genetic Algorithm (GA) |
Hansen et al. [13] | MILP | CPLEX, Genetic Algorithm (GA) |
Castelli and Vanneschi [16] | MINLP | Genetic Algorithm (GA), Genetic Algorithm with Variable Neighborhood Search (GA-VNS) |
Corstjens and Doyle [17] | GP | Signomial geometric programming |
Borin et al. [18] | MINLP | Simulated Annealing (SA) |
Drèze et al. [19] | BILP | Linear optimization package (LINDO, LINGO) |
Murray et al. [20] | MINLP | Branch-and-Bound based MINLP algorithm |
Bai et al. [21] | INLP | Multiple neighborhood approach |
Schaal and Hübner [22] | INLP/BNLP | Specialized heuristics |
Yu et al. [23] | MINLP | Reduced Variable Neighborhood Search-based Hyperheuristic (HyVNS) |
Model Parameters | Value |
---|---|
(M, N) | (2, 6), (2, 8), (2, 10), (3, 4), (3, 6), (4, 4) |
270, 315, 360 | |
0, 1, 2 | |
– | 1, 2, 3 |
~ ; ~ Uniform (200, 7000) and ~ Uniform (0.05, 0.15) | |
~ Uniform (15, 40) |
Problem No. | Parameter | S1 | S2 | S3 | S4 | S5 | Best |
---|---|---|---|---|---|---|---|
1 | 270-1-0 | 30,470 | 30,470 | 30,470 | 30,470 | 30,470 | 30,470 |
2 | 270-2-0 | 58,627 | 58,627 | 58,627 | 58,627 | 58,627 | 58,627 |
3 | 270-3-0 | 81,927 | 81,927 | 83,058 | 83,844 | 83,844 | 83,844 |
4 | 270-2-1 | 70,872 | 70,872 | 70,872 | 70,872 | 70,872 | 70,872 |
5 | 270-3-1 | 82,024 | 82,024 | 83,771 | 83,771 | 83,771 | 83,771 |
6 | 270-4-1 | 89,409 | 89,409 | 89,409 | 89,409 | 89,409 | 89,409 |
7 | 270-3-2 | 93,472 | 93,472 | 93,472 | 93,472 | 93,472 | 93,472 |
8 | 270-4-2 | 88,360 | 92,333 | 94,176 | 94,176 | 94,176 | 94,176 |
9 | 270-5-2 | 82,320 | 82,658 | 82,658 | 83,132 | 83,865 | 83,865 |
10 | 315-1-0 | 32,649 | 32,649 | 32,649 | 32,649 | 32,649 | 32,649 |
11 | 315-2-0 | 66,103 | 66,103 | 66,103 | 66,103 | 66,103 | 66,103 |
12 | 315-3-0 | 97,884 | 97,884 | 96,487 | 97,134 | 97,884 | 97,884 |
13 | 315-2-1 | 51,031 | 51,031 | 50,795 | 51,031 | 51,031 | 51,031 |
14 | 315-3-1 | 73,733 | 73,733 | 72,242 | 73,677 | 73,733 | 73,733 |
15 | 315-4-1 | 108,130 | 108,130 | 102,850 | 103,909 | 108,133 | 108,133 |
16 | 315-3-2 | 92,753 | 96,468 | 92,510 | 96,470 | 96,473 | 96,473 |
17 | 315-4-2 | 103,900 | 104,475 | 99,354 | 104,988 | 107,326 | 107,326 |
18 | 315-5-2 | 101,690 | 103,410 | 98,425 | 102,040 | 102,766 | 103,410 |
19 | 360-1-0 | 31,604 | 31,604 | 31,604 | 31,604 | 31,604 | 31,604 |
20 | 360-2-0 | 50,951 | 50,951 | 50,340 | 50,728 | 50,951 | 50,951 |
21 | 360-3-0 | 89,116 | 89,116 | 89,069 | 89,022 | 89,116 | 89,116 |
22 | 360-2-1 | 77,021 | 77,021 | 69,240 | 77,021 | 77,021 | 77,021 |
23 | 360-3-1 | 88,547 | 88,547 | 82,443 | 88,547 | 88,547 | 88,547 |
24 | 360-4-1 | 137,710 | 137,710 | 121,695 | 134,914 | 137,713 | 137,713 |
25 | 360-3-2 | 80,719 | 80,719 | 80,719 | 80,719 | 80,719 | 80,719 |
26 | 360-4-2 | 99,244 | 99,244 | 102,372 | 103,503 | 105,501 | 105,501 |
27 | 360-5-2 | 100,040 | 100,790 | 100,788 | 100,788 | 100,788 | 100,790 |
Problem No. | Parameter | Solution Method | ||||
---|---|---|---|---|---|---|
S1 | S2 | S3 | S4 | S5 | ||
1 | 270-1-0 | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% |
2 | 270-2-0 | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% |
3 | 270-3-0 | 2.29% | 2.29% | 0.94% | 0.00% | 0.00% |
4 | 270-2-1 | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% |
5 | 270-3-1 | 2.09% | 2.09% | 0.00% | 0.00% | 0.00% |
6 | 270-4-1 | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% |
7 | 270-3-2 | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% |
8 | 270-4-2 | 6.18% | 1.96% | 0.00% | 0.00% | 0.00% |
9 | 270-5-2 | 1.84% | 1.44% | 1.44% | 0.87% | 0.00% |
10 | 315-1-0 | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% |
11 | 315-2-0 | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% |
12 | 315-3-0 | 0.00% | 0.00% | 1.43% | 0.77% | 0.00% |
13 | 315-2-1 | 0.00% | 0.00% | 0.46% | 0.00% | 0.00% |
14 | 315-3-1 | 0.00% | 0.00% | 2.02% | 0.08% | 0.00% |
15 | 315-4-1 | 0.00% | 0.00% | 4.89% | 3.91% | 0.00% |
16 | 315-3-2 | 3.86% | 0.01% | 4.11% | 0.00% | 0.00% |
17 | 315-4-2 | 3.19% | 2.66% | 7.43% | 2.18% | 0.00% |
18 | 315-5-2 | 1.66% | 0.00% | 4.82% | 1.32% | 0.62% |
19 | 360-1-0 | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% |
20 | 360-2-0 | 0.00% | 0.00% | 1.20% | 0.44% | 0.00% |
21 | 360-3-0 | 0.00% | 0.00% | 0.05% | 0.11% | 0.00% |
22 | 360-2-1 | 0.00% | 0.00% | 10.10% | 0.00% | 0.00% |
23 | 360-3-1 | 0.00% | 0.00% | 6.89% | 0.00% | 0.00% |
24 | 360-4-1 | 0.00% | 0.00% | 11.63% | 2.03% | 0.00% |
25 | 360-3-2 | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% |
26 | 360-4-2 | 5.93% | 5.93% | 2.97% | 1.89% | 0.00% |
27 | 360-5-2 | 0.74% | 0.00% | 0.00% | 0.00% | 0.00% |
Solution Method | |||||
---|---|---|---|---|---|
S1 | S2 | S3 | S4 | S5 | |
Maximum of the Performance Gap | 6.18% | 5.93% | 11.63% | 3.91% | 0.62% |
Median of the Performance Gap | 0.00% | 0.00% | 0.46% | 0.00% | 0.00% |
StDev. of the Performance Gap | 1.82% | 1.36% | 3.33% | 0.96% | 0.12% |
(#shelves, #products) (M, N) | Solution Method | ||||
---|---|---|---|---|---|
S1 | S2 | S3 | S4 | S5 | |
Maximum of the Performance Gap | |||||
(2, 6) | 6.18% | 5.93% | 11.63% | 3.91% | 0.62% |
(2, 8) | 14.04% | 14.04% | 23.78% | 15.96% | 0.64% |
(2, 10) | 14.04% | 9.35% | 22.48% | 19.24% | 16.72% |
(3, 4) | 5.63% | 5.63% | 10.44% | 3.82% | 0.00% |
(3, 6) | 7.48% | 5.24% | 29.60% | 12.47% | 0.00% |
(4, 4) | 6.87% | 6.87% | 22.06% | 3.69% | 0.00% |
(M, N) | Median of the Performance Gap | ||||
(2, 6) | 0.00% | 0.00% | 0.46% | 0.00% | 0.00% |
(2, 8) | 0.94% | 0.94% | 6.02% | 2.46% | 0.00% |
(2, 10) | 1.03% | 0.00% | 5.16% | 2.18% | 0.00% |
(3, 4) | 0.00% | 0.00% | 3.01% | 0.00% | 0.00% |
(3, 6) | 0.00% | 0.00% | 12.52% | 1.55% | 0.00% |
(4, 4) | 0.00% | 0.00% | 10.26% | 2.09% | 0.00% |
(M, N) | Standard Deviation of the Performance Gap | ||||
(2, 6) | 1.82% | 1.36% | 3.33% | 0.96% | 0.12% |
(2, 8) | 4.14% | 3.65% | 6.29% | 3.97% | 0.12% |
(2, 10) | 2.91% | 2.10% | 7.52% | 4.72% | 3.80% |
(3, 4) | 1.34% | 1.34% | 3.31% | 1.32% | 0.00% |
(3, 6) | 1.82% | 1.28% | 8.06% | 3.61% | 0.00% |
(4, 4) | 1.91% | 1.91% | 6.44% | 1.23% | 0.00% |
Problem Set (M, N) | S1 | S2 | S3 | S4 | |
---|---|---|---|---|---|
TLBO(S5) | (2, 6) | 0.003 ** | 0.016 * | 0.001 *** | 0.003 ** |
(2, 8) | 2.3 × 10−4 *** | 2.3 × 10−4 *** | 2.7 × 10−5 *** | 5.3 × 10−5 *** | |
(2, 10) | 0.019 * | 0.058 | 1.03 × 10−4 *** | 0.001 *** | |
(3, 4) | 0.043 * | 0.043 * | 8.9 × 10−5 *** | 0.001 *** | |
(3, 6) | 0.018 * | 0.018 * | 1.8 × 10−5 *** | 6 × 10−5 *** | |
(4, 4) | 0.144 | 0.144 | 2.7 × 10−5 *** | 6 × 10−5 *** |
(#shelves, #products) (M, N) | Solution Method | |||||
---|---|---|---|---|---|---|
S1 | S2 | S3 | S4 | S5 | S6 | |
Maximum of the Performance Gap | ||||||
(2, 6) | 6.18% | 5.93% | 26.38% | 11.63% | 0.62% | 0.62% |
(2, 8) | 17.03% | 15.66% | 23.78% | 20.00% | 13.26% | 13.26% |
(2, 10) | 14.04% | 9.35% | 23.03% | 19.81% | 16.91% | 0.00% |
(3, 4) | 5.63% | 5.63% | 10.44% | 3.82% | 0.00% | 0.00% |
(3, 6) | 7.48% | 5.24% | 29.60% | 13.15% | 0.78% | 0.11% |
(4, 4) | 6.87% | 6.87% | 22.06% | 3.69% | 0.00% | 0.00% |
Median of the Performance Gap | ||||||
(2, 6) | 0.00% | 0.00% | 0.86% | 0.46% | 0.00% | 0.00% |
(2, 8) | 0.94% | 0.94% | 8.03% | 1.88% | 0.00% | 0.00% |
(2, 10) | 1.35% | 1.09% | 5.16% | 2.89% | 0.00% | 0.00% |
(3, 4) | 0.00% | 0.00% | 3.01% | 0.00% | 0.00% | 0.00% |
(3, 6) | 0.00% | 0.00% | 12.52% | 1.55% | 0.00% | 0.00% |
(4, 4) | 0.00% | 0.00% | 10.26% | 2.09% | 0.00% | 0.00% |
Standard Deviation of the Performance Gap | ||||||
(2, 6) | 1.82% | 1.36% | 8.70% | 3.33% | 0.12% | 0.12% |
(2, 8) | 5.82% | 5.39% | 6.98% | 6.21% | 3.54% | 3.54% |
(2, 10) | 2.89% | 2.05% | 7.42% | 4.70% | 3.86% | 0.00% |
(3, 4) | 1.34% | 1.34% | 3.31% | 1.32% | 0.00% | 0.00% |
(3, 6) | 1.81% | 1.28% | 8.07% | 3.68% | 0.15% | 0.02% |
(4, 4) | 1.91% | 1.91% | 6.44% | 1.23% | 0.00% | 0.00% |
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Chen, Y.-K.; Weng, S.-X.; Liu, T.-P. Teaching–Learning Based Optimization (TLBO) with Variable Neighborhood Search to Retail Shelf-Space Allocation. Mathematics 2020, 8, 1296. https://0-doi-org.brum.beds.ac.uk/10.3390/math8081296
Chen Y-K, Weng S-X, Liu T-P. Teaching–Learning Based Optimization (TLBO) with Variable Neighborhood Search to Retail Shelf-Space Allocation. Mathematics. 2020; 8(8):1296. https://0-doi-org.brum.beds.ac.uk/10.3390/math8081296
Chicago/Turabian StyleChen, Yan-Kwang, Shi-Xin Weng, and Tsai-Pei Liu. 2020. "Teaching–Learning Based Optimization (TLBO) with Variable Neighborhood Search to Retail Shelf-Space Allocation" Mathematics 8, no. 8: 1296. https://0-doi-org.brum.beds.ac.uk/10.3390/math8081296