MULTI-ITEM SHELF-SPACE ALLOCATION OF BREAKABLE ITEMS

J. Appl. Math. & Computing Vol. 20(2006), No. 1 - 2, pp. 327 - 343

Website: http://jamc.net

MULTI-ITEM SHELF-SPACE ALLOCATION OF BREAKABLE ITEMS VIA GENETIC ALGORITHM MANAS KUMAR MAITI∗ AND MANORANJAN MAITI

Abstract. A general methodology is suggested to solve shelf-space allocation problem of retailers. A multi-item inventory model of breakable items is developed, where items are either complementary or substitute. Demands of the items depend on the amount of stock on the showroom and unit price of the respective items. Also demand of one item decreases (increases) due to the presence of others in case of substitute (complementary) product. For such a model, a Contractive Mapping Genetic Algorithm (CMGA) has been developed and implemented to find the values of different decision variables. These are evaluated to have maximum possible profit out of the proposed system. The system has been illustrated numerically and results for some particular cases are derived. The results are compared with some other heuristic approaches- Simulated Annealing (SA), simple Genetic Algorithm (GA) and Greedy Search Approach (GSA) developed for the present model. AMS Mathematics Subject Classification : 90B05. Key words and phrases : Shelf-space, backroom inventory, contractive mapping genetic algorithm, complementary product, substitute product, crossspace elasticity.

1. Introduction In an inventory management, there are different types of demands considered by researchers. Among them, constant, stock-dependent, time-dependent and price-dependent demands draw more attention. According to Levin et al [10], “ the presence of inventory has a motivational effect on the people around it ”. In the present competitive market, the inventory/stock is decoratively displayed through electronic media to attract the customers and thus to push the sale. Schary and Becker [16] and Wolfe [21] also established the impact of product Received November 12, 2003. Revised August 18, 2004. ∗ Corresponding author. c 2006 Korean Society for Computational & Applied Mathematics and Korean SIGCAM.

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availability for simulating demand. Mandal and Phaujdar [13], Datta and Pal [5] and others considered linear form of stock-dependent demand i.e. D=c+dq whereas Urban [19], Mandal and Maiti [11] and others took the demand of the form D=dqβ . The marketing community has recognized the above relationship and incorporated it into product assortment and shelf-space allocation models. These models are formulated with the demand rate of a product as a function of the shelf-space allocated to the product and, sometimes, to the shelf-space allocated to it’s substitute and/or complementary products also. Urban [18] developed a model to identify which product should be included in a firm’s product line. The model was formulated with the demand rate as a nonlinear function of price, advertisement, and distribution (represented by the number of shelf facings in the empirical application), considering both main and cross elasticities of the marketing variables, and was solved using an iterative search routine. Corstjens and Doyle [4] developed a shelf-space allocation model in which the demand rate is a function of shelf-space allocated to the product, also applying a nonlinear functional form of demand with main and cross- elasticities of shelf space. They utilized Signomial Geometric Programming to solve the model. Zurfryden [22] suggested the use of dynamic programming to solve the shelf -space allocation problem, as it will allow the consideration of general objective-function specification and provide integer solutions. Borin, Farris and Freeland [3] presented an integrated product assortment and shelf-space allocation problem incorporating cross elasticity effects of substitute items as well as the effect on the demand of products when other products are not included in the assortment and suggested Simulated Annealing as a solution methodology. But these propositions put a retailer/manufacturer of the items of glass, ceramic, china- clay etc. in a conflicting situation. He/She is tempted to go for a large number of goods in the form of heaped stock to take advantage of more demand (hence more consumption), less price, transport convenience, etc. and as a result, invites more damage to his/her units, as damageability increases with the increase of piled stock (cf. Mandal & Maiti [11]). Hence in addition to the usual contradiction in the holding and set-up costs, the manufacturer/retailer faces the above contradiction and tries to make an optimal decision for maximum profit. This paper gives an answer to the above problem. One drawback of the existing product assortment and shelf-space allocation models is that they give no explicit consideration to inventory-related decisions. Some include an operating-cost factor, which assumes that the costs are proportional to the sales of the product; however, these costs are independent of the order size, inventory levels, or frequency of ordering. Borin et al. [3] use return on inventory as the objective and incorporate stockouts, but did not include the conventional inventory-control decisions as variables. Another drawback of the models is that they do not give a general solution methodology to solve

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the complete model rather give a solution methodology for the problem of some particular form of parametric values. Also the existing shelf-space allocation models implicitly assume that the entire inventory (i.e. entire order quantity) is displayed [3-5,10-13]. This may be appropriate for some applications, such as used car sales, where the customer can see the entire inventory. But many organizations (e.g. many retail outlets) have a backroom inventory or a warehouse in which the order is received before being placed in the showroom. Thus there is a limited amount of displayed inventory that has an effect on sales. Much of the inventory is not in the customer’s view and has no impact on sales. Very few inventory models have been developed to reflect this type of situation [20]. In the present competitive market, the effect of marketing policies and conditions such as the price variations and advertisement of an item changes its selling rate among the public. In selecting of an item for use, the selling price of an item is one of the decisive factors to the customers. It is commonly seen that lesser selling price causes increasing in the selling rate where as higher selling price has the reverse effect. Hence the selling rate of an item is dependent on the selling price of that item. This selling rate function must be a decreasing function with respect to the selling price. Incorporating the price variations, recently several researchers [1],[2],[17] developed their models for deteriorating and non-deteriorating items. Again, the inventory control problem with the above realistic assumptions is so complex and non-linear that it is very difficult to get the optimum solution via analytical approach and thus researchers are forced to apply numerical optimization techniques for approximate optimum solution. Among these techniques Greedy Search Method, Simulated Annealing [9], Genetic Algorithm [8] play major role. In this paper, an inventory model of breakable items is developed, where demands of the items depend on the amount of stock on the showroom and selling price of the respective items. The model is developed for multi-items, where items may be complementary or substitute. Also demand of one item decreases (increases) due to the presence of others in case of substitute (complementary) products. At the time of replenishment of an item, initially allocated shelf-space is filled-up and then excess amount is stored in the backroom. Items are sold from the showroom and are continuously filled-up from backroom. When inventory level reaches to reorder-point then again replenishment occurs. Since items are sold from the showroom, a minimum inventory level is always maintained in the showroom for all the items. Here initial inventory level, allocated shelfspace, reorder-point and selling price of different items are decision variables. The model is formulated in integral form with the help of profit maximization principle and for a finite time horizon. To solve the system, the evolutionary method CMGA has been developed and implemented to find out the values of different decision variables, so that the total proceeds out of the system is

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maximum. The model has been illustrated through some numerical data. Few other heuristic approaches- SA, GA and GSA are developed for the model and results from these methods are compared with the results observed via CMGA. 2. Genetic algorithm Genetic Algorithms are exhaustive search algorithms based on the mechanics of natural selection and genesis (crossover, mutation etc.) and have been developed by Holland, his colleagues and students at the University of Michigan (c.f. Goldberg [8]). In natural genesis, we know that chromosomes are the main carriers of hereditary information from parent to offspring and that genes, which present hereditary factors, are lined up on chromosomes. At the time of reproduction, crossover and mutation take place among the chromosomes of parents. In this way hereditary factors of parents are mixed-up and carried to their offspring. Again Darwinian principle states that only the fittest animals can survive in nature. So a pair of fittest parent normally reproduces a better offspring. The same phenomenon is followed to create a genetic algorithm for an optimization problem. Here potential solutions of the problem are analogous with the chromosomes and chromosome of better offspring with the better solution of the problem. Crossover and mutation happen among a set of potential solutions to get a new set of solutions and it continues until terminating conditions are encountered. The advantages of GAs are manifold. They (i) deal with continuous or discrete parameters, (ii) do not require derivative information, (iii) simultaneously search from a wide sampling of the cost function , (iv) deal with a large number of parameters, (v) are well suited for parallel computers, (vi) work with parameters having extremely complex surfaces; they can jump out of a local minima, (vii) provides a set of near-optimum solutions, not just a single solution, (viii) may encode the parameters so that the optimization is done with the encoded parameters, (ix) work with numerically generated data, experimental data or analytical functions. A GA for a particular problem must have the following components. (i) A genetic representation for potential solutions to the problem. (ii) A way to create an initial population of potential solutions (chromosomes). (iii) A way to find fitness of each solution. (iv) An evolution function that plays the role of environment, rating solutions in term of their fitness.

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(v) Genetic operators (crossover, mutation) that alter the composition of children. (vi) Values of different parameters that the genetic algorithm uses (Population size, probabilities of applying genetic operators etc.) A simple genetic algorithm in general form is given below. START T ←0 // T is iteration counter Initialize pc , pm // pc ,pm are probability of crossover // and probability of mutation respectively Initialize P(T) // P(T) is the population of potential solutions // for iteration T Evaluate P(T) // This function evaluate fitness of each member of P(T) While (Not termination condition) { Select solutions from P(T) for evolution. Let this set be P 1 (T). Select solutions from P 1 (T) for crossover. Made crossover on selected solutions to get population P 2 (T). Select solutions from P 2 (T) for mutation. Made mutation on selected solutions to get new population P(T+1). T ←T+1 Evaluate (P(T)) } END 2.1. Contractive mapping genetic algorithm In CMGA, movement from old population to new population takes place only when average fitness of new population is better than the old one. Michalewicz [14] proposed the algorithm and proved the asymptotic convergence of the algorithm by Banach fixed point theorem. A CMGA in general form is given below. START T ←0 // T is iteration counter Initialize pc , pm // pc , pm are probability of crossover // and probability of mutation respectively Initialize P(T) // P(T) is the population of potential solutions // for iteration T Evaluate P(T) // This function evaluate fitness of each member

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of P(T) While (Not termination condition) { Select solutions from P(T) for evolution. Let this set be P 1 (T). Select solutions from P 1 (T) for crossover. Made crossover on selected solutions to get population P 2 (T). Select solutions from P 2 (T) for mutation. Made mutation on selected solutions to get new population P(T+1). Evaluate (P(T+1)). If average fitness of P(T+1)>average fitness of P(T) then T ←T+1 } END 3. Simulated annealing Consider an ensemble of molecules at a high temperature, which are moving around freely. Since physical systems tend towards lower energy states, the molecules are likely to move to the positions that lower the energy of the ensemble as a whole as the system cool down. However molecules actually move to positions which increase the energy of the system with a probability e−∆E/T , where ∆E is the increase in the energy of the system and T is the current temperature. If the ensemble is allowed to cool down slowly, it will eventually promote a regular crystal, which is the optimal state rather than flawed solid, the poor local minima. In function optimization, a similar process can be defined. This process can be formulated as the problem of finding a solution, among a potentially very large number of solutions, with minimum cost. By considering the cost function of the proposed system as the free energy and the possible solutions as the physical states, a solution method was introduced by Kirkpatrick [9] in the field of optimization based on a simulation of the physical annealing process. This method is called Simulated Annealing. The Simulated Annealing algorithm to solve such problems is given below: 1. 2. 3. 4. 5. 6. 7. 8.

Start with some state, S. T=T0 Repeat { While (not at equilibrium){ Perturb S to get a new state Sn ∆E=E(Sn )-E(S) If ∆E