Multi-item stochastic and fuzzy-stochastic inventory models under two restrictions

doi:10.1016/S0305-0548(03)00120-5

Computers & Operations Research

Volume 31, Issue 11, September 2004, Pages 1793-1806

https://doi.org/10.1016/S0305-0548(03)00120-5 Get rights and content

Abstract

Multi-item stochastic and fuzzy-stochastic inventory models are formulated under total budgetary and space constraints. Here, the inventory costs are directly proportional to the respective quantities, unit purchase/production cost is inversely related to the demand and replenishment/production rate is assumed to vary directly with demand. Shortages are allowed but fully backlogged. Here, for both models, demand and budgetary resource are assumed to be random. In fuzzy-stochastic model, in addition to the above assumptions, available storage space and total expenditure are imprecise in nature. Impreciseness in the parameters have been expressed with the help of linear membership functions. Assuming random variables to be independent and to follow normal distributions, the models have been formulated as stochastic and fuzzy-stochastic non-linear programming problems. The stochastic problem is first reduced to the equivalent single objective or multiple objectives problems following chance-constraint method. The problem with single objective is solved by a gradient-based technique whereas fuzzy technique is applied to the multi-objective one. In the same way, the fuzzy-stochastic programming problem is first reduced to a corresponding equivalent fuzzy non-linear programming problem and then it is solved by fuzzy non-linear programming (FNLP) following Zimmermann technique. The models are illustrated numerically and the results of different models are compared.

Introduction

After the publication of classical lot-size formula by Harris in 1915, many researchers worked on the EOQ model and currently these results are available in reference books and survey papers (e.g. Raymond [1], Hadley and Whitin [2], Naddor [3], Clark [4], etc.).

In classical inventory models, normally the rate of replenishment/production is assumed to be constant. In practice, it is observed that the replenishment/production is influenced by demand. This situation generally arises in the case of highly demandable products. If the demand of an item in the market goes up, consumption of that item obviously will be more and to meet the extra requirement, a retailer increases the replenishment/production of the item. Sometimes, rate of replenishment is also determined depending upon the quantity of idle stock at godown. If the stock at the godown moves very slowly, then the retailer is forced to cut down the replenishment/production rate. Goswami and Chaudhuri [5], Bhunia and Maiti [6], Balki and Benkherouf [7] developed inventory models with variable replenishment/production for multi-item inventory problems.

In real-life situations, the demand and cost of production of an item are not totally independent of each other. When demand of an item is high, a manufacturing firm produces more and can spread the set-up cost, etc. over a large number of produced units and this will result in lower average unit production cost. Hence, it is practical to assume that demand and unit cost are inversely related to each other. With this assumption, Cheng [8], [9] has developed some inventory models and solved by geometric programming technique.

In classical inventory models, inventory costs—holding and set-up costs are assumed to be constant. But, holding cost for an item is supposed to be dependent on the amount put in the storage. Similarly, set-up cost also depends upon the total quantity to be produced in a scheduling period. Amongst the inventory costs, it is very difficult to measure the shortage cost. Not taking the cost of lost goodwill (which cannot be measured numerically) into account, the shortage cost also can be made directly proportional to the amount not supplied to the customer. Hence, to make the model realistic, inventory costs can be made proportional to a power of the respective quantities. Some research papers (cf. Brown et al. [10], Chitgopekar [11], Hariri and Abou-el-ata [12], Abou-el-ata and Kotb [13]) have dealt with inventory models with variable inventory costs solved by geometric programming and other gradient-based technique.

Multi-item classical inventory models under resource constraints such as budgetary cost, limited storage area, number of orders, available set-up times, etc. are presented in well-known books (Naddor [3], Silver and Peterson [14], Churchman et al. [15], Hadely and Whitin [2], Lewis [16], etc.). Recently Ben-daya and Raouf [17] discussed a multi-item inventory model with stochastic demand and Abou-el-ata and Kotb [13] developed a crisp inventory model under two restrictions.

The complexity arises in modelling a realistic decision-making inventory situation mainly due to the presence of some non-deterministic information, in the sense that they are not just capable of being encoded with precision and certainty of classical mathematical reasoning. Actually, a realistic situation is no longer realistic when the imprecise and uncertain information are neglected for the sake of mathematical model building. During the last three decades, considerable developments in the fields of operations research has enabled the theories of probabilistic and fuzzy sets to offer ways for constructing metaphors that represents many aspects of uncertainty and impreciseness. These theories have been extensively applied to model the decision-making situations in their respective environments. Generally, ‘imprecision’ and ‘uncertainty’ are modelled by fuzzy and stochastic approaches, respectively. Now, existence of a mixed environment or the coexistence of imprecision and uncertainty in an inventory model is again a realistic phenomenon and the mathematical realization of this fact is an interesting field. Here, extension of chance constrained programming to fuzzy environment has been investigated through an inventory model. We have developed an inventory model in a mixed environment where some constraints are imprecisely defined and others probabilistically.

Realistically, a retailer/owner of a factory starts the business/production with a fixed amount of cash in hand to purchase the items/materials and a godown of finite area to store the items/products. But, in the course of business/production, the retailer augments the said capital by some amount in the interest of the business, if the situation demands. Similarly, to avail the certain transport facility/concession or to capitalize some production situations, items may be replenished/produced more than the capacity of the available warehouse and in that case, he manages some additional storage space, if it is required and situation is so called for. Hence, for real-life inventory problems, both budgetary amount and storage space are not defined with certainty i.e. may be uncertain in stochastic or non-stochastic (imprecise) sense.

In this paper, we have formulated multi-item stochastic and fuzzy-stochastic inventory problems with demand-dependent unit cost and replenishment under budgetary and storage-space constraints. Inventory costs are dependent on their respective quantities. Shortages are allowed and fully backlogged. Here, for the stochastic model, demand and both resources i.e. budgetary amount and storage area are assumed to be random. The constraints imposed on these resources are to be satisfied in the probabilistic sense i.e. maximum probabilities on the violation of these constraints are specified. For the fuzzy-stochastic model, the assumptions are the same as the stochastic one except that available storage area is imprecise instead of being random and total expenditure of the model also is somewhat vague instead of being certain and precisely defined. In this model, the restriction on the warehouse space is more rigid i.e. to be satisfied deterministically, not in the probabilistic sense. Here, all random parameters/variables are taken as independent and follow normal distribution. Though all fuzzy goals may be represented by linear and/or non-linear membership functions, for simplicity, these are represented here by linear membership functions. In this paper, the models have been formulated as stochastic and fuzzy stochastic programming problems. The stochastic programming problem is solved by chance constraint technique. Fuzzy-stochastic programming problem is first reduced to a corresponding equivalent fuzzy non-linear programming problem and then solved by FNLP technique. The models are illustrated numerically and the results of different models are compared.

Section snippets

Model formulation

The following notations and assumptions are used in developing the model.

Crisp model

Therefore, the objective is to minimize the total average cost of multi-items for finite replenishment problem subject to limitations on total budgetary cost and storage area $Min TC(D,Q,S)= ∑ i=1 n b_{1i} D_{i}^{1−β_{i}} +b_{2i} S_{i}^{2+γ_{i}} Q_{i}^{−1} +b_{3i} θ_{i} Q_{i} −S_{i}^{2+δ_{i}} Q_{i}^{−1} +b_{4i} D_{i} Q_{i}^{α_{i}−1} s.t. ∑ i=1 n b_{1i} D_{i}^{−β_{i}} Q_{i} ⩽B, ∑ i=1 n a_{i} (θ_{i} Q_{i} −S_{i})⩽A,Q_{li} ⩽Q_{i} ⩽Q_{ui},D_{i},S_{i} >0 (i=1,…,n),$ where b_1i=c_0i, b_2i=g_0i2^−(1+γ_i), b_3i=h_0i2^−(1+δ_i), b_4i=u_0i and θ_i=1−1/ν_i.

Stochastic model

With the demands D_i's, maximum allowable budgetary cost, maximum allowable storage area as random, probabilities

Mathematical Analysis

Multi-objective mathematical programming. The problem of optimization concerns with the maximization/minimization of an algebraic or a transcendental function of one or more variables, known as objective function under some available resources which are represented as constraints.

A general multi-objective non-linear programming problem is of the following form: $Find x=(x_{1},x_{2},…,x_{N})^{T} which minimizes/maximizes F(x)=(f_{1} (x),f_{2} (x),…,f_{k} (x))^{T} subject to g_{j} (x)⩽a_{j} (j=1,2,…,l), h_{r} (x)=b_{r} (r=1,2,…,m), x_{i} ⩾0 (i=1,2,…,N),$

Fuzzy programming technique to solve the SNLP E–V model

To solve the multi-objective programming described in (17c), the first step is to assign two values U_k and L_k as upper and lower bounds for each kth (k=1,2) objective, where for the kth objective $L_{k} = aspired level of achievement,$ $U_{k} = acceptable level of achievement .$

Now, the stochastic programming problem (2) has completely defined in crisp environment. Such reduced crisp problem is to be solved using the fuzzy programming technique. The steps are as follows:

Step 1: Solve the multi-objective programming

Stochastic model

Using the above algorithm the probabilistic model (2) is transformed to $Max α s.t. U_{1} −ETC(D ̄,Q,S) U_{1} −L_{1} ⩾α, U_{2} −VTC(D ̄,Q,S) U_{2} −L_{2} ⩾α, ∑ i=1 n b_{1i} D ̄_{i}^{−β_{i}} Q_{i} − B ̄ −s_{1} ∑ i=1 n −β_{i} b_{1i} D ̄_{i}^{−β_{i}−1} Q_{i}^{2} σ_{D_{i}}^{2} +σ_{B}^{2}^{1/2} ⩽0, ∑ i=1 n a_{i} (θ_{i} Q_{i} −S_{i})− A ̄ −s_{2} σ_{A} ⩽0,$ where $ETC(D ̄,Q,S)= ∑ i=1 n b_{1i} D ̄_{i}^{1−β_{i}} +b_{2i} S_{i}^{2+γ_{i}} Q_{i}^{−1} + b_{3i} θ_{i} Q_{i} −S_{i}^{2+δ_{i}} Q_{i}^{−1} +b_{4i} D ̄_{i} Q_{i}^{α_{i}−1},$ $VTC(D ̄,Q,S)= ∑ i=1 n b_{1i} (1−β_{i}) D ̄_{i}^{−β_{i}} +b_{4i} Q_{i}^{α_{i}−1}^{2} σ_{D_{i}}^{2}^{1/2} Q_{li} ⩽Q_{i} ⩽Q_{ui}, D ̄_{i} >0, S_{i} >0.$

It is clear that the stochastic model (19) can be reduced to a crisp model in the absence of randomness, i.e. when D_i's, B and A are

Numerical examples and discussion

To illustrate the above methodology, we consider the following parametric values for two items (n=2) shown in Table 1 and the optimum values for the stochastic and fuzzy-stochastic models are shown in Table 2.

In addition to the parametric values in Table 1 let p₁=0.80, p₂=0.77, $A ̄ =70 ft^{2}$ , $B ̄ =$130$ , $σ_{A} =14 ft^{2}$ , σ_B=$19.5, $D ̂_{1} =(D ̄_{1},0.05 D ̄_{1})$ , $D ̂_{2} =(D ̄_{2},0.05 D ̄_{2})$ . To solve the fuzzy stochastic model (3), also let C_E=$240, C_V=$1.20, $A=40 ft^{2}$ , P_E=$60, P_V=$2.0, $P_{A} =35 ft^{2}$ .

In the case of the stochastic and

Conclusion

In this paper, we have formulated inventory problems in both random and random-fuzzy environments. Till now, stochastic inventory problems have been normally formulated using a probability distribution for an inventory parameter and solved reducing them to equivalent crisp problems by integrating the distribution function. Here, for the first time, we have formulated and solved stochastic inventory problem completely in a different way. We have taken the demand (a decision parameter) involved

Acknowledgements

The authors are grateful to the reviewers for their valuable suggestions and comments for the improvement of the paper.

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An economic order quantity model with demand-dependent unit production cost and imperfect production process
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Multi-item stochastic and fuzzy-stochastic inventory models under two restrictions

Abstract

Introduction

Section snippets

Model formulation

Crisp model

Stochastic model

Mathematical Analysis

Fuzzy programming technique to solve the SNLP E–V model

Stochastic model

Numerical examples and discussion

Conclusion

Acknowledgements

European Journal of Operational Research

Quantity and economic in manufacture

Analysis of inventory systems

Inventory systems

An informal survey of multi-echelon inventory theory

Naval Research Logistics Quaterly

An EOQ model for deteriorating items with shortages and a linear trend in demand

Journal of Operational Research Society

Deterministic inventory models for variable production

Journal of Operational Research Society

A production lot size inventory model for deteriorating items and arbitrary production and demand rates

European Journal of Operational Research

An economic order quantity model with demand-dependent unit production cost and imperfect production process

IIE Transactions