This paper describes a stochastic production planning problem for an environmentally conscious recycling technique to adequately cope with unpredictable recovery rate of end-of-life products towards a more sustainable pattern of production. The model is formulated as a nonlinear integer programming (NIP), which contains two types of nonlinear programming problems: determining (1) the production quantities of end-product and (2) component parts machining in a finite planning horizon with both the stochastic demand rate of end-product and recovery rate of end-of-use product. This decision problem is solved so as to minimize the sum of production costs, inventory holding costs, and backorder costs subject to the capacity constraints, the stochastic demand requirement, and desired component part quantity determining from the end-product scheduling. The Lagrangean relaxation technique is introduced to solve the NIP by developing the two-step hybrid procedure. Finally, a numerical example is demonstrated to explain the effectiveness of optimal solution procedures.
[...] Table 3 shows computational results along with optimal solution procedures. Table 1 Basic data for computational study Table 2 Stochastic demand rate and recovery rate Table 3 Computational results 8. Conclusions This paper describes a stochastic production planning problem for an environmentally conscious recycling technique to adequately cope with unpredictable recovery rate of end-of-life products towards a more sustainable pattern of production. The Lagrangean relaxation technique is introduced to solve the decision problem by developing the two-step hybrid procedure. Finally, a numerical example is [...]
[...] Optimal Solution Procedure The essentials of recycling conscious planning described above is the development of a integrated two types of decision-making problem for a stochastic recovery rate of end-of-use products. The one is recycling planning for determining the production quantity of parts or unit, that should be satisfied forecast demand rate of products. The other decision-making is to determine the production quantity of end-product subject to quantities of part and unit to be produced. This paper develops an optimal solution procedure as two-step hybrid algorithm associated with effective Lagrangean relaxation techniques. [...]
[...] The maximum allowable inventory level of product and part is Wi [pcs] and W j [pcs] respectively Formulating the Objective Function The objective for optimizing the production quantities of product and its part considering recycled part in the specified planning period is to minimize the sum of production costs, inventory holding costs, backorder costs subject to the capacity constraints, the stochastic demand requirement, desired part or unit quantity to meet product demand requirement. The decision-making problem can be formulated as follows: [Primal Problem] Objective function (expected total production costs): Min E ( Z ) xit , x jt I = {ai xit i t T T I T + ci [ xit d it f it it )dd it 0 J xit + si it ) xit k[Var it 2 + Var (rit 2 i t t j x + j x jt LT + c j [ x jt LT jt LT d jt LT f jt LT jt LT )dd jt LT j t + R jt f jt ( R jt ) dR jt J T where x it [pcs] and x jt [pcs] means production quantities for part i and part j at the end of period t. [...]
[...] This paper presents strategies and optimal analysis of the environmental conscious recycling planning to adequately cope with unpredictable recovery rate of end-of-life products The Nature of Recycling Planning The environmentally conscious recycling planning has the following characteristics: 1. Recovery rate of end-of-life products is dependent upon actual production (or sales) volume accumulated during the specified planning periods. According to increase production volume substantially, deviations of errors of a forecast over planning periods tend to being very small relative to the value of beginning period An optimal incentive mechanism is therefore vital for enhancing re-production and re-use of end-of-life products as well as stimulating recycling activities. [...]
[...] An optimal hybrid-algorithm is developed by applying Lagrangean relaxation technique Developing Recycling Conscious Plans To describe the recycling conscious planning in precise mathematical terms, the following assumptions are considered: Dit [pcs] denotes the forecast demand of part i in period t and a random variable having known probability distribution, Fit it ) pr{Dit d it , density functions f it it ) with variance Var it ) . Lead time, LTj [periods] and τ j [periods] requires to manufacture and recycle a part j with processing time y j [hours / pc]. [...]
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