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In this article, we propose novel reformulations for capacitated lot sizing problem. These reformulations are the result of reducing the number of variables (by eliminating the backorder variable) or increasing the number of constraints (time capacity constraints) in the standard problem formulation. These reformulations are expected to reduce the computational time complexity of the problem. Their computational efficiency is evaluated later in this article through numerical analysis on randomly generated problems.

Lot sizing problem aims to optimally utilize the available production resources while meeting the demand targets. It is classified as medium-term planning in production planning taxonomy. Lot sizing problem formulation depends upon the layout and the operating constraints in the production system. In the manufacturing industry, we come across many types of production systems. These production systems further give rise to different types of lot sizing problems (with different constraints and operating conditions) and their solution methodologies. Hence there is a rich literature on lot sizing problem and their solution methods. In this article, we restrict our discussion to general dynamic multi-level capacitated lot sizing problem.

This problem was first proposed by Billington et al. [

A capacitated lot sizing problem is a well-known NP-hard problem. If the capacity constraints of the problem are relaxed, then the problem can be solved in polynomial time [

We next discuss certain problem reformulations from the literature which are computationally efficient. CLSP can be formulated to assign each production quantity to a demand in a specific time period while minimizing the production cost. Shortest route formulation was proposed by Eppen and Martin [

Apart from reformulation, additional inequalities can be added to the problem formulation to tighten the bound while reducing the search space. Important researches in this category are discussed next. Barany et al. [

Research in this article is based on the appropriate reformulation of the standard capacitated lot sizing problem. We state the standard problem formulation and then derive three reformulations of the problem by eliminating the backordering variable or/and adding two capacity constraints. Efficacies of these formulations in terms of reduced computational complexity are demonstrated through numerical analysis of random problems on GAMS.

As stated in the previous section, we intend to evaluate the improvement in computational efficiency of the model when the number of decision variables are decreased, or constraints are added to tighten the bound of the solution space. Model A1 is the reference standard model, which is tinkered to develop model A2, A3, and A4 accordingly. In model A2 (proposed later), we have eliminated the backordering variable; hence it is expected to be computationally efficient when compared with model A1. Similarly, we have added two extra constraints (Equation (27), Equation (28)) in our standard model (A1) which is referred to as model A3. Further, we eliminate backordering variable while adding two constraints in model A1 and refer it to model A4. Hence model A4 is expected to perform best among all. Efficacy of each model is evaluated by performing paired t-test of the computational time of random problem instances on A1, A2, A3, and A4. Branch and bound method is used in GAMS for solving random problem optimally by these models. Finally it is concluded in section 6 that most computationally efficient formulation should be used solving capacitated lot sizing problem.

Indices used | |
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T | Set of time periods. |

t | Particular time period such that t ∈ T . |

I | Set of products to be produced. |

i | Particular product such that i ∈ I . |

Constants | |

C P i t | Unit cost of producing i in the period t. |

C S i t | Unit cost of setup for the item i in the period t. |

C I N V i t | Unit cost of holding inventory of item i for period 1. |

C B O i t | Unit cost of backordering item i demanded during the period t. |

C A P i t | Capacity available to produce item i during the period t. |

C A P T t | Capacity available in time units in the period t. |
---|---|

D i t | Demand for the item i during the period t. |

P T i t | Time required for processing the item i. |

S T i | Time required for setting up the production for the item i. |

Definition of Variables | |

X P i t | Number of items i to be produced in the period t. |

X I N V i t | Number of items i carried as inventory to be produced during the period t. |

X B O i t | Number of item i that will be backordered from the period t. |

Y S i t | Binary setup variable. |

Minimize Z = ∑ i = 1 I ∑ t = 1 T [ C P i t * X P i t + C S i t * Y S i t + C I N V i t * X I N V i t + C B O i t * X B O i t ] (1)

Subject to:

X P i t + X I N V i , t − 1 + X B O i t = D i t + X I N V i t + X B O i , t − 1 ∀ i ∈ I , t ∈ T (2)

∑ i = 1 I ( P T i t * X P i t + S T i * Y S i t ) ≤ C A P T t ∀ t ∈ T (3)

∑ i = 1 I ( P T i t * D i t + S T i * Y S i t ) ≤ C A P T t ∀ t ∈ T (4)

X P i t ≤ C A P i t Y S i t ∀ i ∈ I , t ∈ T (5)

∑ t = 1 T X P i t ≥ ∑ t = 1 T D i t ∀ i ∈ I (6)

X I N V i 0 = 0 ∀ i ∈ I (7)

X I N V i T = 0 ∀ i ∈ I (8)

X B O i 0 = 0 ∀ i ∈ I (9)

X B O i T = 0 ∀ i ∈ I (10)

Y S i t ∈ { 0 , 1 } ∀ i ∈ I , t ∈ T (11)

X I N V i t , X P i t , X B O i t ≥ 0 ∀ i ∈ I , t ∈ T (12)

Minimize Z = ∑ i = 1 I ∑ t = 1 T [ C P i t * X P i t + C S i t * Y S i t + C I N V i t * X I N V i t ] + ∑ t 1 = 1 T ∑ i = 1 I C B O i t 1 * ( ∑ t = 1 t 1 D i t + X I N V i t 1 − ∑ t = 1 t 1 X P i t ) (13)

Subject to:

∑ t = 1 t 1 D i t + X I N V i t 1 − ∑ t = 1 t 1 X P i t ≥ 0 ∀ i ∈ I , t 1 ∈ T (14)

∑ i = 1 I ( P T i t * X P i t + S T i * Y S i t ) ≤ C A P T t ∀ t ∈ T (15)

X P i t ≤ C A P i t Y S i t ∀ i ∈ I , t ∈ T (16)

∑ t = 1 T X P i t ≥ ∑ t = 1 T D i t ∀ i ∈ I (17)

X I N V i 0 = 0 ∀ i ∈ I (18)

X I N V i T = 0 ∀ i ∈ I (19)

Y S i t ∈ { 0 , 1 } ∀ i ∈ I , t ∈ T (20)

X I N V i t , X P i t ≥ 0 ∀ i ∈ I , t ∈ T (21)

Minimize Z = ∑ i = 1 I ∑ t = 1 T [ C P i t * X P i t + C S i t * Y S i t + C I N V i t * X I N V i t + C B O i t * X B O i t ] (22)

Subject to:

X P i t + X I N V i , t − 1 + X B O i t = D i t + X I N V i t + X B O i , t − 1 ∀ i ∈ I , t ∈ T (23)

∑ i = 1 I ( P T i t * X P i t + S T i * Y S i t ) ≤ C A P T t ∀ t ∈ T (24)

∑ i = 1 I ( P T i t * D i t + S T i * Y S i t ) ≤ C A P T t ∀ t ∈ T (25)

X P i t ≤ C A P i t Y S i t ∀ i ∈ I , t ∈ T (26)

∑ i = 1 I ∑ t = 1 T ( P T i t * D i t + S T i * Y S i t ) ≤ ∑ t = 1 T C A P T t (27)

∑ i = 1 I ∑ t = 1 T ( P T i t * X P i t + S T i * Y S i t ) ≤ ∑ t = 1 T C A P T t (28)

∑ t = 1 T X P i t ≥ ∑ t = 1 T D i t ∀ i ∈ I (29)

X I N V i 0 = 0 ∀ i ∈ I (30)

X I N V i T = 0 ∀ i ∈ I (31)

X B O i 0 = 0 ∀ i ∈ I (32)

X B O i T = 0 ∀ i ∈ I (33)

Y S i t ∈ { 0 , 1 } ∀ i ∈ I , t ∈ T (34)

X I N V i t , X P i t , X B O i t ≥ 0 ∀ i ∈ I , t ∈ T (35)

Minimize Z = ∑ i = 1 I ∑ t = 1 T [ C P i t * X P i t + C S i t * Y S i t + C I N V i t * X I N V i t ] + ∑ t 1 = 1 T ∑ i = 1 I C B O i t 1 * ( ∑ t = 1 t 1 D i t + X I N V i t 1 − ∑ t = 1 t 1 X P i t ) (36)

Subject to:

∑ t = 1 t 1 D i t + X I N V i t 1 − ∑ t = 1 t 1 X P i t ≥ 0 ∀ i ∈ I , t 1 ∈ T (37)

∑ i = 1 I ( P T i t * X P i t + S T i * Y S i t ) ≤ C A P T t ∀ t ∈ T (38)

X P i t ≤ C A P i t Y S i t ∀ i ∈ I , t ∈ T (39)

∑ i = 1 I ∑ t = 1 T ( P T i t * D i t + S T i * Y S i t ) ≤ ∑ t = 1 T C A P T t (40)

∑ i = 1 I ∑ t = 1 T ( P T i t * X P i t + S T i * Y S i t ) ≤ ∑ t = 1 T C A P T t (41)

∑ t = 1 T X P i t ≥ ∑ t = 1 T D i t ∀ i ∈ I (42)

X I N V i 0 = 0 ∀ i ∈ I (43)

X I N V i T = 0 ∀ i ∈ I (44)

Y S i t ∈ { 0 , 1 } ∀ i ∈ I , t ∈ T (45)

X I N V i t , X P i t ≥ 0 ∀ i ∈ I , t ∈ T (46)

Problem notations are tabulated in

40 problems each of size 5 × 5 and 6 × 6 are randomly generated in GAMS. 5 × 5, 6 × 6 problems denotes the lot sizing problem to find an optimum production plan of 5 items over 5 time periods, and 6 items over 6 time periods respectively. Only feasible problems are retained for data analysis (6 × 6―29 problems, 5 × 5―31 problems). Value of constants in these problems is randomly generated according to normal distribution (

Constant | Mean | Standard Deviation |
---|---|---|

Unitcost of set up | 100 | 2 |

Unitcost of Back Order | 100 | 2 |

Unit Cost of Production | 200 | 2 |

Unit Cost of Holding Inventory | 200 | 2 |

Capacity (Production resource) | 30 | 2 |

Demand | 10 | 2 |

Capacity (time) | 400 | 2 |

Constant | Lower Limit | Upper Limit |
---|---|---|

Production time | 1 | 5 |

Set up time | 2 | 4 |

All the problems are implemented in GAMS. Solution to these sample problems are tabulated in Appendix. According to t-test performed on data, problem A3 is computationally efficient to problem A1 with a statistical significance of 0.009317 (p-value). Similarly A4 is better than A2 with a statistical significance of 0.003071 (p-value). Model A2 is computationally more efficient than model A1 with a statistical significance of 0.000695 (p-value). Model A4 is computationally more efficient than model A3 with a statistical significance of 0.00473 (p-value).

In this article, we have demonstrated the effect of reducing the number of variables, increasing the number of constraints on the computational time of lot sizing problem through 4 models. We infer from our data analysis that model A4 is the most computationally efficient model, and hence is recommended to be used for solving capacitated lot sizing problem.

Sharma, R.R.K., Sinha, P. and Verma, M.K. (2018) Computationally Efficient Problem Reformulations for Capacitated Lot Sizing Problem. American Journal of Operations Research, 8, 312-322. https://doi.org/10.4236/ajor.2018.84018