Capacitated single item lot sizing problem (CLSP) with setup, backorders and inventory is a well studied problem (see Wolsey  for a detailed literature review). Pochet and Wolsey  gave several valid inequalities of uncapacitated LSP which resulted in a reformulation (linear program) that can be solved much more easily (compared to effort required to solve the 0-1 mixed integer programming formulation of CLSP). We use formulation of Kumar  , and pose capacitated single item lot sizing problem (CLSP) with setup, backorders and inventory as a single item lot sizing problem with set up, production and inventory problem. We can then reformulate it by using valid inequality given in Pochet and Wolsey  .
2. Problem Formulation
t: Set of Time period from .
ft: fixed cost in time period “t”;
pt: per unit variable (production) cost in time period “t”;
ct: production capacity in time period “t”;
Dt: demand in time period “t”;
ht: per unit inventory carrying cost in time period “t”;
sht: per unit shortage cost in time period “t”.
Definition of Variables
xt: amount produced in time period “t”;
yt: 1, if machine setup to produce in time period “t”,
st: shortage in time period “t”;
It: Inventory in time period “t”.
for all (2)
; and (4)
This formulation is based on the formulation given for the location-distributed problem with shortages and inventory by Kumar  . Traditionally the problem is formulated as (Wolsey  , p. 1593) given below:
and (3) & (4).
In model (1), we substitute
in (1) and (2), to get
for all (6)
(6) is substituted in (1) along with xt, = ct * yt to get following:
; and (8)
It can be easily seen that coefficient of It is positive; and coefficient of yt can be positive, negative or zero. It can be easily seen that Model A3 is a lot sizing problem without shortage variables as in (b). Now we can apply the methods of reformulation and valid inequalities developed as given in (1).
Thus we show that a lot sizing problem with set up, production, inventory and shortage costs is reduced to a lot sizing problem with set up, production and inventory costs. This is possible due to new formulation given in Kumar  . Also then reformulation-based methods given in  and  can be fruitfully applied. This is the useful contribution given in this paper.
 Wolsey, L.A. (2002) Solving Multi-Item Lot-Sizing Problems with an MIP Solver Using Classification and Reformulation. Management Science, 48, 1587-1602.
 Kumar, V. (2012) Equal Distribution of Shortages in Supply Chain of Food Corporation of India: Using Lagrangian Relaxation Methodology. Master’s Thesis, Indian Institute of Technology, Kanpur. (Unpublished)