Introduction

Deterioration is a significant factor in inventory analysis and it cannot be ignored its effect in the inventory. Many researchers have been doing their research in both EPQ and EOQ models by considering deterioration effect in inventory. In this connection, the reader can study the works of Cárdenas-Barrón and Sarkar [1], Sarkar [3], Sett et al. [5], Sarkar et al.  [4] and others.

Das et al. [2] developed a production inventory model for a deteriorating item under permissible delay in payments considering that the demand is stock dependent. In the production inventory model, the production rate is partially constant and partially dependent upon on both on-hand inventory and demand. The production inventory model considers that the supplier offers price discount and permissible delay in payment. Das et al. [2] formulated a single objective optimization problem that maximizes the total profit. Then they solved the optimization problem with a real-coded genetic algorithm (GA) with rank-based selection and arithmetic crossover. They illustrated the production inventory model with a numerical example, and a sensitivity analysis was done.

Das et al. [2] said that they found the optimal solutions. However, it is worth mentioning that a genetic algorithm (GA) cannot guarantee to obtain the optimal solution.

Discussion

We have read Das et al. [2]’s paper with a high interest and after going through the paper very carefully, we identified two shortcomings in their paper. The shortcomings of their paper are as follows:

  1. 1.

    The solutions are not optimal

  2. 2.

    The solutions are infeasible

Das et al. [2] stated that they obtained the optimal solution. However, in fact, their solutions are not optimal because they solved the numerical example with a genetic algorithm. It is important to mention that the solutions have another problem because the optimization problem contains two decision variables and the results in all tables only show one decision variable, which is the production time-period \((t_{1})\). There is missing the solution to the cycle time (T). Additionally, we identified that all solutions are wrong because all solutions have inconsistencies. The inconsistencies are as follows:

Considering the results reported in Table 1 in Das et al. [2]’s paper, the following discussion is stated: The results for Case I are wrong because the cash discount is negative for the five solutions. It is worth mentioning that a negative value for cash discount is impossible, therefore, the solutions are incorrect. The results of Case II in the last three solutions the cash discount values are negative, and the payable interest values for the five solutions are negative. It is important to note that the payable interest cannot be negative therefore all five solutions are incorrect. In addition, it was found that all these solutions are infeasible. In Case III both cash discount and interest payable are positive but the solutions are infeasible.

Taking into account the results shown in Table 2 in Das et al. [2] paper, the following argument is given: The results for Case IV are incorrect due to the fact that the cash discount value is negative in all five solutions. A negative value for cash discount does not make sense. With respect to Case V, it was found that all solutions the cash discount and the interest payable are negative. These inconsistencies produce invalid solutions. Because both cash discount and interest payable must be positive. Finally, in Case VI both cash discount and interest payable are positive but the solutions are infeasible.

Das et al. [2]’s production inventory model

The production inventory model considers the following notation, which was given by Das et al. [2].

Symbol

Description

Units

\(C_3 \)

Replenishment cost per order

$/order

\(C_1 \)

Holding cost excluding interest charges

$/unit/unit time

\(P_0 \)

Regular production rate

units/unit time

\(C_p \)

Regular production cost

$/unit

\(C_o \)

Overtime production cost

$/unit

\(q\left( t \right) \)

On hand inventory level

units

D(t)

Demand rate dependent on inventory level

units/unit time

s

Selling price

$/unit

\(I_c \)

Interest paid by the retailer

%/unit time

\(I_d \)

Interest earn by the retailer

%/ unit time

r

Cash discount rate

%/ unit

\(t_1 \)

Production time period (time at which the inventory level reaches its maximum level)

unit time

\(M_1 \)

The period of cash discount for which supplier cannot charge the interest

unit time

\(M_2 \)

Last time of permissible delay period

unit time

T

The length of the cycle

unit time

\(\theta \)

Deterioration rate

%

\(\alpha ,\beta \)

Demand parameters \((\alpha ,\beta >0)\)

 

\(\gamma ,\delta \)

Overtime production parameters (where \(\delta >0\) and \(0\le \gamma \le 1\))

 

Notation

Das et al. [2] considered six cases. For each case, they stated that profit function is a function of the length of the cycle (T). But, actually, the production inventory model has two decision variables: one is the production time period \((t_{1})\) and another one is the length of cycle (T).

Thus, mathematical formulation of Das et al. [2] model is given as follows. The selling price is determined as

$$\begin{aligned} S_T (t_1 ,T)= & {} s\left[ P_0 t_1 +k_1 t_1 +k_2 \left( {e^{-\lambda t_1 }-1} \right) -\frac{\mu \theta }{\lambda }\left\{ {t_1 +\frac{1}{\lambda }\left( {e^{-\lambda t_1 }-1} \right) } \right\} \right. \\&\left. +\frac{\alpha \theta }{\theta +\beta }\left\{ \frac{1}{\theta +\beta }\left( {1-e^{(\theta +\beta )(T-t_1 )}} \right) +(T-t_1 ) \right\} \right] \\ k_1= & {} \alpha \delta +\frac{\mu (\beta \delta -\gamma )}{\lambda }, \quad k_2 =\frac{\mu (\beta \delta -\gamma )}{\lambda ^{2}}, \lambda =\theta +\gamma -\beta (\delta -1)\hbox { and }\\ \mu= & {} P_0 +\alpha (\delta -1) \end{aligned}$$

The total cost for each case is given below

\(TC_i (t_1 ,T)= \hbox {Ordering cost} + \hbox {Holding cost} + \hbox {Production cost} + \hbox {Interest charged} - \hbox {Interest earned} - \hbox {Cash discount}\)

$$\begin{aligned}&TC_i (t_1 ,T)=C_3 +\frac{\mu C_1 }{\lambda }\left\{ {t_1 +\frac{1}{\lambda }\left( {e^{-\lambda t_1 }-1} \right) } \right\} \\&\quad -\,\frac{\alpha C_1 }{\theta +\beta }\left\{ \frac{1}{\theta +\beta }\left( {1-e^{(\theta +\beta )(T-t_1 )}} \right) +(T-t_1 ) \right\} \\&\quad +\,C_p P_0 t_1 +C_o \left\{ {k_1 t_1 +k_2 \left( {e^{-\lambda t_1 }-1} \right) } \right\} \\&\quad +\,C_p (1-r_i )I_c \left\{ {\begin{array}{l} \frac{1}{2}(\alpha +\frac{\beta \mu }{\lambda })\left( {t_1^2 -M_i^2 } \right) +\frac{\beta \mu }{\lambda }\left\{ \frac{1}{\lambda }\left( {e^{-\lambda t_1 }-e^{-\lambda M_i }} \right) +\left( {t_1 e^{-\lambda t_1 }-M_i e^{-\lambda M_i }} \right) \right\} \\ +\frac{\alpha \theta }{2(\theta +\beta )}\left( {T^{2}-t_1^2 } \right) -\frac{\alpha \beta }{(\theta +\beta )^{2}}\left( {T-t_1 e^{(\theta +\beta )(T-t_1 )}} \right) -\frac{\alpha \beta }{(\theta +\beta )^{3}}\left( {1-e^{(\theta +\beta )(T-t_1 )}} \right) \\ \end{array}} \right\} \\&\quad -\,sI_d \left\{ \frac{1}{2}\left( {\alpha +\frac{\beta \mu }{\lambda }} \right) M_i^2 +\frac{\beta \mu }{\lambda ^{2}}M_i e^{-\lambda M_i }+\frac{\beta \mu }{\lambda ^{3}}\left( {e^{-\lambda M_i }-1} \right) \right\} \\&\quad -\,r_i C_p \left\{ {k_1 t_1 +k_2 \left( {e^{-\lambda t_1 }-1} \right) } \right\} \end{aligned}$$

for \(i=1\) or 4 and \(M_{1}= M_{1,} M_{4}=M_{2}, r_{1}=r, r_{4}=0\).

\(TC_j (t_1 ,T)= \hbox {Ordering cost} + \hbox {Holding cost} + \hbox {Production cost} + \hbox {Interest charged} - \hbox {Interest earned} - \hbox {Cash discount}\)

$$\begin{aligned}&TC_j (t_1 ,T)=C_3 +\frac{\mu C_1 }{\lambda }\left\{ {t_1 +\frac{1}{\lambda }\left( {e^{-\lambda t_1 }-1} \right) } \right\} \\&\quad -\frac{\alpha C_1 }{\theta +\beta }\left\{ \frac{1}{\theta +\beta }\left( {1-e^{(\theta +\beta )(T-t_1 )}} \right) +(T-t_1 )\right\} \\&\quad +\,C_p P_0 t_1 +C_o \left\{ {k_1 t_1 +k_2 \left( {e^{-\lambda t_1}-1} \right) } \right\} \\&\quad +\,C_p (1-r_j )I_c \left\{ \frac{\alpha \theta }{2(\theta +\beta )}\left( {T^{2}-M_j^2 } \right) \right. \\&\quad \left. -\,\frac{\alpha \beta }{(\theta +\beta )^{2}}\left( {T-M_j e^{(\theta +\beta )(T-M_j )}} \right) -\frac{\alpha \beta }{(\theta +\beta )^{3}}\left( {1-e^{(\theta +\beta )(T-M_j )}} \right) \right\} \\&\quad -\,sI_d \left\{ {\begin{array}{l} \frac{1}{2}\left( {\alpha +\frac{\beta \mu }{\lambda }} \right) t_1^2 +\frac{\beta \mu }{\lambda ^{2}}t_1 e^{-\lambda t_1 }+\frac{\beta \mu }{\lambda ^{3}}\left( {e^{-\lambda t_1 }-1} \right) +\frac{\alpha \theta }{2(\theta +\beta )}\left( {M_j^2 -t_1^2 } \right) \\ -\frac{\alpha \beta }{(\theta +\beta )^{2}}\left( {M_j e^{(\theta +\beta )(T-M_j )}-t_1 e^{(\theta +\beta )(T-t_1 )}} \right) -\frac{\alpha \beta }{(\theta +\beta )^{3}}\left( {e^{(\theta +\beta )(T-M_j )}-e^{\left( {\theta +\beta } \right) (T-t_1 )}}\right) \\ \end{array}} \right\} \\&\quad -\,r_j C_p \left\{ {k_1 t_1 +k_2 \left( {e^{-\lambda t_1 }-1} \right) } \right\} \end{aligned}$$

for \(j=2\) or 5 and \(M_{2}= M_{1,} M_{5}=M_{2}, r_{2}=r, r_{5}=0\).

\(TC_k (t_1 ,T)= \hbox {Ordering cost} + \hbox {Holding cost} + \hbox {Production cost} - \hbox {Interest earned} - \hbox {Cash discount}\)

$$\begin{aligned}&TC_k (t_1 ,T)=C_3 +\frac{\mu C_1 }{\lambda }\left\{ {t_1 +\frac{1}{\lambda }\left( {e^{-\lambda t_1 }-1} \right) } \right\} \\&\quad -\,\frac{\alpha C_1 }{\theta +\beta }\left\{ \frac{1}{\theta +\beta }\left( {1-e^{(\theta +\beta )(T-t_1 )}} \right) +(T-t_1 ) \right\} \nonumber \\&\quad +\,C_p P_0 t_1 +C_o \left\{ {k_1 t_1 +k_2 \left( {e^{-\lambda t_1 }-1} \right) } \right\} \nonumber \\&\quad -\,sI_d \left\{ {\begin{array}{l} \frac{1}{2}\left( {\alpha +\frac{\beta \mu }{\lambda }} \right) t_1^2 +\frac{\beta \mu }{\lambda ^{2}}t_1 e^{-\lambda t_1 }+\frac{\beta \mu }{\lambda ^{3}}\left( {e^{-\lambda t_1 }-1} \right) +\frac{\alpha \theta }{2(\theta +\beta )}\left( {T^{2}-t_1^2}\right) \\ -\frac{\alpha \beta }{(\theta +\beta )^{2}}\left( {T-t_1 e^{(\theta +\beta )(T-t_1 )}} \right) -\frac{\alpha \beta }{(\theta +\beta )^{3}}\left( {1-e^{\left( {\theta +\beta } \right) (T-t_1 )}} \right) +\left( {M_k -T} \right) \left\{ {\left( {\alpha +\frac{\beta \mu }{\lambda }} \right) t_1 +\frac{\beta \mu }{\lambda ^{2}}\left( {e^{-\lambda t_1 }-1} \right) } \right\} \\ +\left( {M_k -T} \right) \left( {T-t_1 } \right) \frac{\alpha \theta }{\left( {\theta +\beta } \right) }-\left( {M_k -T} \right) \frac{\alpha \beta }{\left( {\theta +\beta } \right) ^{2}}\left( {1-e^{(\theta +\beta )(T-t_1 )}} \right) \\ \end{array}} \right\} \\&\quad -\,r_k C_p \left\{ {k_1 t_1 +k_2 \left( {e^{-\lambda t_1 }-1} \right) } \right\} \end{aligned}$$

for \(k=3\) or 6 and \(M_{3}= M_{1,} M_{6}=M_{2}, r_{3}=r, r_{6}=0\).

Therefore, the total average profit is expressed as follows

$$\begin{aligned} TP_i (t_1 ,T)=\frac{\left[ {S_T (t_1 ,T)-TC_i (t_1 ,T)} \right] }{T} \end{aligned}$$

for \(i=1\) or 4 and \(M_{1}= M_{1,} M_{4}=M_{2}, r_{1}=r, r_{4}=0\).

$$\begin{aligned} TP_j (t_1 ,T)=\frac{\left[ {S_T (t_1 ,T)-TC_j (t_1 ,T)} \right] }{T} \end{aligned}$$

for \(j=2\) or 5 and \(M_{2}= M_{1,} M_{5}=M_{2}, r_{2}=r, r_{5}=0\).

$$\begin{aligned} TP_k (t_1 ,T)=\frac{\left[ {S_T (t_1 ,T)-TC_k (t_1 ,T)} \right] }{T} \end{aligned}$$

for \(k=3\) or 6 and \(M_{3}= M_{1,} M_{6}=M_{2}, r_{3}=r, r_{6}=0\).

Thus, the optimization problem is expressed as follows:

Maximize \(TP_i (t_1 ,T) \quad \forall \quad i=1,\ldots ,6\)

The above optimization problem can be solved optimally using Lingo 10. In next section, the optimal solutions to the numerical example are provided.

Table 1 Optimal solution of the numerical example for Cases I, II and III
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Table 2 Optimal solution of the numerical example for Cases IV, V and VI
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Optimal solution for numerical example in Das et al. [2]’s production inventory model.

The parameters for the numerical example are: \(C_3 = 55,C_1 = 0.25,C_p =2,C_o =2.5,s=3.5,I_c =0.2,I_d =0.15,P_0 =75,\gamma = 0.03,\delta =0.3,r=0.001\) in appropriate units. Thus, the optimal solutions to the six cases are presented in Tables 1 and 2. The sensitivity analysis are reported in Tables 3, 4, 5 and 6.

Table 3 Sensitivity analysis with respect demand parameters when \(\theta =0.1\) and \(M_{1 }=4.9\)
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Table 4 Sensitivity analysis with respect deterioration parameters when \(\alpha =55\) and \(M_{1 }=4.9\)
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Table 5 Sensitivity analysis with respect demand parameters when \(\theta =0.1\) and \(M_{2} = 5.4\)
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Table 6 Sensitivity analysis with respect deterioration parameters when \(\alpha =55\) and \(M_{2} = 5.4\)
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Conclusion

This paper identifies some shortcomings in the solutions of the numerical example in Das et al. [2]’s model. The paper shows that all solutions reported by Das et al. [2] are incorrect and infeasible. Additionally, this paper provides the optimal solutions to all cases of the production inventory model. Now, the Das et al. [2]’s research is valuable and significant because it is corrected.