Keywords

MSC

1 Introduction

For survival and growth of business, proper coordination and communication among supply chain players place an important role in this competitive atmosphere. Firstly, Goyal (1976) formulated an integrated model for single supplier and single customers. Banerjee (1986) made an appropriate price adjustment and obtained joint order so that it is beneficial to both parties. Chung and CÃardenas-BarrÃşn (2013) generated model for deteriorating items for stock dependent demand with two-level trade credits. Chung et al. (2014) extended previous model for exponentially deteriorating items. Shah (2015) derived model with two-level trade credits for items deteriorate constantly. Further, Shah et al. (2015) extended this model by taking price sensitive and time-dependent demand.

Most of the inventory researchers have used constant rate demand. But in the real world, demand is not always constant. It may vary with time. Donaldson (1977) obtained the fundamental result in EOQ model with time-varying linear demand over a known and finite time horizon. Dave and Patel (1981) extended model for deteriorating items. Further, Wee and Wang (1999) considered time varying demand and developed a variable production policy. Mishra and Singh (2011) had taken into account time-dependent holding cost and formulated an inventory model under shortages. Mishra (2013) extended model for time-varying deterioration.

Deterioration is the process in which the items loses its utility and become useless. In classical EOQ model, researcher considered inventory depletes due to demand only. But in the real world, inventory is not only reduce due to demand but also reduced due to deterioration. In earlier literature, Ghare and Schrader (1963) developed model for items those deteriorates exponential. Firstly, Philip and Covert (1973) formulated model for time-dependent deteriorating items which follow Weibill distribution. Further, Philip (1974) generalized this model. Manna and Chaudhuri (2001) derived inventory model under shortages for time-dependent deteriorating items. Bakker et al. (2012) gave up to date review of inventory models for deteriorating items. To reduce deterioration rate, different researcher used preservation technology. Mishra (2013) used preservation technology for time-dependent deteriorating items that follow Weibull Distribution. Chang (2013) used preservation technology for non-instantaneous deteriorating items. Singh and Rathore (2015) extended that model under shortages. Mishra and Talati (2018) derived integrated inventory model and used preservation technology under quantity discount scenario. Mahapatra et al. (2019) formulated inventory model for deteriorating items under fuzzy environment.

In last decades, to optimize the inventory models, researchers used different heuristic search algorithms like ant colony, swarm intelligence and genetic algorithm. Genetic algorithm describes a set of techniques inspired by natural selection like inheritance, mutation, selection and crossover. This technique requires fitness function and genetic representation of solution domain. In each generation, it uses fitness function to select global optimum. This process terminates when the satisfactory fitness level has been reached. Goldberg (1989) used GA for optimization. Then, different researchers like Murata et al. (1996), Goren et al. (2008), Radhakrishnan et al. (2009, 2010), Narmadha et al. (2010), Woarawichai et al. (2012), Mishra and Talati (2015), Talati and Mishra (2019), Alejo-Reyes et al. (2021) used heuristic search algorithm for optimized their models.

2 Notations and Assumptions

2.1 Notations

2.1.1 Inventory Parameters for Manufacturer

figure a

2.1.2 Inventory Parameters for Retailer

figure b

2.2 Assumptions

  1. 1.

    In present model, we have considered two-echelon supply chain model (single manufacturer and single retailer) for single item.

  2. 2.

    Demand is time dependent \(D(t)=a+bt;a,b>0.\)

  3. 3.

    Replenishment rate is infinite.

  4. 4.

    Lead time is zero.

  5. 5.

    Shortages are not allowed.

  6. 6.

    Constant production rate is considered. \(P>D(t)\).

  7. 7.

    Ordering cost is lot size dependent.

  8. 8.

    The inventory deteriorate with respect to time and follow Weibull distribution \(\theta (t)= \alpha \beta t^{\beta }\) where \(\alpha \) is shape parameter \(0<\alpha <1\), and \(\beta \) is scale parameter \(\beta \ge 1\).

  9. 9.

    Preservation technologies are used for reducing the deterioration rate.

  10. 10.

    The salvage value \(\gamma , 0 \le \gamma \le 1\) is associated to deteriorated units.

3 Model Formulation

3.1 Manufacturer’s Total Cost

Here, we considered production dominates demand. Due to preservation technology, the rate of change of inventory during period [0,T] is shown in Fig. 1.

Fig. 1
figure 1

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Inventory level for manufacturer.

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Thus, the on-hand inventory for manufacturer is generated by the following differential equation

$$\begin{aligned} \frac{\text {d}Q_{\text {m}}}{\text {d}t}+\tau _{\text {P}} Q_{\text {m}} =P-D(t); \quad 0\le t \le T \end{aligned}$$
(1)

Solving Eq. (1) using boundary condition \(Q_\text {m} (0)=0\) and \(Q_\text {m} (T)=Q_\text {m}\)

$$\begin{aligned} Q_\text {m} (t)= \left( {(P-a)t\left( 1+\frac{\alpha t^{\beta }}{\beta +1}\right) -(Pm+b-ma)t^{2}\left( \frac{1}{2}+\frac{\alpha t^{\beta }}{\beta +2}\right) }\right. \quad&\\ \left. {-mbt^{3} \left( \frac{1}{3}+\frac{\alpha t^{\beta }}{\beta +3}\right) }\right) (1-\alpha t^{\beta }+mt-m\alpha t^{\beta +1})&\end{aligned}$$

So total quantity by manufacturer per cycle is \(Q_\text {m} (T)=Q_\text {m}\).

Basic Costs

1. Set-up cost

$$\begin{aligned} \text {SC}_\text {m}=A_\text {m} \end{aligned}$$
(2)

2. Inventory holding cost per unit is given by

$$\begin{aligned} \text {HC}_{\text {m}}=h_\text {m}\int \limits _{0}^{T} Q_{\text {m}}(t) \text {d}t \end{aligned}$$
(3)

3. Now number of deteriorating units during cycle time T

$$\begin{aligned} \text {DE}_1(T)=Q_\text {m}-aT-\frac{(bT^2)}{2} \end{aligned}$$
(4)

4. Deteriorating cost is given by

$$\begin{aligned} \text {DC}_\text {m}=b_1 \text {DE}_1(T) \end{aligned}$$
(5)

5. Salvage value is given by

$$\begin{aligned} \text {SV}_\text {m}= \gamma \text {DE}_1(T) \end{aligned}$$
(6)

6. Preservation cost is given by

$$\begin{aligned} \text {PC}_\text {m}=\xi _1 \end{aligned}$$
(7)

Thus, the total cost of manufacturer is

$$\begin{aligned} \text {TC}_\text {m} (T)=\text {SC}_\text {m}+\text {HC}_\text {m}+\text {DC}_\text {m}-\text {SV}_\text {m}+\text {PC}_\text {m} \end{aligned}$$
(8)
Fig. 2
figure 2

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Inventory level for retailer.

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3.2 Retailer’s Total Cost

Retailer’s on-hand inventory depletes with time-dependent demand and deterioration under preservation technology. The rate of change of inventory level due to preservation technology is shown in Fig. 2. So the governing differential equation describes the inventory level at instantaneous time t which is given by

$$\begin{aligned} \frac{\text {d}Q_{\text {r}}}{\text {d}t}+\tau _{\text {P}} Q_{\text {r}} =-D(t); \quad 0\le t \le T \end{aligned}$$
(9)

Solving Eq. (9) using boundary condition \(Q_\text {r} (T)=0\) and \(Q_\text {m} (0)=Q_\text {r}\), we get

$$\begin{aligned} Q_{\text {r}}(t)&=\left[ {a(T-t)+\frac{b}{2}(T^{2}-t^{2})+\frac{\alpha a }{\beta +1}(T^{\beta +2}-t^{\beta +2})-\frac{ma}{2}(T^{2}-t^{2})} \right. \nonumber \\&\left. {\quad -\frac{mb}{3}(T^{3}-t^{3})- \frac{\alpha a}{\beta +2}(T^{\beta +2}-t^{\beta +2})-\frac{m \alpha b}{\beta +3}(T^{\beta +3}-t^{\beta +3})} \right] \nonumber \\&\quad \quad [1-\alpha t^\beta +mt-m \alpha t^{\beta +1}] \end{aligned}$$
(10)

\(\therefore \) Total quantity purchase by retailer per cycle is

$$\begin{aligned} Q_\text {r}=Q_\text {r} (0)=\left[ {a(T)+\frac{b}{2}(T^{2})+\frac{\alpha a }{\beta +1}(T^{\beta +2})-\frac{ma}{2}(T^{2})-\frac{mb}{3}(T^{3})} \right.&\nonumber \quad \\ -\left. {\frac{\alpha a}{\beta +2}(T^{\beta +2})-\frac{m \alpha b}{\beta +3}(T^{\beta +3})} \right]&\end{aligned}$$
(11)

Basic costs associated with retailer total cost are

  1. 1.

    Ordering cost is lot size dependent

    $$\begin{aligned} \text {OC}_\text {r}=C_0 Q_\text {r}^\eta \end{aligned}$$
    (12)
  2. 2.

    Holding cost per unit is given by

    $$\begin{aligned} \text {HC}_{\text {r}}=h_\text {r}\int \limits _{0}^{T} Q_{\text {r}}(t) \text {d}t \end{aligned}$$
    (13)
  3. 3.

    Total number of deteriorating units during cycle time T

    $$\begin{aligned} \text {DE}_2(T)=Q_\text {r}-aT-\frac{(bT^2)}{2} \end{aligned}$$
    (14)
  4. 4.

    The deteriorating cost per time unit is

    $$\begin{aligned} \text {DC}_\text {r}=b_1 \text {DE}_2(T) \end{aligned}$$
    (15)
  5. 5.

    Salvage value per time unit is

    $$\begin{aligned} \text {SV}_\text {r}= \gamma \text {DE}_2(T) \end{aligned}$$
    (16)
  6. 6.

    Preservation cost is given by

    $$\begin{aligned} \text {PC}_\text {r}=\xi _2 \end{aligned}$$
    (17)

The total cost for retailer is by

$$\begin{aligned} \text {TC}_\text {r}=\text {OC}_\text {r}+\text {HC}_\text {r}+\text {DC}_\text {r}+\text {PC}_\text {r}-\text {SV}_\text {r} \end{aligned}$$
(18)

3.3 Joint Total Cost

Total cost for the inventory system is

$$\begin{aligned} \text {TC}=\text {TC}_\text {m}+\text {TC}_\text {r} \end{aligned}$$
(19)

4 Computational Algorithm

4.1 Analytical Approach

  • Set all parameters value in the mathematical model except decision variables.

  • Find optimum T using \(\text {TC}_{\text {m}}\).

  • Used optimal T and \(Q_{\text {m}}\) to find total cost for manufacturer.

  • Optimized T and \(\eta \) simultaneously from \(\text {TC}_{\text {r}}\).

  • Used optimal T, \(\eta \) and \(Q_{\text {r}}\) and obtain total cost for retailer.

  • Find optimal T and \(\eta \) from system total cost.

  • Used optimal T, \(\eta \) and optimal quantity and calculate total system cost.

4.2 Genetic Algorithm Approach

  • Set all parameters value in the fitness function except decision variables.

  • Start G.A. with an initial population of 20 chromosomes.

  • On the basis of their fitness score rank the chromosomes.

  • Chromosomes with good fitness score will enter in mating pool.

  • Perform stochastic uniform crossover for reproduction. We have considered crossover fraction is 0.8 and each generation is 2-Elites.

  • On the basis of their fitness value, rank all members and select members for new generation.

  • Perform step (iii) and step (iv) till absolute difference between two successive members is \(10^{-5}\).

5 Numerical Example and Sensitivity Analysis

5.1 Numerical Example

Consider one integrated production-inventory system with \(P=500,~a=400,~m=0.5,b=2,~\alpha =0.5,~\beta =2,~h_\text {m}=0.2,~\xi _1=500 \$,~\xi _2=500\$,~h_\text {r}=0.2,~C_0=2000,~A_\text {m}=2000.\)

We have optimized this using analytical method by MAPLE18; we get some computational results those are shown in Table 1.

Table 1 Computational results obtained by analytical approach
Full size table

Here, in independent decision, the convexity of the function is given below

For manufacturer

$$\begin{aligned} {\frac{\text {d}^2 \text {TC}_\text {m}}{\text {d}T^2}}|_{(T=T^* )}=2808.527238 \ge 0 \end{aligned}$$

For retailer

$$\begin{aligned} \left. \begin{vmatrix} \frac{\partial ^{2} \text {TC}_\text {r}}{\partial \eta ^2} \frac{\partial ^{2} \text {TC}_\text {r}}{\partial \eta \partial T}\\ \frac{\partial ^{2} \text {TC}_\text {r}}{\partial \eta \partial T} \frac{\partial ^{2} \text {TC}_\text {r}}{\partial T^2} \end{vmatrix} \right. = 5.545177044479552\times 10^2>0 \end{aligned}$$

and

$$\begin{aligned} \frac{\partial ^{2} \text {TC}_\text {r}}{\partial T ^2} =8.256 \times 10^2 \ge 0 \end{aligned}$$

For integrated

$$\begin{aligned} \left. \begin{vmatrix} \frac{\partial ^{2} \text {TC}_\text {r}}{\partial \eta ^2} \frac{\partial ^{2} \text {TC}_\text {r}}{\partial \eta \partial T}\\ \frac{\partial ^{2} \text {TC}_\text {r}}{\partial \eta \partial T} \frac{\partial ^{2} \text {TC}_\text {r}}{\partial T^2} \end{vmatrix} \right. = 5.26352\times 10^2>0 \end{aligned}$$

and

$$\begin{aligned} \frac{\partial ^{2} \text {TC}_\text {r}}{\partial T ^2} =0.8039621 \times 10^3 \ge 0 \end{aligned}$$

Above example is also optimized by genetic algorithm using MATLAB16a. Computational results obtain by genetic algorithm are shown in Table 2. For independent decision, genetic algorithm took 51 for manufacturer, 190 for retailer and 84 for integrated system. Best fitness plot of manufacturer, retailer and the system is shown in Figs. 3, 4 and 5, respectively.

Table 2 Computational results obtained by using genetic algorithm
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Fig. 3
figure 3

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Best fitness solution for manufacturer total cost.

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Fig. 4
figure 4

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Best fitness solution for retailer total cost.

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Fig. 5
figure 5

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Best fitness solution for system.

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The sensitivity analysis for the above example is carried out to check the behaviour of inventory and supply chain parameters related to total cost in joint decision by varying inventory parameters as \(-20\), \(-10\), 10 and 20%. The computational results is shown in Table 3.

Table 3 Sensitivity analysis for inventory and supply chain parameters
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The results obtained in Table 3 can be summarized as follows:

  • As inventory parameters \(a,~b, ~\alpha ,~h_\text {m}\) increase, integrated total cost decreases.

  • As inventory parameters \(m, ~\beta , ~h_\text {r}\) increase, integrated total cost increases.

6 Conclusion

Supply chain management has required models and processes which can find a solution in a fast and efficient way. For comparison purposes, we have found a solution for the same numerical example using gradient-based analytical method and genetic algorithm. Complexity is explained mathematically for analytical techniques and graphically for genetic algorithms. It is shown that the decision taken in an integrated scenario reduces the cost compared to the decision in an isolated scenario in both techniques. Results clearly show that in our model, evolutionary algorithm provides global minimum while the analytical method fails. Future research may be extended into more realistic situations like shortages, random demand and inflation. Additionally, genetic algorithms can be modified to find solutions in a very efficient manner.