The paper studies a kind of deteriorating seasonal product whose deterioration rate can be controlled by investing on the preservation efforts. In contrast to previous studies, the paper considers the seasonal and deteriorating properties simultaneously. A deteriorating inventory model is developed for this problem. We also provide a solution procedure to find the optimal decisions about the preservation technology investment, the market price, and the ordering frequency. Then a case study is used to illustrate the model and the solution procedure. Finally, sensitive analysis of the optimal solution with respect to major parameters is carried out.
The research on deteriorating items has begun from 1963. A model with exponentially decaying inventory was initially proposed by Ghare [
Hence, this will become a very difficult problem to decide the inventory if the product is both deteriorating and seasonal. In this paper, we mainly study the optimal inventory decision of the seasonal deteriorating products. Some researchers have studied such deteriorating inventory model, but they do not consider that the deterioration rate can be controlled.
In reality, the deterioration rate can be controlled through preservation technology investment. For example, the fruit retailer can reduce the rate of product deterioration by adopting the cool supply chain. But the preservation technology investment will lead to additional cost. Hence, a key inventory problem is to find the optimal replenishment and preservation technology investment policy which maximizes the unit time profit.
This paper is the first paper to study both the preservation technology investment and pricing strategies of deteriorating seasonal products. In this paper, a model for deteriorating seasonal products is built, in which deterioration rate can be controlled by preservation technology investment. The decision variables are the market demand, the preservation technology investment parameter, and the ordering frequency. To get the optimal solution, an algorithm is designed. To foster additional managerial insights, we perform extensive sensitivity analyses and illustrate our results with a case study.
The rest of the paper is organized as follows. Section
The property of the deterioration rate is very important in the research of deteriorating inventory. In most literatures till now, it is assumed that deterioration rate is a constant [
In addition to the deterioration rate, market demand is another very important factor considered in this paper. In some situations, the demand rate is assumed to be a constant (see [
For some seasonal deteriorating products, the demand can only exist for a limited time horizon. Since the time horizon is fixed, it is necessary to decide the ordering frequency in a limited time horizon instead of the ordering period length. Some people have considered such situation, such as Sana et al. [
The notation in this paper is listed below.
Ordering frequency
Cost of preservation technology investment per unit time
Market price
Ordering quantity.
Buying cost per unit
Inventory holding cost per unit per time
Inventory level of a time point
Ordering cost per order
Market demand,
Demand scale
Price sensitive parameter
Total profit of the selling season.
The model in this paper is built on the base of the following assumptions.
Market demand is linear related to market price.
Market demand only exists in a limited time horizon
Demand cannot be backlogged.
Ordering lead time is zero.
Deteriorated products have no value, and there is no cost to dispose or store them.
The relationship of deterioration rate and the preservation technology investment parameter satisfies
The cost of preservation technology investment per unit time is restricted to
This study considers a single retailer’s inventory policy in which the deterioration rate is affected by the preservation technology investment. For seasonal products, the decision variables are the market price, the ordering frequency, and the preservation technology investment parameter.
In this model, there are two tradeoffs. The first one is the tradeoff between the ordering frequency and the ordering cost per order. By increasing ordering frequency, we can decrease the deteriorating cost. But the ordering cost increases. The second tradeoff is the preservation technology investment and the deteriorating cost. By increasing the preservation technology investment, deteriorating cost decreases.
According to the assumption, the time length is equal in all the ordering periods. So, we only study the first period. In the first period, according to the modeling of exponential deteriorating inventory in Ghare [
The inventory system.
The boundary condition is
By solving (
The total profit of the season can be formulated as
Sales revenue: The total revenue in time
Buying cost: According to (
Inventory cost: The total inventory quantity
The formulation of the total inventory cost is
Ordering cost: Ordering cost is a constant in every period. The total cost can be formulated as
Preservation cost:
Hence, the profit function is
The problem is to solve the next program
According to the Taylor series theory, for small
When market price
The first and second partial derivatives of the target function
According to (
For known
(1) If
(2) If
(3) If
(
The first and second partial derivatives of the target function
For simplicity, we set
We define
It is obvious that
If
If
If
Proposition
There exists unique
The first and second partial derivatives of the target function
Let
At point
Thus,
Combining Propositions
For fixed
In the subsection, we use an interaction algorithm to solve numerical examples.
Set
Set
Calculate
If
If
If
Set
If
Calculate
Set
If
Set
Calculate corresponding
To better illustrate our conclusions, we proposed four cases. The first one is a normal case, which is a benchmark for the other two. In the second case, there is a constraint for the investment capital. In the third and fourth case, the value of initial deterioration rate and the efficiency parameter is relatively small. Here, we apply the above algorithm to solve the problem.
The initial values of the parameters are listed in Table
Initial values of the parameters.










500  10  0.2  10  1  0.02  10  100  0.5 
By calculating with Matlab 7.1, when
The search process of the problem.





TP 

3  2.6383  38.78  0.0053  81.47  754.4 
4  2.4144  36.62  0.0060  71.88  1337.0 
5  2.1512  35.34  0.0068  62.64  1582.6 






7  1.6681  33.88  0.0087  48.91  1529.2 
8  1.4567  33.43  0.0097  43.93  1347.8 
9  1.2634  33.07  0.0106  39.84  1104.0 
The function with respect to
In this example, the upper bound of investment on preservation cost is sufficiently large. The optimal solution is not on the boundary.
In this example, we set
The search process of the problem.





TP 

3  1.7500  39.03  0.0083  83.31  707.65 
4  1.7500  36.78  0.0083  73.04  1300.1 
5  1.7500  35.42  0.0083  63.20  1553.4 






7  1.6681  33.88  0.0087  48.91  1529.2 
8  1.4567  33.43  0.0097  43.93  1347.8 
9  1.2634  33.07  0.0106  39.84  1104.0 
The initial values of the parameters are shown in Table
Initial values of the parameters for Example










500  10  0.2  10  1  0.001  10  100  0.5 
The initial values of the parameters are shown in Table
Initial values of the parameters for Example










500  10  0.2  10  1  0.02  10  100  0.01 
In this part, we performed the sensitivity analysis on the optimal solution of the model with respect to parameters (
Sensitive analysis results for Example
−50%  −40%  −30%  −20%  −10%  0  +10%  +20%  +30%  +40%  +50%  



9  8  7  7  6  6  5  5  5  5  4 

1.2634  1.4567  1.6681  1.6681  1.8996  1.8996  2.1512  2.1512  2.1512  2.1512  2.4144  

33.07  33.43  33.88  33.88  34.49  34.49  35.34  35.34  35.34  35.34  36.62  

358.56  351.44  342.37  342.37  330.24  330.24  313.20  313.20  313.20  313.20  287.52  
TP  3354.0  2947.8  2579.2  2229.2  1921.9  1621.9  1332.6  1082.6  832.6  582.6  337.0  
 


6  6  6  6  6  6  6  6  6  6  6 

0.5133  0.8779  1.1862  1.4533  1.6889  1.8996  2.0902  2.2642  2.4243  2.5725  2.7105  

34.49  34.49  34.49  34.49  34.49  34.49  34.49  34.49  34.49  34.49  34.49  

330.24  330.24  330.24  330.24  330.24  330.24  330.24  330.24  330.24  330.24  330.24  
TP  1760.5  1724.0  1693.2  1666.5  1642.9  1621.9  1602.8  1585.4  1569.4  1554.6  1540.8  
 


6  6  6  6  6  6  6  6  5  5  5 

0.8172  1.1247  1.3743  1.5806  1.7536  1.8996  2.0232  2.1278  2.4540  2.5220  2.5762  

31.94  32.45  32.96  33.47  33.98  34.49  35.00  35.51  36.88  37.40  37.91  

401.10  384.30  369.36  355.62  342.66  330.24  318.18  306.42  277.75  266.35  255.05  
TP  3439.0  3046.5  2669.7  2307.3  1958.2  1621.9  1297.7  985.4  696.7  424.6  164.0  
 


4  5  5  5  6  6  6  6  6  7  7 

2.8430  2.4120  2.3501  2.2860  1.9537  1.8996  1.8439  1.7866  1.7275  1.4783  1.4277  

33.43  33.30  33.81  34.32  34.06  34.49  34.91  35.34  35.77  35.34  35.71  

351.48  354.00  343.80  333.60  338.70  330.24  321.72  313.20  304.62  313.18  305.83  
TP  3209.2  2837.0  2508.1  2189.4  1883.9  1621.9  1366.9  1119.0  878.26  649.85  443.05  
 


6  6  6  6  6  6  6  6  6  6  6 

0.9388  1.4150  1.6682  1.8032  1.8712  1.8996  1.9036  1.8927  1.8723  1.8461  1.8163  

34.83  34.71  34.63  34.57  34.53  34.49  34.46  34.43  34.41  34.40  34.38  

343.50  339.12  335.94  333.54  331.68  330.24  328.98  327.96  327.12  326.40  325.74  
TP  1511.3  1533.1  1557.0  1580.3  1602.0  1621.9  1640.0  1656.5  1671.6  1685.4  1699.8 
Percent changes of parameter on total profit for Example
From Table
The retailer’s ordering frequency is insensitive to the change of
The retailer’s total ordering quantity
The market price is insensitive to
The preservation cost is increasing in
In this paper, we study a kind of deteriorating seasonal products whose deterioration rate can be controlled by investing on the preservation efforts. Then, we propose an algorithm to solve the nonlinear program problem. By analysis, we can find some properties when parameters changed. Smaller buying cost per unit, holding cost per unit time, and ordering cost can all benefit the company. Besides, when deterioration rate is relatively small or the sensitivity parameter of the investment (
For future research, we can take the backlogged demand into our model. Furthermore, we can assume that the ordering lead time exists and can be controlled by extra investment. Also, we can extend the model to the deteriorating problems in multiechelon supply chains.
The authors thank the valuable comments of the anonymous referees for an earlier version of this paper. Their comments have significantly improved the paper. This work is supported by the National Natural Science Foundation of China (no. 71001025). Also, this research is partly supported by the Program for New Century Excellent Talents in University (no. NCET100327) and the Ministry of Education of China. GrantinAid for Humanity and Social Science Research (no. 11YJCZH139).