# An Inventory Model for Perishable Items with Price-, Stock-, and Time-Dependent Demand Rate considering Shelf-Life and Nonlinear Holding Costs

Authors: Adrián Macías-López; Leopoldo Eduardo Cárdenas-Barrón; Rodrigo E. Peimbert-García; Buddhadev Mandal
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

Nowadays, consumers are more health conscious than before, and their demand of fresh items has intensely increased. In this context, an effective and efficient inventory management of the perishable items is needed in order to avoid the relevant losses due to their deterioration. Furthermore, the demand of products is influenced by several factors such as price, stock, and freshness state, among others. Hence, this research work develops an inventory model for perishable items, constrained by both physical and freshness condition degradations. The demand for perishable items is a multivariate function of price, current stock quantity, and freshness condition. Specific to price, six different price-dependent demand functions are used: linear, isoelastic, exponential, logit, logarithmic, and polynomial. By working with perishable items that eventually deteriorate, this inventory model also takes into consideration the expiration date, a salvage value, and the cost of deterioration. In addition, the holding cost is modelled as a quadratic function of time. The proposed inventory model jointly determines the optimal price, the replenishment cycle time, and the order quantity, which together result in maximum total profit per unit of time. The inventory model has a wide application since it can be implemented in several fields such as food goods (milk, vegetables, and meat), organisms, and ornamental flowers, among others. Some numerical examples are presented to illustrate the use of the inventory model. The results show that increasing the value of the shelf-life results in an increment in price, inventory cycle time, quantity ordered, and profits that are generated for all price demand functions. Finally, a sensitivity analysis is performed, and several managerial insights are provided.


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## Body

## 1. Introduction

Inventory management is a valuable function for companies around the world. It seeks to control the materials from acquisition up to sales-related decision-making (how much and when to buy items) in order to avoid overstock and/or stockout. According to Yavari et al. [1], one of the challenges when managing inventories is the inherent perishability of many items, which means their freshness and quality decrease over time, and these cannot be sold after their expiration date. Tirkolaee et al. [2] noted that the inherent perishability widely occurs in food goods (e.g., milk, vegetables, and meat), organisms, and ornamental flowers. These authors also stated that the time window between preparation and sales of perishable items is very significant for producers and purchasers.Given that the inherent perishability can occur immediately, Pal et al. [3] addressed a production-inventory model for deteriorating products when the production cost depends on both production order quantity and production rate. Later, meanwhile, Mashud et al. [4] determined the optimal replenishment policy of deteriorating goods for the classical newsboy inventory problem by considering multiple just-in-time deliveries. Additionally, Mashud et al. [5] derived an inventory model for deteriorating products that calculates the optimal vales for replenishment time, price, and green investment cost.There are also some items in which the inherent perishability is noninstantaneous. Mashud et al. [6] considered this by developing an inventory model for noninstantaneous deteriorating items, which jointly optimizes the cycle time, price, the spending in preservation technology, and credit financing. Alongside, Mashud et al. [7] proposed another inventory model for noninstantaneous deteriorating goods which determines the cycle length, price, and preservation cost. On the contrary, Hasan et al. [8] introduced a noninstantaneous inventory model for agricultural goods taking into account the effects of the inherent perishability. This model states the optimal pricing and timing inventory policies.Similarly, there is currently an increasing demand for fresh items since consumers are more concerned about their health habits. In this context, consumers want to buy fresh goods far from their expiry date, so these can be stored during longer periods of time. Thus, companies should carefully manage and control their inventories of fresh items in the warehouse. One of the most common approaches to tracking the freshness and quality of perishable items in the warehouse is the radio frequency identification (RFID) smart tags, which allows to decrease the risk of selling items after their expiration date. Some examples have been exemplified by Herbon et al. [9], Herbon et al. [10], and Herbon and Ceder [11], who investigated the effect of implementing the time-temperature-indicator (TTI) in order to make information about expiration dates (for different items) available online.Once items are available on shelves, exhibiting large quantities is one of the factors that often influence consumers to purchase more products. Then, retailers might become more profitable by increasing their goods’ availability. However, this is different for cases where time plays a key role, such as perishable items, whose degradation tends to make them less attractive and unsalable at the end of shelf-life. Another factor is price, which inversely stimulates demand; lower prices result in higher demand, and vice versa. Overall, price, stock availability, time, and shelf-life are the critical factors that affect the demand of perishable products and should, consequently, be considered when developing inventory models.The literature on inventory models with time-, price-, and stock-dependent demand is also abundant. For instance, Mashud et al. [12] constructed a price-sensitive inventory model by considering that the demand follows an exponential or isoelastic price-dependent demand.Avinadav et al. [13] proposed two inventory models under the assumption that the demand function is dependent on both price and stock age and determined the optimal pricing and inventory policies. While the first inventory model assumes a multiplicative demand function, the second assumes an additive demand function. During the same year, Qin et al. [14] formulated an inventory model that determines the pricing and inventory policies for a perishable item considering stock-dependent demand and the effects of item’s quality degradation. Later, Chen et al. [15] derived the optimal inventory policy and shelf-space size for fresh products considering the expiration time and a demand rate that depends on freshness and stock. The demand function treated by Chen et al. [15] grows in the on-hand stock level and diminishes with respect to the age of the item. Afterwards, Feng et al. [16] resolved the inventory model of Chen et al. [15] considering the demand rate jointly dependent of price, stock, and age. Meanwhile, Dobson et al. [17] built an economic order quantity (EOQ) inventory model for a perishable item with fixed shelf-life when the demand rate decreases linearly with respect to the age of the stock.Other authors that work on a similar problem are Herbon and Khmelnitsky [18], who developed an inventory model that determines the optimal replenishment time and price when the demand rate actually depends on both time and price. Alongside, Banerjee and Agrawal [19] defined the optimal discounting and ordering policies for deteriorating products including a demand rate that depends on price and freshness. Hsieh and Dye [20] also provided the optimal pricing policy for deteriorating goods by considering the impact of reference prices under the assumption that stocks stimulate demand. Finally, Li and Teng [21] obtained the lot sizing and pricing strategies for a perishable item when the demand rate is dependent on reference price, exhibited inventory, and item freshness.Recently, Agi and Soni [22] developed an inventory model for joint optimal pricing and inventory management for a perishable item under stock-, age-, and price-dependent demand, allowing surplus inventory at the end of cycle. These authors stated that finishing the inventory cycle period with a positive inventory level results in a benefit because the demand increases when large quantities are ordered and exhibited in the shelf space. Conversely, having inventory of perishable products at the end of their cycle is not desirable, as these cannot be stored for the next inventory cycle. Hence, these must be sold at a salvage price.Although most inventory models consider a constant holding cost, different factors that affect inventories on the storage place make the holding cost intrinsically variable. There are common scenarios in the real world where the holding cost increases over time because longer storage periods require more sophisticated and costly warehouse facilities. For instance, a longer storage of fresh products needs refrigeration and some specific conditions in place to prevent damages. Alfares and Ghaithan [23] provided an excellent state-of-the-art review on the economic order quantity (EOQ) and economic production quantity (EPQ) inventory models with variable holding costs. Specifically, Alfares and Ghaithan [23] classified the variable holding cost into three categories: time-dependent holding cost, stock-dependent holding cost, and multiple dependence cost variability. The types of holding cost functions are constant, linear, nonlinear, step, or general. For modelling the case of variable time-dependent holding cost, there exist several functions. One is the quadratic holding cost function that is exemplified by Valliathal and Uthayakumar [24], who constructed two EPQ inventory models for deteriorating items by including the holding cost as nonlinear time dependent. Years later, Pal et al. [25] investigated the single-period newsvendor model and obtained the optimal lot size when the customers’ balking happens by taking into consideration a nonlinear holding cost that depends on both lot size and the stock level. Tripathi and Mishra [26] studied two inventory models, with time-varying holding cost, that obtain both the optimal order quantity and the optimal replenishment cycle time. While the first inventory model presents a linearly time-dependent holding cost, the second one has a quadratic time dependence, carrying inventory cost. In the same year, Sivashankari [27] also presented a comparative study of three EPQ inventory models under the assumption that the carrying inventory cost is either constant, a linear function of time, or a quadratic function of time.Since closing the stock cycle time with zero inventory is the most appropriate strategy when dealing with perishable items, this research work builds an inventory model for perishable items with zero inventory at the end of stock cycle. On the one hand, these perishable items are subject to physical deterioration and freshness degradation over time, and on the other hand, the demand is a multivariate function of price, on-hand stock quantity, and freshness state. For the demand that is related to price, six distinct price-dependent demand functions are considered: linear, isoelastic, exponential, logit, logarithmic, and polynomial. The inventory model also considers the expiration date, a salvage value, and a deterioration cost of the perishable item. Moreover, the holding cost is considered as nonlinear with a quadratic function of time. The proposed inventory model conjointly derives the optimal policy for the price, the replenishment cycle time, and the order quantity, which together maximize the total profit per unit of time.The rest of this research work is comprised of several sections as follows. Section2 introduces the notation and assumptions. Section 3 develops the inventory model with price-, stock-, and time-dependent demand with nonlinear holding cost. Section 4 develops the solution procedure to determine the optimal solution. Section 5 presents the solution to six different price-dependent demand functions and solves some numerical examples. Section 6 performs a sensitivity analysis and proposes some managerial insights. Finally, Section 7 provides the conclusions and outlines several areas for further research.


## 2. Notation and Assumptions

### 2.1. Notation

The notation used to develop the inventory model with price-, stock-, and age-dependent demand, with zero inventory at the end of cycle, is given as follows. The symbols of Agi and Soni [22] are used, and some more additional symbols are defined here, in order to have a standard notation (Table 1).Table 1
Notation.
ParametersDescriptionsSalvage cost of deteriorated item (currency symbol/unit)cPurchase cost (currency symbol/unit)cdDeterioration cost (currency symbol/unit)hHolding cost (currency symbol/unit/unit of time)h1Holding cost (currency symbol/unit/unit of time2)h2Holding cost (currency symbol/unit/unit of time3)KOrdering cost (currency symbol/per cycle)nShelf-life of the item upon which the remaining amount, if any, is withdrawn immediately from the storage (unit of time)WMaximum shelf space (units)ηSalvage coefficient0≤η≤1θInventory deterioration rate0≤θ≤1ωSensitivity parameter to the current level of stockaScale parameter for the part of price-dependent demandbSensitivity parameter for the part of price-dependent demandFunctionsdpPrice-dependent demand for the item, which can be linear, isoelastic, exponential, logit, logarithmic, or polynomial function (units/unit of time)ItStock level at timet (units)Dp,It,tPrice-, stock-, and age-dependent demand for the item at timet (units/unit of time)HtQuadratic holding cost functionπcp,TTotal profit per cycle (currency symbol/unit)πp,TTotal profit per unit of time (currency symbol/unit of time)tAge of the stock, which is the time passed since the last replenishment (unit of time)Decision variablespProduct selling price (currency symbol/unit)TReplenishment cycle length (unit of time)Dependent decision variableQOrder quantity (units)


### 2.2. Assumptions

The inventory model is based on the following assumptions:(1)
The item in the storage is subject to two types of degradations over time: physical degradation at a constant rate and freshness degradation.(2)
The item has a definite and limited shelf-life after which it is not salable.(3)
The demand of the item depends on price, the current stock amount, and its freshness. For the demand that is price-dependent, six different functions of the price-dependent demand are used: linear, isoelastic, exponential, logit, logarithmic, and polynomial.(4)
The number of remaining items at the end of the cycle is zero.(5)
The holding cost is nonlinear and modelled with a quadratic function that depends on time (i.e.,h+h1t+h2t2).(6)
The salvage value and deterioration cost are taken into account for the items deteriorated throughout the inventory cycle.(7)
The time horizon planning is infinite. The lead time is zero, and therefore, the replenishment rate is instantaneous.(8)
At the beginning of the stock periodt = 0, the item is fresh and has no age effect on demand. Thus, the item loses its freshness over time, and its demand consequently decreases.(9)
The length of the stock periodT must not surpass the item’s shelf-life n since the item is unsalable after its expiration date is over T≤n.


## 2.1. Notation

The notation used to develop the inventory model with price-, stock-, and age-dependent demand, with zero inventory at the end of cycle, is given as follows. The symbols of Agi and Soni [22] are used, and some more additional symbols are defined here, in order to have a standard notation (Table 1).Table 1
Notation.
ParametersDescriptionsSalvage cost of deteriorated item (currency symbol/unit)cPurchase cost (currency symbol/unit)cdDeterioration cost (currency symbol/unit)hHolding cost (currency symbol/unit/unit of time)h1Holding cost (currency symbol/unit/unit of time2)h2Holding cost (currency symbol/unit/unit of time3)KOrdering cost (currency symbol/per cycle)nShelf-life of the item upon which the remaining amount, if any, is withdrawn immediately from the storage (unit of time)WMaximum shelf space (units)ηSalvage coefficient0≤η≤1θInventory deterioration rate0≤θ≤1ωSensitivity parameter to the current level of stockaScale parameter for the part of price-dependent demandbSensitivity parameter for the part of price-dependent demandFunctionsdpPrice-dependent demand for the item, which can be linear, isoelastic, exponential, logit, logarithmic, or polynomial function (units/unit of time)ItStock level at timet (units)Dp,It,tPrice-, stock-, and age-dependent demand for the item at timet (units/unit of time)HtQuadratic holding cost functionπcp,TTotal profit per cycle (currency symbol/unit)πp,TTotal profit per unit of time (currency symbol/unit of time)tAge of the stock, which is the time passed since the last replenishment (unit of time)Decision variablespProduct selling price (currency symbol/unit)TReplenishment cycle length (unit of time)Dependent decision variableQOrder quantity (units)


## 2.2. Assumptions

The inventory model is based on the following assumptions:(1)
The item in the storage is subject to two types of degradations over time: physical degradation at a constant rate and freshness degradation.(2)
The item has a definite and limited shelf-life after which it is not salable.(3)
The demand of the item depends on price, the current stock amount, and its freshness. For the demand that is price-dependent, six different functions of the price-dependent demand are used: linear, isoelastic, exponential, logit, logarithmic, and polynomial.(4)
The number of remaining items at the end of the cycle is zero.(5)
The holding cost is nonlinear and modelled with a quadratic function that depends on time (i.e.,h+h1t+h2t2).(6)
The salvage value and deterioration cost are taken into account for the items deteriorated throughout the inventory cycle.(7)
The time horizon planning is infinite. The lead time is zero, and therefore, the replenishment rate is instantaneous.(8)
At the beginning of the stock periodt = 0, the item is fresh and has no age effect on demand. Thus, the item loses its freshness over time, and its demand consequently decreases.(9)
The length of the stock periodT must not surpass the item’s shelf-life n since the item is unsalable after its expiration date is over T≤n.


## 3. Inventory Model with Price-, Stock-, and Age-Dependent Demand, with Zero Inventory at the End of the Cycle

The problem under study is described as follows. A company manages a product that has an inherent perishability, and this item is subject to both physical and freshness degradation. Moreover, it is known that the item has a limited shelf-life, after which it is not marketable. Thus, the length of the stock cycle must not be greater than the item’s shelf-life. Due to the nature of the item, the demand depends on price, the current stock, and its freshness. On the contrary, the target is to have zero inventory at the end of the inventory cycle. The salvage value and deterioration cost are also considered for the deteriorated items throughout the inventory period. Figure1 shows the behavior of the inventory level over time. At the beginning of the stock cycle t = 0, the retailer receives a lot size of Q units, and the inventory level immediately starts to decrease due to both demand and deterioration, until the stock level reaches zero units at t=T.Figure 1
Graphical representation of the inventory level over time.For the duration of the stock cycle0,T, the on-hand stock deteriorates at a constant rate θ, while it also loses its freshness over time. The demand depends on price, stock, and age, according to the following function:(1)Dp,It,t=n−tndp+ωIt,n>0,ω≥0,t≤n,where the part of demand that is dependent on price dp takes one of the following expressions:(2)dp=a−bp,0≤p≤ablinear demand,dp=ap−b,0≤p≤∞iso−elastic demand,dp=ae−bp,0≤p≤∞exponential demand,dp=a1+ebp,0≤p≤∞logit demand,dp=a−blnp,0≤p≤ea/blogarithmic demand,dp=a−bpm,0≤p≤ab1/mpolynomial demand.The six demand functions used in this study, which depend on price, adequately model the scenario where the demand increases as the price decreases, and vice versa.By considering the aforementioned assumptions, the behavior of the on-hand inventory levelIt is modelled by the following differential equation:(3)dItdt=−n−tndp−ωIt−θIt,0≤t≤T,with the boundary condition:(4)IT=0.By solving the differential equation (3), the inventory level It is expressed as follows:(5)It=dpω+θ1+1nω+θeω+θT−t−1−dpnω+θTeω+θT−t−t,0≤t≤T.The order quantityQ, which occurs when I0, is expressed as follows:(6)Q=dpω+θ−1−1nω+θ+dpeω+θTω+θ1+1nω+θ−Tn≤W.The profit is calculated as the difference between the sales revenue of those perfect and deteriorated items and the total costs resulting from sum of the ordering cost, holding cost, purchase cost, and deterioration cost. A brief explanation about the calculation of sales revenue and costs follows. The sales revenue is the product of selling pricep and the total demand occurred within the replenishment time T, where the total demand is obtained with the definite integral from zero to T of the demand function Dp,It,t. The salvage value is computed as the product of salvage cost, salvage coefficient, and the number of deteriorated units per cycle. The ordering cost represents the cost of placing an order. The holding cost is calculated by the definite integral from zero to T, of the product of the quadratic holding cost function Ht, and the stock level function It. The purchase cost is computed by multiplying the unit purchase cost c by the order quantity Q. Finally, the deterioration cost is the product of unit deterioration cost cd by the number of deteriorated units per cycle.The detailed calculation of all components of the profit function is mathematically presented below.(1)
Sales revenueSR per cycle:(7)SR=p∫0Tdp1−tn+ωItdt=dpT1−T2n−ωdpTω+θ1−T2n+1nω+θ+ωdpω+θeω+θT−1ω+θ1+1nω+θ−Tn.(2)
Salvage valueSV of deteriorated items per cycle:(8)SV=sηDeteriorated units per cycle,SV=sηQ−∫0Tdp1−tn+ωItdt,SV=sηdpω+θ−1−1nω+θ+dpeω+θTω+θ1+1nω+θ−Tn−dpT1−T2n−ωdpTω+θ1−T2n+1nω+θ+eω+θT−1ω+θωdpω+θ1+1nω+θ−Tn.(3)
Ordering costOC per cycle:(9)OC=K.(4)
The holding costHC per cycle:(10)HC=∫0THtItdt=∫0Th+h1t+h2t2It,HC=∫0THtItdt=hdpTω+θ−1+T2n−1nω+θ+eω+θT−1ω+θdpω+θ1+1nω+θ−Tn+h1dpT2ω+θ−12+T3n−12nω+θ+eω+θT−ω+θT−1ω+θ2dpω+θ1+1nω+θ−Tn+h2dpT3ω+θ−13+T4n−13nω+θ+2eω+θT−Tω+θTω+θ+2−2ω+θ3dpω+θ1+1nω+θ−Tn.(5)
The purchase costPC per cycle:(11)PC=cdpω+θ−1−1nω+θ+dpeω+θTω+θ1+1nω+θ−Tn.(6)
The deterioration costDC per cycle:(12)DC=cdQ−∫0Tdp1−tn+ωIt,DC=cddpω+θ−1−1nω+θ+dpeω+θTω+θ1+1nω+θ−Tn−dpT1−T2n−ωdpTω+θ1−T2n+1nω+θ+eω+θT−1ω+θωdpω+θ1+1nω+θ−Tn.Therefore, the total profit per cycle is(13)πc=sales revenue+salvage value−ordering cost−holding cost−purchase cost−deterioration cost,πc=SR+SV−OC−HC−PC−DC.And, it is expressed as(14)πcp,T=dpT1−T2n−ωdpTω+θ1−T2n+1nω+θ+ωdpω+θeω+θT−1ω+θ1+1nω+θ−Tn+sηdpω+θ−1−1nω+θ+dpeω+θTω+θ1+1nω+θ−Tn−dpT1−T2n−ωdpTω+θ1−T2n+1nω+θ+eω+θT−1ω+θωdpω+θ1+1nω+θ−Tn−hdpTω+θ−1+T2n−1nω+θ+eω+θT−1ω+θdpω+θ1+1nω+θ−Tn+h1dpT2ω+θ−12+T3n−12nω+θ+eω+θT−ω+θT−1ω+θ2dpω+θ1+1nω+θ−Tn+h2dpT3ω+θ−13+T4n−13nω+θ+2eω+θT−Tω+θTω+θ+2−2ω+θ3dpω+θ1+1nω+θ−Tn−cdpω+θ−1−1nω+θ+dpeω+θTω+θ1+1nω+θ−Tn−cddpω+θ−1−1nω+θ+dpeω+θTω+θ1+1nω+θ−Tn−dpT1−T2n−ωdpTω+θ1−T2n+1nω+θ+eω+θT−1ω+θωdpω+θ1+1nω+θ−Tn.Hence, the total profit per unit of time is expressed as follows:(15)πp,T=1TdpT1−T2n−ωdpTω+θ1−T2n+1nω+θ+ωdpω+θeω+θT−1ω+θ1+1nω+θ−Tn+sηdpω+θ−1−1nω+θ+dpeω+θTω+θ1+1nω+θ−Tn−dpT1−T2n−ωdpTω+θ1−T2n+1nω+θ+eω+θT−1ω+θωdpω+θ1+1nω+θ−Tn−hdpTω+θ−1+T2n−1nω+θ+eω+θT−1ω+θdpω+θ1+1nω+θ−Tn+h1dpT2ω+θ−12+T3n−12nω+θ+eω+θT−ω+θT−1ω+θ2dpω+θ1+1nω+θ−Tn+h2dpT3ω+θ−13+T4n−13nω+θ+2eω+θT−Tω+θTω+θ+2ω+θ3dpω+θ1+1nω+θ−Tn−cdpω+θ−1−1nω+θ+dpeω+θTω+θ1+1nω+θ−Tn−cddpω+θ−1−1nω+θ+dpeω+θTω+θ1+1nω+θ−Tn−dpT1−T2n−ωdpTω+θ1−T2n+1nω+θ+eω+θT−1ω+θωdpω+θ1+1nω+θ−Tn.In general, the optimization problem is formulated as follows:(16)Maxp,Tπp,TSubject toc≤p≤pmaxandT≤n.The total profit per unit of time function given in equation (15) shows that this is highly nonlinear in both selling price p and replenishment cycle time T. Therefore, it is not possible to determine a close form for these decision variables. Nonetheless, the optimal solution to p and T are found through an optimization procedure that is based on the traditional conditions for optimality. The solution procedure is explained in the following section.


## 4. Solution Procedure to Obtain the Optimal Solution

### 4.1. Theoretical Results

The goal is to determine the optimal selling pricep and optimal replenishment cycle time T that maximize the total profit. Since the total profit per unit of time function πp,T is continuous and twice differentiable with respect to both variables p,T on the interval [0, ∞], there exists, then, a global maximum on that interval.The necessary conditions for the total profit per unit of time functionπp,T to be maximized are as follows:(17)∂πp,T∂p=0,(18)∂πp,T∂T=0.Moreover, for the expected total profit per unit of time functionπp,T to be concave, the sufficient conditions are given as follows:(19)∂2πp,T∂p2<0,(20)∂2πp,T∂T2<0,(21)∂2πp,T∂p∂T=∂2πp,T∂T∂p,(22)∂2πp,T∂p2∂2πp,T∂T2−∂2πp,T∂p∂T∂2πp,T∂T∂p>0.The optimal solution is determined by simultaneously solving the first partial derivatives of the total profit per unit of time function given in equation (15), with respect to p and T equalizing to zero.If the solutionp,T satisfies the conditions given by equations (19)–(22), it demonstrates that the function πp,T is strictly concave in both decision variables with a negative-definite Hessian matrix. If so, the solution p,T is optimal.In the optimization problem given in equation (16), it is stated that the selling price has an upper bound of pmax and the replenishment cycle time has an upper bound of n. The component of the demand that corresponds to price dp is modelled with six different functions. For isoelastic, exponential, and logit functions, the interval for the price p is [0, ∞]. On the contrary, for linear, logarithmic, and polynomial, there exists a maximum permissible value pmax. The upper bound pmax is a/b,ea/b and a/b1/m for the linear, logarithmic, and polynomial, respectively. For these demand functions, the solution for selling price is p=pmax when the solution for the selling price p is greater than pmax. This allows avoiding a negative value for the demand that is dependent on price. This is mathematically right, but it does not make sense in any business of the real world.Notice that the replenishment cycle timeT is on the interval [0, ∞]. When the solution for the replenishment cycle time T is greater than n, the solution for the replenishment cycle time, then, is T=n due to the shelf-life constrain.


### 4.2. Algorithm for Finding the Optimal Solution

Considering the theoretical results presented in the previous section, the following algorithm is proposed (Algorithm1).Algorithm 1:Algorithm for finding the optimal solution.
Step 1. Input the inventory parameters.Step 2. Calculatepmax.Step 3. Solve simultaneously equations (17) and (18) to obtain the values for p and T.Step 4. If the conditions (19)–(22) are satisfied, then the solution is optimal, and go to step 5. Otherwise, the solutions are infeasible, and go to step 14.Step 5. If bothc≤p≤pmax and T≤n are satisfied, then set p∗=p and T∗=T, and go to step 11.Else, go to step 6.Step 6. If bothp≤c and T≤n are satisfied, then set p∗=c and T∗=T, and go to step 11.Else, go to step 7.Step 7. If bothp≤c and T>n are satisfied, then set p∗=c and T∗=n, and go to step 11.Else, go to step 8.Step 8. If bothp>pmax and T≤n are satisfied, then set p∗=pmax and T∗=T, and go to step 11.Else, go to step 9.Step 9. If bothp≤pmax and T>n are satisfied, then set p∗=p and T∗=n, and go to step 11.Else, go to step 10.Step 10. Setp=pmax and T∗=n.Step 11. Calculate the lot sizeQ∗ with equation (6).Step 12. Compute the total profit per unit of timeπ∗p∗,T∗ with equation (15).Step 13. Report the optimal solutionπ∗p∗,T∗, p∗, T∗, and Q∗.Step 14. Stop.


## 4.1. Theoretical Results

The goal is to determine the optimal selling pricep and optimal replenishment cycle time T that maximize the total profit. Since the total profit per unit of time function πp,T is continuous and twice differentiable with respect to both variables p,T on the interval [0, ∞], there exists, then, a global maximum on that interval.The necessary conditions for the total profit per unit of time functionπp,T to be maximized are as follows:(17)∂πp,T∂p=0,(18)∂πp,T∂T=0.Moreover, for the expected total profit per unit of time functionπp,T to be concave, the sufficient conditions are given as follows:(19)∂2πp,T∂p2<0,(20)∂2πp,T∂T2<0,(21)∂2πp,T∂p∂T=∂2πp,T∂T∂p,(22)∂2πp,T∂p2∂2πp,T∂T2−∂2πp,T∂p∂T∂2πp,T∂T∂p>0.The optimal solution is determined by simultaneously solving the first partial derivatives of the total profit per unit of time function given in equation (15), with respect to p and T equalizing to zero.If the solutionp,T satisfies the conditions given by equations (19)–(22), it demonstrates that the function πp,T is strictly concave in both decision variables with a negative-definite Hessian matrix. If so, the solution p,T is optimal.In the optimization problem given in equation (16), it is stated that the selling price has an upper bound of pmax and the replenishment cycle time has an upper bound of n. The component of the demand that corresponds to price dp is modelled with six different functions. For isoelastic, exponential, and logit functions, the interval for the price p is [0, ∞]. On the contrary, for linear, logarithmic, and polynomial, there exists a maximum permissible value pmax. The upper bound pmax is a/b,ea/b and a/b1/m for the linear, logarithmic, and polynomial, respectively. For these demand functions, the solution for selling price is p=pmax when the solution for the selling price p is greater than pmax. This allows avoiding a negative value for the demand that is dependent on price. This is mathematically right, but it does not make sense in any business of the real world.Notice that the replenishment cycle timeT is on the interval [0, ∞]. When the solution for the replenishment cycle time T is greater than n, the solution for the replenishment cycle time, then, is T=n due to the shelf-life constrain.


## 4.2. Algorithm for Finding the Optimal Solution

Considering the theoretical results presented in the previous section, the following algorithm is proposed (Algorithm1).Algorithm 1:Algorithm for finding the optimal solution.
Step 1. Input the inventory parameters.Step 2. Calculatepmax.Step 3. Solve simultaneously equations (17) and (18) to obtain the values for p and T.Step 4. If the conditions (19)–(22) are satisfied, then the solution is optimal, and go to step 5. Otherwise, the solutions are infeasible, and go to step 14.Step 5. If bothc≤p≤pmax and T≤n are satisfied, then set p∗=p and T∗=T, and go to step 11.Else, go to step 6.Step 6. If bothp≤c and T≤n are satisfied, then set p∗=c and T∗=T, and go to step 11.Else, go to step 7.Step 7. If bothp≤c and T>n are satisfied, then set p∗=c and T∗=n, and go to step 11.Else, go to step 8.Step 8. If bothp>pmax and T≤n are satisfied, then set p∗=pmax and T∗=T, and go to step 11.Else, go to step 9.Step 9. If bothp≤pmax and T>n are satisfied, then set p∗=p and T∗=n, and go to step 11.Else, go to step 10.Step 10. Setp=pmax and T∗=n.Step 11. Calculate the lot sizeQ∗ with equation (6).Step 12. Compute the total profit per unit of timeπ∗p∗,T∗ with equation (15).Step 13. Report the optimal solutionπ∗p∗,T∗, p∗, T∗, and Q∗.Step 14. Stop.


## 5. Optimal Inventory Policy for Six Different Price-Dependent Demands

### 5.1. Optimal Inventory Policy Using the Price-Dependent Linear Demand Functiondp=a−bp

The first partial derivative ofπp,T with respect to p is(23)∂πp,T∂p=−bp+dp1−T2n+ωω+θ−1+T2n−1nω+θ+ωTω+θeω+θT−1ω+θ1+1nω+θ−ωTnω+θTeω+θT−1ω+θ+sη−b1Tω+θ−1−1nω+θ+eω+θTTω+θ1+1nω+θ−Tn−1+T2n+ωω+θ1−T2n+1nω+θ+ωTω+θeω+θT−1ω+θ−1−1nω+θ+Tn−−hb1ω+θ−1+T2n−1nω+θ+1Tω+θeω+θT−1ω+θ1+1nω+θ−Tn+−h1bTω+θT3n−12−12nω+θ+1Tω+θeω+θT−Tω+θ−1ω+θ21+1nω+θ−Tn+−h2bT2ω+θT4n−13−13nω+θ+1Tω+θ2eω+θT−Tω+θTω+θ+2−2ω+θ31+1nω+θ−Tn+−cb1Tω+θ1+1nω+θ+eω+θTTω+θ−1−1nω+θ+Tn−cd−b1Tω+θ−1−1nω+θ+eω+θTTω+θ1+1nω+θ−Tn−1+T2n+ωω+θ1−T2n+1nω+θ+ωTω+θeω+θT−1ω+θ−1−1nω+θ+Tn=0.The first partial derivative ofπp,T with respect to T is(24)∂πp,T∂T=pdp2nωω+θ−1+ωdpω+θeω+θTωT+θT−1+1ω+θT21+1nω+θ−ωdpeω+θTnω+θ+sηdpω+θT21+1nω+θ+dpω+θeω+θTωT+θT−1+1T21+1nω+θ−dpnω+θω+θeω+θT+dp2n1−ωω+θ−ωdpω+θeω+θTωT+θT−1+1ω+θT21+1nω+θ+ωdpnω+θeω+θT+KT2−hdp2nω+θ+dpω+θeω+θTωT+θT−1+1ω+θT21+1nω+θ−dpnω+θeω+θT+h1−dp2ω+θ+2dpT3nω+θ−dp2nω+θ2+dpω+θωTeω+θT+θTeω+θT−eω+θT+1ω+θ2T21+1nω+θ−dpnω+θeω+θT−1ω+θ+h2−2dpT3ω+θ+3dpT24nω+θ−2dpT3nω+θ2+dpω+θ−θ2T2+2ωθT2−2θTeω+θT−2ωTeω+θT+2eω+θT+ω2T2−2ω+θ3T21+1nω+θ−dpnω+θ2eω+θT−2Tω+θ−2ω+θ2−cdpω+θT21+1nω+θ+dpω+θeω+θTωT+θT−1T21+1nω+θ−dpnω+θω+θeω+θT−cddpω+θT21+1nω+θ+dpω+θeω+θTωT+θT−1+1T21+1nω+θ−dpnω+θω+θeω+θT+dp2n1−ωω+θ−ωdpω+θeω+θTωT+θT−1+1ω+θT21+1nω+θ+ωdpnω+θeω+θT=0.Example 1.
In order to represent a real scenario, let us consider a place that sells a fresh item. Assume that the price-dependent component of the demand for the fresh item is linear as follows:dp=600−20p (i.e., a=600 and b=20). The shelf-life of the fresh item is n=1 week. There is a maximum shelf space of W=500 units. The cost for placing an order to the supplier is K=250 euros per order, the purchase cost is c=5 euros per unit, and the salvage value of the deteriorated item is s=4 euros per unit. The sensitivity coefficient to the level of stock is ω=0.5, and the stock deterioration rate is θ=0.05. The aforementioned data were taken from Agi and Soni [22]. Additional data are still needed to solve the numerical examples. The values of the holding cost are h=1.75 euros per unit per week, h1=0.15 euros per unit per week2, and h2=0.25 euros per unit per week3. The deterioration cost of the item is cd=2 euros per unit, and the salvage coefficient is η=0.8. Since n=1 week, the replenishment cycle time must be T≤1 week. For the price linear demand, the upper bound for price is pmax=a/b=600/20=30. This means that the price must satisfy 5≤p≤30.
By applying the proposed algorithm, the following optimal solution for the inventory system is calculated:p∗=17.69124 euros per unit, T∗=0.4395923 weeks, Q∗=94.42941 units, and π∗p∗,T∗=2049.903 euros per week. This solution satisfies all conditions for the optimality: ∂2πp,T/∂p2=−34.56784<0, ∂2πp,T/∂T2=−49710.46<0, ∂2πp,T/∂p∂T=∂2πp,T/∂T∂p=13.01441, and ∂2πp,T/∂p2∂2πp,T/∂T2−∂2πp,T/∂p∂T∂2πp,T/∂T∂p=0.1718214×107>0. As the Hessian determinant is greater than zero, the total profit function then is strictly concave with a negative-definite Hessian matrix. Consequently, the solution is optimal.
By plotting the total profit per unit of time functionπp,T with distinct values of p (taking values between 7 and 27) given T and distinct values of T (taking values between 0 and 1) given p, it is observed that πp,T is strictly concave with respect to p when T is fixed (see Figure 2(a)) and with respect to T when p is given (see Figure 2(b)). Additionally, πp,T is also strictly concave with respect to both p and T (see Figure 2(c)). Thus, it is ensured that the solution corresponds a global maximum.Figure 2
Concave property of the profit functionπp,T when the price-dependent demand function is linear.
(a)(b)(c)


### 5.2. Optimal Inventory Policy Using the Price-Dependent Isoelastic Demand Functiondp=ap−b

The first partial derivative ofπp,T with respect to p is(25)∂πp,T∂p=−abpb+1p+dp1−T2n+ωω+θ−1+T2n−1nω+θ+ωTω+θeω+θT−1ω+θ1+1nω+θ−ωTnω+θTeω+θT−1ω+θ+sη−abpb+11Tω+θ−1−1nω+θ+eω+θTTω+θ1+1nω+θ−Tn−1+T2n+ωω+θ1−T2n+1nω+θ+ωTω+θeω+θT−1ω+θ−1−1nω+θ+Tn−h−abpb+11ω+θ−1+T2n−1nω+θ+1Tω+θeω+θT−1ω+θ1+1nω+θ−Tn+h1−abpb+1Tω+θT3n−12−12nω+θ+1Tω+θeω+θT−Tω+θ−1ω+θ21+1nω+θ−Tn+h2−abpb+1T2ω+θT4n−13−13nω+θ+1Tω+θ2eω+θT−Tω+θTω+θ+2−2ω+θ31+1nω+θ−Tn+c−abpb+11Tω+θ1+1nω+θ+eω+θTTω+θ−1−1nω+θ+Tn−cd−abpb+11Tω+θ−1−1nω+θ+eω+θTTω+θ1+1nω+θ−Tn−1+T2n+ωω+θ1−T2n+1nω+θ+ωTω+θeω+θT−1ω+θ−1−1nω+θ+Tn=0.The first partial derivative ofπp,T with respect to T is(26)∂πp,T∂T=pdp2nωω+θ−1+ωdpω+θeω+θTωT+θT−1+1ω+θT21+1nω+θ−ωdpeω+θTnω+θ+sηdpω+θT21+1nω+θ+dpω+θeω+θTωT+θT−1+1T21+1nω+θ−dpnω+θω+θeω+θT+dp2n1−ωω+θ−ωdpω+θeω+θTωT+θT−1+1ω+θT21+1nω+θ+ωdpnω+θeω+θT+KT2−hdp2nω+θ+dpω+θeω+θTωT+θT−1+1ω+θT21+1nω+θ−dpnω+θeω+θT+h1−dp2ω+θ+2dpT3nω+θ−dp2nω+θ2+dpω+θωTeω+θT+θTeω+θT−eω+θT+1ω+θ2T21+1nω+θ−dpnω+θeω+θT−1ω+θ+h2−2dpT3ω+θ+3dpT24nω+θ−2dpT3nω+θ2+dpω+θ−θ2T2+2ωθT2−2θTeω+θT−2ωTeω+θT+2eω+θT+ω2T2−2ω+θ3T21+1nω+θ−dpnω+θ2eω+θT−2Tω+θ−2ω+θ2−cdpω+θT21+1nω+θ+dpω+θeω+θTωT+θT−1T21+1nω+θ−dpnω+θω+θeω+θT−cddpω+θT21+1nω+θ+dpω+θeω+θTωT+θT−1+1T21+1nω+θ−dpnω+θω+θeω+θT+dp2n1−ωω+θ−ωdpω+θeω+θTωT+θT−1+1ω+θT21+1nω+θ+ωdpnω+θeω+θT=0.Example 2.
Let us consider the same data as in Example1. Suppose now that the price-dependent element of demand for the perishable good has an isoelastic function as follows: dp=30,000p−1.4 (i.e., a=30,000 and b=1.4).
Sincen=1 week, the replenishment cycle time must be T≤1 week. For the price isoelastic demand, the upper bound for price is pmax=∞. Therefore, the price must be in the interval 5≤p≤∞. By using the algorithm proposed, the following optimal solution for the inventory model is determined: p∗=18.47849 euros per unit, T∗=0.3096932 weeks, Q∗=143.5169 units, and π∗p∗,T∗=5266.004 euros per week. This solution satisfies all conditions for the optimality: ∂2πp,T/∂p2=−9.960414<0, ∂2πp,T/∂T2=−271515.0<0, ∂2πp,T/∂p∂T=∂2πp,T/∂T∂p=28.89635, and ∂2πp,T/∂p2∂2πp,T/∂T2−∂2πp,T/∂p∂T∂2πp,T/∂T∂p=0.2703567×107>0 Given that the Hessian determinant is greater than zero, the total profit function is strictly concave with a negative-definite Hessian matrix. So, the solution is optimal.
By drawing the total profit per unit of time functionπp,T, with some values of p between 8 and 90, and considering T as given and some values of T between 0 and 1 when p is fixed, it is clear that πp,T is strictly concave with respect to p when T is given (see Figure 3(a)) and with respect to T when p is fixed (see Figure 3(b)). Moreover, πp,T is also strictly concave with respect to both decision variables: p and T (see Figure 3(c)). This confirms the concavity property in the total profit πp,T, and the solution corresponds to a global maximum.Figure 3
Concave property of the profit functionπp,T when the price-dependent demand function is isoelastic.
(a)(b)(c)


### 5.3. Optimal Inventory Policy Using the Price-Dependent Exponential Demand Functiondp=ae−bp

The first partial derivative ofπp,T with respect to p is(27)∂πp,T∂p=−abebpp+dp1−T2n+ωω+θ−1+T2n−1nω+θ+ωTω+θeω+θT−1ω+θ1+1nω+θ−ωTnω+θTeω+θT−1ω+θ+sη−abebp1Tω+θ−1−1nω+θ+eω+θTTω+θ1+1nω+θ−Tn−1+T2n+ωω+θ1−T2n+1nω+θ+ωTω+θeω+θT−1ω+θ−1−1nω+θ+Tn−h−abebp1ω+θ−1+T2n−1nω+θ+1Tω+θeω+θT−1ω+θ1+1nω+θ−Tn+h1−abebpTω+θT3n−12−12nω+θ+1Tω+θeω+θT−Tω+θ−1ω+θ21+1nω+θ−Tn+h2−abebpT2ω+θT4n−13−13nω+θ+1Tω+θ2eω+θT−Tω+θTω+θ+2−2ω+θ31+1nω+θ−Tn+c−abebp1Tω+θ1+1nω+θ+eω+θTTω+θ−1−1nω+θ+Tn−cd−abebp1Tω+θ−1−1nω+θ+eω+θTTω+θ1+1nω+θ−Tn−1+T2n+ωω+θ1−T2n+1nω+θ+ωTω+θeω+θT−1ω+θ−1−1nω+θ+Tn=0.The first partial derivative ofπp,T with respect to T is(28)∂πp,T∂T=pdp2nωω+θ−1+ωdpω+θeω+θTωT+θT−1+1ω+θT21+1nω+θ−ωdpeω+θTnω+θ+sηdpω+θT21+1nω+θ+dpω+θeω+θTωT+θT−1+1T21+1nω+θ−dpnω+θω+θeω+θT+dp2n1−ωω+θ−ωdpω+θeω+θTωT+θT−1+1ω+θT21+1nω+θ+ωdpnω+θeω+θT+KT2−hdp2nω+θ+dpω+θeω+θTωT+θT−1+1ω+θT21+1nω+θ−dpnω+θeω+θT+h1−dp2ω+θ+2dpT3nω+θ−dp2nω+θ2+dpω+θωTeω+θT+θTeω+θT−eω+θT+1ω+θ2T21+1nω+θ−dpnω+θeω+θT−1ω+θ+h2−2dpT3ω+θ+3dpT24nω+θ−2dpT3nω+θ2+dpω+θ−θ2T2+2ωθT2−2θTeω+θT−2ωTeω+θT+2eω+θT+ω2T2−2ω+θ3T21+1nω+θ−dpnω+θ2eω+θT−2Tω+θ−2ω+θ2−cdpω+θT21+1nω+θ+dpω+θeω+θTωT+θT−1T21+1nω+θ−dpnω+θω+θeω+θT−cddpω+θT21+1nω+θ+dpω+θeω+θTωT+θT−1+1T21+1nω+θ−dpnω+θω+θeω+θT+dp2n1−ωω+θ−ωdpω+θeω+θTωT+θT−1+1ω+θT21+1nω+θ−dpnω+θω+θeω+θT+dp2n1−ωω+θ−ωdpω+θeω+θTωT+θT−1+1ω+θT21+1nω+θ+ωdpnω+θeω+θT=0.Example 3.
Using the data of Example1 with the price-dependent part of the demand modelled with an exponential function, it follows dp=2000e−0.2p (i.e., a=2000 and b=0.2). The replenishment cycle time must satisfy T≤1 week because n=1 week. For the price exponential demand, the upper bound for price is pmax=∞. As a result, the price must be into the interval 5≤p≤∞.
By employing the algorithm, the optimal solution is found:p∗=10.50583 euros per unit, T∗=0.6187657 weeks, Q∗=121.3688 units, and π∗p∗,T∗=564.3379 euros per week. This solution satisfies all conditions for the optimality: ∂2πp,T/∂p2=−38.73461<0, ∂2πp,T/∂T2=−17613.81<0, ∂2πp,T/∂p∂T=∂2πp,T/∂T∂p=23.85985, and ∂2πp,T/∂p2∂2πp,T/∂T2−∂2πp,T/∂p∂T∂2πp,T/∂T∂p=681694.7>0. The Hessian determinant is greater than zero, with which the total profit function is strictly concave with a negative-definite Hessian matrix. Consequently, the solution is optimal.
By drawing the total profit per unit of time functionπp,T with different values of p between 7 and 20, and considering T as fixed and different values of T between 0 and 1 when p is given, πp,T is strictly concave with respect to p when T is fixed (see Figure 4(a)) and with respect to T when p is given (see Figure 4(b)). Besides, πp,T is also strictly concave with respect to both decision variables: p and T (see Figure 4(c)). This ratifies that total profit πp,T has the concavity property, and the solution is a global maximum.Figure 4
Concave property of the profit functionπp,T when the price-dependent demand function is exponential.
(a)(b)(c)


### 5.4. Optimal Inventory Policy Using the Price-Dependent Logit Demand Functiondp=a/1+ebp

The first partial derivative ofπp,T w.r.t. p is(29)∂πp,T∂p=−abebp1+ebp2p+dp1−T2n+ωω+θ−1+T2n−1nω+θ+ωTω+θeω+θT−1ω+θ1+1nω+θ−ωTnω+θTeω+θT−1ω+θ+sη−abebp1+ebp21Tω+θ−1−1nω+θ+eω+θTTω+θ1+1nω+θ−Tn−1+T2n+ωω+θ1−T2n+1nω+θ+ωTω+θeω+θT−1ω+θ−1−1nω+θ+Tn−h−abebp1+ebp21ω+θ−1+T2n−1nω+θ+1Tω+θeω+θT−1ω+θ1+1nω+θ−Tn+h1−abebp1+ebp2Tω+θT3n−12−12nω+θ+1Tω+θeω+θT−Tω+θ−1ω+θ21+1nω+θ−Tn+h2−abebp1+ebp2T2ω+θT4n−13−13nω+θ+1Tω+θ2eω+θT−Tω+θTω+θ+2−2ω+θ31+1nω+θ−Tn+c−abebp1+ebp21Tω+θ1+1nω+θ+eω+θTTω+θ−1−1nω+θ+Tn−cd−abebp1+ebp21Tω+θ−1−1nω+θ+eω+θTTω+θ1+1nω+θ−Tn−1+T2n+ωω+θ1−T2n+1nω+θ+ωTω+θeω+θT−1ω+θ−1−1nω+θ+Tn=0.The first partial derivative ofπp,T w.r.t. T is(30)∂πp,T∂T=pdp2nωω+θ−1+ωdpω+θeω+θTωT+θT−1+1ω+θT21+1nω+θ−ωdpeω+θTnω+θ+sηdpω+θT21+1nω+θ+dpω+θeω+θTωT+θT−1+1T21+1nω+θ−dpnω+θω+θeω+θT+dp2n1−ωω+θ−ωdpω+θeω+θTωT+θT−1+1ω+θT21+1nω+θ+ωdpnω+θeω+θT+KT2−hdp2nω+θ+dpω+θeω+θTωT+θT−1+1ω+θT21+1nω+θ−dpnω+θeω+θT+h1−dp2ω+θ+2dpT3nω+θ−dp2nω+θ2+dpω+θωTeω+θT+θTeω+θT−eω+θT+1ω+θ2T21+1nω+θ−dpnω+θeω+θT−1ω+θ+h2−2dpT3ω+θ+3dpT24nω+θ−2dpT3nω+θ2+dpω+θ−θ2T2+2ωθT2−2θTeω+θT−2ωTeω+θT+2eω+θT+ω2T2−2ω+θ3T21+1nω+θ−dpnω+θ2eω+θT−2Tω+θ−2ω+θ2−cdpω+θT21+1nω+θ+dpω+θeω+θTωT+θT−1T21+1nω+θ−dpnω+θω+θeω+θT−cddpω+θT21+1nω+θ+dpω+θeω+θTωT+θT−1+1T21+1nω+θ−dpnω+θω+θeω+θT+dp2n1−ωω+θ−ωdpω+θeω+θTωT+θT−1+1ω+θT21+1nω+θ+ωdpnω+θeω+θT=0.Example 4.
Consider the input parameters of Example1 with the price-dependent term of the demand for the fresh article following a logit demand function: dp=9000/1+e0.3p (i.e., a=9000 and b=0.3).
The replenishment cycle time is constrained toT≤1 week given that the shelf-life is n=1 week. For the price logit demand, the upper bound for price is pmax=∞. Therefore, the price is constrained by the interval 5≤p≤∞.
By running the algorithm, the optimal solution is obtained:p∗=8.963560 euros per unit, T∗=0.4682024 weeks, Q∗=231.1214 units, and π∗p∗,T∗=1205.467 euros per week. This solution satisfies all conditions for the optimality: ∂2πp,T/∂p2=−146.5883<0, ∂2πp,T/∂T2=−87888.64<0, ∂2πp,T/∂p∂T=∂2πp,T/∂T∂p=100.7604, and ∂2πp,T/∂p2∂2πp,T/∂T2−∂2πp,T/∂p∂T∂2πp,T/∂T∂p=0.128733×108>0. The Hessian determinant is greater than zero, and the total profit function is strictly concave with a negative-definite Hessian matrix. Thus, the solution is optimal.
By doing some graphs for the total profit per unit of time functionπp,T, with different values of p between 6 and 18, considering also T as fixed, and different values of T between 0 and 1 when p is given, πp,T is strictly concave with respect to p when T is fixed (see Figure 5(a)) and with respect to T when p is given (see Figure 5(b)). Besides, πp,T is strictly concave with respect to both decision variables p and T (see Figure 5(c)). This proves that total profit πp,T has the concavity property, so the solution is a global maximum.Figure 5
Concave property of the profit functionπp,T when the price-dependent demand function is logit.
(a)(b)(c)


### 5.5. Optimal Inventory Policy Using the Price-Dependent Logarithmic Demand Functiondp=a−blnp

The first partial derivative ofπp,T w.r.t. p is(31)∂πp,T∂p=−bpp+dp1−T2n+ωω+θ−1+T2n−1nω+θ+ωTω+θeω+θT−1ω+θ1+1nω+θ−ωTnω+θTeω+θT−1ω+θ+sη−bp1Tω+θ−1−1nω+θ+eω+θTTω+θ1+1nω+θ−Tn−1+T2n+ωω+θ1−T2n+1nω+θ+ωTω+θeω+θT−1ω+θ−1−1nω+θ+Tn−h−bp1ω+θ−1+T2n−1nω+θ+1Tω+θeω+θT−1ω+θ1+1nω+θ−Tn+h1−bpTω+θT3n−12−12nω+θ+1Tω+θeω+θT−Tω+θ−1ω+θ21+1nω+θ−Tn+h2−bpT2ω+θT4n−13−13nω+θ+1Tω+θ2eω+θT−Tω+θTω+θ+2−2ω+θ31+1nω+θ−Tn+c−bp1Tω+θ1+1nω+θ+eω+θTTω+θ−1−1nω+θ+Tn−cd−bp1Tω+θ−1−1nω+θ+eω+θTTω+θ1+1nω+θ−Tn−1+T2n+ωω+θ1−T2n+1nω+θ+ωTω+θeω+θT−1ω+θ−1−1nω+θ+Tn=0.The first partial derivative ofπp,T w.r.t. T is(32)∂πp,T∂T=pdp2nωω+θ−1+ωdpω+θeω+θTωT+θT−1+1ω+θT21+1nω+θ−ωdpeω+θTnω+θ+sηdpω+θT21+1nω+θ+dpω+θeω+θTωT+θT−1+1T21+1nω+θ−dpnω+θω+θeω+θT+dp2n1−ωω+θ−ωdpω+θeω+θTωT+θT−1+1ω+θT21+1nω+θ+ωdpnω+θeω+θT+KT2−hdp2nω+θ+dpω+θeω+θTωT+θT−1+1ω+θT21+1nω+θ−dpnω+θeω+θT+h1−dp2ω+θ+2dpT3nω+θ−dp2nω+θ2+dpω+θωTeω+θT+θTeω+θT−eω+θT+1ω+θ2T21+1nω+θ−dpnω+θeω+θT−1ω+θ+h2−2dpT3ω+θ+3dpT24nω+θ−2dpT3nω+θ2+dpω+θ−θ2T2+2ωθT2−2θTeω+θT−2ωTeω+θT+2eω+θT+ω2T2−2ω+θ3T21+1nω+θ−dpnω+θ2eω+θT−2Tω+θ−2ω+θ2−cdpω+θT21+1nω+θ+dpω+θeω+θTωT+θT−1T21+1nω+θ−dpnω+θω+θeω+θT−cddpω+θT21+1nω+θ+dpω+θeω+θTωT+θT−1+1T21+1nω+θ−dpnω+θω+θeω+θT+dp2n1−ωω+θ−ωdpω+θeω+θTωT+θT−1+1ω+θT21+1nω+θ+ωdpnω+θeω+θT=0.Example 5.
Using the information from Example1, now the price-dependent factor of the demand for the fresh produce follows a logarithmic demand function: dp=95−21lnp (i.e., a=95 and b=21).
The replenishment cycle time is subjected toT≤1 week, as the shelf-life is n=1 week. For the price logarithmic demand, the upper bound for price is pmax=ea/b=e95/21=92.18612. Therefore, the price is subjected to be into the interval 5≤p≤92.18612.
By executing the algorithm, the optimal solution is computed:p∗=39.15353 euros per unit, T∗=0.8729460 weeks, Q∗=10.65368 units, and π∗p∗,T∗=116.4864 euros per week. This solution satisfies all conditions for the optimality: ∂2πp,T/∂p2=−0.4098830<0, ∂2πp,T/∂T2=−1389.405<0, ∂2πp,T/∂p∂T=∂2πp,T/∂T∂p=0.1036674, and ∂2πp,T/∂p2∂2πp,T/∂T2−∂2πp,T/∂p∂T∂2πp,T/∂T∂p=569.4828>0. The Hessian determinant is greater than zero, and the total profit function is strictly concave with a negative-definite Hessian matrix. Thus, the solution is optimal.
By making some diagrams for the total profit per unit of time functionπp,T, with different values of p between 20 and 60, considering also T as fixed, and different values of T between 0 and 1 when p is given, πp,T is strictly concave with respect to p when T is fixed (see Figure 6(a)) and with respect to T when p is given (see Figure 6(b)). Besides, πp,T is strictly concave w.r.t. both decision variables: p and T (see Figure 6(c)). This demonstrates that total profit πp,T has the concavity property. So, the solution is a global maximum.Figure 6
Concave property of the profit functionπp,T when the price-dependent demand function is logarithmic.
(a)(b)(c)


### 5.6. Optimal Inventory Policy Using the Price-Dependent Polynomial Demand Functiondp=a−bpm

The partial derivative ofπp,T with respect to p is(33)∂πp,T∂p=−mbpm−1p+dp1−T2n+ωω+θ−1+T2n−1nω+θ+ωTω+θeω+θT−1ω+θ1+1nω+θ−ωTnω+θTeω+θT−1ω+θ+sη−mbpm−11Tω+θ−1−1nω+θ+eω+θTTω+θ1+1nω+θ−Tn−1+T2n+ωω+θ1−T2n+1nω+θ+ωTω+θeω+θT−1ω+θ−1−1nω+θ+Tn−h−mbpm−11ω+θ−1+T2n−1nω+θ+1Tω+θeω+θT−1ω+θ1+1nω+θ−Tn+h1−mbpm−1Tω+θT3n−12−12nω+θ+1Tω+θeω+θT−Tω+θ−1ω+θ21+1nω+θ−Tn+h2−mbpm−1T2ω+θT4n−13−13nω+θ+1Tω+θ2eω+θT−Tω+θTω+θ+2−2ω+θ31+1nω+θ−Tn+c−mbpm−11Tω+θ1+1nω+θ+eω+θTTω+θ−1−1nω+θ+Tn−cd−mbpm−11Tω+θ−1−1nω+θ+eω+θTTω+θ1+1nω+θ−Tn−1+T2n+ωω+θ1−T2n+1nω+θ+ωTω+θeω+θT−1ω+θ−1−1nω+θ+Tn=0.The partial derivative ofπp,T with respect to T is(34)∂πp,T∂T=pdp2nωω+θ−1+ωdpω+θeω+θTωT+θT−1+1ω+θT21+1nω+θ−ωdpeω+θTnω+θ+sηdpω+θT21+1nω+θ+dpω+θeω+θTωT+θT−1+1T21+1nω+θ−dpnω+θω+θeω+θT+dp2n1−ωω+θ−ωdpω+θeω+θTωT+θT−1+1ω+θT21+1nω+θ+ωdpnω+θeω+θT+KT2−hdp2nω+θ+dpω+θeω+θTωT+θT−1+1ω+θT21+1nω+θ−dpnω+θeω+θT+h1−dp2ω+θ+2dpT3nω+θ−dp2nω+θ2+dpω+θωTeω+θT+θTeω+θT−eω+θT+1ω+θ2T21+1nω+θ−dpnω+θeω+θT−1ω+θ+h2−2dpT3ω+θ+3dpT24nω+θ−2dpT3nω+θ2+dpω+θ−θ2T2+2ωθT2−2θTeω+θT−2ωTeω+θT+2eω+θT+ω2T2−2ω+θ3T21+1nω+θ−dpnω+θ2eω+θT−2Tω+θ−2ω+θ2−cdpω+θT21+1nω+θ+dpω+θeω+θTωT+θT−1T21+1nω+θ−dpnω+θω+θeω+θT−cddpω+θT21+1nω+θ+dpω+θeω+θTωT+θT−1+1T21+1nω+θ−dpnω+θω+θeω+θT+dp2n1−ωω+θ−ωdpω+θeω+θTωT+θT−1+1ω+θT21+1nω+θ+ωdpnω+θeω+θT=0.Example 6.
Utilizing the information given in Example1 with the price-dependent portion of the demand for the fresh produce following a polynomial demand function: dp=4000−2p3 (i.e., a=4000, b=2, and m=3).
The replenishment cycle time is subjected toT≤1 week because n=1 week. For the price polynomial demand, the upper bound for price is pmax=a/b1/m=4000/21/3=12.59921. Therefore, the price is limited to the interval 5≤p≤12.59921.
By carrying out the algorithm, the optimal solution isp∗=9.475246 euros per unit, T∗=0.2256880 weeks, Q∗=488.7249 units, and π∗p∗,T∗=8083.700 euros per week. This solution satisfies all conditions for the optimality:∂2πp,T/∂p2=−1464.246<0, ∂2πp,T/∂T2=−3018587<0, ∂2πp,T/∂p∂T=∂2πp,T/∂T∂p=439.8788, and ∂2πp,T/∂p2∂2πp,T/∂T2−∂2πp,T/∂p∂T∂2πp,T/∂T∂p=0.4419761×1010>0. The Hessian determinant is greater than zero, and the total profit function is strictly concave with a negative-definite Hessian matrix, which means the solution is optimal.
By carrying out some graphs for the total profit per unit of time functionπp,T, with different values of p between 6 and 12, considering T as fixed, and different values of T between 0 and 1, when p is given, πp,T is strictly concave with respect to p when T is fixed (see Figure 7(a)) and with respect to T when p is given (see Figure 7(b)). Besides, πp,T is strictly concave w.r.t. both decision variables: p and T (see Figure 7(c)). This validates both total profit πp,T has the concavity property, and the solution is a global maximum.Figure 7
Concave property of the profit functionπp,T when the price-dependent demand function is polynomial.
(a)(b)(c)


## 5.1. Optimal Inventory Policy Using the Price-Dependent Linear Demand Functiondp=a−bp

The first partial derivative ofπp,T with respect to p is(23)∂πp,T∂p=−bp+dp1−T2n+ωω+θ−1+T2n−1nω+θ+ωTω+θeω+θT−1ω+θ1+1nω+θ−ωTnω+θTeω+θT−1ω+θ+sη−b1Tω+θ−1−1nω+θ+eω+θTTω+θ1+1nω+θ−Tn−1+T2n+ωω+θ1−T2n+1nω+θ+ωTω+θeω+θT−1ω+θ−1−1nω+θ+Tn−−hb1ω+θ−1+T2n−1nω+θ+1Tω+θeω+θT−1ω+θ1+1nω+θ−Tn+−h1bTω+θT3n−12−12nω+θ+1Tω+θeω+θT−Tω+θ−1ω+θ21+1nω+θ−Tn+−h2bT2ω+θT4n−13−13nω+θ+1Tω+θ2eω+θT−Tω+θTω+θ+2−2ω+θ31+1nω+θ−Tn+−cb1Tω+θ1+1nω+θ+eω+θTTω+θ−1−1nω+θ+Tn−cd−b1Tω+θ−1−1nω+θ+eω+θTTω+θ1+1nω+θ−Tn−1+T2n+ωω+θ1−T2n+1nω+θ+ωTω+θeω+θT−1ω+θ−1−1nω+θ+Tn=0.The first partial derivative ofπp,T with respect to T is(24)∂πp,T∂T=pdp2nωω+θ−1+ωdpω+θeω+θTωT+θT−1+1ω+θT21+1nω+θ−ωdpeω+θTnω+θ+sηdpω+θT21+1nω+θ+dpω+θeω+θTωT+θT−1+1T21+1nω+θ−dpnω+θω+θeω+θT+dp2n1−ωω+θ−ωdpω+θeω+θTωT+θT−1+1ω+θT21+1nω+θ+ωdpnω+θeω+θT+KT2−hdp2nω+θ+dpω+θeω+θTωT+θT−1+1ω+θT21+1nω+θ−dpnω+θeω+θT+h1−dp2ω+θ+2dpT3nω+θ−dp2nω+θ2+dpω+θωTeω+θT+θTeω+θT−eω+θT+1ω+θ2T21+1nω+θ−dpnω+θeω+θT−1ω+θ+h2−2dpT3ω+θ+3dpT24nω+θ−2dpT3nω+θ2+dpω+θ−θ2T2+2ωθT2−2θTeω+θT−2ωTeω+θT+2eω+θT+ω2T2−2ω+θ3T21+1nω+θ−dpnω+θ2eω+θT−2Tω+θ−2ω+θ2−cdpω+θT21+1nω+θ+dpω+θeω+θTωT+θT−1T21+1nω+θ−dpnω+θω+θeω+θT−cddpω+θT21+1nω+θ+dpω+θeω+θTωT+θT−1+1T21+1nω+θ−dpnω+θω+θeω+θT+dp2n1−ωω+θ−ωdpω+θeω+θTωT+θT−1+1ω+θT21+1nω+θ+ωdpnω+θeω+θT=0.Example 1.
In order to represent a real scenario, let us consider a place that sells a fresh item. Assume that the price-dependent component of the demand for the fresh item is linear as follows:dp=600−20p (i.e., a=600 and b=20). The shelf-life of the fresh item is n=1 week. There is a maximum shelf space of W=500 units. The cost for placing an order to the supplier is K=250 euros per order, the purchase cost is c=5 euros per unit, and the salvage value of the deteriorated item is s=4 euros per unit. The sensitivity coefficient to the level of stock is ω=0.5, and the stock deterioration rate is θ=0.05. The aforementioned data were taken from Agi and Soni [22]. Additional data are still needed to solve the numerical examples. The values of the holding cost are h=1.75 euros per unit per week, h1=0.15 euros per unit per week2, and h2=0.25 euros per unit per week3. The deterioration cost of the item is cd=2 euros per unit, and the salvage coefficient is η=0.8. Since n=1 week, the replenishment cycle time must be T≤1 week. For the price linear demand, the upper bound for price is pmax=a/b=600/20=30. This means that the price must satisfy 5≤p≤30.
By applying the proposed algorithm, the following optimal solution for the inventory system is calculated:p∗=17.69124 euros per unit, T∗=0.4395923 weeks, Q∗=94.42941 units, and π∗p∗,T∗=2049.903 euros per week. This solution satisfies all conditions for the optimality: ∂2πp,T/∂p2=−34.56784<0, ∂2πp,T/∂T2=−49710.46<0, ∂2πp,T/∂p∂T=∂2πp,T/∂T∂p=13.01441, and ∂2πp,T/∂p2∂2πp,T/∂T2−∂2πp,T/∂p∂T∂2πp,T/∂T∂p=0.1718214×107>0. As the Hessian determinant is greater than zero, the total profit function then is strictly concave with a negative-definite Hessian matrix. Consequently, the solution is optimal.
By plotting the total profit per unit of time functionπp,T with distinct values of p (taking values between 7 and 27) given T and distinct values of T (taking values between 0 and 1) given p, it is observed that πp,T is strictly concave with respect to p when T is fixed (see Figure 2(a)) and with respect to T when p is given (see Figure 2(b)). Additionally, πp,T is also strictly concave with respect to both p and T (see Figure 2(c)). Thus, it is ensured that the solution corresponds a global maximum.Figure 2
Concave property of the profit functionπp,T when the price-dependent demand function is linear.
(a)(b)(c)


## 5.2. Optimal Inventory Policy Using the Price-Dependent Isoelastic Demand Functiondp=ap−b

The first partial derivative ofπp,T with respect to p is(25)∂πp,T∂p=−abpb+1p+dp1−T2n+ωω+θ−1+T2n−1nω+θ+ωTω+θeω+θT−1ω+θ1+1nω+θ−ωTnω+θTeω+θT−1ω+θ+sη−abpb+11Tω+θ−1−1nω+θ+eω+θTTω+θ1+1nω+θ−Tn−1+T2n+ωω+θ1−T2n+1nω+θ+ωTω+θeω+θT−1ω+θ−1−1nω+θ+Tn−h−abpb+11ω+θ−1+T2n−1nω+θ+1Tω+θeω+θT−1ω+θ1+1nω+θ−Tn+h1−abpb+1Tω+θT3n−12−12nω+θ+1Tω+θeω+θT−Tω+θ−1ω+θ21+1nω+θ−Tn+h2−abpb+1T2ω+θT4n−13−13nω+θ+1Tω+θ2eω+θT−Tω+θTω+θ+2−2ω+θ31+1nω+θ−Tn+c−abpb+11Tω+θ1+1nω+θ+eω+θTTω+θ−1−1nω+θ+Tn−cd−abpb+11Tω+θ−1−1nω+θ+eω+θTTω+θ1+1nω+θ−Tn−1+T2n+ωω+θ1−T2n+1nω+θ+ωTω+θeω+θT−1ω+θ−1−1nω+θ+Tn=0.The first partial derivative ofπp,T with respect to T is(26)∂πp,T∂T=pdp2nωω+θ−1+ωdpω+θeω+θTωT+θT−1+1ω+θT21+1nω+θ−ωdpeω+θTnω+θ+sηdpω+θT21+1nω+θ+dpω+θeω+θTωT+θT−1+1T21+1nω+θ−dpnω+θω+θeω+θT+dp2n1−ωω+θ−ωdpω+θeω+θTωT+θT−1+1ω+θT21+1nω+θ+ωdpnω+θeω+θT+KT2−hdp2nω+θ+dpω+θeω+θTωT+θT−1+1ω+θT21+1nω+θ−dpnω+θeω+θT+h1−dp2ω+θ+2dpT3nω+θ−dp2nω+θ2+dpω+θωTeω+θT+θTeω+θT−eω+θT+1ω+θ2T21+1nω+θ−dpnω+θeω+θT−1ω+θ+h2−2dpT3ω+θ+3dpT24nω+θ−2dpT3nω+θ2+dpω+θ−θ2T2+2ωθT2−2θTeω+θT−2ωTeω+θT+2eω+θT+ω2T2−2ω+θ3T21+1nω+θ−dpnω+θ2eω+θT−2Tω+θ−2ω+θ2−cdpω+θT21+1nω+θ+dpω+θeω+θTωT+θT−1T21+1nω+θ−dpnω+θω+θeω+θT−cddpω+θT21+1nω+θ+dpω+θeω+θTωT+θT−1+1T21+1nω+θ−dpnω+θω+θeω+θT+dp2n1−ωω+θ−ωdpω+θeω+θTωT+θT−1+1ω+θT21+1nω+θ+ωdpnω+θeω+θT=0.Example 2.
Let us consider the same data as in Example1. Suppose now that the price-dependent element of demand for the perishable good has an isoelastic function as follows: dp=30,000p−1.4 (i.e., a=30,000 and b=1.4).
Sincen=1 week, the replenishment cycle time must be T≤1 week. For the price isoelastic demand, the upper bound for price is pmax=∞. Therefore, the price must be in the interval 5≤p≤∞. By using the algorithm proposed, the following optimal solution for the inventory model is determined: p∗=18.47849 euros per unit, T∗=0.3096932 weeks, Q∗=143.5169 units, and π∗p∗,T∗=5266.004 euros per week. This solution satisfies all conditions for the optimality: ∂2πp,T/∂p2=−9.960414<0, ∂2πp,T/∂T2=−271515.0<0, ∂2πp,T/∂p∂T=∂2πp,T/∂T∂p=28.89635, and ∂2πp,T/∂p2∂2πp,T/∂T2−∂2πp,T/∂p∂T∂2πp,T/∂T∂p=0.2703567×107>0 Given that the Hessian determinant is greater than zero, the total profit function is strictly concave with a negative-definite Hessian matrix. So, the solution is optimal.
By drawing the total profit per unit of time functionπp,T, with some values of p between 8 and 90, and considering T as given and some values of T between 0 and 1 when p is fixed, it is clear that πp,T is strictly concave with respect to p when T is given (see Figure 3(a)) and with respect to T when p is fixed (see Figure 3(b)). Moreover, πp,T is also strictly concave with respect to both decision variables: p and T (see Figure 3(c)). This confirms the concavity property in the total profit πp,T, and the solution corresponds to a global maximum.Figure 3
Concave property of the profit functionπp,T when the price-dependent demand function is isoelastic.
(a)(b)(c)


## 5.3. Optimal Inventory Policy Using the Price-Dependent Exponential Demand Functiondp=ae−bp

The first partial derivative ofπp,T with respect to p is(27)∂πp,T∂p=−abebpp+dp1−T2n+ωω+θ−1+T2n−1nω+θ+ωTω+θeω+θT−1ω+θ1+1nω+θ−ωTnω+θTeω+θT−1ω+θ+sη−abebp1Tω+θ−1−1nω+θ+eω+θTTω+θ1+1nω+θ−Tn−1+T2n+ωω+θ1−T2n+1nω+θ+ωTω+θeω+θT−1ω+θ−1−1nω+θ+Tn−h−abebp1ω+θ−1+T2n−1nω+θ+1Tω+θeω+θT−1ω+θ1+1nω+θ−Tn+h1−abebpTω+θT3n−12−12nω+θ+1Tω+θeω+θT−Tω+θ−1ω+θ21+1nω+θ−Tn+h2−abebpT2ω+θT4n−13−13nω+θ+1Tω+θ2eω+θT−Tω+θTω+θ+2−2ω+θ31+1nω+θ−Tn+c−abebp1Tω+θ1+1nω+θ+eω+θTTω+θ−1−1nω+θ+Tn−cd−abebp1Tω+θ−1−1nω+θ+eω+θTTω+θ1+1nω+θ−Tn−1+T2n+ωω+θ1−T2n+1nω+θ+ωTω+θeω+θT−1ω+θ−1−1nω+θ+Tn=0.The first partial derivative ofπp,T with respect to T is(28)∂πp,T∂T=pdp2nωω+θ−1+ωdpω+θeω+θTωT+θT−1+1ω+θT21+1nω+θ−ωdpeω+θTnω+θ+sηdpω+θT21+1nω+θ+dpω+θeω+θTωT+θT−1+1T21+1nω+θ−dpnω+θω+θeω+θT+dp2n1−ωω+θ−ωdpω+θeω+θTωT+θT−1+1ω+θT21+1nω+θ+ωdpnω+θeω+θT+KT2−hdp2nω+θ+dpω+θeω+θTωT+θT−1+1ω+θT21+1nω+θ−dpnω+θeω+θT+h1−dp2ω+θ+2dpT3nω+θ−dp2nω+θ2+dpω+θωTeω+θT+θTeω+θT−eω+θT+1ω+θ2T21+1nω+θ−dpnω+θeω+θT−1ω+θ+h2−2dpT3ω+θ+3dpT24nω+θ−2dpT3nω+θ2+dpω+θ−θ2T2+2ωθT2−2θTeω+θT−2ωTeω+θT+2eω+θT+ω2T2−2ω+θ3T21+1nω+θ−dpnω+θ2eω+θT−2Tω+θ−2ω+θ2−cdpω+θT21+1nω+θ+dpω+θeω+θTωT+θT−1T21+1nω+θ−dpnω+θω+θeω+θT−cddpω+θT21+1nω+θ+dpω+θeω+θTωT+θT−1+1T21+1nω+θ−dpnω+θω+θeω+θT+dp2n1−ωω+θ−ωdpω+θeω+θTωT+θT−1+1ω+θT21+1nω+θ−dpnω+θω+θeω+θT+dp2n1−ωω+θ−ωdpω+θeω+θTωT+θT−1+1ω+θT21+1nω+θ+ωdpnω+θeω+θT=0.Example 3.
Using the data of Example1 with the price-dependent part of the demand modelled with an exponential function, it follows dp=2000e−0.2p (i.e., a=2000 and b=0.2). The replenishment cycle time must satisfy T≤1 week because n=1 week. For the price exponential demand, the upper bound for price is pmax=∞. As a result, the price must be into the interval 5≤p≤∞.
By employing the algorithm, the optimal solution is found:p∗=10.50583 euros per unit, T∗=0.6187657 weeks, Q∗=121.3688 units, and π∗p∗,T∗=564.3379 euros per week. This solution satisfies all conditions for the optimality: ∂2πp,T/∂p2=−38.73461<0, ∂2πp,T/∂T2=−17613.81<0, ∂2πp,T/∂p∂T=∂2πp,T/∂T∂p=23.85985, and ∂2πp,T/∂p2∂2πp,T/∂T2−∂2πp,T/∂p∂T∂2πp,T/∂T∂p=681694.7>0. The Hessian determinant is greater than zero, with which the total profit function is strictly concave with a negative-definite Hessian matrix. Consequently, the solution is optimal.
By drawing the total profit per unit of time functionπp,T with different values of p between 7 and 20, and considering T as fixed and different values of T between 0 and 1 when p is given, πp,T is strictly concave with respect to p when T is fixed (see Figure 4(a)) and with respect to T when p is given (see Figure 4(b)). Besides, πp,T is also strictly concave with respect to both decision variables: p and T (see Figure 4(c)). This ratifies that total profit πp,T has the concavity property, and the solution is a global maximum.Figure 4
Concave property of the profit functionπp,T when the price-dependent demand function is exponential.
(a)(b)(c)


## 5.4. Optimal Inventory Policy Using the Price-Dependent Logit Demand Functiondp=a/1+ebp

The first partial derivative ofπp,T w.r.t. p is(29)∂πp,T∂p=−abebp1+ebp2p+dp1−T2n+ωω+θ−1+T2n−1nω+θ+ωTω+θeω+θT−1ω+θ1+1nω+θ−ωTnω+θTeω+θT−1ω+θ+sη−abebp1+ebp21Tω+θ−1−1nω+θ+eω+θTTω+θ1+1nω+θ−Tn−1+T2n+ωω+θ1−T2n+1nω+θ+ωTω+θeω+θT−1ω+θ−1−1nω+θ+Tn−h−abebp1+ebp21ω+θ−1+T2n−1nω+θ+1Tω+θeω+θT−1ω+θ1+1nω+θ−Tn+h1−abebp1+ebp2Tω+θT3n−12−12nω+θ+1Tω+θeω+θT−Tω+θ−1ω+θ21+1nω+θ−Tn+h2−abebp1+ebp2T2ω+θT4n−13−13nω+θ+1Tω+θ2eω+θT−Tω+θTω+θ+2−2ω+θ31+1nω+θ−Tn+c−abebp1+ebp21Tω+θ1+1nω+θ+eω+θTTω+θ−1−1nω+θ+Tn−cd−abebp1+ebp21Tω+θ−1−1nω+θ+eω+θTTω+θ1+1nω+θ−Tn−1+T2n+ωω+θ1−T2n+1nω+θ+ωTω+θeω+θT−1ω+θ−1−1nω+θ+Tn=0.The first partial derivative ofπp,T w.r.t. T is(30)∂πp,T∂T=pdp2nωω+θ−1+ωdpω+θeω+θTωT+θT−1+1ω+θT21+1nω+θ−ωdpeω+θTnω+θ+sηdpω+θT21+1nω+θ+dpω+θeω+θTωT+θT−1+1T21+1nω+θ−dpnω+θω+θeω+θT+dp2n1−ωω+θ−ωdpω+θeω+θTωT+θT−1+1ω+θT21+1nω+θ+ωdpnω+θeω+θT+KT2−hdp2nω+θ+dpω+θeω+θTωT+θT−1+1ω+θT21+1nω+θ−dpnω+θeω+θT+h1−dp2ω+θ+2dpT3nω+θ−dp2nω+θ2+dpω+θωTeω+θT+θTeω+θT−eω+θT+1ω+θ2T21+1nω+θ−dpnω+θeω+θT−1ω+θ+h2−2dpT3ω+θ+3dpT24nω+θ−2dpT3nω+θ2+dpω+θ−θ2T2+2ωθT2−2θTeω+θT−2ωTeω+θT+2eω+θT+ω2T2−2ω+θ3T21+1nω+θ−dpnω+θ2eω+θT−2Tω+θ−2ω+θ2−cdpω+θT21+1nω+θ+dpω+θeω+θTωT+θT−1T21+1nω+θ−dpnω+θω+θeω+θT−cddpω+θT21+1nω+θ+dpω+θeω+θTωT+θT−1+1T21+1nω+θ−dpnω+θω+θeω+θT+dp2n1−ωω+θ−ωdpω+θeω+θTωT+θT−1+1ω+θT21+1nω+θ+ωdpnω+θeω+θT=0.Example 4.
Consider the input parameters of Example1 with the price-dependent term of the demand for the fresh article following a logit demand function: dp=9000/1+e0.3p (i.e., a=9000 and b=0.3).
The replenishment cycle time is constrained toT≤1 week given that the shelf-life is n=1 week. For the price logit demand, the upper bound for price is pmax=∞. Therefore, the price is constrained by the interval 5≤p≤∞.
By running the algorithm, the optimal solution is obtained:p∗=8.963560 euros per unit, T∗=0.4682024 weeks, Q∗=231.1214 units, and π∗p∗,T∗=1205.467 euros per week. This solution satisfies all conditions for the optimality: ∂2πp,T/∂p2=−146.5883<0, ∂2πp,T/∂T2=−87888.64<0, ∂2πp,T/∂p∂T=∂2πp,T/∂T∂p=100.7604, and ∂2πp,T/∂p2∂2πp,T/∂T2−∂2πp,T/∂p∂T∂2πp,T/∂T∂p=0.128733×108>0. The Hessian determinant is greater than zero, and the total profit function is strictly concave with a negative-definite Hessian matrix. Thus, the solution is optimal.
By doing some graphs for the total profit per unit of time functionπp,T, with different values of p between 6 and 18, considering also T as fixed, and different values of T between 0 and 1 when p is given, πp,T is strictly concave with respect to p when T is fixed (see Figure 5(a)) and with respect to T when p is given (see Figure 5(b)). Besides, πp,T is strictly concave with respect to both decision variables p and T (see Figure 5(c)). This proves that total profit πp,T has the concavity property, so the solution is a global maximum.Figure 5
Concave property of the profit functionπp,T when the price-dependent demand function is logit.
(a)(b)(c)


## 5.5. Optimal Inventory Policy Using the Price-Dependent Logarithmic Demand Functiondp=a−blnp

The first partial derivative ofπp,T w.r.t. p is(31)∂πp,T∂p=−bpp+dp1−T2n+ωω+θ−1+T2n−1nω+θ+ωTω+θeω+θT−1ω+θ1+1nω+θ−ωTnω+θTeω+θT−1ω+θ+sη−bp1Tω+θ−1−1nω+θ+eω+θTTω+θ1+1nω+θ−Tn−1+T2n+ωω+θ1−T2n+1nω+θ+ωTω+θeω+θT−1ω+θ−1−1nω+θ+Tn−h−bp1ω+θ−1+T2n−1nω+θ+1Tω+θeω+θT−1ω+θ1+1nω+θ−Tn+h1−bpTω+θT3n−12−12nω+θ+1Tω+θeω+θT−Tω+θ−1ω+θ21+1nω+θ−Tn+h2−bpT2ω+θT4n−13−13nω+θ+1Tω+θ2eω+θT−Tω+θTω+θ+2−2ω+θ31+1nω+θ−Tn+c−bp1Tω+θ1+1nω+θ+eω+θTTω+θ−1−1nω+θ+Tn−cd−bp1Tω+θ−1−1nω+θ+eω+θTTω+θ1+1nω+θ−Tn−1+T2n+ωω+θ1−T2n+1nω+θ+ωTω+θeω+θT−1ω+θ−1−1nω+θ+Tn=0.The first partial derivative ofπp,T w.r.t. T is(32)∂πp,T∂T=pdp2nωω+θ−1+ωdpω+θeω+θTωT+θT−1+1ω+θT21+1nω+θ−ωdpeω+θTnω+θ+sηdpω+θT21+1nω+θ+dpω+θeω+θTωT+θT−1+1T21+1nω+θ−dpnω+θω+θeω+θT+dp2n1−ωω+θ−ωdpω+θeω+θTωT+θT−1+1ω+θT21+1nω+θ+ωdpnω+θeω+θT+KT2−hdp2nω+θ+dpω+θeω+θTωT+θT−1+1ω+θT21+1nω+θ−dpnω+θeω+θT+h1−dp2ω+θ+2dpT3nω+θ−dp2nω+θ2+dpω+θωTeω+θT+θTeω+θT−eω+θT+1ω+θ2T21+1nω+θ−dpnω+θeω+θT−1ω+θ+h2−2dpT3ω+θ+3dpT24nω+θ−2dpT3nω+θ2+dpω+θ−θ2T2+2ωθT2−2θTeω+θT−2ωTeω+θT+2eω+θT+ω2T2−2ω+θ3T21+1nω+θ−dpnω+θ2eω+θT−2Tω+θ−2ω+θ2−cdpω+θT21+1nω+θ+dpω+θeω+θTωT+θT−1T21+1nω+θ−dpnω+θω+θeω+θT−cddpω+θT21+1nω+θ+dpω+θeω+θTωT+θT−1+1T21+1nω+θ−dpnω+θω+θeω+θT+dp2n1−ωω+θ−ωdpω+θeω+θTωT+θT−1+1ω+θT21+1nω+θ+ωdpnω+θeω+θT=0.Example 5.
Using the information from Example1, now the price-dependent factor of the demand for the fresh produce follows a logarithmic demand function: dp=95−21lnp (i.e., a=95 and b=21).
The replenishment cycle time is subjected toT≤1 week, as the shelf-life is n=1 week. For the price logarithmic demand, the upper bound for price is pmax=ea/b=e95/21=92.18612. Therefore, the price is subjected to be into the interval 5≤p≤92.18612.
By executing the algorithm, the optimal solution is computed:p∗=39.15353 euros per unit, T∗=0.8729460 weeks, Q∗=10.65368 units, and π∗p∗,T∗=116.4864 euros per week. This solution satisfies all conditions for the optimality: ∂2πp,T/∂p2=−0.4098830<0, ∂2πp,T/∂T2=−1389.405<0, ∂2πp,T/∂p∂T=∂2πp,T/∂T∂p=0.1036674, and ∂2πp,T/∂p2∂2πp,T/∂T2−∂2πp,T/∂p∂T∂2πp,T/∂T∂p=569.4828>0. The Hessian determinant is greater than zero, and the total profit function is strictly concave with a negative-definite Hessian matrix. Thus, the solution is optimal.
By making some diagrams for the total profit per unit of time functionπp,T, with different values of p between 20 and 60, considering also T as fixed, and different values of T between 0 and 1 when p is given, πp,T is strictly concave with respect to p when T is fixed (see Figure 6(a)) and with respect to T when p is given (see Figure 6(b)). Besides, πp,T is strictly concave w.r.t. both decision variables: p and T (see Figure 6(c)). This demonstrates that total profit πp,T has the concavity property. So, the solution is a global maximum.Figure 6
Concave property of the profit functionπp,T when the price-dependent demand function is logarithmic.
(a)(b)(c)


## 5.6. Optimal Inventory Policy Using the Price-Dependent Polynomial Demand Functiondp=a−bpm

The partial derivative ofπp,T with respect to p is(33)∂πp,T∂p=−mbpm−1p+dp1−T2n+ωω+θ−1+T2n−1nω+θ+ωTω+θeω+θT−1ω+θ1+1nω+θ−ωTnω+θTeω+θT−1ω+θ+sη−mbpm−11Tω+θ−1−1nω+θ+eω+θTTω+θ1+1nω+θ−Tn−1+T2n+ωω+θ1−T2n+1nω+θ+ωTω+θeω+θT−1ω+θ−1−1nω+θ+Tn−h−mbpm−11ω+θ−1+T2n−1nω+θ+1Tω+θeω+θT−1ω+θ1+1nω+θ−Tn+h1−mbpm−1Tω+θT3n−12−12nω+θ+1Tω+θeω+θT−Tω+θ−1ω+θ21+1nω+θ−Tn+h2−mbpm−1T2ω+θT4n−13−13nω+θ+1Tω+θ2eω+θT−Tω+θTω+θ+2−2ω+θ31+1nω+θ−Tn+c−mbpm−11Tω+θ1+1nω+θ+eω+θTTω+θ−1−1nω+θ+Tn−cd−mbpm−11Tω+θ−1−1nω+θ+eω+θTTω+θ1+1nω+θ−Tn−1+T2n+ωω+θ1−T2n+1nω+θ+ωTω+θeω+θT−1ω+θ−1−1nω+θ+Tn=0.The partial derivative ofπp,T with respect to T is(34)∂πp,T∂T=pdp2nωω+θ−1+ωdpω+θeω+θTωT+θT−1+1ω+θT21+1nω+θ−ωdpeω+θTnω+θ+sηdpω+θT21+1nω+θ+dpω+θeω+θTωT+θT−1+1T21+1nω+θ−dpnω+θω+θeω+θT+dp2n1−ωω+θ−ωdpω+θeω+θTωT+θT−1+1ω+θT21+1nω+θ+ωdpnω+θeω+θT+KT2−hdp2nω+θ+dpω+θeω+θTωT+θT−1+1ω+θT21+1nω+θ−dpnω+θeω+θT+h1−dp2ω+θ+2dpT3nω+θ−dp2nω+θ2+dpω+θωTeω+θT+θTeω+θT−eω+θT+1ω+θ2T21+1nω+θ−dpnω+θeω+θT−1ω+θ+h2−2dpT3ω+θ+3dpT24nω+θ−2dpT3nω+θ2+dpω+θ−θ2T2+2ωθT2−2θTeω+θT−2ωTeω+θT+2eω+θT+ω2T2−2ω+θ3T21+1nω+θ−dpnω+θ2eω+θT−2Tω+θ−2ω+θ2−cdpω+θT21+1nω+θ+dpω+θeω+θTωT+θT−1T21+1nω+θ−dpnω+θω+θeω+θT−cddpω+θT21+1nω+θ+dpω+θeω+θTωT+θT−1+1T21+1nω+θ−dpnω+θω+θeω+θT+dp2n1−ωω+θ−ωdpω+θeω+θTωT+θT−1+1ω+θT21+1nω+θ+ωdpnω+θeω+θT=0.Example 6.
Utilizing the information given in Example1 with the price-dependent portion of the demand for the fresh produce following a polynomial demand function: dp=4000−2p3 (i.e., a=4000, b=2, and m=3).
The replenishment cycle time is subjected toT≤1 week because n=1 week. For the price polynomial demand, the upper bound for price is pmax=a/b1/m=4000/21/3=12.59921. Therefore, the price is limited to the interval 5≤p≤12.59921.
By carrying out the algorithm, the optimal solution isp∗=9.475246 euros per unit, T∗=0.2256880 weeks, Q∗=488.7249 units, and π∗p∗,T∗=8083.700 euros per week. This solution satisfies all conditions for the optimality:∂2πp,T/∂p2=−1464.246<0, ∂2πp,T/∂T2=−3018587<0, ∂2πp,T/∂p∂T=∂2πp,T/∂T∂p=439.8788, and ∂2πp,T/∂p2∂2πp,T/∂T2−∂2πp,T/∂p∂T∂2πp,T/∂T∂p=0.4419761×1010>0. The Hessian determinant is greater than zero, and the total profit function is strictly concave with a negative-definite Hessian matrix, which means the solution is optimal.
By carrying out some graphs for the total profit per unit of time functionπp,T, with different values of p between 6 and 12, considering T as fixed, and different values of T between 0 and 1, when p is given, πp,T is strictly concave with respect to p when T is fixed (see Figure 7(a)) and with respect to T when p is given (see Figure 7(b)). Besides, πp,T is strictly concave w.r.t. both decision variables: p and T (see Figure 7(c)). This validates both total profit πp,T has the concavity property, and the solution is a global maximum.Figure 7
Concave property of the profit functionπp,T when the price-dependent demand function is polynomial.
(a)(b)(c)


## 6. Sensitivity Analysis

This section presents a sensitivity analysis that was carried out in order to examine the influence of the inventory model’s input parameters, over the total profitπp,T and decision variables p,T,Q. The sensitivity analysis is performed for each of the six price-dependent demands. Distinct values for each input parameter are used, while the other input data are fixed. The results are shown in Tables 2–7. Some observations and managerial insights are also provided below.Table 2
Sensitivity analysis of the optimal solution for the price-dependent linear demand function.
ParameterpTQProfitn0.517.620650.289550354.613951457.2270.617.635860.321145662.929781627.6490.717.650280.351536270.962421762.8360.817.664190.381190278.83291874.1330.917.67780.410451786.631231968.361117.691240.439592294.429412049.9031.117.704630.4688389102.28872121.7281.217.718070.4983923110.26432185.9281.317.731650.5284379118.40822244.0321.417.745460.5591541126.77152297.1871.517.759560.5907188135.40612346.277K25017.691240.439592294.429412049.90330017.704970.4766712100.67981940.78635017.716960.5101394106.04481839.46540017.727580.5407686110.71591744.32145017.737070.5690932114.82531654.22850017.745620.5955007118.46881568.368ω0.2517.682490.409521984.914831920.0880.3517.68580.420769788.450351970.1980.4517.689360.433041592.338542022.6890.517.691240.439592294.429412049.9030.5517.693170.446435596.62652077.807θ0.0317.683520.439994694.16712052.5150.0417.687370.439794394.298372051.2110.0517.691240.439592294.429412049.9030.0617.69510.439388194.56022048.5930.0717.698980.439182294.690742047.279c417.181220.425619895.763672269.8094.517.436150.432479295.105162158.583517.691240.439592294.429412049.9035.517.946470.446974193.735571943.772618.201860.454642193.022751840.193s3.617.692730.43944494.392032049.2423.817.691990.43951894.410712049.572417.691240.43959294.429412049.9034.217.690490.43966594.44812050.2344.417.689740.43973994.46682050.565h,h1,h21.65, 0.05, 0.1517.680380.44092094.746122054.7291.70, 0.10, 0.2017.685820.44025594.587532052.3141.75, 0.15, 0.2517.691240.43959294.429412049.9031.80, 0.20, 0.3017.696640.43893094.271762047.4961.85, 0.25, 0.3517.702030.43827194.114582045.093cd117.686550.44005294.546332051.9721.517.688890.43982294.487852050.937217.691240.43959294.429412049.9032.517.693580.43936294.371012048.87317.695920.43913294.312662047.837η0.617.694980.43922494.335992048.250.717.693110.43940894.382692049.0760.817.691240.43959294.429412049.9030.917.689360.43977694.476162050.731117.687480.43996094.522932051.558a,bfixed36011.793610.77009469.96747231.056348014.732380.55497885.21757956.045260017.691240.43959294.429412049.90372020.662640.365883100.81733508.11584023.641650.314190105.57545328.862b,afixed562.600450.21741958.9337614329.261032.637710.30469276.550086005.9212017.691240.43959294.429412049.9032514.714250.50248799.051321313.2283012.736330.566856101.7333847.1779Table 3
Sensitivity analysis of the optimal solution for the price-dependent isoelastic demand function.
ParameterpTQProfitn0.518.12070.200602187.688954386.0670.618.195690.223185499.392924638.5650.718.26790.2450845110.66744839.0740.818.338530.2666374121.69345004.3620.918.408490.2881024132.60665144.51118.47850.3096932143.51695266.0041.118.54920.3315992154.51975373.2371.218.621160.3539997165.70285469.3151.318.694940.3770726177.15155556.5091.418.771110.4010032188.95245636.5331.518.850280.4259907201.1975710.711K25018.47850.3096932143.51695266.00430018.556450.3368009153.68765111.34735018.625910.3613822162.57874968.13540018.688670.3839697170.474833.98345018.745980.4049329177.55474707.23450018.798730.4245407183.97264586.683ω0.2518.402960.2805749127.67085051.1890.3518.430830.2912172133.45475133.8260.4518.461760.303162139.9565220.7560.518.47850.3096932143.51695266.0040.5518.496190.3166393147.31025312.546θ0.0318.439950.3101287143.68655270.350.0418.459220.309911143.60185268.1770.0518.47850.3096932143.51695266.0040.0618.497790.3094752143.43195263.8310.0718.517090.3092571143.34675261.658c414.907780.2935993184.78685805.1864.516.694230.3021076161.81045515.016518.47850.3096932143.51695266.0045.520.261060.3165394128.63915049.084622.042240.3227805116.32464857.783s3.618.48610.3095581143.37855264.9533.818.48230.3096256143.44775265.478418.47850.3096932143.51695266.0044.218.47470.3097608143.58635266.5294.418.47090.3098285143.65575267.055h,h1,h21.65, 0.05, 0.1518.425840.3107838144.54445273.3431.70, 0.10, 0.2018.452220.3102363144.02845269.6671.75, 0.15, 0.2518.47850.3096932143.51695266.0041.80, 0.20, 0.3018.50470.3091542143.00995262.3541.85, 0.25, 0.3518.530820.3086195142.50735258.717cd118.454720.3101169143.95155269.2931.518.466620.3099047143.73395267.647218.47850.3096932143.51695266.0042.518.490370.3094822143.30075264.363318.502230.3092719143.08535262.724η0.618.497490.309356143.17145263.3790.718.4880.3095244143.34395264.6910.818.47850.3096932143.51695266.0040.918.4690.3098624143.69045267.318118.459480.3100319143.86445268.634a,bfixed36019.013430.507996969.673331350.35648018.658040.3728992111.09053266.69560018.47850.3096932143.51695266.00472018.364770.2709628171.09397308.27184018.28420.244065195.51069377.999b,afixed531.326210.2405638108.478110666.251022.749590.2742978131.41567419.4312018.47850.3096932143.51695266.0042515.931540.3474883148.26343776.6483014.248020.3882774147.90292718.869Table 4
Sensitivity analysis of the optimal solution for the price-dependent exponential demand function.
ParameterpTQProfitn0.510.310470.428176168.1642205.86620.610.356510.470655580.00251311.09820.710.397870.51036891.08083393.54490.810.436020.5479941101.5864460.55930.910.471810.5840098111.6502516.5487110.505820.6187656121.3688564.33791.110.538460.6525299130.8162605.83241.210.570.6855155140.0512642.37081.310.60070.7178965149.1221674.92521.410.630730.7498184158.069704.22091.510.660230.7814064166.9269730.8121K25010.505820.6187656121.3688564.337930010.538440.6740886127.360748735010.565320.7243232132.0419415.497740010.587410.7705352135.6953348.607345010.605390.8134684138.5174285.480550010.619710.8536671140.6506225.5008ω0.2510.503970.5998278111.1458498.0510.3510.504590.6069651115.0211523.70450.4510.505370.6146843119.1772550.49250.510.505820.6187656121.3688564.33790.5510.50630.622996123.6407578.4992θ0.0310.485140.6194522121.2816567.93720.0410.495470.6191109121.3257566.13890.0510.505820.6187656121.3688564.33790.0610.51620.6184164121.4107562.53430.0710.52660.6180633121.4516560.7281c49.4586120.5636089140.2348785.87654.59.9819140.5905406130.5493668.5296510.505820.6187656121.3688564.33795.511.030240.6483456112.6667471.9582611.555020.6793453104.418390.1792s3.610.50980.6185865121.2488563.5473.810.507810.618676121.3088563.9424410.505820.6187656121.3688564.33794.210.503830.6188552121.4288564.73374.410.501850.6189449121.4888565.1296h,h1,h21.65, 0.05, 0.1510.475530.6210162122.4024570.47791.70, 0.10, 0.2010.49070.6198884121.8838567.40161.75, 0.15, 0.2510.505820.6187656121.3688564.33791.80, 0.20, 0.3010.520890.6176479120.8572561.28671.85, 0.25, 0.3510.535910.6165353120.3493558.2478cd110.493390.6193265121.7445566.81441.510.499610.6190458121.5564565.5753210.505820.6187656121.3688564.33792.510.512030.6184858121.1815563.1023310.518240.6182064120.9946561.8685η0.610.515760.6183181121.0693562.36180.710.510790.6185417121.2189563.34930.810.505820.6187656121.3688564.33790.910.500850.6189897121.5189565.3277110.495880.6192141121.6693566.3185a,bfixed36010.619710.853667170.32532112.750448010.556930.708069797.95558329.076460010.505820.6187656121.3688564.337972010.465930.5567266142.0386811.74284010.434080.5103661160.74231067.793b,afixed515.328650.3701014143.33813165.2931012.08510.4884015136.22421328.0542010.505820.6187656121.3688564.3379259.5880890.7720412102.6994200.5329308.9768810.963388381.8371219.17279Table 5
Sensitivity analysis of the optimal solution for the price-dependent logit demand function.
ParameterpTQProfitn0.58.835360.331024142.0055765.34440.68.8655110.3625493162.432896.27420.78.8928120.3915668181.2635998.01640.88.9179710.4186151198.84121080.0290.98.9414480.4440691215.40261147.97218.963560.4682024231.12141205.4671.18.984540.4912223246.12921254.9531.29.0045630.5132901260.52861298.1411.39.0237670.5345347274.40131336.2711.49.042260.5550609287.81421370.2651.59.0601330.5749555300.82251400.829K2508.963560.4682024231.12141205.4673008.9938960.5132866246.23051103.5813509.0205620.554725258.98511009.9494009.0442240.5932662269.8674922.84034509.0653440.6294328279.2142841.05435009.0842550.663607287.2716763.7174ω0.258.9523280.4436449210.76491111.3620.358.9565610.4528506218.38991147.8060.458.9611380.4628661226.69071185.8230.58.963560.4682024231.12141205.4670.558.9660730.4737715235.75491225.56θ0.038.949140.4699697231.73291213.0750.048.9563530.4690874231.42841209.270.058.963560.4682024231.12141205.4670.068.970760.4673149230.81191201.6650.078.9779540.4664249230.51197.863c47.987320.4095817271.75551780.7994.58.4719510.4376222251.071471.91358.963560.4682024231.12141205.4675.59.461640.501547211.9723976.762569.9656780.5379028193.6657781.43s3.68.9663690.4678248230.79041203.8663.88.9649650.4680135230.95581204.66648.963560.4682024231.12141205.4674.28.9621540.4683915231.28721206.2694.48.9607470.4685807231.45321207.071h,h1,h21.65, 0.05, 0.158.9433140.4718563233.88351217.3211.70, 0.10, 0.208.9534740.4700191232.4941211.3761.75, 0.15, 0.258.963560.4682024231.12141205.4671.80, 0.20, 0.308.9735730.466406229.76561199.5921.85, 0.25, 0.358.9835150.4646297228.42631193.751cd18.9547580.469387232.1611210.4851.58.9591630.4687938231.64021207.97428.963560.4682024231.12141205.4672.58.9679480.4676128230.60461202.96638.9723280.4670249230.08981200.469η0.68.9705770.4672598230.29551201.4670.78.9670710.4677306230.70781203.4650.88.963560.4682024231.12141205.4670.98.9600430.4686754231.53631207.47218.9565210.4691495231.95251209.48a,bfixed3608.9829960.4968576214.10931013.9594808.9729150.4818983222.74411109.266008.963560.4682024231.12141205.4677208.954850.455602239.2631302.5068408.9467150.4439588247.18741400.313b,afixed510.859350.3217571270.41183869.684109.6981030.3895725253.82812170.566208.963560.4682024231.12141205.467258.4741520.5625861204.7163636.0077308.1381760.680003176.082295.6301Table 6
Sensitivity analysis of the optimal solution for the price-dependent logarithmic demand function.
ParameterpTQProfitn0.538.887630.54.977036−168.60880.638.944280.66.078415−80.010930.739.00090.77.218579−15.075010.839.055940.76710338.38081735.865790.939.106350.82048219.52320279.0227139.153530.87294610.65368116.48641.139.198490.92489111.78091149.5841.239.241930.97664112.91222179.25191.339.284351.0284714.05402206.17351.439.326111.08061915.21214230.86191.539.367511.13330216.39207253.7117K25039.153530.87294610.65368116.486430039.165970.935185110.8258861.1997835039.170680.990424710.886849.28129840039.17079110.88822−40.7630745039.17079110.88822−90.7630750039.17079110.88822−140.7631ω0.2539.170.88682519.80537482.895890.3539.163280.880534310.1319595.805910.4539.156730.875244810.47529109.41220.539.153530.87294610.65368116.48640.5539.150360.87086410.83679123.7472θ0.0339.130380.872758710.58556116.47720.0439.141930.872852210.61954116.48180.0539.153530.87294610.65368116.48640.0639.165170.873040210.68797116.49090.0739.176860.873134710.72242116.4954c438.253830.863045710.90399128.90434.538.705990.867978710.77778122.6416539.153530.87294610.65368116.48645.539.596640.877948710.53156110.4366640.035490.882987610.41132104.4901s3.639.157910.872968910.65237116.42623.839.155720.872957510.65302116.4563439.153530.87294610.65368116.48644.239.151330.872934610.65433116.51654.439.149140.872923210.65499116.5467h,h1,h21.65, 0.05, 0.1539.117310.872848610.66483116.9841.70, 0.10, 0.2039.135420.872897210.65925116.73511.75, 0.15, 0.2539.153530.87294610.65368116.48641.80, 0.20, 0.3039.171630.872995110.64811116.23791.85, 0.25, 0.3539.189720.873044310.64254115.9895cd139.139810.872874810.65777116.67471.539.146670.872910410.65572116.5806239.153530.87294610.65368116.48642.539.160380.872981710.65163116.3923339.167240.873017410.64958116.2982η0.639.16450.873003110.6504116.33590.739.159010.872974610.65204116.41110.839.153530.87294610.65368116.48640.939.148040.872917510.65532116.5617139.142550.87288910.65695116.637a,bfixed36033.220580.95022510.5243437.5275248036.049360.910540610.6141574.4294960039.153530.87294610.65368116.486472042.561690.837248310.6493164.229184046.305110.803282510.60662218.2455b,afixed5212.45930.46321785.8504052049.14210103.67340.6026437.845838847.53212059.919410.73883229.513484351.49082539.153530.87294610.65368116.48643032.792270.940142610.9796846.17976Table 7
Sensitivity analysis of the optimal solution for the price-dependent polynomial demand function.
ParameterpTQProfitn0.59.4526450.1573383319.15327122.2710.69.4577810.1728178357.22097406.720.79.4625360.1871819392.7217628.5440.89.4669960.2006759426.22517807.9640.99.4712180.21347458.13027957.09619.4752460.225688488.72498083.71.19.4895290.22915965008191.5521.29.5066260.22776615008280.1371.39.5204880.2266085008353.8671.49.5319550.22563025008416.2061.59.5415980.2247945008469.614K2509.4752460.225688488.72498083.73009.5030190.23293335007867.4513509.5383320.23506485007653.7684009.5738490.23726545007442.0454509.6095690.23953835007232.3085009.6454930.24188685007024.585ω0.259.4698760.2061207438.40967840.1590.359.4718510.2132807456.78267934.2930.459.474050.2213019477.41698032.7150.59.4752460.225688488.72498083.70.559.4770040.22999975008135.968θ0.039.4727830.2271689490.89978101.0750.049.4740170.2264268489.81168092.380.059.4752460.225688488.72498083.70.069.4764690.2249528487.63968075.0340.079.4776850.224221486.55598066.381c49.1341910.2089559489.858710340.614.59.3019640.2168661489.66479189.81159.4752460.225688488.72498083.75.59.654220.2356095486.90417025.40469.8390880.2468774484.03936018.295s3.69.4757390.2253742488.03618080.033.89.4754920.225531488.38028081.86449.4752460.225688488.72498083.74.29.4749990.2258454489.07028085.5374.49.4747520.2260029489.41618087.375h,h1,h21.65, 0.05, 0.159.4719410.2280448493.85198108.6431.70, 0.10, 0.209.4736010.2268579491.27058096.1391.75, 0.15, 0.259.4752460.225688488.72498083.71.80, 0.20, 0.309.4768750.2245349486.21458071.3261.85, 0.25, 0.359.4784890.2233981483.73868059.016cd19.4736980.2266757490.89248095.2021.59.4744730.2261806489.80588089.44529.4752460.225688488.72498083.72.59.4760150.2251981487.64968077.96739.4767820.2247108486.57988072.246η0.69.4764760.2249054487.0078074.5330.79.4758620.2252959487.86428079.1130.89.4752460.225688488.72498083.70.99.4746280.2260818489.58928088.29519.4740090.2264773490.45718092.897a,bfixed3608.7925540.2813326427.1284446.0234809.1487220.2496727459.5586171.7026009.4752460.225688488.72498083.77209.7865950.200634550010164.2884010.081290.177287850012395.47b,afixed511.476390.1919163454.510313544.771010.246190.2103775475.190810165.04209.4752460.225688488.72498083.7258.9304350.2392571498.1566637.54308.5206370.24936675005559.827Tables2–7 present the impact of the input parameters on the optimal solution of p, T, Q, and the total profit for the price-dependent linear, isoelastic, exponential, logit, logarithmic, and polynomial demand functions, respectively.Table8 is built from the numerical experimentation shown in Tables 2–7. Table 8 shows the behavior of the variables and the total profit when the inventory model parameters are increased. The following observations and managerial insights are derived from Tables 2–8.(i)
The fresh item’s maximum shelf-lifen and the ordering cost K have the exact same impact over the inventory policy and total profit in all price demand functions. A longer product’s maximum shelf-life n results in a higher optimal price, a longer optimal cycle time, and a larger order quantity. This means that a fresh product with extended life span (it takes longer to lose its attractiveness) allows the marketer to offer it at a higher price. As the ordering cost K increases, the price and inventory cycle time increase as well, but the overall profit tends to be lower. Then, managers should implement actions to lower the ordering costs, so higher profits can be generated.(ii)
Higher values for the parameter of demand sensitivity to the stock level (ω), result in longer inventory cycle time, higher prices, and higher profits. This behavior is observed in five price demand functions except in the logarithmic function, where the optimal price and inventory cycle time tend to decrease as this parameter increases. For all price demand functions, it is found that higher values of the parameter of demand sensitivity to the stock level (ω) result in a larger order quantity. In this case, since ω parameter does not directly depend on the decision maker, the suggestion would be to take high values of this sensitivity parameter to generate high values of profits. This means that managers are more interested in exhibiting larger quantities of stock on shelf space.(iii)
Increasing values for the inventory deterioration rateθ result in significant variations in price demand functions. On the one hand, in linear and exponential price demand functions, slightly lower inventory cycle time, slightly higher prices, and lower profits are observed. On the other hand, isoelastic, logit, and polynomial price demand functions present lower inventory cycles and profits, with slightly higher prices. Therefore, if the fresh item has one of these previous five price-dependent demands, the suggestion for the buyer would be to buy products whose rate of deterioration is small so that a higher profit is generated. When using the logarithmic price demand function, the inventory deterioration rate and the optimal price are higher, and the inventory cycle time and profit are both optimal.(iv)
As the purchase costc grows, it results in slightly higher price and inventory cycle time. Conversely, the profit is lower as this cost increases. This behavior is presented in all price demand functions. Therefore, the indication would be to develop low-cost purchasing policies, so profits are higher regardless of the function of demand depending on price that is used. One example would be to look for alternative suppliers that offer products at a lower cost without neglecting their quality.(v)
The increment in the salvage value (s) always results in slightly lower optimal price and slightly higher inventory cycle time and profits, particularly in the isoelastic, exponential, logit, and polynomial price demand functions. In linear price demand function, slightly higher price and inventory cycle time are found, and the profit turns out to be higher as this parameter increases. In logarithmic price demand function, a larger value results in a lower optimal price and inventory cycles, but higher profits. If linear price demand function is followed, managers should implement actions to assign higher salvage values in order to increase the profits that are generated.(vi)
Higher coefficients for the holding cost functionh,h1,h2 and deterioration cost cd get a higher result in slightly higher selling price, but a lower profit. This behavior is observed for all price demand functions. On the contrary, the inventory cycle time increases for the logarithmic function and decreases for the rest price demand functions. It is generally advised to decision makers to implement policies to have low deterioration costs in the products in order to increase the overall profits. Likewise, lower values for the components of the holding cost function must be considered.(vii)
As the salvage coefficientη increases, a slightly higher inventory cycle time is observed, but slightly lower selling prices and profits are obtained in linear, isoelastic, exponential, logit, and polynomial price demand functions. In the logarithmic function, a slightly lower values in the inventory cycle time can be observed. Here, the person in charge should consider high salvage coefficients to guarantee higher profits for all cases.(viii)
By increasing the values of the scale parameter of the demanda and having fixed the parameter of the demand sensitivity to the price b, higher prices and lower inventory cycle times are obtained in the linear, logarithmic, and polynomial functions. On the contrary, lower prices and lower inventory cycles are observed as scale parameter increases in the isoelastic, exponential, and logit price demand functions. In all price demand functions, increasing the value of a while keeping the value of b fixed results in larger benefits. The suggestion is, then, that this parameter is maintained with higher values.(ix)
Higher values of the demand sensitivity to priceb while keeping the scale parameter of the demand a fixed together result in lower selling prices and profits, but higher inventory cycle time in all price demand functions. Hence, low values of this parameter b should be used, so the overall profits are always higher regardless of the price demand used.Table 8
Behavior of the variables and the total profit when the parameters are increased.
Price-dependent demand functionParameterpTQProfitLinearnKωθcsh,h1,h2cdηa,bfixedb,afixedIsoelasticnKωθcsh,h1,h2cdηa,bfixedb,afixedExponentialnKωθcsh,h1,h2cdηa,bfixedb,afixedLogitnKωθcsh,h1,h2cdηa,bfixedb,afixedLogarithmicnKωθcsh,h1,h2cdηa,bfixedb,afixedPolynomialnKωθcsh,h1,h2cdηa,bfixedb,afixed


## 7. Conclusions

This research work develops an inventory model with price-, stock-, and time-dependent demand. The physical deterioration and condition of freshness degradation over time are both considered, and zero-ending inventory is assumed. Six different types of price-dependent demand functions are studied: linear, isoelastic, exponential, logit, logarithmic, and polynomial. When working with perishable products, a salvaged value and a deterioration cost are considered in the entire cycle. A nonlinear time-dependent holding cost is included, specifically with a quadratic-type function.Through an algorithm, the inventory model determines the optimal values for price, the inventory cycle time, and the order quantity. Some numerical examples are provided, and a sensitivity analysis is presented for all the input parameters. By observing the behavior of the decision variables and total profits, it was found that an increase in the ordering costK, purchasing cost c, and shelf-life n results in a similar pattern in the selling price, the inventory cycle time, the quantity to order, and the total profit. Furthermore, an increase in the value of the shelf-life n results in an increment in price, inventory cycle time, quantity ordered, and profits generated for all functions. In addition, as the ordering cost increases K, price, the inventory cycle time, and quantity ordered also increase for all functions. Nonetheless, the profits show a decreasing trend. Finally, by escalating the purchasing cost for all functions, there is an increase in both the price and the inventory cycle time; however, the quantity to order and total profits tend to decrease.This research work extends and widely contributes to the state-of-the-art on the inventory field, with focus on perishable items with price-stock-time-dependent demand. The inventory model studied here has some limitations from where several directions for extension and further research are highlighted. First, an inventory model can be built with the same characteristics and demand pattern, but including the sustainability elements, so the effects of the carbon-tax and cap-and-trade mechanisms can be assessed. Second, a model that allows shortages with full or partial backlogging should be explored. Third, the trade-off and benefits of investing on preservation technology should be also studied. Fourth, the noninstantaneous item’s freshness degradation can be integrated into the proposed inventory model. Finally, other components such as incorporating discount policies or advertising efforts can also be investigated.



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*Source: 6630938-2021-04-10.xml*