Economic production quantity (EPQ) inventory model for trended demand has been analyzed with rework facility and stochastic preventive machine time. Due to the complexity of the model, search method is proposed to determine the best optimal solution. A numerical example and sensitivity analysis are carried out to validate the proposed model. From the sensitivity analysis, it is observed that the rate of change of demand has significant impact on the optimal inventory cost. The model is very sensitive to the production and demand rate.
An item that does not satisfy quality standards but can be attained after reprocess is termed as a recoverable item and the process is known as rework. It is observed that in an industrial sector, the rework reduced production cost and maintained quality standard of the item. Schrady [
Meller and Kim [
In this paper, we analyze an economic production quantity (EPQ) model with rework and random preventive maintenance time together when demand is increasing function of time. The consideration of random preventive maintenance time, rework, and trended demand in the model increases its applicability in the electronic and automobile industries. In this production system, produced items are inspected immediately. Defective items are stocked and reworked at the end of the production uptime. We will call these items as recoverable items. Out of these recoverable items, the fraction of the items will be labeled as “new” and rest will be scrapped. Preventive maintenance is performed at the end of the rework process, and the maintenance time is assumed to be random. When demand is increasing, shortages may occur which will be treated as lost sales in this study. It is observed that the rate of change of demand has significant impact on the optimal inventory cost. It is suggested that when demand is trended, preventive maintenance time should be controlled by recruiting wellqualified technicians. The uniform distribution and exponential distribution for preventive maintenance time are explored. The paper is organized as follows: Section
The status of the serviceable inventory is depicted in Figure
Inventory status of serviceable items with lost sales.
From the above description, the inventory level in a production uptime period is governed by the differential equation
Under the assumption of LIFO production system, the inventory level of good items depletes at a constant rate during rework uptime and downtime. The inventory level is governed by
Using
Now, let us analyze the inventory level of recoverable items (Figure
Inventory status of recoverable items.
The inventory level of recoverable items in a production uptime is governed by the differential equation
The inventory level at the beginning of the production downtime is equal to the inventory level at the end of the production uptime; that is,
The optimal production uptime for the EPQ system without lost sales can be obtained by setting
Define the probability distribution function
Calculate (
If
Set
Calculate (
If
If
Define the probability distribution function
Arguing as in (Section
Consider, following parametric values to study the working of the proposed problem. Let
Convexity of total cost.
The sensitivity analysis is carried out by changing each of the parameters by
Sensitivity analysis of
Parameter  Percentage change  Uniform distribution  Exponential distribution 








−40%  0.1118836  2217.023407  0.165363168  3105.82179 
−20%  0.1127598  2396.444313  0.168010377  3225.883231  
0  0.1136351  2574.499976  0.170676999  3344.12915  
20%  0.1145095  2751.208003  0.173362159  3460.599558  
40%  0.1153831  2926.585673  0.176064945  3575.33395  
 

−40%  0.119  0.224  0.873011973  3769.650118 
−20%  0.116  0.220356527  0.280588937  3224.896639  
0  0.1136351  2574.499976  0.170676999  3344.12915  
20%  0.0784111  2671.944386  0.123903512  3421.121046  
40%  0.0598742  2748.060997  0.09758743  3473.297234  
 

−40%  0.1252239  2840.779326  0.179184671  3667.154126 
−20%  0.117738  2670.650277  0.17374104  3462.440106  
0  0.1136351  2574.499976  0.170676999  3344.12915  
20%  0.1110459  2512.591082  0.168712009  3266.965218  
40%  0.1093473  2469.539689  0.167344624  3212.637461  
 

−40%  0.0468611  2285.690117  0.081365729  2483.982152 
−20%  0.0736194  2419.311543  0.118822843  2928.274545  
0  0.1136351  2574.499976  0.170676999  3344.12915  
20%  0.1801641  2784.239588  0.248801609  3743.389279  
40%  0.313401  3147.605249  0.385620567  4166.830717  
 

−40%  0.1133496  2572.975288  0.170020919  3336.335153 
−20%  0.113492  2573.73768  0.170347969  3340.233353  
0  0.1136351  2574.499976  0.170676999  3344.12915  
20%  0.1137789  2575.262185  0.171008033  3348.022553  
40%  0.1139235  2576.024314  0.1713411  3351.913594  
 

−40%  0.1075459  2440.729188  0.165578265  3180.040146 
−20%  0.1105098  2506.778232  0.168074836  3261.520565  
0  0.1136351  2574.499976  0.170676999  3344.12915  
20%  0.1169347  2644.037111  0.173391457  3427.954424  
40%  0.1204229  2715.549502  0.176225505  3513.094599  
 

−40%  0.1131507  2451.633648  0.170247776  3220.242525 
−20%  0.1133924  2513.000758  0.170462207  3282.119979  
0  0.1136351  2574.499976  0.170676999  3344.12915  
20%  0.1138787  2636.131758  0.17089215  3406.270464  
40%  0.1141232  2697.89656  0.171107662  3468.544359  
 

−40%  0.1163864  2035.802579  0.196138784  2497.162768 
−20%  0.114993  2306.766649  0.18157314  2935.554059  
0  0.1136351  2574.499976  0.170676999  3344.12915  
20%  0.1123112  2839.068303  0.162020977  3729.597492  
40%  0.11102  3100.535386  0.15486872  4096.262909  
 

−40%  0.113733  2555.165605  0.171388527  3315.223296 
−20%  0.113684  2564.8349  0.171031253  3329.691775  
0  0.1136351  2574.499976  0.170676999  3344.12915  
20%  0.1135862  2584.160837  0.170325715  3358.53567  
40%  0.1135373  2593.817484  0.169977356  3372.911591  
 

−40%  0.1119752  2570.475444  0.15263831  3147.25954 
−20%  0.1129966  2572.958931  0.162618107  3255.743808  
0  0.1136351  2574.499976  0.170676999  3344.12915  
20%  0.1140719  2575.549497  0.1774544  3418.940914  
40%  0.1143897  2576.310317  0.183312722  3483.929611  
 

−40%  0.113643  2453.561123  0.170776446  3223.952673 
−20%  0.113639  2514.030563  0.170726744  3284.041215  
0  0.1136351  2574.499976  0.170676999  3344.12915  
20%  0.1136311  2634.969362  0.17062721  3404.216467  
40%  0.1136271  2695.438718  0.170577378  3464.303169 
Figures
Sensitivity analysis of production uptime for uniform distribution.
The optimal total cost per unit time is slightly sensitive to changes in
Sensitivity analysis of total cost for uniform distribution.
Sensitivity analysis of production uptime for exponential distribution.
Sensitivity analysis of total cost for exponential distribution.
In this research, rework of imperfect quality and random preventive maintenance time are incorporated in economic production quantity model when demand increases with time. The random preventive maintenance time is distributed uniformly and exponentially. The models are validated by the example. The sensitivity analysis suggests that the optimal total cost per unit time is sensitive to changes in the production rate, the demand rate, and the product defect rate in both the uniform and the exponential distributed preventive maintenance time. To combat increasing demand, the management should adopt the latest machinery which decreases defective production rate, reducing rework, and as a consequence, the machine’s production uptime can be utilized to its utmost. Further research can be carried out to study the effect of deterioration of units.
Serviceable inventory level in a production uptime
Serviceable inventory level in a production downtime
Serviceable inventory level in a rework uptime
Serviceable inventory level from rework uptime
Serviceable inventory level from rework process in rework downtime
Recoverable inventory level in a production uptime
Recoverable inventory level in a rework uptime
Total serviceable inventory in a production uptime
Total serviceable inventory in a production downtime
Total serviceable inventory in a rework uptime
Total serviceable inventory from a rework uptime
Total serviceable inventory from rework process in a rework downtime
Total recoverable inventory level in a production uptime
Total recoverable inventory level in a rework uptime
Production uptime
Production downtime
Rework uptime
Rework downtime
Total production downtime
Production uptime when the total production downtime is equal to the upper bound of uniform distribution parameter
Inventory level of serviceable items at the end of production uptime
Maximum inventory level of recoverable items in a production uptime
Total recoverable inventory
Production rate
Rework process rat
Demand rate;
Product defect rate
Product scrap rate
Production setup cost
Scrap cost
Lost sales cost
Total inventory cost
Cycle time
Serviceable items holding cost
Recoverable items holding cost
Total inventory cost per unit time for lost sales model
Total inventory cost per unit time for without lost sales model
Total inventory cost per unit time for lost sales model with uniform distribution preventive maintenance time
Total inventory cost per unit time for lost sales model with exponential distribution preventive maintenance time.