Data for continuousreview model
1. Introduction
The production planning problem has received much attention, and many sophisticated models and procedures have been developed to deal with this problem. Many other components of production systems have also been taken into account by researchers in so called integrated systems, in order to achieve a more effective control over the system.
In this work, optimal control theory is used to derive the optimal production rate in a manufacturing system presenting the following features: the demand rate during a certain period depends on the demand rate of the previous period (dependent demand), the demand rate depends on the inventory level, items in inventory are subject to deterioration, and the firm can adopt a periodic or a continuous review policy. Also, we are using the fact that the current demand is related to the previous demand in order to integrate the forecasting component into the production planning problem. The forecast of future demand for the products being produced is needed to plan future activities. Forecasting information is an important input in several areas of manufacturing activity. This problem has been considered in the literature. The proposed approach is different from that of other authors which is mainly based on timeseries. In [1], the authors deal with the interaction between forecasting and stock control in the case of nonstationary demand. In [2], the authors assume a distribution for the unknown demand, estimate its parameters and replace the unknown demand parameters by these estimates in the theoretically correct model. In [3], the authors propose an approach to evaluate the impact of interaction between demand forecasting method and stock control policy on the inventory system performances. In [4], the authors present a supply chain management framework based on model predictive control (MPC) and time series forecasting. In [5], the authors consider a datadriven forecasting technique with integrated inventory control for seasonal data and compare it to the traditional HoltWinters algorithm for random demand with a seasonal trend. In [6], the authors assess the empirical stock control performance of intermittent demand estimation procedures. In [7], the authors study two modifications of the normal distribution, both taking nonnegative values only.
Many researchers have investigated the situation where the demand rate is dependent on the level of the onhand inventory. In [8], the authors consider an inventory model under inflation for deteriorating items with stockdependent consumption rate and partial backlogging shortages. In [9], the authors examine an inventory model for deteriorating items under stockdependent demand and twolevel trade credit. The reference [10] deals with a supply chain model for deteriorating items with stockdependent consumption rate and shortages under inflation and permissible delay in payment. In [11], the authors deal with the optimal replenishment policies for noninstantaneous deteriorating items with stockdependent demand. In [12], the authors investigate an inventory model with stock–dependent demand rate and dual storage facility. In [13], the authors develop a two warehouse inventory model for single vendor multiple retailers with price and stock dependent demand. In [14], the authors asses an integrated vendorbuyer model with stockdependent demand. In [15], the authors study an EOQ model for perishable items with stock and price dependent demand rate. In [16], the authors develop the optimal replenishment policy for perishable items with stockdependent selling rate and capacity constraint. In [17], the authors consider an inventory model for Weibull deteriorating items with price dependent demand and timevarying holding cost. In [18], the authors study fuzzy EOQ models for deteriorating items with stock dependent demand and nonlinear holding costs. In [19], the authors approach an extended twowarehouse inventory model for a deteriorating product where the demand rate has been assumed to be a function of the onhand inventory. In [20], the authors investigate a channel who sells a perishable item that is subject to effects of continuous decay and fixed shelf lifetime, facing a price and stocklevel dependent demand rate. In [21], the authors develop a mathematical model to formulate optimal ordering policies for retailer when demand is practically constant and partially dependent on the stock, and the supplier offers progressive credit periods to settle the account. The literature on stockdependent demand rate is abundant. We have reported some of it here but only a comprehensive survey can summarize and classify it efficiently.
In [22], the authors review the most recent literature on deteriorating inventory models, classifying them on the basis of shelflife characteristic and demand variations. In [23], the authors introduce an orderlevel inventory model for a deteriorating item, taking the demand to be dependent on the sale price of the item to determine its optimal selling price and net profit. In [18], the authors formulate an inventory model with imprecise inventory costs for deteriorating items under inflation. Shortages are allowed and the demand rate is taken as a ramp type function of time as well. In [10], the authors model the retailer's cost minimization retail strategy when he confronts with the supplier trade promotion offer of a credit policy under inflationary conditions and inflationinduced demand. In [24], the authors develop two deterministic economic production quantity (EPQ) models for Weibulldistributed deteriorating items with demand rate as a ramp type function of time.
The goal of this chapter is to study the same problem in periodic and continuous review policy context, knowing that the inventory can be reviewed continuously or periodically. In a continuousreview model, the inventory is monitored continually and production/order can be started at any time. In contrast, in periodicreview models, there is a fixed time when the inventory is reviewed and a decision is made whether to produce/order or not.
We assume that the firm has set an inventory goal level, a demand goal rate and a production goal rate, to build the objective function of our model. The inventory goal level is a safety stock that the company wants to keep on hand. The demand goal rate is the amount that the company wishes to sell per unit of time. The production goal rate is the most efficient rate desired by the firm. The objective is to determine the optimal production rate that will keep the inventory level, the demand rate, and the production rate as close as possible to the inventory goal level, the demand goal rate, and the production goal rate, respectively.
Therefore, we deal with a dynamic problem and the solution sought, the optimal production rate, is a function of time. The problem is then represented as an optimal control problem with two state variables, the inventory level and the demand rate, and one control variable, the rate of manufacturing.
The rest of this chapter is organized as follows. In section 2, the notation used is introduced and the dynamics of the system are described for both periodic and continuous review systems. In section 3, the optimal solution is computed for each case. Simulations are conducted in section 4 to verify the results obtained theoretically in section 3. Section 5 summarizes the chapter and outlines some future research directions.
2. Model formulation
2.1. Continuous review integrated production model
Consider a manufacturing firm producing units of an item over some time interval
In (1), the expression
To use a matrix notation, which is more convenient, let
where
Two state equations are used to describe the dynamics of our system. The variations of the inventory level and demand rate are governed by the following state equations
with known initial inventory level
with known initial demand rate
and
The state equations (6)(7) can also be written in matrix form as
where
2.2. Periodic review integrated production model
In the periodic review model, the time interval
Assuming that the initial inventory level is
where
Also, and as mentioned in the introduction and previous paragraph, assuming a dependent demand rate and a stockdependent demand rate with initial value
where
It is assumed that the firm has set for each period
where the shift operator Δ is defined by
Since the target variables satisfy the dynamics (9) and (10), these can be rewritten using the shift operator Δ to get:
It is more convenient to write the model using a matrix notation. To this end, let
The objective function (11) becomes
where
where
Thus we need to determine the production rates
3. Optimal control
3.1. Optimal control of continuous review model
Given the preceding definitions, the optimal control problem is to minimize the objective function (3) subject to the state equation (8):
To use Pontryagin principle, see for example the reference [25], we introduce the Hamiltonian
where
The second condition is the adjoint equation
The next condition is the state equation
Note that, using the control equation (17), the state equation (8) becomes
We propose the following two equivalent solution approaches to solve the optimal control problem
3.1.1. First solution approach
In this approach, we need to solve a Riccati equation as we will see below. To use the backward sweep method of Bryson and Ho [26], we let
where
Differentiating (20) with respect to
Also, using the change of variable (20), the adjoint equation (18) becomes
Equating expressions (21) and (22), we obtain the following Riccati equation
To solve Riccati equation (23), we use a change of variable to reduce it to a pair of linear matrix equations. Let
The Riccati equation (23) becomes
Multiplying this expression from the right by
Now, set
Then, we have
Multiplying this expression from the left by
Equations (27) and (29) now give two sets of linear equations
Call
where
The boundary conditions
We recall that for a given a constant matrix
In the next solution approach, which also leads to a set of homogeneous equations, we will show how the constant term
where
with initial condition
and
3.1.2. Second solution approach
This approach avoids the Riccati equation as we will see. The adjoint equation (18) and the state equation (19) are equivalent to the vectormatrix state equation
Let
where
Expression (35) is a set of 4 firstorder homogeneous differential equations with constant coefficients. It is similar to (31) and it has a solution similar to (33). We give here the explicit solution.
The matrix
Note that the first two components of
To determine
where
To determine
which can be rewritten as
Using the terminal condition
we have
from which, it follows
Finally, the optimal solutions of the problem
where
and
3.2. Optimal control of periodic review model
Here also we assume that the system state is available during each period
Then, the Lagrangian function is given by
The necessary optimality conditions are the control equation
and the adjoint equation
In order to solve these equations, we use the backward sweep method of Bryson and Ho (1975), who treat extensively in their book the problem of optimal control and estimation. They detail two methods for solving the Riccati equation arising in linear optimal control problem, the first one being the transition matrix method and the second being the backward sweep method. Let
The control equation (40) becomes
Solving for
where
Now the adjoint equation (41) becomes
so that
Finally, the matrices
The boundary condition
so that
Also, from the dynamics (15), we have the optimal production rates
where the optimal state vector
Parameters  Values  
Nonmonetary parameters  Length of planning horizon  
Initial inventory level  
Initial demand rate  
Demand rate coefficient  
Inventory level coefficient  
Deterioration rate  
Monetary parameters  Penalty for production rate deviation  
Penalty for inventory level deviation  
Penalty for demand rate deviation  
Inventory salvage value  
Demand salvage value 
4. Simulation results
4.1. Simulation of continuous review model
To illustrate numerically the results obtained, firstly we present some simulations for optimal control of the continuousreview integrated productionforecasting system with stockdependent demand and deteriorating items. The data used in this simulation is presented in Table 1.
Using the MATLAB software, we implemented the results of the previous section and obtained the graphs below. Figures 1, 2 and 3 show the variations of
Using equation (3), the optimal cost is found to be
0.1  5822.40  23426.21  3047.93 
0.2  489.77  7648.73  463.79 
0.3  485.62  5560.00  461.53 
0.4  481.57  740.12  459.30 
0.5  477.58  659.34  457.09 
0.6  473.67  600.85  454.91 
0.7  469.84  556.02  452.76 
0.8  466.08  520.24  450.63 
0.9  462.38  490.83  448.52 
As can be seen, the objective function decreases as any of the three parameters increases.
4.2. Simulation of periodic review model
In this second part of the simulation, we illustrate the results obtained on the optimal control of the periodic review integrated productionforecasting system with stockdependent demand and deteriorating items. Thus, consider the production planning problem for a firm for the next
where the sign function of a real number
We have to note that the goal inventory level and the goal demand rate were constant in the continuous review case.
The goal production rate is then computed using
where we assume that the inventory goal level is constant over a certain range. The penalties for deviating from these targets are



10 

51 

0.1 

0.2 

0.01 

20 

15 

0.01 
For the periodic review case, the simulation results are shown in the graphs below. Figure 4 shows the variations of the optimal inventory level and the inventory goal level. We observe that except for the early transient periods,
Figure 5 shows the variations of the optimal demand rate and the demand goal rate. We observe that except for the early transient periods,
Finally, Figure 6 shows the variations of the optimal production rate and the production goal rate. We again observe that except for the early transient periods,
The optimal cost is found to be





10.3081  10.0807  10.3072 

10.3173  10.3081  10.3064 

10.3339  10.6868  10.3056 

10.3601  11.2166  10.3050 

10.4004  11.8973  10.3044 

10.4630  12.7284  10.3036 

10.5651  13.7098  10.3040 

10.7444  14.8413  10.3033 

11.0880  16.1226  10.3032 
The second column of Table 4 shows that the optimal cost increases as
5. Conclusion
In this chapter we have used optimal control theory to derive the optimal production rate in a manufacturing system presenting the following features: the demand rate during a certain period depends on the demand rate of the previous period (dependent demand), the demand rate depends on the inventory level, items in inventory are subject to deterioration, and the firm adopts either a continuous or periodic review policy. In contrast to most of the existing research which uses time series forecasting models, we propose a new model, namely, the demand dynamics equation. This model approaches realistic problems by integrating the forecasting component into the production planning problem with deteriorating items and stock dependent demand under continuousreview policy. Simulations were conducted in order to show the performance of the obtained solution. The theoretical and the simulations results allow gaining insights into operational issues and demonstrating the scope for improving stock control systems.
Of course, as with any research work, this study is not without limitations. The main contribution of our model is equation (5) where we use the demand from the previous period to predict the demand in the current period. The main limitation of that equation is that it involves two coefficients. We have assumed in this chapter that the parameters
Another research direction would be to use a predictive control strategy where, given the current inventory level, the optimal production rates to be implemented at the beginning of each of the following periods over the control horizon, are determined. Model predictive control (or recedinghorizon control) strategies have gained widespread acceptance in industry. It is also wellknown that these models are interesting alternatives for realtime control of industrial processes. In the case where the above parameters
Note that our state equations are linear and thus linear model predictive control (LMPC), which is widely used both in academic and industrial fields, can be used. Nonlinear model predictive control (NMPC) can be used in case one of the state equations is nonlinear, for example, if equation (5) were of the form
NMPC has gained significant interest over the past decade. Various NMPC strategies that lead to stability of the closedloop have been developed in recent years and key questions such as the efficient solution of the occurring openloop control problem have been extensively studied.
The case combining unknown coefficients and a nonlinear relationship between the demand rate and the onhand inventory yields a very complex, highly nonlinear process for which there is no simple mathematical model. The use of fuzzy control seems particularly well appropriate. Fuzzy control is a technique that should be seen as an extension to existing control methods and not their replacement. It provides an extra set of tools which the control engineer has to learn how to use where it makes sense. Nonlinear and partially known systems that pose problems to conventional control techniques can be tackled using fuzzy control.
Acknowledgement
This work has been supported by the Research Center of College of Computer and Information Sciences, King Saud University.
References
 1.
Strijbosch, L.W.G., Syntetos, A.A., Boylan, J.E., and Janssen, E. On the interaction between forecasting and stock control: The case of nonstationary demand. International Journal of Production Economics2011; 133:1, 470480.  2.
Janssen, E., Strijbosch, L., and Brekelmans, R. Assessing the effects of using demand parameters estimates in inventory control and improving the performance using a correction function.International Journal of Production Economics 2009; 118, 34–42.  3.
Tiacci, L. and Saetta, S. An approach to evaluate the impact of interaction between demand forecasting method and stock control policy on the inventory system performances 2009; International Journal of Production Economics; 118, 6371.  4.
Doganis, P., Aggelogiannaki, E., and Sarimveis, H. A combined model predictive control and time series forecasting framework for productioninventory systems, International Journal of Production Research 2008; 46:24, 68416853.  5.
Metan, G. and Thiele, A. Integrated forecasting and inventory control for seasonal demand: A comparison with the HoltWinters approach. Technical Report. Department of Industrial and Systems Engineering, Lehigh University, Bethlehem, PA; 2007.  6.
Syntetos, A.A. and Boylan, J.E. On the stock control performance of intermittent demand estimators 2006; International Journal of Production Economics, 103, 3647.  7.
Strijbosch, L.W.G. and Moors, J.J.A. Modified normal demand distributions in (R,S)inventory control. European Journal of Operational Research 2006; 172, 201212.  8.
Yang, H.L., Teng, J.T., and Chern, M.S. An inventory model under inflation for deteriorating items with stockdependent consumption rate and partial backlogging shortages. International Journal of Production Economics 2010; 123:1, 819.  9.
Min, J., Zhou, Y.W., and Zhao, J. An inventory model for deteriorating items under stockdependent demand and twolevel trade credit. Applied Mathematical Modelling 2010; 34:11, 32733285.  10.
Jaggi, C.K. and Khanna, A. Supply chain model for deteriorating items with stockdependent consumption rate and shortages under inflation and permissible delay in payment. International Journal of Mathematics in Operational Research 2010; 2:4, 491514.  11.
Chang, C.T., Teng, J.T., and Goyal, S.K Optimal replenishment policies for noninstantaneous deteriorating items with stockdependent demand. International Journal of Production Economics. 2010; 123:1, 6268.  12.
Choudhury, K.D. and Datta, T.K. An inventory model with stock–dependent demand rate and dual storage facility. Assam University Journal of Science & Technology: Physical Sciences and Technology 2010; 5(2) 5363.  13.
Panda, D., Maiti, M.K., and Maiti, M. Twowarehouse inventory models for single vendor multiple retailers with price and stock dependent demand. Applied Mathematical Modelling. 2010; 34(11) 35713585.  14.
Sajadieh, M.S., Thorstenson, A., and Jokar, M.R.A. An integrated vendorbuyer model with stockdependent. Transportation Research Part E: Logistics and Transportation Review 2010; 46(6) 963974.  15.
Khanra, S., Sana, S.S., and Chaudhuri, K. An EOQ model for perishable items with stock and price dependent demand rate, International Journal of Mathematics in Operational Research 2010; 2(3) 320335.  16.
Hsieh, T.P. and Dye, C.Y. Optimal replenishment policy for perishable items with stockdependent selling rate and capacity constraint. Computers & Industrial Engineering 2010; 59(2) 251258.  17.
Tripathy, C.K. and Mishra, U. An inventory model for Weibull deteriorating items with price dependent demand and timevarying holding cost. Applied Mathematical Sciences 2010; 4(44) 21712179.  18.
Mahata, G. C. and Goswami, A. Fuzzy EOQ models for deteriorating items with stock dependent demand and nonlinear holding costs, International Journal of Applied Mathematics and Computer Sciences 2009; 5(2) 9498.  19.
Gayen, M. and Pal A.K. A twowarehouse inventory model for deteriorating items with stock dependent demand rate and holding cost, Journal of Operational Research 2009; 9(2) 153165.  20.
Chen, L.T. and Wei, C.C. Coordinated supply chain mechanism for an item with price and stock dependent demand subject to fixed lifetime and deterioration. In: International Conference on Business and Information, 68 July 2009, 6(1) ISSN: 17299322.  21.
Soni, H. and Shah, N.H. Ordering policy for stockdependent demand rate under progressive payment scheme. International Journal of Systems Science. 2009; 40(1) 8189.  22.
Goyal, S.K. and Giri, B.C. Recent trends in modelling of deteriorating inventory. European Journal of Operational Research 2001; 134, 116.  23.
Roy, T. and Chaudhuri, K.S. An inventory model for a deteriorating item with pricedependent demand and special sale. International Journal of Operational Research 2007; 2(2) 173187.  24.
Manna, S.K. and Chiang, C. Economic production quantity models for deteriorating items with ramp type demand. International Journal of Operational Research 2010; 7(4) 429444.  25.
Sethi, S.P. and Thompson, G.L. Optimal Control Theory: Applications to Management Science and Economics. 2nd ed., Kluwer Academic Publishers, Dordrecht; 2000.  26.
Bryson, A.E. and Ho, Y.C. Applied Optimal Control. Halsted Press. Washington D.C.;1975.