- Email: [email protected]
Production Planning Problem with Inventory Boundary Affected by Demand Uncertainty
Copyright @ IFAC Management and Control of Production and Logistics, Grenoble, France, 2000
PRODUCTION PLANNING PROBLEM WITH INVENTORY BOUNDARY AFFECTED BY DEMAND UNCERTAINTY
Oscar S. Silva Filho Fundariio cn - Automation Institute Rod. D. Pedro I (SP 65) Km. 143,6 13081-970 Campinas SP Brazil Email: [email protected]
Abstract: A sub-optimal policy for a stochastic production planning problem with probabilistic-constraint in which inventory variable is analysed. This kind of solution is provided from an equivalent model known as the Mean Value problem. In this model, the uncertainties related to demand are considered through a safety-stock function which is concave and growing over the periods. It also depends on the inventory's variance and the probability measure provided by the manager. Some properties of this function and their effects over the mean optimal solution are discussed. An example is used to illustrate such results. Copyright© 2000 IFAC Keywords: production, planning" probability, suboptimal, dynamic programming
I. INTRODUCTION
provide a sub-optimal policy for stochastic production planning problems, see, for example: (Lassere et aI., 1985), (Parlar, 1985), and (Silva Filho, 1999). Comparing the current paper with other similar papers from literature, the basic difference is that the objective here is exclusively to investigate the effect of probabilistic constraints of the stochastic problem on the sub-optimal solution provided.
Aggregated production planning is often used to help managers make appropriate decisions related to the use of the company's material resources. Indeed, the aggregated information allows managers to state a production plan that will be used as goal to be reached in the detailed production levels (Hill, 1991). This paper considers the problem of finding an optimal policy that minimizes a linear quadratic stochastic production planning problem with chance constraint on inventory variable. Since providing an optimal solution from the above problem is a difficult task, sub-optimal solutions are usually preferred.
During the transformation of the stochastic problem to the mean value problem, the probabilistic constraint on inventory variable is converted to a deterministic equivalent constraint that preserves the main statistic characteristics of the original constraint. In this constraint, a lower boundary to mean inventory level is provided. This lower bound can be interpreted as a safety stock model that evolves over the periods, i.e., a dynamic safety stock model. Particularly, this model is defined by a concave function that increases over the periods. Such function depends on a probability measure,
In this paper, an equivalent deterministic model is considered. It provides a mean approximate solution to the stochastic problem that gives important managerial insight to managers. Note that, in the literature, this kind of approach has been used to
651
provided by the manager and the variance of demand. This fact permits to the manager exploit different scenarios related to customer satisfaction. For example, assuming that customers never accept delays on the delivery of the products ordered, it is strongly recommended to maintain a minimum safety stock level for these products. Using the probability measure managers can evaluate different scenarios to guarantee customer satisfaction. In this way, this paper investigates the effect of a safety stock function on the solution of the mean value problem, verifying its relation to customer satisfaction through using the probability measure.
quadratic. This permits that both posItIve and negative value of inventory and production variables can be penalized. Besides. the average inventory cost can be easily statistically manipulated.
3.1. Mean Problem To derive an optimal production policy for ( I) is not a trivial task, unless in the very simple single product problems by using stochastic dynamic programming. Approximate solutions are usually used to solve multiproduct problems. In this context. an equivalent mean value problem can be formulated to generate a sub-optimal solution to (I). Note that, the equivalent mean problem is a consequence of the application of the certainty-equivalence principle (Bertesekas, 1995). This principle states that the behavior of the stochastic system can be measured approximately by the evolution of the first static moments (i.e .. the average values) of the random decision variables involved. As a result. the equivalent problem can be formulated by changing these random variables by their average values. Note, also. that, in the formulation proposed here, besides the average values, the second statistic moments (i.e.. the variance values) are used (Silva Filho. 1999). As a result, this formulation is more generic than the one provided by using exclusively the above principle. This problem is given as follows:
2. GENERAL NOTATION h·x(k)2 = inventory (holding) cost in period k; h>O. c'u(k)2 = production cost in period k; c>O. x(k) = inventory level in period k. u(k) = production level in period k. v(k) = demand level (random variable) in period k with mean v(k) and standard deviation cry ~O. Xo = initial inventory, and x = final inventory. N N = planning horizon. and a = probability measure. E{.} = average operator, and F•. k = inventory probability distribution function entirely determined from the inventory mean and variance equations: E{ x(k)} = x(k)= x(k -1) + u(k)- v(k} x(O)= x" E{(x(k)- X(k»2 }=cr:(k)=k'cr~;
cr~(O)=O
E(u(k» = u(k)=u(k) with cr~ (k) = O. s.t.
E{x(kY}= x(kY + k· cr~
3. STOCHASTIC PRODUCTION PROBLEM The objective is to determine an optimal production policy {u( I) u(2) ... u(N)} for the following problem:
N
where K=h·Lk·cr~ =h·«N2+N)/2)·cr~ k=l)
Comments: a) Problem (2) preserves the linearity and convexity properties of (1). b) An optimal solution can be provided by any applicable mathematical programming method.
s.t.
x(k)= x(k -1)+ u(k)- v(k); x(O) = x"
Pr{x(k)~ O}~ u (k ) ~O
I-a} _
(1)
k -1.2,...• N 4. SAFETY-STOCK CONSIDERATION
Comments: 1. The demands {v(1), v(2)•..., v(N)} are Gaussian non-negatives and independents random variables. Since the demand fluctuation affects the linear inventory balance equation, the inventory variable x(k) is also a Gaussian random variable. 2. Because of randomness of inventory variable. the inventory constraint is considered in probability. The probability measure a is fixed by the manager. This probability has an important meaning, for example if the manager chooses a = 0.95, it means a 95% chance of customer satisfaction. The criterion is
An important characteristic of problem (2) is observed from the equivalent inventory probabilistic constraint, defined as follows:
where raCk) is the safety stock function. It is worth noting that such function is directly proportional to
Jk, this
652
cr v' and F;' (a). Some important features of function are discussed In following:
(a)
(b)
10 r(k)
10
5
r(k)
5
0
0
2
4
6
8
10
12
2
4
k
6
8
IQ
12
k
Figure I. function ra(k): (a) a=O.1 fix with a, = 1.0 and a, = 2.0, and (b) a v = 2.0 fix with a=O.1 and a=0.3. 5. SAFETY STOCK EFFECT OVER THE OPTIMAL SOLUTION
(a) The contour of the function raCk) is defined by square root of the period k which is a concave function that increases over the periods (see figure I). (b) The proportionality with the demand standard deviation av~O means that any uncertainty about the future behavior of the demand can be absorbed by the inventory constraint (3) through the evolution of safety stock function ra(k). This evolution reaches its maximum value at the end period N, that is, a.(N) ~ a.(N-I) ~ ... ~ a.(l) (see figure I. (a». Therefore, the larger the value of av, more accentuate will be the contour of the function ra . (c) The probability measure a E [0,1) has an important impact on the function raCk). To understand such effect, consider that the inverse probability distribution function F,-I (a) has the
For single product production problems, an optimal solution can be provided by dynamic programming algorithm (PD) (Bertesekas, 1995). In following the application of PD algorithm to the problem (2) is discussed along with the influence of the safety stock function on the optimal solution. The classical procedure used by PD in order to guarantee that the constraint x(k)~ra (k) will not be violated for any period k, is to make the production level u (k) takes values of a set Q(x(k -I) ~0 . For the problem (2) this set is given by: u(k)EQ (x(k -l)={u Iu(k)=max (0, ra (k)+v(k)-x (k -I»}
following characteristic: for aE [1/2, I) ~ F,-' (a)~O;
(4)
on the other hand, aE (0, 1/2) ~ F,-' (a)
Knowing the constraint (4), the recursive equation of the PD algorithm can be written as follows:
result, the function ra(k) can be interpreted in accordance with two opposite focuses: Focus 1) raCk) > 0 (safety stock), if a E [y2, I) Focus 2) ra(k)::; 0 (backlogging), if a E (0, '12) Focus I is the primary interest in this study. In fact, it guarantees that mean inventory level will be always positive over the periods. Besides, the dynamic safety stock raCk) is used to prevent the production process against stockout situations that lead to low levels of customer satisfaction. On the other hand, focus 2 is unrealistic. Indeed, in this focus, there is a very strong possibility of demand not being met during a certain period k (i.e. stockout occurrence)
J N (x(N»=hx(N)' J k (x(k»= Min {(hx(k)2 +CU(k)2)+J k+1 (x(k + l))} (5) u(k )en
with k = N-I , ... , 1,0 (backward shift). The optimal production level derived from this algorithm is:
u(k)=Jl~ (x(k -1», kE[I,N] where Jl (.) denotes a function that relates
(6) u(k) to
x(k). The total production cost is determined in the last stage of the recursion (5), for a given value of initial inventory x (0), that is, J' =J 11 (x 11) •
Two important properties of the function raCk) are: Property 1) For a given value of av>O, the safety stock area (vide figure I.(b» generated by raCk) rows with aE [1/2, I). For example, let the probabilities al and az, assuming that al ::; az ~ ra, ::; ra, .
Characteristics of the optimal solution:
Property 2) For a v = 0 or for a = '12 (with av:;t{) implies that raCk) = O. In the first case, since the standard deviation is null, the fluctuation of demand is exactly known and the problem (I) is deterministic. Therefore, it doesn't make sense to consider the inventory constraint in probability. In the second case, the probability a=112 means that there is 50% of risk of stockout occurrence.
(a) The optimal solution (6) and total production cost depend on the probability a, via constraint (4). In fact, since the value of a affects of function ra (k) (see property 1 and figure I.(b», the set Q will be also directly affected by a. Thus, from (4), it is possible to verify that a.Qa,. (b) The two most important effects of (4) on the behavior of the optimal solution (6) are:
653
20
c:::J Inventory
- - Stock r(k)
15
Production
10
5
o ~~ljEIjD4C1Dr;op;omO o 2 4 6 8 10 12
3
5
7
9
11
Fig. 6: Scenario 2.2 (a=75%)
20~
Production
Inventory
15
10
~
•
.'
o
-jl.ID-4-1-+1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-+1-11 2
4
6
8
10
12
3
5
7
9
11
Fig. 7 Scenario 2.3 (ex=0.50%)
7. CONCLUSION
Scenario 2.2 a=O.75: There is a 25% chance of demand not being met. Total production cost here is 1"=1.378. Figure 6 shows the inventory and production levels.
In this paper a stochastic production problem was formulated. Due to the complexity of dealing with it, an equivalent mean value problem was introduced. The uncertainties about the fluctuation of demand are included in the inventory constraint of the equivalent problem through a concave function that represents the safety stock. The safety area provided by this function is completely determined by a probability measure. As a result, the manager has the possibility to gain important insight concerning the use of the aggregated material resources of the firm.
Scenario 2.3 a=O.5: This means a strong risk of backorder. In this particular case the safety stock function ra(k) = 0, V k. The safety stock is completely ignored, and, as a consequence, can cause delays on delivery products to the customer (see property 2 in section 4). In practical terms, the manager believes that it is possible to re-negotiate new periods for delivering the products with the customers. The optimal cost obtained with this operation is 1"= 1.221. Figure 7 illustrates the behavior of inventory and production levels.
Acknowledgements: This work has been supported by CNPq and FAPESP under grant 520196/96-1 and 98/15550-4, respectively.
General comments of Case 2: the analysis of the three scenarios above show that the probability ex strongly affects the optimal solution of the mean value problem (2). Thus, the higher the value of ex, the large will be the safety stock function ra(k), and consequently greater chances of improving the customer service level. However, there is a price to maintain a large safety stock, and that is the increase of the total production cost. Indeed, note that
REFERENCES Bertesekas, D. P. (1995):Dynamic Programming and Stochastic Control (Vol. I), Athena Scientific. Hill, T. (1991) Production and Operations Management: Text and Cases", Prentice Hall. Lassere, J. B., Bes c., and Roubelat T. F. (1985): The Stochastic Discrete Dynamic Lot Size Problem: An Open-Loop Solution, Operations Research, Vol. 3, No. 3, pp.: 684-689. Parlar M. (1985): The Stochastic Production Planning Model with a Dynamic Chance Constriant, European Jrn of Opr. Research, 20, pp. 255-260. Silva Filho, O. S (1999): An Aggregate Production Planning Model with Demand under Uncertainty, Prod. Planning & Control, Vol. 10,8,745-756.
J~15<1;1.25 <1;J.U5 . Particularly, scenario 3 shows that if the production policy, given in figure 7, were applied to schedule the production process, there would be a risk of 50% of chances of stockout. In fact, for this to occur, it is only necessary that actual demand level v(k) be greater than the mean level of demand (k) for any period k of planning horizon.
v
656
PRODUCTION PLANNING PROBLEM WITH INVENTORY BOUNDARY AFFECTED BY DEMAND UNCERTAINTY
Oscar S. Silva Filho Fundariio cn - Automation Institute Rod. D. Pedro I (SP 65) Km. 143,6 13081-970 Campinas SP Brazil Email: [email protected]
Abstract: A sub-optimal policy for a stochastic production planning problem with probabilistic-constraint in which inventory variable is analysed. This kind of solution is provided from an equivalent model known as the Mean Value problem. In this model, the uncertainties related to demand are considered through a safety-stock function which is concave and growing over the periods. It also depends on the inventory's variance and the probability measure provided by the manager. Some properties of this function and their effects over the mean optimal solution are discussed. An example is used to illustrate such results. Copyright© 2000 IFAC Keywords: production, planning" probability, suboptimal, dynamic programming
I. INTRODUCTION
provide a sub-optimal policy for stochastic production planning problems, see, for example: (Lassere et aI., 1985), (Parlar, 1985), and (Silva Filho, 1999). Comparing the current paper with other similar papers from literature, the basic difference is that the objective here is exclusively to investigate the effect of probabilistic constraints of the stochastic problem on the sub-optimal solution provided.
Aggregated production planning is often used to help managers make appropriate decisions related to the use of the company's material resources. Indeed, the aggregated information allows managers to state a production plan that will be used as goal to be reached in the detailed production levels (Hill, 1991). This paper considers the problem of finding an optimal policy that minimizes a linear quadratic stochastic production planning problem with chance constraint on inventory variable. Since providing an optimal solution from the above problem is a difficult task, sub-optimal solutions are usually preferred.
During the transformation of the stochastic problem to the mean value problem, the probabilistic constraint on inventory variable is converted to a deterministic equivalent constraint that preserves the main statistic characteristics of the original constraint. In this constraint, a lower boundary to mean inventory level is provided. This lower bound can be interpreted as a safety stock model that evolves over the periods, i.e., a dynamic safety stock model. Particularly, this model is defined by a concave function that increases over the periods. Such function depends on a probability measure,
In this paper, an equivalent deterministic model is considered. It provides a mean approximate solution to the stochastic problem that gives important managerial insight to managers. Note that, in the literature, this kind of approach has been used to
651
provided by the manager and the variance of demand. This fact permits to the manager exploit different scenarios related to customer satisfaction. For example, assuming that customers never accept delays on the delivery of the products ordered, it is strongly recommended to maintain a minimum safety stock level for these products. Using the probability measure managers can evaluate different scenarios to guarantee customer satisfaction. In this way, this paper investigates the effect of a safety stock function on the solution of the mean value problem, verifying its relation to customer satisfaction through using the probability measure.
quadratic. This permits that both posItIve and negative value of inventory and production variables can be penalized. Besides. the average inventory cost can be easily statistically manipulated.
3.1. Mean Problem To derive an optimal production policy for ( I) is not a trivial task, unless in the very simple single product problems by using stochastic dynamic programming. Approximate solutions are usually used to solve multiproduct problems. In this context. an equivalent mean value problem can be formulated to generate a sub-optimal solution to (I). Note that, the equivalent mean problem is a consequence of the application of the certainty-equivalence principle (Bertesekas, 1995). This principle states that the behavior of the stochastic system can be measured approximately by the evolution of the first static moments (i.e .. the average values) of the random decision variables involved. As a result. the equivalent problem can be formulated by changing these random variables by their average values. Note, also. that, in the formulation proposed here, besides the average values, the second statistic moments (i.e.. the variance values) are used (Silva Filho. 1999). As a result, this formulation is more generic than the one provided by using exclusively the above principle. This problem is given as follows:
2. GENERAL NOTATION h·x(k)2 = inventory (holding) cost in period k; h>O. c'u(k)2 = production cost in period k; c>O. x(k) = inventory level in period k. u(k) = production level in period k. v(k) = demand level (random variable) in period k with mean v(k) and standard deviation cry ~O. Xo = initial inventory, and x = final inventory. N N = planning horizon. and a = probability measure. E{.} = average operator, and F•. k = inventory probability distribution function entirely determined from the inventory mean and variance equations: E{ x(k)} = x(k)= x(k -1) + u(k)- v(k} x(O)= x" E{(x(k)- X(k»2 }=cr:(k)=k'cr~;
cr~(O)=O
E(u(k» = u(k)=u(k) with cr~ (k) = O. s.t.
E{x(kY}= x(kY + k· cr~
3. STOCHASTIC PRODUCTION PROBLEM The objective is to determine an optimal production policy {u( I) u(2) ... u(N)} for the following problem:
N
where K=h·Lk·cr~ =h·«N2+N)/2)·cr~ k=l)
Comments: a) Problem (2) preserves the linearity and convexity properties of (1). b) An optimal solution can be provided by any applicable mathematical programming method.
s.t.
x(k)= x(k -1)+ u(k)- v(k); x(O) = x"
Pr{x(k)~ O}~ u (k ) ~O
I-a} _
(1)
k -1.2,...• N 4. SAFETY-STOCK CONSIDERATION
Comments: 1. The demands {v(1), v(2)•..., v(N)} are Gaussian non-negatives and independents random variables. Since the demand fluctuation affects the linear inventory balance equation, the inventory variable x(k) is also a Gaussian random variable. 2. Because of randomness of inventory variable. the inventory constraint is considered in probability. The probability measure a is fixed by the manager. This probability has an important meaning, for example if the manager chooses a = 0.95, it means a 95% chance of customer satisfaction. The criterion is
An important characteristic of problem (2) is observed from the equivalent inventory probabilistic constraint, defined as follows:
where raCk) is the safety stock function. It is worth noting that such function is directly proportional to
Jk, this
652
cr v' and F;' (a). Some important features of function are discussed In following:
(a)
(b)
10 r(k)
10
5
r(k)
5
0
0
2
4
6
8
10
12
2
4
k
6
8
IQ
12
k
Figure I. function ra(k): (a) a=O.1 fix with a, = 1.0 and a, = 2.0, and (b) a v = 2.0 fix with a=O.1 and a=0.3. 5. SAFETY STOCK EFFECT OVER THE OPTIMAL SOLUTION
(a) The contour of the function raCk) is defined by square root of the period k which is a concave function that increases over the periods (see figure I). (b) The proportionality with the demand standard deviation av~O means that any uncertainty about the future behavior of the demand can be absorbed by the inventory constraint (3) through the evolution of safety stock function ra(k). This evolution reaches its maximum value at the end period N, that is, a.(N) ~ a.(N-I) ~ ... ~ a.(l) (see figure I. (a». Therefore, the larger the value of av, more accentuate will be the contour of the function ra . (c) The probability measure a E [0,1) has an important impact on the function raCk). To understand such effect, consider that the inverse probability distribution function F,-I (a) has the
For single product production problems, an optimal solution can be provided by dynamic programming algorithm (PD) (Bertesekas, 1995). In following the application of PD algorithm to the problem (2) is discussed along with the influence of the safety stock function on the optimal solution. The classical procedure used by PD in order to guarantee that the constraint x(k)~ra (k) will not be violated for any period k, is to make the production level u (k) takes values of a set Q(x(k -I) ~0 . For the problem (2) this set is given by: u(k)EQ (x(k -l)={u Iu(k)=max (0, ra (k)+v(k)-x (k -I»}
following characteristic: for aE [1/2, I) ~ F,-' (a)~O;
(4)
on the other hand, aE (0, 1/2) ~ F,-' (a)
Knowing the constraint (4), the recursive equation of the PD algorithm can be written as follows:
result, the function ra(k) can be interpreted in accordance with two opposite focuses: Focus 1) raCk) > 0 (safety stock), if a E [y2, I) Focus 2) ra(k)::; 0 (backlogging), if a E (0, '12) Focus I is the primary interest in this study. In fact, it guarantees that mean inventory level will be always positive over the periods. Besides, the dynamic safety stock raCk) is used to prevent the production process against stockout situations that lead to low levels of customer satisfaction. On the other hand, focus 2 is unrealistic. Indeed, in this focus, there is a very strong possibility of demand not being met during a certain period k (i.e. stockout occurrence)
J N (x(N»=hx(N)' J k (x(k»= Min {(hx(k)2 +CU(k)2)+J k+1 (x(k + l))} (5) u(k )en
with k = N-I , ... , 1,0 (backward shift). The optimal production level derived from this algorithm is:
u(k)=Jl~ (x(k -1», kE[I,N] where Jl (.) denotes a function that relates
(6) u(k) to
x(k). The total production cost is determined in the last stage of the recursion (5), for a given value of initial inventory x (0), that is, J' =J 11 (x 11) •
Two important properties of the function raCk) are: Property 1) For a given value of av>O, the safety stock area (vide figure I.(b» generated by raCk) rows with aE [1/2, I). For example, let the probabilities al and az, assuming that al ::; az ~ ra, ::; ra, .
Characteristics of the optimal solution:
Property 2) For a v = 0 or for a = '12 (with av:;t{) implies that raCk) = O. In the first case, since the standard deviation is null, the fluctuation of demand is exactly known and the problem (I) is deterministic. Therefore, it doesn't make sense to consider the inventory constraint in probability. In the second case, the probability a=112 means that there is 50% of risk of stockout occurrence.
(a) The optimal solution (6) and total production cost depend on the probability a, via constraint (4). In fact, since the value of a affects of function ra (k) (see property 1 and figure I.(b», the set Q will be also directly affected by a. Thus, from (4), it is possible to verify that a.
653
20
c:::J Inventory
- - Stock r(k)
15
Production
10
5
o ~~ljEIjD4C1Dr;op;omO o 2 4 6 8 10 12
3
5
7
9
11
Fig. 6: Scenario 2.2 (a=75%)
20~
Production
Inventory
15
10
~
•
.'
o
-jl.ID-4-1-+1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-+1-11 2
4
6
8
10
12
3
5
7
9
11
Fig. 7 Scenario 2.3 (ex=0.50%)
7. CONCLUSION
Scenario 2.2 a=O.75: There is a 25% chance of demand not being met. Total production cost here is 1"=1.378. Figure 6 shows the inventory and production levels.
In this paper a stochastic production problem was formulated. Due to the complexity of dealing with it, an equivalent mean value problem was introduced. The uncertainties about the fluctuation of demand are included in the inventory constraint of the equivalent problem through a concave function that represents the safety stock. The safety area provided by this function is completely determined by a probability measure. As a result, the manager has the possibility to gain important insight concerning the use of the aggregated material resources of the firm.
Scenario 2.3 a=O.5: This means a strong risk of backorder. In this particular case the safety stock function ra(k) = 0, V k. The safety stock is completely ignored, and, as a consequence, can cause delays on delivery products to the customer (see property 2 in section 4). In practical terms, the manager believes that it is possible to re-negotiate new periods for delivering the products with the customers. The optimal cost obtained with this operation is 1"= 1.221. Figure 7 illustrates the behavior of inventory and production levels.
Acknowledgements: This work has been supported by CNPq and FAPESP under grant 520196/96-1 and 98/15550-4, respectively.
General comments of Case 2: the analysis of the three scenarios above show that the probability ex strongly affects the optimal solution of the mean value problem (2). Thus, the higher the value of ex, the large will be the safety stock function ra(k), and consequently greater chances of improving the customer service level. However, there is a price to maintain a large safety stock, and that is the increase of the total production cost. Indeed, note that
REFERENCES Bertesekas, D. P. (1995):Dynamic Programming and Stochastic Control (Vol. I), Athena Scientific. Hill, T. (1991) Production and Operations Management: Text and Cases", Prentice Hall. Lassere, J. B., Bes c., and Roubelat T. F. (1985): The Stochastic Discrete Dynamic Lot Size Problem: An Open-Loop Solution, Operations Research, Vol. 3, No. 3, pp.: 684-689. Parlar M. (1985): The Stochastic Production Planning Model with a Dynamic Chance Constriant, European Jrn of Opr. Research, 20, pp. 255-260. Silva Filho, O. S (1999): An Aggregate Production Planning Model with Demand under Uncertainty, Prod. Planning & Control, Vol. 10,8,745-756.
J~15<1;1.25 <1;J.U5 . Particularly, scenario 3 shows that if the production policy, given in figure 7, were applied to schedule the production process, there would be a risk of 50% of chances of stockout. In fact, for this to occur, it is only necessary that actual demand level v(k) be greater than the mean level of demand (k) for any period k of planning horizon.
v
656