Dynamic Optimization Technique for Distribution of Goods with Stochastic Shortages

Abstract

This work considers the distribution of goods with stochastic shortages from factories to stores. It is assumed that in the process of shipping the goods to various stores, some proportion of the goods will be damaged (which will lead to shortage of goods in transit). The cost of the damaged goods is added to the cost of the shipment. A proportion of the total expected cost of the shortage goods is assumed to be recovered and should be deducted from the total cost of the shipment. In order to determine the minimum transportation costs for the operation, we adopt dynamic optimization principles. The optimal transportation cost and optimal control policies of shipping the goods from factories to stores were obtained. We find that the optimal costs of the goods recovered could be determined. It was further found that the optimum costs of distributing the goods with minimum and maximum error bounds coincide only at infinity.

1. Introduction

A distribution company plans to minimize the cost of distributing 𝑞 kinds of products from 𝑚 number of factories to 𝑛 number of stores. For the case of a company distributing a particular product from one factory to all the stores (a single product), that is, at factory 𝑆₁, product 𝐴 is distributed to all the 𝑛 stores and factory 𝑆₂ produce 𝐵 products and are distributed to all the 𝑛 stores, and so on; see Nwozo and Nkeki [1]. It is also expected that the goods that leave the factories will not come back to the factories (in the case of defective, damaged, etc., items). The company considered 𝑛 number of control policies 𝜇={𝜋¹,𝜋²,…,𝜋^𝑛} to determine which of them will yield the optimum control policy. They also estimated that certain percentage of the products are to reach their final destination successfully at minimum costs. It is assumed that some proportion of the shortage goods should be recovered. The recovered costs of the shortage goods should be deducted from the total costs continuously over time horizon.

In the related literature, Powell [2] considered the problem of a stochastic fleet assets management problem and used the postdecision state variable implicitly to determine this problem. Closely related to this on the extensive work on stochastic fleet assets management problem is the work by Cheung and Powell [3]. Godfrey and Powell [4] further used postdecision state variable explicitly to determine similar problem on a stochastic fleet management problem. Van Roy et al. [5] proposed the idea of using a parsimonious sufficient statistics in an application of approximate dynamic programming to inventory management. Mulvey and Vladimirou [6] used the stochastic programming technique in financial asset allocation problems for designing low-risk portfolios, and Mulvey and Ruszczynski [7] designed variety of specialized algorithms that help to solve the problems which involved management of homogeneous resources. They further emphasized that for more realistic problems, resources are heterogeneous and that the resulting optimization problems do not generally exhibit pure network structure. Powell et al. [8] described an algorithm for computing parameter values to fit linear and separable concave approximation to the value function to tackle large-scale problems in transportation and logistics. Furthermore, Powell and Topaloglu [9] described a more complicated variation of the algorithm that improved on execution time and memory requirements and such improvement is often critical for practical application to realistic large-scale problems. They further emphasized that the movement of freight over long distances is subject to random delays. Powell and Van Roy [10] studied and applied approximate dynamic programming to high-dimensional resource allocation problems in area of managing a fleet of trucks. Powell [22] worked extensively on an approximate dynamic programming for large-scale asset management problems for both single and multiple assets. Guestrin et al. [11] used an approximate dynamic programming approach to deal with high-dimensional decision spaces. For now, no other applications of approximate dynamic programming have dealt with such large numbers of decision variables (see, Powell and Van Roy [10]). Spivey and Powell [12] studied the dynamic assignment problem, where a resource can serve only one task at a time. They further proposed a general class of dynamic assignment models that involved adaptive and nonmyopic algorithm. Cogill et al. [13] considered the problem of computing decentralized policies for stochastic systems with finite state and action spaces. They presented an algorithm based on linear programming and used this algorithm to obtain a decentralized policy from a function with special structure that approximates the optimal centralized Q-function. Topaloglu and Kunnumkal [14] proposed two approximate dynamic programming methods to optimize the distribution operations of a company manufacturing a certain product at multiple production plants and ship to different customer location for sale. Hauskrecht and Singliar [15] considered real-world distributions and networks which are unreliable and subject to random failures of its components. They investigated MonteCarlo solutions in stochastic networks in which the expected value of resources allocated before and after the occurrence of stochastic failures needs to be optimized. Ozbay et al. [16] proposed mathematical programming models with probabilistic constraints in order to address incident response and resource allocation problems for traffic incident management. They considered the resource allocation problem by assuming that the stochastic distribution of incidents over a network is given and introduced a mathematical model to determine the number of service vehicles allocated to each depot to meet the requirements of the potential incidents by taking into account the stochastic nature of the resource requirement and incident occurrence probabilities. Jacko [17] considered resource allocation problem in telecommunication system. They optimized packet losses by choosing preferred paths in the routing system. They assumed that there are a number of independent competitors with the aim of maximizing the expected return. Elmaghraby and Ramachandra [18] considered the problem of optimally allocating a single resource under uncertainty to the various activities of a project to optimize a certain economic objective composed of resource utilization cost and tardiness cost. Nwozo and Nkeki [19], used DPP to consider the allocation of buses from single station to different routes in Nigeria for profit maximization. Nkeki [20] considered the allocation of vehicles from different (multiple) stations to different routes putting into consideration the rate of breakdown of the vehicles during the allocation but failed to put into consideration the depreciation of the vehicles and effect tax on the returns of the investment. Nwozo and Nkeki [21] studied the allocation of resource to different tasks in a production company so as to maximize returns. They assumed that the company produces the same kinds of products and allocate 𝑚 number of tasks to 𝑛 number of machines. They also assumed that the machines are subject to random breakdown.

This paper discussed the distribution of goods from different factories to stores with stochastic shortage over time horizon.

The highlights of this paper are(a)modeling the distribution of goods from factories to stores in the presence of(i)stochastic shortages of the goods in transit and(ii)deterministic case (i.e., when there are no shortages) was also considered, using a dynamic optimization technique,(b)numerical determination of optimal cost and optimal control policy for the operations was obtained.

The remaining parts of the paper are structured as follows: Section 2 presents the problem formulation and multiperiod expected cost function. In Section 3, we present the dynamic programming formulation of the problem. The optimization of the cost function is determined in this section. Section 4 presents the dynamic programming principles with error bounds. In Section 5, numerical simulations of the problem are discussed. Section 6 concludes the paper.

2. Problem Formulation

The paper studies a problem where there are 𝑛 states and 𝑛 control policies. Let 𝑆^factory={𝑆^𝑓1,𝑆^𝑓2,…,𝑆^𝑓𝑚} be the set of goods produced in the factories and let 𝑆^store={𝑆^𝑠1,𝑆^𝑠2,…,𝑆^𝑠𝑛} be the set of goods that are to be distributed to the stores, where 𝑆^factory,𝑆^store∈𝑆 and 𝑆 is the state space of the system. The transition probability matrix corresponding to the control policies 𝜋¹,𝜋²,…,𝜋^𝑛 is given as 𝑃𝜋^𝑗=𝑃_𝑖,𝑗𝜋^𝑗,𝑖=1,2,…,𝑚;𝑗=1,2,…,𝑛. (1) The transition costs are given by Φ^𝑖,𝑗(𝑆^𝑓𝑖_𝑡,𝜋^𝑗_𝑡(𝑆^𝑓𝑖_𝑡)), 𝑡∈[0,𝑇], 𝑆^𝑓𝑖_𝑡∈𝑆, where 𝑇 is the terminal time period in the planning horizon, 𝑆^𝑓𝑖_𝑡 represents the state of goods produced in factory 𝑖 at period 𝑡, 𝑆^𝑠𝑗_𝑡 represents the state of goods to be distributed to store 𝑗 at period 𝑡, and the discount factor 𝛽, (0≤𝛽<1). We define the function 𝑄^𝜋𝑗𝑡_𝑡(𝑆^𝑗_𝑡) as the cost of the distribution of the goods corresponding to the control policies 𝜋^𝑗_𝑡, 𝑗=1,2,…,𝑛. The function 𝑥^𝜋𝑗𝑡_𝑝,𝑡(𝑆^𝑓𝑖) represents the number of 𝑝 different kinds of goods produced in factory 𝑖 at period 𝑡 under policy 𝜋^𝑗_𝑡 and 𝑥^𝜋𝑗𝑡_𝑝,𝑡(𝑆^𝑠𝑗) represents the number of 𝑝 different kinds of goods distributed to store 𝑗 at period 𝑡 under the policy 𝜋^𝑗_𝑡.

2.1. Multiperiod Expected Cost Function

Suppose that the costs of distributing the goods from factory 𝑖 to store 𝑗, are Φ_𝑡^𝑖,𝑗(𝑥^𝜋𝑗𝑡_𝑝,𝑡(𝑆^𝑓𝑖_𝑡)) at period 𝑡, where the state of goods in factory 𝑖 is 𝑆^𝑓𝑖_𝑡∈𝑆_𝑡 at period 𝑡 and the state of goods to store 𝑗 is 𝑆^𝑠𝑗_𝑡∈𝑆_𝑡 at period 𝑡. Then, the costs over 𝑇-horizon are ^𝑇∑^𝑚_𝑡=0∑^𝑞_𝑖=1∑_𝑝=1Φ_𝑡^𝑖,𝑗𝑥^𝜋𝑗𝑡_𝑝,𝑡𝑆^𝑓𝑖_𝑡,𝑆^𝑓𝑖_𝑡∈𝑆_𝑡,𝑗=1,2,…,𝑛. (2) Since 𝑥^𝜋𝑗𝑡_𝑝,𝑡(𝑆^𝑓𝑖) is the number of 𝑝 different kinds of goods produced in factory 𝑖 at period 𝑡 under policy 𝜋^𝑗_𝑡 and 𝑥^𝜋𝑗𝑡_𝑝,𝑡(𝑆^𝑠𝑗) and the number of 𝑝 different kinds of goods distributed successfully to store 𝑗 at period 𝑡 under policy 𝜋^𝑗_𝑡, then ^𝑚𝑞_𝑖=1𝑝=1𝑥^𝜋𝑗𝑡_𝑝,𝑡𝑆^𝑓𝑖≥^𝑛𝑞_𝑗=1𝑝=1𝑥^𝜋𝑗𝑡_𝑝,𝑡𝑆^𝑠𝑗,𝑥^𝜋𝑗𝑡_𝑝,𝑡𝑆^𝑓𝑖,𝑥^𝜋𝑗𝑡_𝑝,𝑡𝑆^𝑠𝑗∈𝑋,𝑡∈[0,𝑇]. (3)

The expected minimum discounted cost function obtained under control policy 𝜋^𝑗_𝑡, at period 𝑡, is given as follows: 𝑄^𝜋𝑗𝑡_𝑡𝑆^𝑗_𝑡=𝐸_𝑡min_{𝑥𝜋𝑗𝑡𝑝,𝑡𝑆𝑓𝑖,𝑥𝜋𝑗𝑡𝑝,𝑡𝑆𝑠𝑗}^𝑇_∈𝑋^𝑚_𝑡=0^𝑞_𝑖=1𝑝=1𝛽^𝑡Φ_𝑡^𝑖,𝑗𝑥^𝜋𝑗𝑡_𝑝,𝑡𝑆^𝑓𝑖_𝑡,𝑗=1,2,…,𝑛, (4) subject to ^𝑚𝑞_𝑖=1𝑝=1𝑥^𝜋𝑗𝑡_𝑝,𝑡𝑆^𝑓𝑖≥^𝑛𝑞_𝑗=1𝑝=1𝑥^𝜋𝑗𝑡_𝑝,𝑡𝑆^𝑠𝑗,𝑥^𝜋𝑗𝑡_𝑝,𝑡𝑆^𝑓𝑖,𝑥^𝜋𝑗𝑡_𝑝,𝑡𝑆^𝑠𝑗∈𝑋,𝑡∈[0,𝑇_𝑞𝑝=1],𝑥^𝜋𝑗𝑡_𝑝,𝑡𝑆^𝑓𝑖≥^𝑞_𝑝=1𝑥^𝜋𝑗𝑡_𝑝,𝑡𝑆^𝑠𝑗,𝑥^𝜋𝑗𝑡_𝑝,𝑡𝑆^𝑓𝑖,𝑥^𝜋𝑗𝑡_𝑝,𝑡𝑆^𝑠𝑗∈𝑋,𝑡∈[0,𝑇], (5) where 𝑥^𝜋𝑗𝑡_𝑝,𝑡(𝑆^𝑓𝑖),𝑥^𝜋𝑗𝑡_𝑝,𝑡(𝑆^𝑠𝑗)∈𝑋 is the set of feasible solutions of problem (4). We can express (4) as the expected discounted minimal cost from period 𝑡 onward as an optimization over {𝑥^𝜋𝑗1_𝑝,1,𝑥^𝜋𝑗2_𝑝,2,…,𝑥^𝜋𝑗𝑇_𝑝,𝑇} condition on 𝑆^𝑗_𝑡=𝑠_𝑡 as follows: 𝑄^𝜋𝑗𝑡_𝑡𝑆^𝑗_𝑡=𝐸min_{𝑥𝜋𝑗1𝑝,1,𝑥𝜋𝑗2𝑝,2,…,𝑥𝜋𝑗𝑇−1𝑝,𝑇−1}^𝑇_𝑡′^𝑚_=𝑡^𝑞_𝑖=1𝑝=1𝛽^𝑡′Φ_𝑡^𝑖,𝑗_′×𝑥^𝜋𝑗𝑡_𝑝,𝑡′𝑆^𝑗_𝑡′−1∣𝑆^𝑗_𝑡=𝑠_𝑡,𝑆^𝑗_𝑡∈𝑆,𝑥_𝑡∈𝑋,𝑗=1,2,…,𝑛, (6) subject to the constraints (5).

For a function Φ^𝑖,𝑗∈ℝ^𝑛×𝑚,𝑖=1,2,…,𝑚;𝑗=1,2,…,𝑛, if we accumulate the cost of the first 𝑇-stage and add to it the terminal cost Φ_𝑇(𝑆_𝑇)=∑^𝑚_𝑖=1∑^𝑞_𝑝=1Φ_𝑇^𝑖,𝑗(𝑥^𝜋𝑗𝑇_𝑝,𝑇(𝑆^𝑗_𝑇)),𝑗=1,2,…𝑛, then (6) becomes 𝑄^𝜋𝑗𝑡_𝑡𝑆^𝑗_𝑡=𝐸min_𝑥𝑡^𝑇−1_𝑡′^𝑚_=𝑡^𝑞_𝑖=1𝑝=1𝛽^𝑡′Φ_𝑡^𝑖,𝑗_′𝑥^𝜋𝑗𝑡_𝑝,𝑡′𝑆^𝑗_𝑡′−1+𝛽^𝑇Φ_𝑇𝑆^𝑗_𝑇∣𝑆^𝑗_𝑡=𝑠_𝑡,𝑆^𝑗_𝑡∈𝑆,𝑥^𝜋𝑗𝑡_𝑝,𝑡∈𝑋,𝑗=1,2,…,𝑛, (7) subject to the constraints (5).

3. Dynamic Programming Formulation

In this section, we formulate the problem using dynamic programming technique. Since 𝑆_𝑡 is the state variable at period 𝑡 and 𝑆 the state space, we formulate the problem as a dynamic program. Let the cost of the goods that is to be distributed from factory 𝑖 to store 𝑗 at period 𝑡 under policy 𝜋^𝑗_𝑡 be given by 𝑃_𝑖,𝑗𝑥^𝜋𝑗𝑡_𝑝,𝑡(𝑆^𝑓𝑖), where 𝑃_𝑖,𝑗 is the transition probability of the goods from factory 𝑖 to store 𝑗. Hence, the total expected cost of the goods distributed from factory 𝑖 to store 𝑗 at period 𝑡>0 is given by ^𝑚∑^𝑞_𝑖=1∑_𝑝=1𝑃_𝑖,𝑗𝑥^𝜋𝑗𝑡_𝑝,𝑡𝑆^𝑓𝑖,𝑡=1,2,…,𝑇;𝑗=1,2,…,𝑛. (8) Let 𝛾 be the proportion of the total expected cost of goods lost in transit (shortage) from factory 𝑖 to store 𝑗. Then, (8) becomes 𝛾∑^𝑚𝑞_𝑖=1∑_𝑝=1𝑃_𝑖,𝑗𝑥^𝜋𝑗𝑡_𝑝,𝑡𝑆^𝑓𝑖,𝑡=1,2,…,𝑇;𝑗=1,2,…,𝑛. (9) Equation (9) represents the total gross shortage cost incurred from the goods lost in transit.

Let 𝑆^𝑗_𝑡−1 be the cost of distributing the goods from the factories to store 𝑗 in period 𝑡−1 and 𝑆^𝑗_𝑡 the expected cost of distributing the goods from the factories to store 𝑗 at period 𝑡 and let 𝛼 be the estimated proportion of the goods that are to be recovered from the lost ones, which are expected to go to the stores at period 𝑡, so that the net shortage cost incurred from the goods lost in transit will be (1−𝛼)𝛾^𝑚𝑞_𝑖=1𝑝=1𝑃_𝑖,𝑗𝑥^𝜋𝑗𝑡_𝑝,𝑡𝑆^𝑓𝑖,𝑡=1,2,…,𝑇;𝑗=1,2,…,𝑛. (10) It then follows that 𝑆^𝑗_𝑡=𝑆^𝑗_𝑡−1+𝛾(1−𝛼)^𝑚𝑞_𝑖=1𝑝=1𝑃_𝑖,𝑗𝑥^𝜋𝑗𝑡_𝑝,𝑡𝑆^𝑓𝑖,𝑗=1,2,…,𝑛;𝑡=1,2,…,𝑇. (11) Equation (11) is the transformation equation and is stochastic. We now have the following propositions.

Proposition 1. If 𝛾=0 or 𝛼=1, then 𝑆^𝑗_𝑡=𝑆^𝑗_𝑡−1. (12)

This proposition tells us that the transformation equation (11) is deterministic if 𝛾=0 or 𝛼=1. It implies that all the goods distributed from the factories to stores get to their various destination successfully without shortages.

Proposition 2. Suppose that there is no recovery of any of the goods lost in transit and 𝛾>0; then 𝛼 must be zero and (11) becomes 𝑆^𝑗_𝑡=𝑆^𝑗_𝑡−1+𝛾^𝑚𝑞_𝑖=1𝑝=1𝑃_𝑖,𝑗𝑥^𝜋𝑗𝑡_𝑝,𝑡𝑆^𝑓𝑖,𝑗=1,2,…,𝑛;𝑡=1,2,…,𝑇. (13)

The optimal policy can be found by computing the value functions through the optimality equation 𝑍^𝜋𝑗𝑡_𝑡𝑆^𝑗_𝑡=min_𝑥𝑡^𝑚𝑞_𝑖=1𝑝=1Φ_𝑡^𝑖,𝑗𝑥^𝜋𝑗𝑡_𝑝,𝑡𝑆^𝑖_𝑡−1+𝛽𝐸𝑍^𝑗_𝑡+1𝑆^𝑗_𝑡+1∣𝑆^𝑗_𝑡=𝑠_𝑡,𝑖=1,2,…,𝑚;𝑡=1,2,…,𝑇, (14) subject to the constraints (5).

Equation (14) can be rewritten as follows: 𝑍^𝜋𝑗𝑡_𝑡𝑆^𝑗_𝑡=min_𝑥𝑡^𝑚𝑞_𝑖=1𝑝=1Φ_𝑡^𝑖,𝑗𝑥^𝜋𝑗𝑡_𝑝,𝑡𝑆^𝑗_𝑡−1+𝛽𝐸_𝑡𝑍^𝑗_𝑡−1𝑆^𝑗_𝑡, (15) subject to ^𝑚𝑞_𝑖=1𝑝=1𝑥^𝜋𝑗𝑡_𝑝,𝑡𝑆^𝑓𝑖≥^𝑛𝑞_𝑗=1𝑝=1𝑥^𝜋𝑗𝑡_𝑝,𝑡𝑆^𝑠𝑗,𝑥^𝜋𝑗𝑡_𝑝,𝑡𝑆^𝑓𝑖,𝑥^𝜋𝑗𝑡_𝑝,𝑡𝑆^𝑠𝑗∈𝑋,𝑡∈[0,𝑇],𝑝=1,2,…,𝑞._𝑞𝑝=1𝑥^𝜋𝑗𝑡_𝑝,𝑡𝑆^𝑓𝑖≥^𝑞_𝑝=1𝑥^𝜋𝑗𝑡_𝑝,𝑡𝑆^𝑠𝑗,𝑥^𝜋𝑗𝑡_𝑝,𝑡𝑆^𝑓𝑖,𝑥^𝜋𝑗𝑡_𝑝,𝑡𝑆^𝑠𝑗∈𝑋,𝑡∈[0,𝑇],𝑝=1,2,…,𝑞. (16)(17) Equivalently, ^𝑚𝑞_𝑖=1𝑝=1𝑥^𝜋𝑗𝑡_𝑝,𝑡𝑆^𝑓𝑖=^𝑛𝑞_𝑗=1𝑝=1𝑥^𝑠𝑗_𝑝,𝑡+𝛿_𝑡,𝑥^𝑓,𝑥^𝑠𝑥∈𝑋,𝑡∈𝑇,𝑖=1,2,…,𝑚;𝑗=1,2,…,𝑛,_{𝑓𝑝,𝑡},𝑥^𝑠_𝑝,𝑡≥0,𝑝=1,…,𝑚._𝑚𝑖=1𝑥^𝜋𝑗𝑡_𝑝,𝑡𝑆^𝑓𝑖=^𝑛_𝑗=1𝑥^𝑠𝑗_𝑝,𝑡+𝛿_𝑡^𝑖,𝑗,𝑥^𝑓,𝑥^𝑠𝑥∈𝑋,𝑡∈𝑇,𝑖=1,2,…,𝑚;𝑗=1,2,…,𝑛,_{𝑓𝑝,𝑡},𝑥^𝑠_𝑝,𝑡≥0,𝑝=1,…,𝑚, (18)(19) where 𝛿_𝑡^𝑖,𝑗∈ℝ^𝑚 is the quantity of products that is lost in transit from factory 𝑖 to store 𝑗 at period 𝑡 and 𝛿_𝑡=∑^𝑚_𝑖=1𝛿_𝑡^𝑖,𝑗∈ℝ. If 𝛿_𝑡^𝑖,𝑗 is a zero vector, it implies that all the quantities of products that left the factories get to the stores successfully without damages or shortage. It can be shown that (6) is equal to (14) (see Powell, [22]). We may use (6) and (14) interchangeably. We now find the best control policy, 𝜇, which minimizes our problem; that is, we search for 𝑍^∗_𝑡𝑆_𝑡=min_{𝜋𝑗𝑡∈𝜇}𝑍^𝜋𝑗𝑡_𝑡𝑆_𝑡,𝑡=1,2,…,𝑇;𝑆_𝑡=𝑆^𝑗_𝑡. (20) We do that by solving the optimality equation 𝑍^𝜋𝑗𝑡_𝑡𝑆^𝑗_𝑡=min_{𝑥𝑓𝑡,𝑥𝑠𝑡}^𝑚_∈𝑋^𝑞_𝑖=1𝑝=1Φ_𝑡^𝑖,𝑗𝑥^𝜋𝑗𝑡_𝑝,𝑡𝑆^𝑗_𝑡−1+𝛽𝐸_𝑡𝑍^𝜋𝑗𝑡_𝑡−1𝑆^𝑗_𝑡, (21) subject to ^𝑞_𝑝=1𝑥^𝜋𝑗𝑡_𝑝,𝑡𝑆^𝑓𝑖=^𝑞_𝑝=1𝑥^𝜋𝑗𝑡_𝑝,𝑡𝑆^𝑠𝑗+𝛿_𝑡^𝑖,𝑗,𝑥^𝜋𝑗𝑡_𝑝,𝑡𝑆^𝑓𝑖,𝑥^𝜋𝑗𝑡_𝑝,𝑡𝑆^𝑠𝑗𝑥∈𝑋,𝑡∈𝑇,𝑖=1,2,…,𝑚,𝑗=1,2,…,𝑛,_{𝑓𝑝,𝑡},𝑥^𝑠_𝑝,𝑡≥0,𝑝=1,…,𝑚. (22) Let Φ(𝑆^𝑓_𝑡,𝜋^𝑗_𝑡,𝑆^𝑠_𝑡)=∑^𝑚_𝑖=1∑^𝑞_𝑝=1Φ_𝑡^𝑖,𝑗(𝑥^𝜋𝑗𝑡_𝑝,𝑡(𝑆^𝑗_𝑡−1)) be the expected cost per stage. Using the mapping Υ∶𝑍→ℝ^𝑚 and Υ_𝜇∶𝑍^𝜇→ℝ^𝑚, we have (Υ𝑍)𝑆^𝑗_𝑡=min_{𝑆𝜋∈𝜇𝑡}Φ𝑆^𝑓_𝑡,𝜋^𝑗_𝑡,𝑆^𝑠_𝑡+𝛽𝐸_𝑡𝑍𝑆^𝑗_𝑡Υ,𝑗=1,2,…,𝑛,_𝜇𝑍𝑆^𝑗_𝑡=min_{𝑆𝜋∈𝜇𝑡}Φ𝑆^𝑓_𝑡,𝜋^𝑗_𝑡,𝑆^𝑠_𝑡+𝛽𝐸_𝑡𝑍^𝜇𝑆^𝑗_𝑡,𝑗=1,2,…,𝑛. (23)

Theorem 3. Consider the following.(i)Let 𝐵(𝑆) be the set of all bounded real-valued functions 𝑍∶𝑆→ℝ^𝑚. The mapping Υ∶𝐵(𝑆)→𝐵(𝑆) is a contraction.(ii)The operator Υ has a unique fixed point (given 𝑍^∗).(iii)For any 𝑍, Υ^∞𝑍=𝑍^∗.(iv)For any 𝑍, if Υ𝑍≤𝑍, then Υ^𝑡𝑍≥𝑍^∗, ∀𝑡={1,2,…}.

Proof. (See [22]).

We now make the following assumptions.

Assumption 4. Consider the following. (i)𝐸^𝑚𝑞_𝑖=1𝑝=1Φ_𝑡^𝑖,𝑗𝑥^𝜋𝑗𝑡_𝑝,𝑡𝑆^𝑗_𝑡≤Λ_𝑗,𝑗=1,2,…,𝑛. (24) (ii) The discount factor 0<𝛽<1.

Theorem 5. Let the bounded optimal cost function 𝑍∶𝑆→ℝ^𝑚 be 𝑚-dimensional vectors. Then 𝑍 satisfies 𝑍^∗𝑆^𝑗₀=lim_𝑇→∞Υ^𝑇𝑍𝑆^𝑗₀,∀𝑆^𝑗₀∈𝑆. (25)

Proof. Let 𝑦 be a positive integer, 𝑆^𝑗₀∈𝑆 and policy 𝜇={𝜋¹,𝜋²,…,𝜋^𝑛}; we can decompose the return 𝑍^𝜇𝑆^𝑗₀=lim_𝑇→∞𝐸^𝑇−1𝑚_𝑡=0^𝑞_𝑖=1𝑝=1𝛽^𝑡Φ_𝑡^𝑖,𝑗𝑥^𝜋𝑗𝑡_𝑝,𝑡𝑆^𝑗_𝑡 (26) into the portion received over the first 𝑦 stages and over the remaining stages: 𝑍^𝜇𝑆^𝑗₀=lim_𝑇→∞𝐸^𝑦−1𝑚_𝑡=0^𝑞_𝑖=1𝑝=1𝛽^𝑡Φ_𝑡^𝑖,𝑗𝑥^𝜋𝑗𝑡_𝑝,𝑡𝑆^𝑗_𝑡+lim_𝑇→∞𝐸^𝑇−1𝑚_𝑡=𝑦^𝑞_𝑖=1𝑝=1𝛽^𝑡Φ_𝑡^𝑖,𝑗𝑥^𝜋𝑗𝑡_𝑝,𝑡𝑆^𝑗_𝑡. (27) But lim_𝑇→∞𝐸^𝑇−1𝑚_𝑡=𝑦^𝑞_𝑖=1𝑝=1𝛽^𝑡Φ_𝑡^𝑖,𝑗𝑥^𝜋𝑗𝑡_𝑝,𝑡𝑆^𝑗_𝑡≤Λ_𝑗^∞_𝑡=𝑦𝛽^𝑡=𝛽^𝑦Λ_𝑗.1−𝛽 (28) Therefore, 𝑍^𝜇𝑆^𝑗₀≤lim_𝑇→∞𝐸^𝑦−1𝑚_𝑡=0^𝑞_𝑖=1𝑝=1𝛽^𝑡Φ_𝑡^𝑖,𝑗𝑥^𝜋𝑗𝑡_𝑝,𝑡𝑆^𝑗_𝑡+𝛽^𝑦Λ_𝑗.1−𝛽 (29) It therefore follows that 𝑍^𝜇𝑆^𝑗₀−𝛽^𝑦Λ_𝑗1−𝛽−𝛽^𝑦inf_{𝑆𝑗0∈𝑆}Φ𝑆^𝑗₀≤𝐸𝛽^𝑦Φ𝑆^𝑗₀+^𝑦−1𝑚_𝑡=0^𝑞_𝑖=1𝑝=1𝛽^𝑡Φ_𝑡^𝑖,𝑗𝑥^𝜋𝑗𝑡_𝑝,𝑡𝑆^𝑗_𝑡≤𝑍^𝜇𝑆^𝑗₀+𝛽^𝑦Λ_𝑗1−𝛽+𝛽^𝑦inf_{𝑆𝑗0∈𝑆}Φ𝑆^𝑗₀. (30) By taking the infimum over 𝜇, we obtain for all 𝑆^𝑗₀ and 𝑦𝑍^𝜇𝑆^𝑗₀−𝛽^𝑦Λ_𝑗1−𝛽−𝛽^𝑦inf_{𝑆𝑗0∈𝑆}Φ𝑆^𝑗₀≤Υ^𝑦𝑍𝑆^𝑗₀≤𝑍^𝜇𝑆^𝑗₀+𝛽^𝑦Λ_𝑗1−𝛽+𝛽^𝑦inf_{𝑆𝑗0∈𝑆}Φ𝑆^𝑗₀. (31) and by taking the limit as 𝑦→∞, we obtain 𝑍^∗𝑆^𝑗₀≤Υ^𝑦𝑍𝑆^𝑗₀≤𝑍^∗𝑆^𝑗₀. (32) Hence, 𝑍^∗𝑆^𝑗₀=lim_𝑦→∞Υ^𝑦𝑍𝑆^𝑗₀,∀𝑆^𝑗₀∈𝑆. (33) This result shows that our optimization problem converges to a fixed point 𝑍^∗ in an infinite horizon. It therefore follows that 𝑍^∗𝑆^𝑗₀=min_{𝑥𝜋𝑗𝑡𝑝,𝑡∈𝑋}𝐸^𝑚𝑞_𝑖=1𝑝=1Φ_𝑡^𝑖,𝑗𝑥^𝜋𝑗𝑡_𝑝,𝑡𝑆^𝑗₀+𝛽𝑍^∗𝑥^𝜋𝑗𝑡_𝑝,𝑡𝑆^𝑗₀,∀𝑆^𝑗₀∈𝑆. (34)

4. Dynamic Programming Principles with Error Bounds

In this section, the optimality equation with error bounds is considered. We established the minimum and maximum error bounds of our problem. We show that at infinity the minimum and maximum error bounds cancelled out. Let 𝑍^𝜇𝑆^𝑗_𝑡=Φ_𝜇𝑆^𝑗_𝑡,𝜇𝑆^𝑗_𝑡+^∞_𝑡=1𝛽^𝑡𝐸Φ𝑆^𝑗_𝑡,𝜇𝑆^𝑗_𝑡∣𝑆^𝑗₀. (35) But Λ^′_𝑗≤Φ𝑆^𝑗_𝑡,𝜇𝑆^𝑗_𝑡≤Λ^′′_𝑗,𝑗=1,2,…,𝑛,lim_𝑇→∞𝛽^𝑡𝐸Φ𝑆^𝑗_𝑡,𝜇𝑆^𝑗_𝑡≤Λ∑_𝑗^∞_𝑡=𝑇𝛽^𝑡=𝛽^𝑇Λ_𝑗.1−𝛽 (36) It then follows that Φ_𝜇+𝛽Λ^′_𝑗1−𝛽≤𝑍^𝜇≤Φ_𝜇+𝛽Λ^′′_𝑗,1−𝛽 (37) where Λ^′_𝑗 and Λ^′′_𝑗 are such that Λ^′_𝑗=min_𝑆𝑗𝑡Φ𝑆^𝑗_𝑡,𝜇𝑆^𝑗_𝑡.Λ^′′_𝑗=max_𝑆𝑗𝑡Φ𝑆^𝑗_𝑡,𝜇𝑆^𝑗_𝑡. (38) Equation (37) can be improved upon as follows: Λ^′_𝑗1−𝛽≤Φ_𝜇+𝛽Λ^′_𝑗1−𝛽≤𝑍^𝜇≤Φ_𝜇+𝛽Λ^′′_𝑗≤Λ1−𝛽^′′𝑗.1−𝛽 (39) For a vector 𝑍, we compute Υ_𝜇𝑍=Φ_𝜇+𝛽𝑃_𝜇𝑍𝑍,_𝜇=Φ_𝜇+𝛽𝑃_𝜇𝑍^𝜇. (40)(41) By subtracting (40) and (41), we have 𝑍^𝜇=Υ_𝜇𝑍+𝛽𝑃_𝜇𝑍^𝜇−𝑍. (42)𝑍^𝜇 is the cost vector associated with the control policy 𝜇 and 𝑍 is the cost per stage vector. It then follows from (37) that for 𝜂^′=min_𝑆𝑗𝑡Υ_𝜇𝑍𝑆^𝑗_𝑡−𝑍𝑆^𝑗_𝑡,𝜂^′′=max_𝑆𝑗𝑡Υ_𝜇𝑍𝑆^𝑗_𝑡−𝑍𝑆^𝑗_𝑡, (43) we have 𝜂^′1−𝛽≤Υ_𝜇𝑍−𝑍+𝛽𝜂^′1−𝛽≤𝑍^𝜇−𝑍≤Υ_𝜇𝑍−𝑍+𝛽𝜂^′′≤𝜂1−𝛽^′′.1−𝛽 (44) Equivalently, for the vector 𝑍, we have 𝜗𝑍+^′𝛽≤Υ_𝜇𝑍+𝜗^′≤𝑍^𝜇≤Υ_𝜇𝑍+𝜗^′′𝜗≤𝑍+^′′𝛽, (45) where 𝜗^′=𝛽𝜂^′/(1−𝛽) and 𝜗^′′=𝛽𝜂^′′/(1−𝛽). Equation (45) can be improved upon as follows. From (40), we have 𝜂^′=min_𝑆^𝑗_𝑡((Υ𝑍)(𝑆^𝑗_𝑡)−𝑍(𝑆^𝑗_𝑡)). It follows that 𝑍+𝜂^′≤Υ𝑍. (46) From (46), we obtain ΥΥ𝑍−𝜂^′=min_𝜋𝑗𝑡Φ^𝜋𝑗𝑡_𝑡+𝛽𝑃_𝜋^𝑗_𝑡Υ𝑍−𝜂^′=min_𝜋𝑗𝑡Φ^𝜋𝑗𝑡_𝑡+𝛽𝑃_𝜋^𝑗_𝑡Υ𝑍−𝛽𝜂^′=Υ²𝑍−𝛽𝜂^′. (47)(48)(49) Now, combining (46) and (47), we have 𝑍+1−𝛽𝛽𝜂^′≤Υ²𝑍+𝛽²𝜂^′≤Υ³𝑍. (50) Substituting (45) into (50), we have 𝑍+1+𝛽+𝛽²𝜂^′≤Υ𝑍+𝛽+𝛽²𝜂^′≤Υ²𝑍+𝛽²𝜂^′≤Υ³𝑍. (51) After 𝑘 steps, we have 𝑍+^𝑘_𝑖=0𝛽^𝑖𝜂^′≤Υ𝑍+^𝑘_𝑖=1𝛽^𝑖𝜂^′≤Υ²𝑍+^𝑘_𝑖=2𝛽^𝑖𝜂^′≤⋯≤Υ^𝑘+1𝑍. (52) As 𝑘→∞, we have 𝑍+𝛽𝜂^′𝛽1−𝛽≤Υ𝑍+𝛽𝜂^′1−𝛽≤Υ²𝛽𝑍+²𝜂^′1−𝛽≤⋯≤𝑍^∗. (53) Let 𝜗^′₁=𝛽𝜂^′/(1−𝛽); we have 𝜗𝑍+^′1𝛽≤Υ𝑍+𝜗^′₁≤Υ²𝑍+𝛽𝜗^′₁≤𝑍^∗. (54) Let 𝑍=Υ^𝑘+1𝑍 in (54); we have Υ^𝑘+1𝑍+𝜗^′_𝑘+1≤𝑍^∗. (55) From (46), we have that min_𝑆^𝑗_𝑡((Υ²𝑍)(𝑆^𝑗_𝑡)−(Υ𝑍)(𝑆^𝑗_𝑡))≥𝛽𝜂^′ and 𝛽𝜗^′₁=(𝛽²𝜂^′/(1−𝛽))≤𝜗^′₂. Therefore, we have Υ𝑍+𝜗^′₁≤Υ²𝑍+𝜗^′₂≤𝑍^∗. (56) It implies that Υ^𝑘𝑍𝑆^𝑗_𝑡+𝜗^′_𝑘≤Υ^𝑘+1𝑍𝑆^𝑗_𝑡+𝜗^′_𝑘+1≤𝑍^∗𝑆^𝑗_𝑡. (57) Similarly, 𝜂^′′=‖Υ𝑍−𝑍‖_∞. We therefore have that Υ^𝑘𝑍𝑆^𝑗_𝑡+𝜗^′_𝑘≤Υ^𝑘+1𝑍𝑆^𝑗_𝑡+𝜗^′_𝑘+1≤𝑍^∗𝑆^𝑗_𝑡≤Υ^𝑘+1𝑍𝑆^𝑗_𝑡+𝜗^′′_𝑘+1≤Υ^𝑘𝑍𝑆^𝑗_𝑡+𝜗^′′_𝑘. (58) This is the dynamic programming (value iteration) that is bounded above and below. The value (Υ^𝑘𝑍)(𝑆^𝑗_𝑡)+𝜗^′_𝑘, is the value iteration with minimum error bound and (Υ^𝑘𝑍)(𝑆^𝑗_𝑡)+𝜗^′′_𝑘 is the value iteration with maximum error bound. Now taking the limit as 𝑘 tends to infinity, we have that 𝜗^′_∞=𝜗^′′_∞=0; it now implies that lim_𝑘→∞Υ^𝑘𝑍𝑆^𝑗_𝑡+𝜗^′_𝑘=lim_𝑘→∞Υ^𝑘𝑍𝑆^𝑗_𝑡+𝜗^′′_𝑘=Υ^∞𝑍𝑆^𝑗, (59) and for 𝑘<∞, lim_𝑘→𝑘Υ^𝑘𝑍𝑆^𝑗_𝑡+𝜗^′_𝑘≠lim_𝑘→𝑘Υ^𝑘𝑍𝑆^𝑗_𝑡+𝜗^′′_𝑘. (60)

5. Numerical Simulation

In this section, we present the numerical simulation of the company problem. The state transition diagram of the distribution of goods from factories to stores, under the control policies 𝜋¹,𝜋²,𝜋³,𝜋⁴,𝜋⁵,𝜋⁶,𝜋⁷,𝜋⁸, is given by the transition diagram in Figure 1.

Figure 1 

Transition diagram for the distribution of goods from factories to stores.

The transition probability matrix of distributing the goods from factory 𝑖 to store 𝑗 under the control policies 𝜋¹,𝜋²,𝜋³,𝜋⁴,𝜋⁵,𝜋⁶,𝜋⁷,𝜋⁸, is presented below: 𝑃_𝑖𝑗𝜋^𝑗=734014014051110120418181411201105134074014057140110120418134014011081431401405714032014057140418111012014014031014081720140514141120817403401401105112014057340140418110. (61)

The initial estimate of the cost of distributing the goods from factory 𝑖 to store 𝑗 in 100000 is given below.Φ₀^𝑖𝑗=120.0456.81007580.5909299.0411080.91206065707588.978.91056413284.2939587.0959.814092112.8626975.378.110810185.572.0576.566.465.399.05105.6112.690.180.479.567.55975.9112.03109.280.25807775.269.977.08113111.0197.879.989.560.892.0277.09. (62)

The company estimates 𝛾=5% of the total goods as shortage in transit and 𝛼=70% of the total expected shortage to be recovered and should be deducted from the total cost of the shipment. The cost of the damaged goods is added to the cost of the shipment. MATLAB was used to solve the problem. The results are presented in the tables.

The optimal discounted costs of distributing the goods from factories to stores with shortages are presented in Tables 1 to 7 and Figures 2 and 3, while optimal discounted costs of distributing the goods from factories to stores without shortages are presented in Tables 8 to 10 and Figure 5. Table 1 presents the minimum costs of distributing the goods from various factories to all the stores. Here, error bound is not considered. It was obtained that the minimum costs of distributing the goods from factory 1 to all the eight stores are 5,716,200 naira. The minimum costs of distributing the goods from factory 2 to the eight stores are obtained as 6,034,000 naira, from factory 3 to the eight stores are 6,428,400 naira, and so on. It was observed that the costs of distributing the goods from factory 7 to all the stores yield the highest amount. This may be due to longer distance, nature of the roads, or more shortage costs.

Table 1 

ODCDG without error bound (EB).

Figure 2 

The bar chart representing the discounted minimum costs of the distribution with 8 different policies for 𝛽=99%, 𝛼=70%, and 𝛾=5%.

Figure 3 

The bar chart representing the discounted minimum cost of the distribution with 8 different policies for 𝛽=99%, 𝛼=0.0%, and 𝛾=5%. This occurs when there is no recovery of the goods.

Table 2 presents the optimal costs of distributing goods with minimum error bound from the factories to stores and Table 3 presents the optimal costs of distributing the goods with maximum error bound from the factories to stores. It was observed that the minimum costs of distributing the goods with minimum and maximum error bounds coincide only at infinity.

Table 2 

ODCDG with minimum EB.

Table 3 

ODCDG with maximum EB.

Table 4 shows the optimal costs of distributing the goods without error bound and where there is no recovery of some damaged goods (i.e., when 𝛼=0). It is observed from Tables 1 and 4 that the costs of distributing the goods will increase when the value of 𝛼 decreases and vice versa. This is an expected result. Tables 5 and 6 show the optimal costs of distributing the goods with minimum and maximum error bounds, respectively, and where there is no recovery of some damaged goods.

Table 4 

ODCDG where there is no recovery; that is, when 𝛼=0.

Table 5 

ODCDG with minimum EB and where there is no recovery.

Table 6 

ODCDG with maximum EB where there is no recovery.

Table 7 

Optimal discounted costs of recovered goods.

Table 8 

ODCDG without shortages.

The additional costs incurred and the corresponding percentage increase in the additional costs incurred when 𝛼=0 are shown in Table 7.

In Table 7, it was found that the costs of recovered goods in transit from factory 1 to the stores are 228,600 naira with 3.85% when there is no EB and 220000 naira with 3.70% when there is EB from factory 2 to the stores 230,800 with 3.68% when there is no EB, and 240,000 with 3.83% when there is EB and so on.

Figure 2 shows the bar chart representing the minimum costs of distributing the goods from factories to stores by setting 𝛽=99%, 𝛼=70%, and 𝛾=5%. It was observed from Figure 2 that optimum costs can be attained under policy 1. Hence, the optimum policy for the operation is 𝜋¹. We therefore deduce from Figure 2 that 𝜋¹<𝜋⁶<𝜋⁴<𝜋²<𝜋⁸<𝜋³<𝜋⁵<𝜋⁷. Figure 3 shows the bar chart representing the minimum costs of distributing the goods from factories to stores by setting 𝛽=99%, 𝛼=0.0%, and 𝛾=5%. This is when there are no goods recovered from the expected lost ones in transit. We also found that policy 1 yields the optimum costs. Figure 4 shows the relationship between the minimum and maximum error bound. It was observed that at infinity, the minimum and maximum error bounds coincide. We referred to this point as the point of harmony. Comparing the optimal discounted costs of the distribution of goods in Tables 1, 8, 2, 9, 3, and 10 and Figures 2 and 5, we found that the optimal cost is at minimal when there are no shortages, which is an expected result. It is obvious that the higher the shortages, the higher the cost of the shipment. This may lead to the high cost of the goods on the final consumers.

Table 9 

ODCDG without shortage with minimum EB.

Table 10 

ODCDG without shortage with maximum EB.

Figure 4 

The maximum and minimum error bound showing the point of their intersection “point of harmony” at 𝛽=99%, 𝛼=70%, and 𝛾=5%.

Figure 5 

The bar chart representing the discounted minimum cost of the distribution with 8 different policies for 𝛽=99%, 𝛼=0.0%, and 𝛾=0.0%. This is when the goods get to their various stores successfully without shortages.

6. Conclusion

We have considered distribution of goods with stochastic shortages from factories to stores. This paper modeled the distribution of goods from factories to stores in the presence of stochastic shortages of the goods in transit, and the deterministic case (i.e., when there are no shortages) was also considered, using a dynamic optimization technique. Numerical determination of optimal cost and optimal control policy for the operations were obtained. We obtained the optimal transportation cost of shipping goods from factories to stores. The optimal control policies were also obtained. We found that the optimal costs of the goods recovered could be determined. It was found that the optimum costs of distributing the goods with minimum and maximum error bounds coincide only at infinity. We referred to this point as a “point of harmony.”