Joint pricing and production planning of multiple products

Many industries are beginning to use innovative pricing techniques to improve inventory control, capacity utilisation, and ultimately the profit of the firm. In manufacturing, the coordination of pricing and production decisions offers significant opportunities to improve supply chain performance by better matching supply and demand. This integration of pricing, production and distribution decisions in retail or manufacturing environments is still in its early stages in many companies. Importantly it has the potential to radically improve supply chain efficiencies in much the same way as revenue management has changed the management of the airline, hotel and car rental industries. These developments raise the need and interest of having models that integrate production decisions, inventory control and pricing strategies.In this thesis, we focus on joint pricing and production planning, where prices and production values are determined in coordination over a multiperiod horizon with non-perishable inventory. We specifically look at multiproduct systems with either constant or dynamic pricing. The fundamental problem is: when the capacity limitations and other parameters like production, holding, and backordering costs are given, what the optimal values are for production quantities, and inventory and backorder levels for each item as well as a price at which the firm commits to sell the products over the total planning horizon. Our aim is to develop models and solution strategies that are practical to implement for real sized problems.We initially formulate the problem of time-varying pricing and production planning of multiple products over a multiperiod horizon as a nonlinear programming problem. When backorders are not allowed, we show that if the demand/price function is linear, as a special case of the without backorders model, the problem becomes a Quadratic Programming problem which has only linear constraints. Existing solution methods for Quadratic Programming problem are discussed. We then present the case of allowed backorders. This assumption makes the problem more difficult to handle, because the constraint set changes to a non-convex set. We modify the nonlinear constraints to obtain an alternative formulation with a convex set of constraints. By this modification the problem becomes a Mixed Integer Nonlinear Programming problem over a linear set of constraints. The integer variables are all binary variables. The limitation of obtaining the optimal solution of the developed models is discussed. We describe our strategy to overcome the computational difficulties to solve the models.We tackle the main nonlinear problem with backorders through solving an easier case when prices are constant. This resulting model involves a nonlinear objective function and some nonlinear constraints. Our strategy to reduce the level of difficulty is to utilise a method that solves the relaxed problem which considers only linear constraints. However, our method keeps track of the feasibility with respect to the nonlinear constraints in the original problem. The developed model which is a combination of Linear Programming (LP) and Nonlinear Programming (NLP) is solved iteratively. The solution strategy for the constant pricing case constructs a tree search in breadth-first manner. The detailed algorithm is presented. This algorithm is practical to implement, as we demonstrate through a small but practical size numerical example.The algorithm for the constant pricing case is extended to the more general problem. More specifically, we reformulate the timevariant problem in which there are multi blocks of constant pricing problems. The developed model is a combination of Linear Programming (LP) and linearly constrained Nonlinear Programming (NLP) which is solved iteratively. Iterations consist of two main stages: finding the value of LP’s objective function for a known basis, solving a very smaller size NLP problem. The detailed algorithm is presented and a practical size numerical example is used to implement the algorithm. The significance of this algorithm is that it can be applied to large scale problems which are not easily solved with the existing commercial packages

A heuristic algorithm for optimal fleet composition with vehicle routing considerations

This paper proposes a fast heuristic algorithm for solving a combined optimal fleet composition and multi-period vehicle routing problem. The aim of the problem is to determine an optimal fleet mix, together with the corresponding vehicle routes, to minimize total cost subject to various customer delivery requirements and vehicle capacity constraints. The total cost includes not only the fixed, variable, and transportation costs associated with operating the fleet, but also the hiring costs incurred whenever vehicle requirements exceed fleet capacity. Although the problem under consideration can be formulated as a mixed-integer linear program (MILP), the MILP formulation for realistic problem instances is too large to solve using standard commercial solvers such as the IBM ILOG CPLEX optimization tool. Our proposed heuristic decomposes the problem into two tractable stages: in the first (outer) stage, the vehicle routes are optimized using cross entropy; in the second (inner) stage, the optimal fleet mix corresponding to a fixed set of routes is determined using dynamic programming and golden section search. Numerical results show that this heuristic approach generates high-quality solutions and significantly outperforms CPLEX in terms of computational speed

Cargo scheduling decision support for offshore oil and gas production: a case study

Woodside Energy Ltd (Woodside), Australia’s largest independent oil and gas company, operates multiple oil and gas facilities off the coast of Western Australia. These facilities require regular cargo shipments from supply vessels based in Karratha, Western Australia. In this paper, we describe a decision support model for scheduling the cargo shipments to minimize travel cost and trip duration, subject to various operational restrictions including vessel capacities, cargo demands at the facilities, time windows at the facilities, and base opening times. The model is a type of non-standard vehicle routing problem involving multiple supply vessels—a primary supply vessel that visits every facility during a round trip taking at most 1 week, and other supply vessels that are used on an ad hoc basis when the primary vessel cannot meet all cargo demands. We validate the model via test simulations using real data provided by Woodside

Adapted cross entropy method to investigate costly price-changes in pricing and production planning

In this paper, we propose an adaptation of the cross entropy (CE) method, called ACE, to solve an integrated production planning and pricing problem in which price change is not costless. More specifically, we consider a firm with a common production capacity shared amongst multiple products. We show that by using a chance constrained approach we can convert the case of uncertain price dependent demand to a model similar to the deterministic case. Both systems of fixed and variable price change costs are studied. This problem arises in the management of manufacturing systems where it is necessary to find a policy that is both economical and operational from the production perspective. The above problem is mathematically formulated as a mixed integer nonlinear program. Solving such problems is algorithmically very challenging. In fact, commercial codes fail to solve or even find a feasible solution to realistic size problems. The challenge originates from the fact that an optimum should be found despite the difficulty of finding even a feasible solution. The ACE method shows promise in solving optimisation problems regardless of continuity or other assumptions. In our approach, we sample the integer variables using the CE mechanism, and solve simplified nonlinear programming problems (NLP) to obtain corresponding continuous variables. Numerical results, on a range of test problems with sufficient complexity to reflect the difficulty of practical size problems demonstrate the effectiveness of our methodology

Joint Pricing and Production Planning of Multi-period Multi-product Systems with Uncertainty in Demand

In this paper, we present a multiperiod model for production planning in a make-to-stock manufacturing system with constant pricing under uncertainty. We consider a multiproduct capacitated setting and introduce a demand-based model where the demand is a function of the price. There is an assumption that the production setup costs are negligible. A key part of the model is that the uncertain price / demand function is chosen from a discrete set of scenarios. As a result of this, the problem becomes a non linear programming problem with the nonlinearities only in the objective function. We develop a robust optimization model for this problem that considers the optimality and feasibility of all scenarios. The robust solution is obtained by solving a series of nonlinear programming problems. We illustrate our methodology with detailed numerical examples

Optimal Pricing and Production Planning for Multi-product Multi-period Systems with Backorders

In this paper, we develop models for production planning with coordinated dynamic pricing. The application that motivated this research is manufacturing pricing, where the products are non-perishable assets and can be stored to fulfill the future demands. We assume that the firm does not change the price list very frequently. However, the developed model and its solution strategy have the capability to handle the general case of manufacturing systems with frequent time-varying price lists. We consider a multi-product capacitated setting and introduce a demand-based model, where the demand is a function of the price. The key parts of the model are that the planning horizon is discrete-time multi-period, and backorders are allowed. As a result of this, the problem becomes a nonlinear programming problem with the nonlinearities in both the objective function and some constraints. We develop an algorithm which computes the optimal production and pricing policy on a finite time horizon. We illustrate the application of the algorithm through a detailed numerical example

Joint Pricing and Production Planning for Fixed Priced Multiple Products with Backorders

In this paper, we present a multiperiod model for production planning in a make-to-stock manufacturing system with constant pricing. We consider a multiproduct capacitated setting and introduce a demand-based model where the demand is a function of the price. There is an assumption that the production setup costs are negligible. A key part of the model is that backorders are allowed. As a result of this, the problem becomes a nonlinear programming problem with the nonlinearities in both the objective function and some constraints. We develop an algorithm that computes the global optimal production and pricing policy on a finite time horizon. We illustrate the application of the algorithm through a detailed numerical example

Factors and scenarios affecting a farmer’s grain harvest logistics

This paper explores how changes in Australia’s grain industry supply chains are likely to impact on the nature and profitability of an Australian farmer’s grain harvest logistics. A simulation model is used to show how receival site rationalisation, cheaper on-farm storage, larger trucks, higher-yielding crops and new harvest technologies, separately and in combination, affect the nature and profitability of a farmer’s grain harvest logistics. Applying the model to a typical Australian grain farm shows that many of these changes unambiguously advantage the farm business, and often, the combination of these changes increases a farmer’s harvest profits by at least 10 per cent. For many farmers, the task of efficiently designing and managing harvest logistics will be an increasingly difficult yet important series of choices due to the range of storage options, grain pathways, crop portfolios and market opportunities that are arising. A farmer’s decisions about cost-effective on-farm storage and transport, and their judicious use, will be a key contributor to additional profit in future years

Robust optimisation model for the cold food chain logistics problem under uncertainty

In the last two decades, food safety has become one of the main concerns in the area of logistics and supply chain management and also in cold chain. Safety is a critically sensitive area in this category as if the required safety conditions are not satisfied during the logistics process, foods will soon deteriorate and probably become unsafe to use by customers. Thus, the problem of cold food safety has encouraged serious attentions among the logistics practitioners. However, because of the complexity in nature of such problems, research so far is limited to the quantitative models with deterministic parameters and the robustness of this nature still remains unanswered. In this paper, a robust optimisation model has been developed aiming to maximise the food safety aspects and thus to minimise the logistics cost of the cold chain system under various uncertainties and customers time windows restrictions. The model has been solved by artificial bee colony intelligence algorithm through MATLAB 8 software. Finally, the results are analysed for possible real world considerations in order to propose some key practical highlights

Approximate dynamic programming for an energy-efficient parallel machine scheduling problem

In this paper, we propose an approximate dynamic programming approach for an energy-efficient unrelated parallel machine scheduling problem. In this scheduling problem, jobs arrive at the system randomly, and each job’s ready and processing times become available when an order is placed. Therefore, we consider the online version of the problem. Our objective is to minimize a combination of makespan and the total energy costs. The energy costs include cost of energy consumption of machines for switching on, processing, and idleness. We propose a binary program to solve the optimization problem at each stage of the approximate dynamic program. We compare the results of the approximate programming approach against an integer linear programming formulation of the offline version of the scheduling problem and an existing heuristic method suitable for scheduling problem with ready times. The results show that the approximate dynamic programming algorithm outperforms the two off-line methods in terms of solution quality and computational time