## 1 Introduction  
Currently, autonomous robotic systems are used in various processes in manufacturing. In the food industry, robotic systems are mainly used for food processing and packaging in mass-production plants [1], while cooking and meal preparation at home or in restaurants is one of the least automated areas. In restaurants that offer a variety of choices on their menu, efficient job assignments are critical to serving orders on time as task scheduling greatly affects efficiency [2]. Attempts have been made to automate cooking tasks using robots. Cooking motions, such as cutting and peeling vegetables [3], mixing and chopping [4], and rocking and flipping of pans [5], have been studied using robots. Another research [6] proposes the collaboration of two robots to cook, but is limited to making a single type of dish. In general, to make a recipe executable by a robot, the recipe has to be decomposed into sub-tasks. In the case of simple recipes, the order of operation can be specified by breaking down a written recipe into keywords, and the robot could cook according to the recipes [4]. However, for more complicated recipes, such as those with verbose descriptions, includingcomments from recipe creators [7], more sophisticated solutions are needed. Furthermore, since most of the available cooking robots perform a single task at a time [8; 9], multi-tasking, which is an essential part of efficiently cooking, remains an open problem. Such a task scheduling strategy is necessary to produce an optimal schedule that robots can follow to concurrently perform multiple tasks and complete a dish on time. In general, there are several methods to solve a task-planning problem. Specifically, planning domain description language (PDDL) [10; 11; 12] is one of the most popular task-planning languages. By predefining the object and its corresponding predicates(e.g., the affordances of an object), a domain can be established to describe its actions and the corresponding results. However, one disadvantage of PDDL is that the number of symbols and type are fixed [13], which makes it harder to generate domains in which the robot can discover or reconsider the type of objects, which is an essential requirement for many tasks such as cooking. Recently, attempts have been made to solve the task-planning problem with reinforcement learning or deep learning [14; 15]. However, machine learning relies on handcrafted heuristics for making decisions or has an expensive training cost [16]. In addition, deep learning usually requires tremendous training data, and optimal solutions are not always guaranteed. On the other hand, research to solve the decision-making process using Mixed Integer Programming(MIP) has been actively conducted [17; 18; 19; 20]. As a well-known flexible and powerful method for solving large problems consisting of integer constraints, MIP has seen recent developments as a computational resource in terms of speed and memory, and new and improved algorithms and preprocessing techniques [21] have further advanced its ability to be applied to more complex applications such as task planning. Naturally, attempts have been made to solve these scheduling problems using MIP, but most studies have focused on cases where the same operation is repeated [22; 23; 24]. Since tasks are cyclic, the optimized solution of a cycle needs to be calculated before its operation. Therefore, a preprogrammed task sequence cannot deal with situations where modifications are needed on the fly depending on changing circumstances. Moreover, while MIPs have been widely applied to other robotics applications, to the best of our knowledge, the applications of MIP in cooking robotics have been few.