Purpose To Assess Your Ability To Apply Linear Programming

Purposeto Assess Your Ability To Apply Linear Programming Techniques T

Purpose To assess your ability to apply linear programming techniques to solve a non-uniform distribution business problem. Overview This is a statistical scheduling problem to be completed individually. Action Items 1. Complete Problem 8-3 in Quantitative Analysis. 8-3 ISM 6407 Fall (Restaurant work scheduling problem). The famous Y. S. Chang Restaurant is open 24hours a day. Waiters and busboys report for duty at 3AM., 7 AM., 11 AM., 3 P.M., 7 P.M., or 11P.M., and each works an 8-hour shift. The following table shows the minimum number of workers needed during the six periods into which the day is divided. Chang’s scheduling problem is to determine how many waiters and busboys should report for work at the start of each time period to minimize the total staff required for one day’s operation. (Hint: Let Xi equal the number of waiters and busboys beginning work in time period i, where i = 1, 2,3,4,5,6.) 2. Individually complete your assignment, responding to the listed questions. Submission Instructions · Complete and submit your assignment per your professor's instructions. 1-Purpose To assess your ability to describe the principles of inventory control. Action Items 1. Complete Discussion Questions 6-1 and 6-2 on page 238 in Quantitative Analysis. Completely answer the questions in a one- to two-page composition format. 6-1 Why is inventory an important consideration for managers? 6-2 what is the purpose of inventory control? 2- Purpose To assess your ability to calculate economic order quantity (EOQ) and re-order point (ROP). Action Items 2. Review Case 6-21 in Chapter 6 of Quantitative Analysis. 3. Completely answer the following questions listed at the end for the problem exercise: Compute the EOQ. Compute the ROP. Compute the optimal number of orders per year. Compute the total annual cost for Western valves. 6-21 Barbara Bright is the purchasing agent for West Valve Company. West Valve sells industrial valves and fluid control devices. One of the most popular valves is the Western, which has an annual demand of 4,000 units. The cost of each valve is $90, and the inventory carrying cost is estimated to be 10% of the cost of each valve. Barbara has made a study of the costs involved in placing an order for any of the valves that West Valve stocks, and she has concluded that the average ordering cost is $25 per order. Furthermore, it takes about two weeks for an order to arrive from the supplier, and during this time the demand per week for West valves is approximately 80. (a) What is the EOQ? (b) What is the ROP? (c) What is the average inventory? What is the annual holding cost? (d) How many orders per year would be placed? What is the annual ordering cost?

Paper For Above instruction

Linear programming (LP) is an essential mathematical technique used nationally and internationally by organizations to optimize resource allocation under constraints. This methodology has been instrumental in solving complex operational problems, including scheduling, inventory management, and logistics planning, which require maximizing or minimizing an objective function subject to a set of linear constraints (Winston, 2004). The application of LP facilitates decision-makers in developing optimal strategies by providing quantitative insights into complex, multi-variable problems, thus supporting efficient and cost-effective operations.

One of the practical business problems suited for linear programming techniques is employee scheduling, as illustrated in the case of the Y. S. Chang Restaurant, which operates 24 hours daily. The challenge is determining how many waiters and busboys should report for duty at specific times to meet minimal staffing requirements while minimizing overall staffing costs. This problem can be effectively modeled using LP, where the decision variables represent the number of staff starting shifts in each period, and the objective function minimizes total staff needed. Constraints include staffing minimums per period and logical relationships that ensure staff scheduled at a later time is adequate to cover previous shifts, thus maintaining continuous coverage (Hillier & Lieberman, 2015).

The LP model for Chang's schedule involves defining six decision variables, each representing the number of workers starting work in one of the six scheduled periods (3 AM, 7 AM, 11 AM, 3 PM, 7 PM, 11 PM). The constraints ensure that the sum of staff from current and previous shifts satisfies the minimum workforce needs for each time interval (McConnell et al., 2017). Solving this LP model traditionally involves using simplex algorithm implementations in optimization software, which efficiently provides the optimal number of employees to report for each shift, thereby minimizing total staff costs.

Beyond scheduling, inventory control is another domain where linear programming proves valuable. Inventory management involves balancing holding costs, ordering costs, and stockout risks. The economic order quantity (EOQ) model exemplifies this, calculating the optimal order size to minimize total inventory costs (Harris, 1913). For example, in the case of West Valve Company, the EOQ formula calculates the ideal order quantity that minimizes combined ordering and holding expenses, given demand, ordering costs, and carrying costs (Silver, Pyke, & Peterson, 1998). Once EOQ is determined, the reorder point (ROP) indicates when new stock should be ordered based on lead time demand, ensuring that stock is replenished timely without excess inventory.

Calculating EOQ involves using the classic formula: EOQ = √(2DS / H), where D is annual demand, S is ordering cost per order, and H is holding cost per unit annually (Harris, 1913). Applying this to West Valve, with demand of 4,000 units, ordering cost of $25, and holding cost of 10% of the unit cost ($9), the EOQ is approximately 471 units. The ROP accounts for demand during lead time: ROP = demand per week × lead time in weeks, resulting in an ROP of 160 units, considering a two-week lead time at a demand of 80 units per week. These calculations help streamline inventory replenishment, reducing both stockouts and excess stock (Nahmias, 2013).

Furthermore, linear programming aids in cost minimization for batch or lot sizing, considering multiple constraints. When applied to employee shifts, inventory replenishment, or transportation logistics, LP offers structured solutions that save costs and optimize resource use (Taha, 2017). These techniques underscore the importance of mathematical modeling in enhancing managerial decisions, reducing operational costs, and improving service levels, demonstrating their applicability across diverse operational domains.

References

• Harris, F. W. (1913). How many parts to make at once. Factory, The Magazine of Management, 10(2), 135-136.
• Hillier, F. S., & Lieberman, G. J. (2015). Introduction to Operations Research (10th ed.). McGraw-Hill Education.
• McConnell, C. R., Brue, S. L., & Flynn, S. M. (2017). Economics: Principles, Problems, and Policies (20th ed.). McGraw-Hill Education.
• Nahmias, S. (2013). Production and Operations Analysis (6th ed.). Waveland Press.
• Silver, E. A., Pyke, D. F., & Peterson, R. (1998). Inventory Management and Production Planning and Scheduling. Wiley.
• Taha, H. A. (2017). Introduction to Operations Research (10th ed.). Pearson Education.
• Winston, W. L. (2004). Operations Research: Applications and Algorithms. Duxbury Press.