Dr Luc Muyldermans During the next four months, Company ABC aims to meet the demands (shown in Table 1) for four types of products as much as possible on time. Backordering (i.e., satisfying the period demand by production in later periods) is not permitted and shortages (i.e., not being able to meet some of the period demand) may occur. Two machines are available to produce these products. Machine 1 can produce products 1, 2 and 3 and machine 2 can produce products 2, 3 and 4. Each machine can be used for up to 40 hours in months 1 and 4 and up to 30 hours in months 2 and 3. Table 2 shows the time required to produce one unit of each product (independent of the type of machine); the cost of producing one unit of each product on each type of machine; the cost of holding one unit of each product in inventory for one month; and the shortage cost (or penalty cost) per unit for each product for not being able to meet demand. There are no starting inventories and there are also no ending inventory requirements. Table 1: Product demand 

Month	Product 1	Product 2	Product 3	Product 4
1	50	20	70	80
2	60	30	90	120
3	45	40	50	150
4	70	60	40	60

Table 2: Production time and cost characteristics

Product	Production Time (minutes)	Production Cost ()	Holding Cost ()	Shortage Cost ()
Machine 1	Machine 2
1	30	4	--	0.15	15
2	20	4.3	6	0.10	15
3	16	7	3	0.8	10
4	15	--	5.5	0.5	14

Tasks:

2. Formulate a linear programming model that can be used to minimize the total cost of meeting demands as much as possible on time. Clearly explain the decision variables, objective function and constraints. (Note: this part is completely independent of the setup of the model in Excel.)
4. Enter your model in MS Excel and solve through Solver. Discuss the optimal production plan and indicate where and when inventory is carried over between months and/or when and for what products shortages occur. You must also provide a screenshot of the models layout (with solution) in Excel. A good layout in Excel will clearly show the structure of the LP formulation (all variables, constraints and coefficients).
6. Instead of minimizing the total cost, company ABC has formulated the following two ranked goals:

Rank (1) Minimise the total number of units short but ensure that the shortages for each product type separately are no more than 20% of the total demand of that product Rank (2) Minimise the total cost Modify your model and reformulate as a ranked goal programming model (only show in your report what changes are required); solve the goal programming model and compare its solution with the solution obtained from part (1). Please submit your report before the due date (25^th November, 3pm) electronically through Moodle. Your report should be submitted as a word or pdf document (do not submit Excel files). Note that this is an individual assignment and that you have to write up your report individually. References are not required for this coursework. Write your report concisely but make sure you add explanation for those parts/sections that need explanation. Good luck!