ΠΜΣ: Μαθηματικά των Υπολογιστών και των Αποφάσεων. Μάθημα: Επιχειρησιακή Έρευνα Ακαδημαϊκό Έτος: 2012-13 Διδάσκων: Ν. Τσάντας Homework 4 1. Ασκήσεις: δείτε τις σελίδες 2-5 του παρόντος. 2. Πρόκειται για ατομικές ασκήσεις οι οποίες συνεισφέρουν το 25% του τελικού σας βαθμού. 3. Το παραδοτέο είναι ολόσωμο κείμενο Word όλων των ασκήσεων μαζί σε ελληνική γλώσσα μαζί με τα αρχεία δεδομένων των μοντέλων που κατασκευάσατε στο QSB ή στο POM-QM. 4. Όλα τα παραπάνω αρχεία θα πρέπει να συμπτυχθούν σε ένα zip αρχείο και να αποσταλούν ηλεκτρονικά (e-mail: tsantas@upatras.gr) στον διδάσκοντα μέχρι την 24.00 της 28ης Ιουλίου 2013. Στο μήνυμα πρέπει να αναφέρετε απαραίτητα το ονοματεπώνυμό σας. Εργασίες που παραλαμβάνονται εκπρόθεσμα επισύρουν βαθμολογικές κυρώσεις (0,5 βαθμό για κάθε ημερολογιακή ημέρα καθυστέρησης). Εργασίες που υποβάλλονται με καθυστέρηση μεγαλύτερη από 3 ημέρες δεν γίνονται δεκτές. 5. Το αρχείο Word θα το ονομάσετε με λατινικούς χαρακτήρες "eponymo_onoma_hw4.doc". Τα αρχεία δεδομένων θα τα ονομάσετε με τον αύξοντα αριθμό της αντίστοιχης άσκησης. 6. Στο κείμενο της εργασίας, για κάθε άσκηση, πρέπει να υπάρχει μια ενότητα με την εννοιολογική προσέγγιση, δηλαδή την περιγραφή του προβλήματος και των υποθέσεών του. Στη συνέχεια μια ενότητα με την ανάπτυξη του κατάλληλου μοντέλου και τέλος, μια ενότητα με τις απαντήσεις στα υπόλοιπα ερωτήματα της άσκησης. 7. Πίνακες αποτελεσμάτων του QSB ή του POM-QM πρέπει να είναι μέρος του κειμένου ως εικόνες. 8. Στην πρώτη σελίδα της εργασίας πρέπει να αναφέρετε το όνομά σας. Οι σελίδες είναι Α4, όλα τα margins 2.5, το spacing 1.5 και η γραμματοσειρά Arial 12. 1
1. PRODUCTION SCHEDULING EZ Trailers, Inc., manufactures a variety of general purpose trailers, including a complete line of boat trailers. Two of their best-selling boat trailers are the EZ-190 and he EZ-250. The EZ-190 is designed for boats up to 19 feet in length, and the EZ-250 can be used for boats up to 25 feet in length. EZ Trailers would like to schedule production for the next two months for these two models. Each unit of EZ-190 requires 4 hours of production time, and each unit of the EZ-250 uses 6 hours of production time. The following orders have been received for March and April. Model March April EZ-190 800 600 EZ-250 1100 1200 The ending inventory from February was 200 units of the EZ-190 and 300 units of the EZ-250. The total number of hours of production time used in February was 6300 hours. The management of EZ Trailers is concerned about being able to satisfy existing orders for the EZ- 250 for both March and April. In fact, it believes that this goal is the most important one that a production schedule should meet. Next in importance is satisfying existing orders for the EZ-190. In addition, management doesn t want to implement any production schedule that would involve significant labor fluctuations from month to month. In this regard, its goal is to develop a production schedule that would limit fluctuations in labor hours used to a maximum of 1000 hours from one month to the next. Perform an analysis of EZ Trailers s production scheduling problem, and prepare a report for EZ s president that summarizes your findings. Include a discussion and analysis of the following items in your report. 1. The production schedule that best achieves the goals as specified by management. 2. Suppose that EZ Trailers s storage facilities would accommodate only a maximum of 300 trailers in any one month. What effect would this have on the production schedule? 3. Suppose that EZ Trailers can store only a maximum of 300 trailers in any one month. I addition, suppose management would like to have an ending inventory in April of at least 100 units of each model. What effect would both changes have on the production schedule? 4. What changes would occur in the production schedule if the labor fluctuation goal was the highest priority goal? 2
2. PRODUCTION DESIGN Industrial Chemicals produces two adhesives used in the manufacturing process for airplanes. The two adhesives, which have different bonding strengths, require different amounts of production time: the IC-100 adhesive requires 20 minutes of production time per gallon of finished product, and the IC-200 adhesive uses 30 minutes of production time per gallon. Both products use 1 pound of a highly perishable resign for each gallon of finished product. Inventory contains 300 pounds of the resign, and more can be obtained if necessary. However, because of the shelf life of the material, any amont not used in the next 2 weeks will be discarded. The firm has existing orders for 100 gallons of IC-100 and 120 gallons of IC-200. Under normal conditions, the production process operates 8 hours per day, 5 days per week. Management wants to schedule production for the next 2 weeks to achieve the following goals: Priority Level 1 Goals Goal 1: Avoid underutilization of the production process. Goal 2: Avoid overtime in excess of 20 hours for the 2 weeks. Priority Level 2 Goals Goal 3: Satisfy existing orders for the IC-100 adhesive; that is, produce at least 100 gallons of IC-100. Goal 4: Satisfy existing orders for the IC-200 adhesive; that is, produce at least 120 gallons of IC-120. Priority Level 3 Goals Goal 5: Use all the available resign. 1. Formulate a goal programming model for the Industrial Chemicals problem. Assume that both priority level 1 goals and that both priority level 2 goals are equally important. 2. Use the graphical goal programming procedure to develop a solution for the model formulated above. 3
3. R&D Infocomp Systems Lab is a research and development company that develops computer systems and software primarily for the medical industry. The lab has proposals from its own researchers for eight new projects. Each of the proposed research projects requires limited recourses, and it is not possible to undertake all of them. The following table shows the development budget, the number of researchers, and the expected annual sales from each project if successfully developed and implemented. Project Developmental Budget ($ millions) Number of Research Personnel Expected Annual Sales ($ millions) 1 0.675 6 0.82 2 1.050 5 1.75 3 0.725 7 1.60 4 0.430 8 1.90 5 1.240 10 0.93 6 0.890 6 1.70 7 1.620 7 1.30 8 1.200 6 1.80 The lab has developed the following set of prioritized goals for selecting which projects to initiate. a. The company would like to remain within a total developmental budget of $5000000. b. The number of available research personnel is 27, and Infocomp would like to avoid obtaining extra researchers. c. The company would like the expected future annual sales from the implemented projects to be at least $6500000. d. Projects 1, 3, 4 and 6 are considered offensive in that they represent new product initiatives, while projects 2, 5, 7 and 8 are existing product upgrades and thus defensive in nature. The lab would like to select at least two projects from each group. e. Projects 2, 3, 5, 6 and 7 are considered the most risky of the projects and the company would prefer not to select any more than three of those projects. f. The lab s owner has indicated she would like to see projects 5 and 6 initiated if it does not interfere with the achievement of any of the more important goals determined by the lab s top management. 1. Formulate a goal programming model to determine which projects Infocomp Systems Lab should select in order to best achieve its goals. 2. Solve this model using the computer. 4
4. GOTHAM CITY Gotham City is trying to determine the type and location of recreation facilities to be built during the next decade. Four types of facilities are under consideration: golf courses, swimming pools, gymnasiums and tennis court. Six sites are under consideration. If a golf course is built, it must be built at either site 1 or site 6. Other facilities may be built at sites 2-5. The available land, in thousands of square feet, at each site is as follows: Site 2 3 4 5 Land 70 80 95 120 The cost of building each facility, in thousands of dollars, for each facility, and the land, in thousands of square feet, required for each facility are given bellow: Site Construction Cost Maintenance Cost Land Required Golf 340 80 Not relevant Swimming 300 36 29 Gymnasium 840 50 38 Tennis courts 85 17 45 The number of user days (in thousands) for each type of facility depends on where it is built. This dependence follows: Site 1 2 3 4 5 6 Golf 31 X X X X 27 Swimming X 25 21 32 32 X Gymnasium X 37 29 28 38 X Tennis courts X 20 23 22 20 X 1. Consider the following set of priorities: Priority 1: Limit land use at each site to the land available. Priority 2: Construction costs should not exceed $1.2 million. Priority 3: User days should exceed 200000. Priority 4: Annual maintenance costs should not exceed $200000. For this set of priorities, use goal programming to determine the type and location of recreation facilities in Gotham City. Solve this model using the computer. 2. Consider the following set of priorities: Priority 1: Limit land use at each site to the land available. Priority 2: User days should exceed 200000. Priority 3: Construction costs should not exceed $1.2 million. Priority 4: Annual maintenance costs should not exceed $200000. For this set of priorities, use goal programming to determine the type and location of recreation facilities in Gotham City. Solve this model using the computer. 5