- Decision Models in Supply Chains Directions: Answer all questions. Show all work for full credit. Problem # 1: The daily demand for a product at a retailer is normally distributed with a mean of 50 and a standard deviation of 10. Answer the following questions based on the above data: 1. What is the safety stock at a cycle service level of 90% if the lead time is 2 days? 2. If the safety stock is 85, what is the associated cycle service level (CSL) with a lead time of 2 days? 3. If the reorder interval is 3 days and lead time is 2 days, what is the safety stock at a cycle service level of 90%? 4. What is the order up to level for the scenario in question # 3? 5. What is the safety stock if the lead-time is also variable with a mean of 2 days and standard deviation of 0.5 days with a CSL of 90%? Problem # 2 Armco Corporation has developed the following demand forecast for digital cameras for next year. Quarter 1 2 3 4 Sales 7,000 12,000 6,000 4,000 Following is the data the company has for developing an aggregate production plan: Beginning Inventory = 3000 units Beginning workforce = 70 employees Hiring cost per employee = $300 Firing cost per employee = $350 Inventory carrying cost = $10 per unit per quarter of ending inventory Regular payroll = $4,000 per employee per quarter Overtime cost = $30 per unit Subcontract cost = $50 per unit Each employee can produce 150 units in regular time production per quarter and 35 units in over time production. Formulate a linear programming problem that incorporates these factors in identifying an optimal production plan. Define and list the decision variables, and express the objective function and constraints in mathematical form. EXTRA PAGE FOR WORK Problem # 3 Ryan Electronics produces electrical components at two production facilities in Denver and Atlanta. Components produced at either facility may be shipped to either of the firm’s regional warehouses, which are located in Kansas City and Louisville. From the regional warehouses, the firm supplies retail outlets in Detroit, Miami, Dallas, and New Orleans. Following tables show the transportation unit costs, supply and demand at each of the nodes. Unit Transportation Costs, Capacities: Plants and Warehouses Plant/WH Denver Atlanta Kansas $2 $3 Louisville $3 $1 Capacity 700 800 Unit Transportation Costs, Demands: Warehouses and Retail Outlets WH/Retail Detroit Miami Dallas Kansas Louisville Demand $2 $4 200 $6 $4 150 $3 $6 350 New Orleans $6 $5 300 a. Clearly list the decision variables and represent the objective function and constraints in mathematical format b. How does the model change if there is added information that direct shipments from Denver to Detroit ($3/unit) and Atlanta to Dallas ($5)? Just show the revised constraints and the objective function. EXTRA PAGE FOR WORK Problem # 4 A personnel director for a manufacturing company has collected the following data on the salary (Y) earned by each machinist in the factory along with the average performance rating (X1) over the past three years, years of service (X2), and whether they are cross-trained to work on multiple machines (X3). Note that X3 is considered as a binary variable (1 – cross trained and 0 – not cross-trained). Answer the following questions based on the regression models considered by the personnel director: Salary ($1,000s) Y 48.2 55.3 53.7 61.8 56.4 52.5 54.0 55.7 45.1 67.9 53.2 46.8 58.3 59.1 57.8 Avg. Performance Rating X1 3.5 5.3 5.1 5.8 4.2 6.0 6.8 5.5 3.1 7.2 4.5 4.9 8.0 6.5 6.6 Years of Experience X2 9 12 14 19 15 10 11 9 7 14 8 5 5 14 16 CrossTrained X3 0 1 0 1 1 0 0 1 0 1 0 0 1 1 1 (a) What can you conclude about the regression model Y vs. X1? Is it significant? Explain what R2 means for this model. (b) Is it useful to include X2 in the model when X1 is already in the model? Explain your reasoning. (c) What is the model corresponding to the regression Y vs. X1, X2. Interpret the slopes and the Y-intercept in the context of the problem. (d) Of the four regressions considered, which would you suggest as the best model? And why? (e) Develop and interpret a 95% prediction confidence interval for the salary of a machinist with a performance rating of 4.5 with 7 years of experience and who is crosstrained. SUMMARY OUTPUT FOR Y vs X1 Regression Statistics Multiple R 0.71116187 R Square 0.50575121 Adjusted R Square 0.46773207 Standard Error 4.24672306 Observations 15 ANOVA MS 239.9068 18.03466 F 13.30254 Standard Error 4.70525348 0.826935105 t Stat 8.153549 3.647265 P-value 1.81E-06 0.002953 Regression Residual df 1 13 SS 185.5827 288.774634 MS 185.5827 22.21343 F 8.354526 Total 14 474.357333 Regression Residual Total Intercept X1 df 1 13 14 SS 239.9067952 234.4505382 474.3573333 Coefficients 38.3645147 3.01605157 Significance F 0.002952711 SUMMARY OUTPUT FOR Y vs X2 Regression Statistics Multiple R 0.6254836 R Square 0.3912297 Adjusted R Square 0.3444013 Standard Error 4.7131129 Observations 15 ANOVA Intercept X2 Coefficients 45.171588 Standard Error 3.62891687 t Stat 12.44768 P-value 1.35E-08 0.8822987 0.30524933 2.89042 0.012638 Significance F 0.012638 SUMMARY OUTPUT FOR Y vs. X1, X2 Regression Statistics Multiple R 0.873907 R Square 0.763714 Adjusted R Square 0.724333 Standard Error 3.056198 Observations 15 ANOVA Regression Residual df 2 12 SS 362.273166 112.084167 Total 14 474.357333 Coefficients 32.35034 2.629669 0.727872 Intercept X1 X2 MS 181.1366 9.340347 F 19.39292 Standard Error 3.77189116 0.60461091 t Stat 8.576689 4.349357 P-value 1.83E-06 0.000946 0.2010969 3.619508 0.003518 Significance F 0.00017403 SUMMARY OUTPUT FOR Y vs X1, X2, X3 Regression Statistics Multiple R 0.913572 R Square 0.834614 Adjusted R Square 0.789508 Standard Error 2.670583 Observations 15 ANOVA Regression Residual Total Intercept X1 X2 X3 df 3 11 14 SS 395.905179 78.4521548 474.357333 Coefficients 35.85264 2.025096 0.529387 3.873802 Standard Error 3.66941377 0.59719028 0.19807352 1.78388574 MS 131.968 7.13201 t Stat 9.77067 3.39104 2.67268 2.17155 F 18.504 Pvalue 9E-07 0.006 0.0217 0.0526 Significance F 0.00013169