document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1170863>Question 1170863</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A famous restaurant chain decided to open three new branches in your city: one in downtown area (DT), one in a residential area (RA), and one in a touristic area (TA). All branches will open daily from 12pm to 10pm. According to a study done by the analyst team, the minimum number of employees needed in each branch at each hour is shown in table 1.<BR>\n" );
document.write( "Table 1<BR>\n" );
document.write( "Time	Branch 1 (DT)	Branch 2 (RA)	Branch 3 (TA)<BR>\n" );
document.write( "12pm-1pm	12	9	5<BR>\n" );
document.write( "1pm-2pm	12	9	5<BR>\n" );
document.write( "2pm-3pm	11	8	6<BR>\n" );
document.write( "3pm-4pm	9	8	6<BR>\n" );
document.write( "4pm-5pm	7	9	8<BR>\n" );
document.write( "5pm-6pm	6	10	8<BR>\n" );
document.write( "6pm-7pm	6	10	10<BR>\n" );
document.write( "7pm-8pm	5	11	12<BR>\n" );
document.write( "8pm-9pm	5	9	12<BR>\n" );
document.write( "9pm-10pm	4	7	10\r<BR>\n" );
document.write( "\n" );
document.write( "The restaurant chain management offered three types of jobs to hire employees for the three new branches: 2- hours part timers, 4-hours part timers, 8-hours full timers. The 2-hours part timers are only available to work after 4pm.<BR>\n" );
document.write( "The average cost per employee hour is different for each type and each branch (Cost includes salaries, insurance, transportation, etc.). Table 2 shows the average cost per employee hour ($/hour) for each branch.<BR>\n" );
document.write( "Table 2<BR>\n" );
document.write( "Employee type	Branch 1	Branch 2	Branch 3<BR>\n" );
document.write( "A (2-hours)	16	10.5	12<BR>\n" );
document.write( "B (4-hours)	14	9.5	11<BR>\n" );
document.write( "C (8-hours)	12	8	9\r<BR>\n" );
document.write( "\n" );
document.write( "The number of applicants who applied for each type of jobs is 30 for the 2-hours part time jobs, 35 for the 4-hours part time jobs, and 15 for the full-time jobs. All applicants agreed to work in any of the new branches if hired.<BR>\n" );
document.write( "The restaurant chain management has a policy that there should be at least one full time employee available at each branch when the branch closes at the end of the day.<BR>\n" );
document.write( "Your team was chosen by the restaurant chain management to consult them to find the best action for this case. The type of analysis that they are looking for is:<BR>\n" );
document.write( "1-	What would be the number of employees required from each type (i.e. 2-hour part timers, 4-hours part timers, and 8-hours full timers) in each branch that will minimize the total daily cost? What would be the daily cost in each branch in this case?<BR>\n" );
document.write( "2-	How many extra employee hours will be available in each branch at each hour after the required hours are fulfilled?<BR>\n" );
document.write( "3-	How much could the company saves daily if five more full-time employees were available to hire?<BR>\n" );
document.write( "4-	Which branch costs the most for satisfying the policy of having one full time employee available at the end of the day?<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851089 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=851089\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> This is a classic linear programming problem, but with some extra constraints and calculations. Let's break down each part of the analysis.\r<BR>\n" );
document.write( "\n" );
document.write( "**1. Minimizing Total Daily Cost**\r<BR>\n" );
document.write( "\n" );
document.write( "This part requires setting up and solving a linear programming model.\r<BR>\n" );
document.write( "\n" );
document.write( "**Variables:**\r<BR>\n" );
document.write( "\n" );
document.write( "* DT_A: Number of 2-hour part-timers in Downtown (DT)<BR>\n" );
document.write( "* DT_B: Number of 4-hour part-timers in DT<BR>\n" );
document.write( "* DT_C: Number of 8-hour full-timers in DT<BR>\n" );
document.write( "* RA_A: Number of 2-hour part-timers in Residential Area (RA)<BR>\n" );
document.write( "* RA_B: Number of 4-hour part-timers in RA<BR>\n" );
document.write( "* RA_C: Number of 8-hour full-timers in RA<BR>\n" );
document.write( "* TA_A: Number of 2-hour part-timers in Touristic Area (TA)<BR>\n" );
document.write( "* TA_B: Number of 4-hour part-timers in TA<BR>\n" );
document.write( "* TA_C: Number of 8-hour full-timers in TA\r<BR>\n" );
document.write( "\n" );
document.write( "**Objective Function (Minimize Total Cost):**\r<BR>\n" );
document.write( "\n" );
document.write( "Minimize Z = 16(DT_A) + 14(DT_B) + 12(DT_C) + 10.5(RA_A) + 9.5(RA_B) + 8(RA_C) + 12(TA_A) + 11(TA_B) + 9(TA_C)\r<BR>\n" );
document.write( "\n" );
document.write( "**Constraints:**\r<BR>\n" );
document.write( "\n" );
document.write( "* **Employee Availability:**<BR>\n" );
document.write( "    * DT_A + RA_A + TA_A <= 30 (2-hour part-timers)<BR>\n" );
document.write( "    * DT_B + RA_B + TA_B <= 35 (4-hour part-timers)<BR>\n" );
document.write( "    * DT_C + RA_C + TA_C <= 15 (8-hour full-timers)<BR>\n" );
document.write( "* **Hourly Employee Requirements (for each branch and each hour):**<BR>\n" );
document.write( "    * **DT:**<BR>\n" );
document.write( "        * 12pm-1pm: DT_C >= 12<BR>\n" );
document.write( "        * 1pm-2pm: DT_C >= 12<BR>\n" );
document.write( "        * 2pm-3pm: DT_C >= 11<BR>\n" );
document.write( "        * 3pm-4pm: DT_C >= 9<BR>\n" );
document.write( "        * 4pm-5pm: DT_B + DT_C >= 7<BR>\n" );
document.write( "        * 5pm-6pm: DT_A + DT_B + DT_C >= 6<BR>\n" );
document.write( "        * 6pm-7pm: DT_A + DT_B + DT_C >= 6<BR>\n" );
document.write( "        * 7pm-8pm: DT_A + DT_B + DT_C >= 5<BR>\n" );
document.write( "        * 8pm-9pm: DT_A + DT_B + DT_C >= 5<BR>\n" );
document.write( "        * 9pm-10pm: DT_A + DT_B + DT_C >= 4<BR>\n" );
document.write( "    * **RA:**<BR>\n" );
document.write( "        * 12pm-1pm: RA_C >= 9<BR>\n" );
document.write( "        * 1pm-2pm: RA_C >= 9<BR>\n" );
document.write( "        * 2pm-3pm: RA_C >= 8<BR>\n" );
document.write( "        * 3pm-4pm: RA_C >= 8<BR>\n" );
document.write( "        * 4pm-5pm: RA_B + RA_C >= 9<BR>\n" );
document.write( "        * 5pm-6pm: RA_A + RA_B + RA_C >= 10<BR>\n" );
document.write( "        * 6pm-7pm: RA_A + RA_B + RA_C >= 10<BR>\n" );
document.write( "        * 7pm-8pm: RA_A + RA_B + RA_C >= 11<BR>\n" );
document.write( "        * 8pm-9pm: RA_A + RA_B + RA_C >= 9<BR>\n" );
document.write( "        * 9pm-10pm: RA_A + RA_B + RA_C >= 7<BR>\n" );
document.write( "    * **TA:**<BR>\n" );
document.write( "        * 12pm-1pm: TA_C >= 5<BR>\n" );
document.write( "        * 1pm-2pm: TA_C >= 5<BR>\n" );
document.write( "        * 2pm-3pm: TA_C >= 6<BR>\n" );
document.write( "        * 3pm-4pm: TA_C >= 6<BR>\n" );
document.write( "        * 4pm-5pm: TA_B + TA_C >= 8<BR>\n" );
document.write( "        * 5pm-6pm: TA_A + TA_B + TA_C >= 8<BR>\n" );
document.write( "        * 6pm-7pm: TA_A + TA_B + TA_C >= 10<BR>\n" );
document.write( "        * 7pm-8pm: TA_A + TA_B + TA_C >= 12<BR>\n" );
document.write( "        * 8pm-9pm: TA_A + TA_B + TA_C >= 12<BR>\n" );
document.write( "        * 9pm-10pm: TA_A + TA_B + TA_C >= 10<BR>\n" );
document.write( "* **End of Day Full-Time Constraint:**<BR>\n" );
document.write( "    * DT_C >= 1<BR>\n" );
document.write( "    * RA_C >= 1<BR>\n" );
document.write( "    * TA_C >= 1<BR>\n" );
document.write( "* **Non-negativity:**<BR>\n" );
document.write( "    * All variables >= 0\r<BR>\n" );
document.write( "\n" );
document.write( "**Solving this Linear Program:**\r<BR>\n" );
document.write( "\n" );
document.write( "This requires using a linear programming solver (like Excel Solver, Python's `scipy.optimize.linprog`, or a dedicated LP software).\r<BR>\n" );
document.write( "\n" );
document.write( "**2. Extra Employee Hours**\r<BR>\n" );
document.write( "\n" );
document.write( "Once the optimal solution is found, calculate the extra employee hours for each hour in each branch. For example, if the model says DT_C = 13, and the requirement for 12pm-1pm is 12, then there is one extra employee-hour.\r<BR>\n" );
document.write( "\n" );
document.write( "**3. Savings with 5 More Full-Time Employees**\r<BR>\n" );
document.write( "\n" );
document.write( "* Increase the constraint for full-time employees: DT_C + RA_C + TA_C <= 20<BR>\n" );
document.write( "* Re-solve the linear program.<BR>\n" );
document.write( "* Calculate the difference in the total cost between the original solution and the new solution.\r<BR>\n" );
document.write( "\n" );
document.write( "**4. Branch Costing Most for End-of-Day Policy**\r<BR>\n" );
document.write( "\n" );
document.write( "* Examine the optimal solution and determine which branch has the highest cost associated with its full-time employees (DT_C, RA_C, TA_C). Multiply the number of full time employees in each branch by the cost per hour, and then by 8. Then compare the results.\r<BR>\n" );
document.write( "\n" );
document.write( "**Important Notes:**\r<BR>\n" );
document.write( "\n" );
document.write( "* This problem requires a linear programming solver.<BR>\n" );
document.write( "* The results will depend on the optimal solution found by the solver.<BR>\n" );
document.write( "* The exact numbers will depend on the solver, but this process will provide the solution.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );