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Ch 13 - Aggregate Planning

1. Introduction    (p528)
   1. Def - Aggregate Planning - determining the quantity and timing of production for the intermediate future (often 3 - 18 months ahead) in terms of a pseudo-product. Example, a firm produces four different models of PCs, aggregate scheduling done in terms of a generic PC.
   2. Aggregate plans are used to prepare human resource requirements and to facilitate negotiations with subcontractors.
   3. Aggregate Planning Strategies (p531)
      1. Inventory
      2. Hiring and Layoffs
      3. Overtime and Idle time
      4. Subcontracting
      5. Part-time Workers
      6. Back Ordering
      7. Influencing Demand - promotions, discounts
      8. Counterseasonal Product Mix - lawn mowers and snow blowers
   4. Objective - choose a combination of the above strategies to minimize total cost.
2. *Charting Method
   1. Example 1 - Roofing Supplier - p534

      1. Plan 1 - constant workforce with varying inventory - p535
         Inventory carrying cost = $5 per unit per month

         Production rate = (Total demand) / (Total number of days)
                 = 6,200/124 = 50 units/day

         	Production at 50 units/day	Demand	Inventory Change	Ending Inventory	Stock    Outs	Cost    $
         Jan Feb Mar Apr May Jun	1100     900   1050   1050   1100   1000	900     700     800   1200   1500   1100	+200   +200   +250   -150   -400   -100	200   400   650   500   100       0	0   0   0   0   0   0	1000   2000   3250   2500     500         0
         Totals	6200	6200				$9250

      2. Plan 2 - Smaller workforce with subcontracting - see p536
      3. Plan 3 - Hiring and layoffs with no inventory
   2. Time consuming (even with Excel) to investigate several mixed strategies. No guarantee of optimal solution.
                  Do assigned HW
   3. Note: A similar analysis could be done for a service (law firm or accounting firm) using backorders and no inventory.
3. *Transportation Method of Linear Programming           (p538)
   1. Considers overtime, idle time, inventory, subcontracting and backorders. Does give optimal solution.
   2. Farnsworth Tire Company - p538
      1. Input data
         Costs
             r - regular time      = $40/tire
             o - overtime          = $50/tire
             s - subcontracting  = $70/tire
             h - carrying            = $2/tire/month
             b - backordering   = $20/tire/month (assumption)
         Maximum production capacity in period j, j = 1, 2, 3.
             R_j - regular time= 700, 700, 700
             O_j - overtime= 50, 50, 50
             S_j - subcontracting= 150, 150, 130
         Demand in period j
             D_j= 800, 1000, 750
         Initial inventory I₀ = 100
      2. Transportation Costs per tire - this table defines the matrix, a _{i j}.

         	j - Destinations
         i - Origins	Period 1	Period 2	Period 3
         1 - Initial inventory	0.0	h	2h
         2 - Reg-time period 1   3 - Overtime period 1   4 - Subcon period 1	r o s	r+h o+h s+h	r+2h o+2h s+2h
         5 - Reg-time period 2   6 - Overtime period 2   7 - Subcon period 2	r+b o+b s+b	r o s	r+h o+h s+h
         8 - Reg-time period 3   9 - Overtime period 3 10 - Subcon period 3	r+2b o+2b s+2b	r+b o+b s+b	r o s

      3. Define decision variables
             X _{i j} = Number of tires transported from origin i (production method i)
                       to destination j (time period j)
      4. Define objective function
             Minimize transportation cost = S a _{i j} X _{i j}
      5. Define constraints
             Origin 1   S X _{1 j} £  I₀
         
              Origin 2   S X _{2 j} £  R₁
             Origin 3   S X _{3 j} £  O₁
             Origin 4   S X _{4 j} £  S₁
         
             Origin 5   S X _{5 j} £  R₂
             Origin 6   S X _{6 j} £  O₂
             Origin 7   S X _{7 j} £  S₂
         
             Origin 8   S X _{8 j} £  R₃
             Origin 9   S X _{9 j} £  O₃
             Origin 10   S X _{10 j} £  S₃
         
             Destination 1    S X _{i 1} =  D₁
             Destination 2    S X _{i 2} =  D₂
             Destination 3    S X _{i 3} =  D₃
                             Do assigned HW
      6. The Solution - X _{i j}  (Table 13.7, p539)   

         	j - Destinations
         i - Origins	Period 1	Period 2	Period 3
         1 Initial inventory	100	0	0
         2 Reg-time period 1   3 Overtime period 1   4 Subcon period 1	700       0       0	0     50   150	0       0       0
         5 Reg-time period 2   6 Overtime period 2    7 Subcon period 2	0       0       0	700     50     50	0       0       0
         8 Reg-time period 3   9 Overtime period 3 10 Subcon period 3	0       0       0	0       0       0	700     50       0

      7. The Aggregate Plan

         	Period 1	Period 2	Period 3
         Reg-time	S X 2 j =  700	S X 5 j =  700	S X 8 j =  700
         Overtime	S X 3 j =   50	S X 6 j =     50	S X 9 j =     50
         Subcontract	S X 4 j = 150	S X 7 j =    50	S X 10 j =     0

      8. To see Excel Solver solution (for your info only), click here to download worksheet H813ex05.xls.
                 (This page was last edited on October 22, 2009 .)