1. **Problem Statement:** We have demand for 8 quarters and costs for hiring, firing, inventory holding, subcontracting, and overtime. We need to analyze three production strategies: Chase Demand, Level Production, and Mixed Strategy. --- 2. **Given Data:** - Demand per quarter: $D = [260, 210, 470, 650, 430, 270, 210, 360]$ - Hiring cost per unit: $H_c = 150$ - Firing cost per unit: $F_c = 200$ - Inventory holding cost per unit per quarter: $I_c = 50$ - Subcontracting cost per unit: $S_c = 100$ - Overtime cost per unit: $O_c = 25$ - Regular production rate for mixed strategy: $P_r = 50$ - Overtime allowed: 20% of $P_r$ = $10$ units --- ### 1. Pure Strategies #### a) Chase Demand Strategy - Production equals demand each quarter: $P_t = D_t$ - Hiring/firing costs calculated from changes in production between quarters. **Formulas:** - Change in production: $\Delta P_t = P_t - P_{t-1}$ with $P_0=0$ - Hiring cost if $\Delta P_t > 0$: $H_t = H_c \times \Delta P_t$ - Firing cost if $\Delta P_t < 0$: $F_t = F_c \times |\Delta P_t|$ - Total cost = $\sum (H_t + F_t)$ **Calculations:** | Quarter | Demand $D_t$ | Production $P_t$ | $\Delta P_t$ | Hiring Cost | Firing Cost | |---|---|---|---|---|---| | 1 | 260 | 260 | 260 - 0 = 260 | 150*260=39000 | 0 | | 2 | 210 | 210 | 210 - 260 = -50 | 0 | 200*50=10000 | | 3 | 470 | 470 | 470 - 210 = 260 | 150*260=39000 | 0 | | 4 | 650 | 650 | 650 - 470 = 180 | 150*180=27000 | 0 | | 5 | 430 | 430 | 430 - 650 = -220 | 0 | 200*220=44000 | | 6 | 270 | 270 | 270 - 430 = -160 | 0 | 200*160=32000 | | 7 | 210 | 210 | 210 - 270 = -60 | 0 | 200*60=12000 | | 8 | 360 | 360 | 360 - 210 = 150 | 150*150=22500 | 0 | **Totals:** - Hiring cost = 39000 + 39000 + 27000 + 22500 = 127500 - Firing cost = 10000 + 44000 + 32000 + 12000 = 98000 - Total cost = 127500 + 98000 = 225500 --- #### b) Level Production Strategy - Constant production equal to average demand: $$\bar{D} = \frac{260+210+470+650+430+270+210+360}{8} = \frac{2860}{8} = 357.5 \approx 358$$ - Production $P_t = 358$ units each quarter - Inventory and shortage calculated as: $$I_t = I_{t-1} + P_t - D_t$$ with $I_0=0$ - If $I_t < 0$, shortage = $|I_t|$, else inventory = $I_t$ - Inventory holding cost = $I_c \times$ inventory units - Shortage handled by subcontracting at $S_c$ **Calculations:** | Quarter | Demand $D_t$ | Production $P_t$ | Inventory Start $I_{t-1}$ | Inventory End $I_t$ | Inventory Cost | Shortage Cost | |---|---|---|---|---|---|---| | 1 | 260 | 358 | 0 | $0 + 358 - 260 = 98$ | 50*98=4900 | 0 | | 2 | 210 | 358 | 98 | $98 + 358 - 210 = 246$ | 50*246=12300 | 0 | | 3 | 470 | 358 | 246 | $246 + 358 - 470 = 134$ | 50*134=6700 | 0 | | 4 | 650 | 358 | 134 | $134 + 358 - 650 = -158$ | 0 | 100*158=15800 | | 5 | 430 | 358 | -158 | $-158 + 358 - 430 = -230$ | 0 | 100*230=23000 | | 6 | 270 | 358 | -230 | $-230 + 358 - 270 = -142$ | 0 | 100*142=14200 | | 7 | 210 | 358 | -142 | $-142 + 358 - 210 = 6$ | 50*6=300 | 0 | | 8 | 360 | 358 | 6 | $6 + 358 - 360 = 4$ | 50*4=200 | 0 | **Totals:** - Inventory holding cost = 4900 + 12300 + 6700 + 0 + 0 + 0 + 300 + 200 = 24400 - Shortage cost = 0 + 0 + 0 + 15800 + 23000 + 14200 + 0 + 0 = 53000 - Total cost = 24400 + 53000 = 77400 --- ### 2. Mixed Strategy - Constant regular production $P_r = 50$ units - Overtime allowed = 10 units - Total production capacity per quarter = $P_t = 50 + OT_t$ where $0 \leq OT_t \leq 10$ - Demand $D_t$ as given - Shortage $S_t = \max(0, D_t - P_t)$ - Shortage met by cheaper option between hiring and subcontracting: - Hiring cost = 150 per unit - Subcontracting cost = 100 per unit - Choose subcontracting since 100 < 150 - Overtime cost = $O_c \times OT_t$ **Calculations:** | Quarter | Demand $D_t$ | Regular Prod 50 | Overtime $OT_t$ | Total Prod $P_t$ | Shortage $S_t$ | Overtime Cost | Shortage Cost | |---|---|---|---|---|---|---|---| | 1 | 260 | 50 | 10 | 60 | 260 - 60 = 200 | 25*10=250 | 100*200=20000 | | 2 | 210 | 50 | 10 | 60 | 210 - 60 = 150 | 250 | 15000 | | 3 | 470 | 50 | 10 | 60 | 470 - 60 = 410 | 250 | 41000 | | 4 | 650 | 50 | 10 | 60 | 650 - 60 = 590 | 250 | 59000 | | 5 | 430 | 50 | 10 | 60 | 430 - 60 = 370 | 250 | 37000 | | 6 | 270 | 50 | 10 | 60 | 270 - 60 = 210 | 250 | 21000 | | 7 | 210 | 50 | 10 | 60 | 210 - 60 = 150 | 250 | 15000 | | 8 | 360 | 50 | 10 | 60 | 360 - 60 = 300 | 250 | 30000 | **Totals:** - Overtime cost = 8 * 250 = 2000 - Shortage cost = 20000 + 15000 + 41000 + 59000 + 37000 + 21000 + 15000 + 30000 = 238000 - Total cost = 2000 + 238000 = 240000 --- ### 3. Summary Table and Recommendation | Strategy | Hiring Cost | Firing Cost | Inventory Cost | Shortage/Subcontracting Cost | Overtime Cost | Total Cost | |---|---|---|---|---|---|---| | Chase Demand | 127500 | 98000 | 0 | 0 | 0 | 225500 | | Level Production | 0 | 0 | 24400 | 53000 | 0 | 77400 | | Mixed Strategy | 0 | 0 | 0 | 238000 | 2000 | 240000 | **Recommendation:** The Level Production strategy has the lowest total cost (77400) compared to Chase Demand (225500) and Mixed Strategy (240000). It balances production and inventory to minimize costs despite some subcontracting. --- This completes the step-by-step solution with formulas, calculations, and cost comparison.