1. **State the problem**: We need to find the optimal order quantity to minimize total inventory costs given annual demand $D=20000$ units, order cost $S=40$, carrying cost rate $H=0.20$ of unit cost, and unit cost $C$ varying with order quantity. 2. **Define the unit cost ranges**: - For orders $Q$ from 1 to 499, $C=10.00$ - For $Q$ from 500 to 999, $C=9.50$ - For $Q \geq 1000$, $C=8.90$ 3. **Recall the Economic Order Quantity (EOQ) formula:** $$EOQ = \sqrt{\frac{2DS}{H \times C}}$$ where $D$= annual demand, $S$= ordering cost, $H \times C$= holding cost per unit. 4. **Calculate EOQ for each price range**: - For $C=10.00$, holding cost per unit $= 0.20 \times 10 = 2$ $$EOQ_1 = \sqrt{\frac{2 \times 20000 \times 40}{2}} = \sqrt{800000} = 894.43$$ - For $C=9.50$, holding cost per unit $= 0.20 \times 9.50 = 1.9$ $$EOQ_2 = \sqrt{\frac{2 \times 20000 \times 40}{1.9}} = \sqrt{842105.26} = 917.86$$ - For $C=8.90$, holding cost per unit $= 0.20 \times 8.90 = 1.78$ $$EOQ_3 = \sqrt{\frac{2 \times 20000 \times 40}{1.78}} = \sqrt{898876.4} = 948.09$$ 5. **Check if EOQ fits order quantity ranges**: - $EOQ_1 = 894.43$ is not in $1-499$ range, so not valid. - $EOQ_2 = 917.86$ is in $500-999$ range, valid. - $EOQ_3 = 948.09$ is not in $1000+$ range, so not valid. 6. **Calculate total cost at EOQ values and boundaries to find the minimum**: Total cost formula: $$TC = DC + \frac{DS}{Q} + \frac{HQ}{2}$$ where $H= holding cost per unit, C= unit cost, Q= order quantity, D= annual demand, S= ordering cost$. - For $Q=894.43$ (not valid but calculate for cost): $$TC_1 = 20000 \times 10 + \frac{20000 \times 40}{894.43} + \frac{2 \times 894.43}{2} = 200000 + 893.93 + 894.43 = 201788.36$$ - For $Q=917.86$ (valid): $$TC_2 = 20000 \times 9.5 + \frac{20000 \times 40}{917.86} + \frac{1.9 \times 917.86}{2} = 190000 + 871.05 + 871.93 = 191742.98$$ - For $Q=1000$ (boundary for third price): $$TC_3 = 20000 \times 8.9 + \frac{20000 \times 40}{1000} + \frac{1.78 \times 1000}{2} = 178000 + 800 + 890 = 179690$$ - For $Q=948.09$ (not valid for last price range but calculate cost): $$TC_4 = 20000 \times 9.5 + \frac{20000 \times 40}{948.09} + \frac{1.9 \times 948.09}{2} = 190000 + 843.32 + 899.27 = 191742.59$$ 7. **Compare total costs:** Minimum cost is $179690$ at $Q=1000$ units. **Final answer:** Order quantity should be $\boxed{1000}$ units to minimize total cost with unit cost $8.90$ per unit.