Industrial Engineering Test 1

Concept:

The ordering quantity Q* at which holding cost becomes equal to ordering cost and the total inventory cost is minimum is known as Economic order Quantity (EOQ).

At EOQ:

Ordering cost = Holding cost

\(\frac{D}{{{Q^*}}}{C_o} = \frac{{{Q^*}}}{2}{C_h} \Rightarrow {Q^*} = \sqrt {\frac{{2D{C_o}}}{{{C_h}}}} \)

D = Annual or yearly demand of inventory (unit/year)

Q = Quantity to be ordered at each order point (unit/order)

C_o = Cost of placing one order [Rs/order]

C_h = Cost of holding one unit in inventory for one complete year [Rs/unit/year]

Calculation:

D = 6000 units, C_o = Rs. 1000, C_h = 48

\({{\rm{Q}}^{\rm{*}}} = \sqrt {\frac{{2{\rm{D}}{{\rm{C}}_{\rm{o}}}}}{{{{\rm{C}}_{\rm{h}}}}}} = \sqrt {\frac{{2 \times 6000 \times 1000}}{{48}}} = 500{\rm{\;units}}\)

Given:

C₁ = Rs. 2, C₂ = 5 – 2 = Rs. 3

\({\rm{Critical\;ratio\;}} = \frac{{{C_2}}}{{{C_1}\; + \;{C_2}}} = \frac{3}{{2\; + \;3}}\)

∴ Critical ratio = 0.6

Now,

Obtaining cumulative probability:

Critical ratio (= 0.6) lies when 19 & 20.

It becomes less than 0.6 when Q = 19,

∴ Optimum demand = 20

Concept:

\({\rm{Inventory\;turned\;over\;per\;year}} = \frac{{Demand}}{{Average\;order\;quantity + Safety\;stock}}\)

Calculation:

\({\rm{Average\;order\;quantity}} = \frac{{Q + 0}}{2} = \frac{Q}{2}\)

\(\therefore {\rm{Turning\;Turnover}} = \frac{D}{{\left( {\frac{Q}{2}} \right) + 55}}\)

\(\therefore {\rm{Turning\;Turnover}} = \frac{{1000}}{{\left( {\frac{{300}}{2}} \right) + 40}}\)

∴ Turning Turnover = 5.263

Concept:

The economic order quantity is given by:

\(EOQ = \sqrt {\frac{{2D{C_o}}}{{{C_h}}}}\)

C_o = ordering cost, C_h = holding constant, D = demand

The cycle time is related to EOQ as:

EOQ = D × T

Calculation:

D = 15000 units, C_h = 100 × 0.1625, C_o = Rs 3000

\(EOQ = \sqrt {\frac{{2D{C_o}}}{{{C_h}}}} = \sqrt {\frac{{2 \times 15000 \times 3000}}{{100 \times 0.1625}}} = 2353.393\;units\)

Thus:

\(T = \frac{{EOQ}}{D} = \frac{{2353.393}}{{15000}} = 0.15689\;years = 0.15689 \times 12 = 1.882\;months\)

Concept:

For inventory model with shortage cost,

\(EOQ = \sqrt {\frac{{2D{C_0}}}{{{C_h}}}\left[ {\frac{{{C_h} + {C_s}}}{{{C_s}}}} \right]} \)

Where,

C_s = shortage cost, C₀ = setup cost/order cost, C_h = holding cost

\({\rm{Inventory\;cost}} = \sqrt {2d{C_0}{C_h}\left( {\frac{{{C_s}}}{{{C_s} + {C_h}}}} \right)} \)

Calculation:

\({\rm{Inventory\;cost}} = \sqrt {2 \times 18000 \times 400 \times 1.2 \times \left( {\frac{5}{{5 + 1.2}}} \right)} \)

∴ Inventory cost = 3733

Variable cost

• Variable cost is directly proportional to the volume of production.
• If v = variable cost per unit quantity produced and Q = number of quantity produce then, variable cost (V) = v × Q

Total cost = Fixed cost + Variable cost

Calculation:

Given:

Process X:

• Fixed cost (F₁) = Rs. 30,000 per month
• Variable cost per unit (v₁) = Rs. 8

Process Y:

• Fixed cost (F₂) = Rs. 12000 per month
• Variable cost per unit(v₂) = Rs. 17

Let Q is the number of quantity produced.

Now,

(Total cost)_x = (Total cost)_y

⇒ (Fixed cost + Variable cost)_X = (Fixed cost + Variable cost)_Y

⇒ (F₁ + v₁Q) = (F₂ + v₂Q)

⇒ 30000 + 8Q = 12000 + 17Q

⇒ 18000 = 9Q

⇒ \({\rm{Q}} = \frac{{18000}}{9} = 2000\;{\rm{units}}\)

Concept:

Maximum inventory (M*) = Q* - S*

Q* = Economic order quantity

\({Q^*} = \sqrt {\frac{{2D{C_0}}}{{{C_h}}}\left( {\frac{{{C_h}\; +\; {C_b}}}{{{C_b}}}} \right)} \)

\({\rm{Back}} - {\rm{order\;quantity\;}} = {S^*} = {Q^*}\left( {\frac{{{C_h}}}{{{C_h}\; + \;{C_b}}}} \right)\)

Given:

D = 10,000, C₀ = 15, C = 22

Holding cost (C_h) = 20 % of C = 0.2 × 22

∴ C_h = 4.4

Back-ordering cost (C_b) = 25% of C

Back-ordering cost (C_b)  = 0.25 × 22

∴ Back-ordering cost (C_b) = 5.5

Calculation:

\({Q^*} = \sqrt {\frac{{2D{C_0}}}{{{C_h}}} \cdot \left( {\frac{{{C_h} + {C_b}}}{{{C_b}}}} \right)} \)

\({Q^*} = \sqrt {\frac{{2\; \times \;10000\; \times \;15}}{{4.4}} \times \left( {\frac{{4.4\; +\; 5.5}}{{5.5}}} \right)} \)

∴ Q* = 350.32

Now,

\({S^*} = {Q^*} = \left( {\frac{{{C_h}}}{{{C_h}\; + \;{C_b}}}} \right)\)

\({S^*} = 350.32\left( {\frac{{4.4}}{{4.4\; + \;5.5}}} \right)\)

∴ S* = 155.69

Now,

Maximum inventory level (M*)  = Q* - S*

M* = 350.32 – 155.69

∴ M* = 194.63

Explanation:

Given:

D = 816, C_h = Rs. 4 per case per year

C_o = Rs. 12 per case

\(EOQ\;\left( {{Q_0}} \right) = \sqrt {\frac{{2D{C_0}}}{{{C_h}}}} \)

Where D = Annual demand

\(\begin{array}{l} {C_0} = Ordering\;cost/order\\ {C_h} = Holding\;cost\;per\;unit\;per\;year \end{array}\)

Now,

\(EOQ = \sqrt {\frac{{2D{C_0}}}{{{C_h}}}} = \sqrt {\frac{{2 \times 816 \times 18}}{{12}}} = {70\;cases} \)

⇒ EOQ lies in 50 to 79 range, which is not the range corresponding to the lowest cost. Thus, it is not optimal EOQ. Thus

\(Total\;cost\;{\left( {TC} \right)_{70}} = \left( {\frac{{{Q_0}}}{2}} \right) \times {C_h} + \left( {\frac{D}{{{Q_0}}}} \right){C_o} + D \times C\)

Where C = Unit cost

\({\left( {TC} \right)_{70}}= \left( {\frac{{70}}{2}} \right) \times 4 + \left( {\frac{{816}}{{70}}} \right) \times 12 + 18 \times 816\)

∴ (TC)70 = Rs. 14968

Now,

To check the minimum total cost lowest price breakpoints should be considered with the least quantity at which discounted price is availed.

Thus,

\({\left( {TC} \right)_{80}} = \left( {\frac{{80}}{2}} \right) \times 4 + \left( {\frac{{816}}{{80}}} \right) \times 12 + 816 \times 17\)

∴ (TC)₈₀ = Rs. 14154

\({\left( {TC} \right)_{100}} = \left( {\frac{{100}}{2}} \right) \times 4 + \left( {\frac{{816}}{{100}}} \right) \times 12 + 816 \times 16\)

∴ (TC)100 = Rs. 13354

∴ 100 cases yield the lowest total cost.

Thus, 100 cases are the optimal order quantity.

Given:

D = 1600, C₀ = 5, C_h = 10% of C

Since, the lowest unit price is 0.98, so computing economic order quantity for Q₂

\(Q_2^* = \sqrt {\frac{{2D{C_0}}}{{{C_h}}}} = \sqrt {\frac{{2\; \times \;1600\; \times \;5}}{{0.1\; \times \;0.98}}} \)

\(\therefore Q_2^* = 404.06\;{\rm{units}}\) 

This is not feasible as Q₂ > 800 units.

Now,

Calculating for Q₁, C = Rs. 1

\(Q_1^* = \sqrt {\frac{{2D{C_0}}}{{{C_n}}}} = \sqrt {\frac{{2\; \times \;1600\; \times \;5}}{{0.1\; \times \;1}}} \)

\(\therefore Q_1^* = 400\;{\rm{units}}\) 

This is feasible, now we will calculate and compare optimum cost at higher price break point.

\({\rm{Total\;cost\;}}\left( {T.C.} \right)\; = D.C + \frac{D}{Q}{C_0} + \frac{Q}{2}{C_n}\)

\(T.C.\;\left( {at\;Q_1^* = 400} \right) = \left( {1600 \times 1} \right) + \left( {\frac{{1600}}{{400}} \times 5} \right) + \left( {\frac{{400}}{2} \times 0.1 \times 1} \right)\)

\(T.C.\;\left( {at\;Q_1^* = 400} \right) = {\rm{\;Rs}}.{\rm{\;}}1640\)

Now,

at Q = 800, (C = 0.98)

\(T.C.\;\left( {at\;Q = 800} \right) = \left( {1600 \times 0.98} \right) + \left( {\frac{{1600}}{{800}} \times 5} \right) + \left( {\frac{{800}}{2} \times 0.98 \times 0.1} \right)\)

∴ T.C. (at Q=800) = Rs. 1617.2

Since,

Total cost at Q = 800 is less than total cost at Q*= 400

∴ Optimal order quantity = 800.