Chapter 8, Problem 49AP

a)

Summary Introduction

Interpretation: Determine the ordering policy is recommended by the silver-Meal heuristic.

Concept Introduction: Silver-Meal heuristic mainly focused on the manufacture planning in the production companies. It determines that within the minimum cost level, the quantities of the products will be produced by the company.

Answer to Problem 49AP

The order policy according to silver meal heuristic method is $\boxed{\left(56,0,52,0,0,114,0,0,106,0\right)}$

Explanation of Solution

Given information: The anticipated demand for an inventory is as follows:

Week	1	2	3	4	5	6	7	8	9	10
Demand	22	34	32	12	8	44	54	16	76	30

The inventory costs 65 cents each and holding cost (h) is calculated on annual interest rate at 0.5% per week. The set up cost (K) is $200.

The order policy of the inventory under silver meal method can be calculated as follows:

According to silver meal method the average cost per period C (T) is a function of the average holding and set up cost per period for T number of Periods. The production in period 1 is equal to the demand in that period 1 to incur the order cost K.

Hence $C\left(1\right)=K$

And $C\left(2\right)=\left(K+h{r}_2\right)/2$

  $C\left(3\right)=\left(K+h{r}_2+2h{r}_3\right)/3$

And general equation is $C\left(j\right)=\left(K+h{r}_2+2h{r}_3+\mathrm{...}+\left(j-1\right)h{r}_j\right)/j$

Once $C\left(j\right)\succ C\left(j-1\right),$ stop the function and set $y=r_1+r_2+r_3+\mathrm{....}{r}_{j-1}$

Now, calculate the order policy using the above formula as follows:

Starting in period 1:

  $K=\$200$

  $h=\left(0.005\right)\left(0.65\right)$

(Since h is calculated on 0.5% annual interest rate with each unit cost at 65cents)

  $h=3.25$ Per thousand

  $r=\left(22,34,32,12,8,44,54,16,76,30\right)$

Starting from period 1

  $C\left(1\right)=200$

  $C\left(2\right)=\left(200+\left(3.24\right)\left(34\right)\right)/2=155.25$

  $C\left(3\right)=\frac{\left[200+\left(3.24\right)\left(34\right)+\left(2\right)\left(3.24\right)\left(32\right)\right]}{3}=172.8$

Stop the process since $C\left(3\right)\succ C\left(2\right),$

  $y_1=r_1+r_2$

  $y_1=22+34=56$

Starting in period 3:

  $C\left(1\right)=200$

  $C\left(2\right)=\frac{\left[200+\left(3.25\right)\left(12\right)\right]}{2}=119.5$

  $C\left(3\right)=\frac{\left[200+\left(3.25\right)\left(12\right)+\left(2\right)\left(3.25\right)\left(8\right)\right]}{3}=97$

  $C\left(4\right)=\frac{\left[200+\left(3.25\right)\left(12\right)+\left(2\right)\left(3.25\right)\left(8\right)+\left(3\right)\left(3.25\right)\left(44\right)\right]}{4}=180$

Stop the process since $C\left(4\right)\succ C\left(3\right),$

  $y_3=r_3+r_4+r_5$

  $y_3=32+12+8=52$

Starting in period 6:

  $C\left(1\right)=200$

  $C\left(2\right)=\frac{\left[200+\left(3.25\right)\left(54\right)\right]}{2}=187.75$

  $C\left(3\right)=\frac{\left[200+\left(3.25\right)\left(54\right)+\left(2\right)\left(3.25\right)\left(16\right)\right]}{3}=159.8$

  $C\left(4\right)=\frac{\left[200+\left(3.25\right)\left(54\right)+\left(2\right)\left(3.25\right)\left(16\right)+\left(3\right)\left(3.25\right)\left(76\right)\right]}{4}=305.12$

Stop the process since $C\left(4\right)\succ C\left(3\right),$

  $y_6=r_6+r_7+r_8$

  $y_6=44+54+16=114$

Starting in period 9:

  $y_9=r_9+r_{10}$

  $y_9=76+30=106$

b)

Summary Introduction

Interpretation: Determine the ordering policy is recommended by the part period balancing heuristic.

Concept Introduction: one of the lot-sizing technique is to implementing the lowest-cost with the interconnection between costs of order and carrying cost for the inventory is generally known as Part period balancing.it is more effective for the product whose demand is unstable.

Answer to Problem 49AP

The order policy according to the part period balancing heuristic method will be

is $\boxed{\left(56,0,52,0,0,98,0,92,0,30\right)}$

Explanation of Solution

Given information: The anticipated demand for an inventory is as follows:

Week	1	2	3	4	5	6	7	8	9	10
Demand	22	34	32	12	8	44	54	16	76	30

The inventory costs 65 cents each and holding cost (h) is calculated on annual interest rate at 0.5% per week. The set up cost (K) is $200.

The order policy according to part period balancing method can be calculated as follows:

In this method the order horizon that equates holding and setup cost over that period has to be calculated as follows:

  $K=\$200$

  $h=\left(0.005\right)\left(0.65\right)$

(Since h is calculated on 0.5% annual interest rate with each unit cost at 65cents)

  $h=3.25$ Per thousand

  $r=\left(22,34,32,12,8,44,54,16,76,30\right)$

Starting from period 1:

Period	Holding Cost
2	110.5
3	318.5

Since the setup cost $200 is closer at period 4, stop and equate

  $y_1=r_1+r_2$

  $y_1=22+34=56$

Starting in period 3:

Period	Holding Cost
2	25
3	65
4	520

Since the setup cost $200 is closer to 4 than 3, stop and equate

  $y_3=r_3+r_4+r_5$

  $y_3=32+12+8=52$

Starting in period 6:

Period	Holding Cost
2	175.5
3	279.5

Since the setup cost $200 is closer to 3, stop and equate

  $y_6=r_6+r_7$

  $y_6=44+54=98$

Starting in period 8:

Period	Holding Cost
2	247

Since the setup cost $200 is closer to 2, stop and equate

  $y_8=r_8+r_9$

  $y_8=16+76=92$

c)

Summary Introduction

Interpretation: Determine the ordering policy is recommended by the least unit cost heuristic.

Concept Introduction: Least unit cost heuristic technique is one of the time-varying demand pattern technique.one of the dynamic lot-sizing concept is that interrelated between the order cost and carrying cost of the inventory for the every lot size is defined as

Least Unit Cost (LUC).

Answer to Problem 49AP

The order policy according to the least unit cost heuristic method will be

is $\boxed{\left(56,0,44,0,106,0,0,92,0,30\right)}$

Explanation of Solution

Given information: The anticipated demand for an inventory is as follows:

Week	1	2	3	4	5	6	7	8	9	10
Demand	22	34	32	12	8	44	54	16	76	30

The inventory costs 65 cents each and holding cost (h) is calculated on annual interest rate at 0.5% per week. The set up cost (K) is $200.

The order policy according to lease unit cost(LUC) method can be calculated as follows:

LUC divides the average cost per period C (T) by the total number of units demanded. Hence $C\left(1\right)=K/r_1$

And $C\left(2\right)=\left(K+h{r}_2\right)+\left(r_1+r_2\right),$

And general equation is $C\left(j\right)=\left(K+h{r}_2+2h{r}_3+\mathrm{...}+\left(j-1\right)h{r}_j\right)/\left(r_1+r_2\mathrm{...}+r_j\right)$

Once $C\left(j\right)\succ C\left(j-1\right),$ stop the function and set $y=r_1+r_2+r_3+\mathrm{....}{r}_{j-1}$

  $r=\left(22,34,32,12,8,44,54,16,76,30\right)$

Starting from period 1

  $C\left(1\right)=200/22=0.09$

  $C\left(2\right)=\left(200+\left(3.25\right)\left(34\right)\right)/\left(22+34\right)=5.54$

  $C\left(3\right)=\frac{\left[200+\left(3.25\right)\left(34\right)+\left(2\right)\left(32\right)\right]}{22+34+32}=5.89$

Stop the process since $C\left(3\right)\succ C\left(2\right),$

  $y_1=r_1+r_2$

  $y_1=22+34=56$

Starting in period 3:

  $C\left(1\right)=200/32=6.25$

  $C\left(2\right)=\frac{\left[200+\left(3.25\right)\left(12\right)\right]}{\left(32+12\right)}=5.43$

  $C\left(3\right)=\frac{\left[200+\left(3.25\right)\left(12\right)+\left(2\right)\left(3.25\right)\left(8\right)\right]}{\left(32+12+8\right)}=5.6$

Stop the process since $C\left(3\right)\succ C\left(2\right),$

  $y_3=r_3+r_4$

  $y_3=32+12=44$

Starting in period 5:

  $C\left(1\right)=200/8=25$

  $C\left(2\right)=\frac{\left[200+\left(3.25\right)\left(44\right)\right]}{\left(8+44\right)}=6.60$

  $C\left(3\right)=\frac{\left[200+\left(3.25\right)\left(44\right)+\left(2\right)\left(3.25\right)\left(54\right)\right]}{\left(8+44+54\right)}=6.54$

  $C\left(4\right)=\frac{\left[200+\left(3.25\right)\left(44\right)+\left(2\right)\left(3.25\right)\left(54\right)+\left(3\right)\left(3.25\right)\left(16\right)\right]}{\left(8+44+54+16\right)}=7.97$

Stop the process since $C\left(4\right)\succ C\left(3\right),$

  $y_5=r_5+r_6+r_7$

  $y_5=8+44+54=106$

Starting in period 8:

  $C\left(1\right)=200/16=12.5$

  $C\left(2\right)=\frac{\left[200+\left(3.25\right)\left(76\right)\right]}{\left(16+76\right)}=4.85$

  $C\left(3\right)=\frac{\left[200+\left(3.25\right)\left(76\right)+\left(2\right)\left(3.25\right)\left(30\right)\right]}{\left(16+76+30\right)}=5.2$

Stop the process since $C\left(3\right)\succ C\left(2\right),$

  $y_8=r_8+r_9$

  $y_8=16+76=92$

  $y_{10}=r_{10}=30$

d)

Summary Introduction

Interpretation: Determine the method resulted in the lowest-cost policy for the given problem.

Concept Introduction: Least unit cost method makes the present period of the demands and evaluates the period of the future. Adding the carrying cost for the setup and the period of the inventory cost, the least unit cost will be occur.

Answer to Problem 49AP

The least expensive method is the least unit cost method with a total cost of $\boxed{\$1351}$

Explanation of Solution

Given information: The anticipated demand for an inventory is as follows:

Week	1	2	3	4	5	6	7	8	9	10
Demand	22	34	32	12	8	44	54	16	76	30

The inventory costs 65 cents each and holding cost (h) is calculated on annual interest rate at 0.5% per week. The set up cost (K) is $200.

Comparison of the three methods can be calculated as follows:

Calculate the total cost of the inventory for the three months. The method which yields less cost is the lowest cost policy.

The total cost under silver meal method can be calculated as follows:

The method required is 4 set ups

The total set up cost is $4\times 200=800$

The total holding cost is $\left(3.25\right)\left(22+32+44+76\right)=565.5$

The total cost $800+565.5=1365.5$

The total cost under silver-meal heuristic is $\boxed{\$1365.5}$

The total cost under least unit cost method can be calculated as follows:

The method required is 5 set ups

The total set up cost is $5\times 200=1000$

The total holding cost is $\left(3.25\right)\left(22+32+8+16+30\right)=351$

The total cost $1000+351=1351$

The total cost under least unit cost method is $\boxed{\$1351}$

The total cost under part period balancing method can be calculated as follows:

The method required is 5 set ups

The total set up cost is $5\times 200=1000$

The total holding cost is $\left(3.25\right)\left(22+32+44+16+30\right)=468$

The total cost $1000+468=1468$

The total cost under part period balancing method is $\boxed{\$1468}$