Answer to Question #264138 in Operations Research for james

Question

b) The table below shows a transportation problem. Solve it using the Vogel approximation

method.

Destinations

Sources 1 2 3 4 Supply

A 2 2 12 21 80

B 4 3 13 7 60

C 5 12 1 6 60

Demand 50 60 30 40

c) State five objectives of inventory management.

d) The following table shows a pay-off matrix for a game. Use it to solve the game.

e) State four characteristics of 2-persons zero sum game.

f) A company’s ordering cost is sh 25 and the holding cost is 10% of average inventory.

Given that the total cost is shs. 750, find the demand.

g) State the two categories of inventory models.

Answer 1

b)

Here Total Demand = 180 is less than Total Supply = 200. So We add a dummy demand constraint with 0 unit cost and with allocation 20.

Now, The modified table is

The maximum penalty, 11, occurs in column D₃.

The minimum cij in this column is c₃₃=1.

The maximum allocation in this cell is min(60,30) = 30.

It satisfy demand of D3 and adjust the supply of S3 from 60 to 30 (60 - 30=30).

The maximum penalty, 5, occurs in row S₃.

The minimum c_ij in this row is c₃₅=0.

The maximum allocation in this cell is min(30,20) = 20.

It satisfy demand of D_dummy and adjust the supply of S₃ from 30 to 10 (30 - 20=10).

c)

• Material Availability.
• Better Level of Customer Service.
• Keeping Wastage and Losses to a Minimum.
• Maintaining Sufficient Stock.
• Cost-Effective Storage

d)

Select minimum from the maximum of columns

Column MiniMax = (6)

Select maximum from the minimum of rows

Row MaxiMin = [2]

Here, Column MiniMax ≠ Row MaxiMin

∴ This game has no saddle point.

$P_{Adj}=\begin{bmatrix} 2 & -7 \\ -6 & 1 \end{bmatrix}$

$P_{Cof}=\begin{bmatrix} 2 & -6 \\ -7 & 1 \end{bmatrix}$

Player A's optimal strategies = $\frac{\begin{bmatrix} 1 & 1 \\ \end{bmatrix}\times P_{Adj}}{\begin{bmatrix} 1 & 1 \\ \end{bmatrix}\times P_{Adj}\times \begin{bmatrix} 1 \\ 1 \end{bmatrix}}=\begin{bmatrix} 2/5 & 3/5 \\ \end{bmatrix}$

p₁=2/5 and p₂=3/5, where p₁ and p₂ represent the probabilities of player A's, using his strategies A1 and A2 respectively.

Player B's optimal strategies = $\frac{\begin{bmatrix} 1 & 1 \\ \end{bmatrix}\times P_{Cof}}{\begin{bmatrix} 1 & 1 \\ \end{bmatrix}\times P_{Adj}\times \begin{bmatrix} 1 \\ 1 \end{bmatrix}}=\begin{bmatrix} 1/2 & 1/2 \\ \end{bmatrix}$

q₁=1/2 and q₂=1/2, where q₁ and q₂ represent the probabilities of player A's, using his strategies B1 and B2 respectively.

Hence, Value of the game V = (Player A's optimal strategies) × (Payoff matrix P_ij) × (Player B's optimal strategies):

$V=\begin{bmatrix} 2/5 & 3/5 \\ \end{bmatrix}\begin{bmatrix} 1& 7 \\ 6 & 2 \end{bmatrix} \begin{bmatrix} 1/2 \\ 1/2 \end{bmatrix}=4$

e)

• the sum of the payoffs to each player is constant for all possible outcomes of the game
• pay-off matrix
• outcomes of player A
• outcomes of player A

f)

Total cost = holding cost + ordering cost

holding cost = $750-25=725$ = 10% x average inventory

average inventory = $725/0.1=7250$

Demand = average inventory/ordering cost

Demand = $7250/25=290$

g)

Fixed Reorder Quantity System:

the maximum and minimum of standard inventory quantity are defined in advance, and the quantity of inventory gradually decreases, and when the number reaches ROP (Reorder Point), an order of EOQ (Economic Order Quantity) is placed.

Fixed Reorder Period System:

an inventory control method where orders are periodically placed, but the order quantity is different every time.