EVALUATING NETWORK DESIGN DECISIONS USING DECISION TREES

When designing a supply chain, a manager is making several decisions. Examples are:

• Whether the firm enter into a long-term agreement for warehousing or get space from the spot market as and when needed?

• What should be the mix of long-term agreement for warehousing or get space from the spot market as and when needed?

• What should be the mix of long-term and spot market be in the portfolio of transportation capacity?

• What should be the capacity of each facility? And what fraction of this capacity should be flexible?

If a manager does not consider uncertainty, he is sure of signing the long-term contract, and avoids flexible capacity, since flexible capacity is more expensive. It may harm the firm in the future due to unutilization of part of the capacity, if the forecast demand does not materialize. At the same time, if the firm has a flexible capacity, they can move it to dedicated capacity only when they are sure of accuracy in their forecast. Thus, it is suggested that mangers when they design network, they need to use a methodology that allows them to estimate the uncertainty in their forecast of demand and price and then incorporate this uncertainty in the decision-making process. In this section we describe such a methodology and show that how uncertainty impact on the value of network design decisions.

A decision tree is a graphic tool used to evaluate decisions under uncertainty. The uncertainty in price, demand, exchange rates and inflation are incorporated using DCF technique in the decision tree to solve such problems. The first and fore most thing that is considered in decision tree is the time horizon. The time horizon may be a day, month, a quarter, or any other time period. Normally planning period is used as time period as ‟N‟.

The next step is to identify factors that will affect the value of the decision and are likely to fluctuate over the next „N‟ periods. Naturally these factors are: demand, price, exchange rate and inflation. Then we should evaluate the probability of each of these factors fluctuate from one period to the next. The next is to identify a periodic discount rate „K‟. This is considered to account the inherent risk associated with the investment. The decision is now computed using a decision tree, which contains the present and „N‟ future periods.

Within each period a node must be defined for every possible combination of factor value.

Arrows are used to connect the nodes between periods. The probability of transitioning from one node to the other is indicated on the arrow. The decision tree is evaluated starting from nodes in period N and working back to period „O‟. For each node, the decision is optimized taking into account the present and future values of each factor. The analysis is based on Bellman‟s Principle, which states that for any choice of strategy in a given state, the optimal strategy in the next period is the one that is selected if entire analysis is assumed

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to begin in the next period. This principle allows the optimal strategy to be solved in a backward direction starting from the last period. Expected future cash are discounted and brought to the present period value. The value of the node in period „o‟ given the value of investment as well as the decisions taken during each time period.

Let us illustrate the above decision tree process using the lease decision facing the manager at a typical logistics center. The decision to be taken by the manager is whether to lease warehouse space for the coming three years and the amount to lease. Let us assume that long-term lease is cheaper than the spot market rate for warehouse s pace. Also assume that the demand and spot prices vary over the coming three years. It is to be noted that if future demand is high the spot market cost will be high. There are three options before the manager, they are:

Option 1: Use spot market strategy

Option 2: Sign for three years contract and use Spot market for additional requirements.

Option 3:Sign a flexible contract with a minimum charge that allows variable usage of warehouse space of to a certain limit with additional requirement from the spot market.

Let us see how the manager makes decisions taking uncertainty into account.

Five hundred square feet of warehouse space is required for every five hundred units of demand and the current demand at the logistics is for 50000 units per year. The manager decides to use a multiplicative binomial representation of uncertainty for both demand and price. From one year to next, demand may go by 20 present with a probability of the two out comes are unchanged from one year to the next.

The manager can sign a three-year lease at a price of Rs 8 per square foot per year.

Warehouse space is currently available on the spot market for Rs10 per square foot per year. From one year to the next, spot prices for warehouse space may go up by 15 percent with probability 0.50 or go down by 15 percent with probability 0.50 according to a binomial process. The probabilities of the two outcomes are unchanged from one year to the next. The manager feels that prices of warehouse space and demand for the products fluctuate independently. Each unit logistics handles results in revenue of Rs 14 and the logistics is com mitted to handling all demand that arises. The logistics uses a discount rate of K=0.10 for each year and thus constructs a decision tree with N=2. The guideline for constricting tree is given in fig 3.6. The decision tree is shown in fig 3.7 with each node representing for the problem demand (D) in thousands of units and price (p) in rupees. The probability of each transition is 0.20 because price and demand fluctuate independently.

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Figure. 3.6 Guidelines for constructing decision tree

D = 7 2

Figure. 3.7 Decision tree for Logistics considering demand and price fluctuations. * Repetitions - can be ignored.

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Let us first analyze the option not signing a lease and obtaining all warehouse space from the spot market. Let us start with period 2 and evaluate the profit for the logistics at each node. At node D=72, P= Rs.12.1, the logistics must satisfy a demand of 72000 and faces a spot price of Rs.121 per square foot for warehouse space in period 2. The cost incurred by the logistics in period 2 at the node D=72, P=Rs.12.1/- is equal to 72000 X 12.1=Rs.871200. The revenue at the node D=72 and P=12.1 is 72000 X 14 = Rs.

1008000. Therefore profit at the mode D=72 and P=12 is equal to 1008000 -

8711200 = Rs.137000/-. The profit for the logistics at each of the other nodes in period 2 is evaluated similarly and given in table 3.17.

Table 3.17 Period 2 calculations for spot market option

Node Revenue Cost Profit

D=72, P=12.1 72000x14 = 1008000 72000x12.1=871200 137000 D-72, P=9.9 72000x14 = 1008000 72000x9.9=712800 295200 D=48, P=12.1 48000x14=672000 48000x12.1=580800 92000

D=48, P=9.9 48000x14=672000 48000x9.9=475200 197000

D=72, P=8.1 72000x14=1008000 72000x8.1=583200 424800

D=48, P=8.1 48000x14=672000 48000x8.1=388800 283200

D=32, P=12.1 32000x14=448000 32000x12.1=3387200 60800

D=32, P=9.9 32000x14=448000 32000x9.9=316800 131200

D=32, P=8.1 32000x14=448000 32000x8.1=259200 188800

Let us now evaluate the expected profit at each node in period 1. Profit at period 1 is equal to the profit during period 1 plus the present value (at the time of period 1) of the expected profit in period 2. The expected profit from period 2 for the node D=60, P=11 is equal to 0.25 [137000 + 295200 + 92000 + 197000] = 180300 and for the node D=60 and P=9 is equal to 0.25 [295200 + 424800 + 197000 + 283200] = 3,00,000 and for the node D=40, P=11 is = 0.25 [92000 + 197000 + 60800 + 131200] = 120250 and for the node D = 40, P = 9 is = 0.25 [197000 + 283200 + 131200 + 188800] = 2,00,050.

Therefore the present value for the expected profit from period 2 for period 1 are 180300/

1.1 = 163909, 300000/1.1 = 272727, 120250/1.1 = 109318 and 200050/1.1 = 181864.

The profit for period 1 is computed and given in table 3.18

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Table 3.18 Profit for period 1.

Node Revenue Cost Period 1 Profi

D=60, P=11 60000x14=840000 60000x11=660000 180000 D=60, P=9 60000x14=840000 60000x9=540000 300000 D=40, P=11 40000x14=560000 40000x11=440000 120000 D=40, P=11 40000x14=560000 40000x9=360000 200000

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Therefore the expected profit at period 1 is equal to 0.25 [180000 + 300000 + 120000 + 200000] = 200000. The expected present value from period 2 is equal to 0.25 [163909 + 272727 + 109318 + 181864] = 181955. Therefore the total expected profit for period 1 is equal to expected profit from period 1 plus expected present value of profits from period 2, which is equal to 200000+181955 = 381955/-

Next, let us compute the profit for the period „0‟. At period zero the profit is sum of the profit at period zero and the net present value (NPV) of profit expected from period 1.

The NPV of period 1 profit = 381955/1.1 = 347232. Profit expected at period zero operation is equal to (D=50, P=10); (50000x14-50000x10) = Rs. 200000/-.

Therefore the total profit for not signing a lease is equal to :200000+347232 = 547232.

Let us now evaluate the alternative where the lease for 50000 sq. ft. of warehouse space is signed. The evaluation procedure is very similar to the previous case but the outcome in terms of profit changes. For example, at the node D=72, P=12.1, the manager has to obtain 22000 sq. ft. of warehouse space from spot market at Rs. 12.10 per square foot because only 50000 sq. ft has been leased at Rs. 8 per square foot. If demand happens to be less than 50000 units, the logistics still has to pay for the entire 50000 sq. ft leased space. For period 2 the profit at each of the nine nodes are worked out and given in table 3.19

Table 3.19 Period 2 profit calculations at logistics for lease option Node Leased Warehouse space Profit D x 14 -

space at (50000x8+SxP)

Spot price (S)

D = 72, P = 12.1 50000 22000 341800

D = 72, P = 9.9 50000 22000 390200

D = 48, P = 12.1 50000 0 608000

D = 48, P = 9.9 50000 0 608000

D = 72, P = 8.1 50000 22000 429800

D = 48, P = 8.1 50000 0 608000

D = 32, P = 12.1 50000 0 608000

D = 32, P = 9.9 50000 0 608000

D = 32, P = 8.1 50000 0 608000

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The expected profit for period 1 from period 2:

D=60, P=11 = 0.25 [341800+390200+608000+608000] = 487000 D=60, P=9 = 0.25 [390200+429800+608000+608000] = 509000 D=40, P=11 = 0.25 [608000+608000+608000+608000] = 608000 D=40, P=9 = 0.25 [608000+608000+608000+608000] = 608000 TOTAL EXPECTED

PROFIT AT PERIOD 2 = 0.25 [487000 + 509000 + 608000 + 608000]

= 53000

NPV of Total expected profit from period 2 = 553000/1.1

= 502727 Profit generated at period 1

Node Leased Space from Profit

space spot Dx14-[50000x8+SP]

D = 60, P = 11 50000 10000 330000

D = 60, P = 9 50000 10000 350000

D = 40, P = 11 50000 0 440000

D = 40, P = 9 50000 0 440000

Expected Value = 0.25[330000+350000+440000+440000]

= 0.25x1560000

= 390000

Total expected profit for period 1 = 390000+502727

= 892727 Profit generated at period ‘0’

Demand = 50000 Price = Rs. 10

Profit = 50000x14-50000x8 = 300000

Total profit = profit at „0‟ period + NPV of period 2

= 300000 + 892727/1.1

= 300000+811570 NPV (lease) = 1111570

It can be seen that the presence of uncertainty in demand and price reduces the value of the lease but does not affect the value of the spot market price option. It is recomm ended

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that the manager can sign the three years lease for 50000 sq. ft. because this option has a higher expected profit.

In document Supply Chain Management (Page 70-76)